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Hamiltonian structure and stabiltiy of relativistic gravitational theories
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O'Neill, Eric, 1964-
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Thesis (Ph.D.)--University of Florida, 2000.
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Includes bibliographical references (leaves 145-148).
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Printout.
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by Eric O'Neill.

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Full Text

HAMILTONIAN STRUCTURE AND STABILITY OF RELATIVISTIC GRAVITATIONAL THEORIES

By

ERIC O'NEILL

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2000

ACKNOWLEDGEMENTS

I wish to express my sincere gratitude and thanks to my doctoral supervisor, Professor Henry E. Kandrup, for his guidance and input into my thesis project during the years that I worked on it. He initiated me into the study of stability problems in relativistic systems by means of phase-preserving perturbations. During later stages of the project his commentary and text editing were a great source of help to me. I have learned a lot from him in terms of his approach to theoretical physics research.

I also wish to express my thanks towards my family for financial support and encouragement during the years spent working on my degree.

Also, I would like to express my thanks to Dr. Samuel Mikaelian for providing me with thesis formatting tex macros.

page

ACK NOW LED G EM ENTS.............................................................................................. ii

A B ST R A C T .................................................................................................................... v

CHAPTERS

1 IN T R O D U C T IO N ........................................................................................................ 1

1.1 Theory of Casim ir Invariants.................................................................................. 2
1.2 Constrained Hamiltonian Dynamics...................................................................... 11
1.3 Covariant Poisson Brackets for Relativistic Field Theories.................................. 20
1.4 Derivation of the Plasma Bracket.......................................................................... 28

2 NEWTONIAN COSMOLOGY AND VLASOV-MAXWELL SYSTEM.................... 34

2.1 The Transformed Vlasov-Poisson Equation.......................................................... 35
2.2 The Vlasov-Poisson Hamiltonian Structure.......................................................... 37
2.3 Stability of Collisionless Vlasov-Poisson Equilibria............................................... 42
2.4 Introduction to the Curved Space Vlasov-Maxwell System.................................. 45
2.5 The Vlasov-Maxwell Hamiltonian formulation...................................................... 51
2.6 Perturbations of Time-Independent Vlasov-Maxwell Equilibria.......................... 58
2.7 Stability Criteria for Time-Independent Vlasov-Maxwell Equilibria.................... 65

3 VLASOV-EINSTEIN SYSTEM.................................................................................. 69

3.1 The Vlasov-Einstein Hamiltonian Formulation...................................................... 73
3.2 All Vlasov-Einstein Equilibria are Energy Extremals............................................ 79
3.3 The Second Hamiltonian Variation 6(2)H .............................................................. 80

4 VLASOV-BRANS-DICKE SYSTEM......................................................................... 88

4.1 The Vlasov-Brans-Dicke Hamiltonian Formulation............................................... 92
4.2 The Second Hamiltonian Variation 6(2)H .............................................................. 102
4.3 T he Q uestion of Stability....................................................................................... 110

5 VLASOV-KIBBLE SYSTEM .................................................................................... 114

5.1 Kibble Gauge Invariant Theory of Gravity .......................................................... 115
5.2 Vlasov-Kibble Hamiltonian Structure and Formulation........................................ 117
5.3 Initial Variation of the Vlasov-Kibble Hamiltonian.............................................. 129
5.4 Second Hamiltonian variation 6(2)H ...................................................................... 132

SUMMARY..................................................................................................................... 141

REFERENCES............................................................................................................... 145

BIOGRAPHICAL SKETCH......................................................................................... 149

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

HAMILTONIAN STRUCTURE AND STABILITY OF RELATIVISTIC GRAVITATIONAL THEORIES

By

Eric O'Neill

May 2000

Chairman: Henry E. Kandrup Major Department: Physics

The research contained within this thesis is concerned with the Hamiltonian structure and stability of several relativistic physical systems. Its primary focus involves a comparison of the Hamiltonian structure and stability properties of two alternative gravitational theories with General Relativity. Specifically, the Brans-Dicke and Kibble theories of gravity are the subject of study.

Working in the context of an Arnowitt-Deser-Misner splitting into space plus time, it may be demonstrated that the Vlasov-Einstein system, i.e., the collisionless Boltzmann equation of General Relativity, is Hamiltonian, and then this Hamiltonian character may be used to derive nontrivial criteria for linear and nonlinear stability of time-independent equilibria. Unlike all earlier work on the problem of stability, the formulation provided here is completely general,

incorporating no assumptions regarding spatial symmetries or the form of the equilibrium. The two alternative gravitational theories are studied by using the same methods.

The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function f, the spatial metric hab, and the conjugate momentum 11ab. The Hamiltonian formulation entails the identification of a Lie bracket (F, G), defined for pairs of functionals F[hab, rlab, f] and G[hab, IHab, f], and a Hamiltonian function H[hab, 11ab, f], so chosen that the equations of motion OtF = (F, H) for arbitrary F, with Ot a coordinate time derivative, are equivalent to the Vlasov-Einstein system. An analogous mathematical structure is deduced for the Vlasov-Brans-Dicke and Vlasov-Kibble Hamiltonian systems, where the gravitational field variables differ from General Relativity.

An explicit expression is derived for the most general dynamically accessible perturbation 6X = {6f, 6hab, 6flab} which satisfies the Hamiltonian and momentum field constraints and the matter constraints associated with conservation of phase, and it is shown that all equilibria are energy extremals with respect to such 6X, i.e., 6(')H[6X] = 0. The sign of the second variation, 6(2)H, which is also computed, is thus related directly to the problem of linear stability. If 6(2)H > 0 for all dynamically accessible perturbations 6X, the equilibrium is guaranteed to be linearly stable. The existence of some perturbation 6X for which 6(2)H[6X] < 0 does not necessarily signal a linear instability.

CHAPTER 1
INTRODUCTION

Throughout this dissertation the Hamiltonian structure, and energetic stability, of several relativistic collisionless plasma systems will be studied. In chapter 2, the collisionless Vlasov-Newtonian cosmology and the fixed curved background spacetime collisionless Vlasov-Maxwell system will be examined. In the remaining three chapters, the Hamiltonian structure of three relativistic theories of gravitation will be analyzed in detail. Chapter 3 consists of a study of the Hamiltonian structure of the Vlasov-Einstein system (General Relativity), while chapter 4 is a corresponding study of the Vlasov-Brans-Dicke Hamiltonian system. Finally, chapter 5 deals with the Vlasov-Kibble Hamiltonian system.

The Hamiltonian of each gravitational theory will be written in terms of the ADM formalism (Arnowitt, Deser, and Misner (1962)). The use of this formalism facilitates a convenient 3+1 split into space and time variables. From a dynamical standpoint, this makes the analysis of static equilibrium matter configurations particularly straightforward. For each theory, there are field variables and corresponding conjugate momenta. Following Dirac (1950), one can construct a bracket out of these canonically conjugate dynamical variable pairs. The equations of dynamics for the conjugate variables may be derived from the bracket. This applies to the field variables of the gravitation theories.

A Boltzmann distribution f will be used to describe the matter distribution. This distribution corresponds to the case of a collisionless matter distribution and it will satisfy the Liouville, i.e., collisionless Boltzmann equation. This variable f does not possess a conjugate variable. Therefore, it is not straightforward to construct a Dirac bracket for it.

1

1.1 Theory of Casimir Invariants

Early work with the theory of Casimir invariants dates back to the late 1950's with the research of W. Newcomb (cf. the appendix in Bernstein (1958)). Newcomb considered a perturbation of a plasma Hamiltonian for which the entropy, serving as a Casimir invariant, remains fixed. His idea, which went as follows, was developed to show that general small motions of a collisionless plasma about thermal equilibrium cannot exhibit exponential growth in time. Newcomb utilized the fact that entropy is a constant of the motion for collisionless systems.

We will let
) 3/2 - my21
fo(v) = N( 2m - exp[ (1.1) 2exkLT 2kBT

denote the Maxwell distribution. In equation (1.1) the particle density is denoted by N and Boltzmann's constant is denoted by kB. The mass and velocity of the plasma particles are denoted, respectively, as m and v. The temperature at thermal equilibrium is denoted by T. Newcomb worked with the following expression for the entropy S = -ksB d3rd3v[flnf - fo In fo], (1.2) where the summation is taken over all species of particle present in the plasma. Since S = 0 for thermal equilibrium, the expression for entropy (1.2) is normalized.

For a departure from thermal equilibrium, that is spatially bounded, the integral (1.2) is well defined. For spatial disturbances of the form e+i'kF, the integration may be taken over a cube of side length 27r/k in which one face is perpendicular to the wave vector . One may compute the time rate of change of S to be dS _ k d3rdv + lnf] . (1.3) dt kB9jt r v at

The term O f/lot will vanish due to the conservation of the total number of particles (factor the partial time derivative out of the integral). For the rest of the integral, it is necessary to utilize the Boltzmann equation (Chapman and Cowling (1939)) Of .Of q x Of Of f+ v- + - + - O = -- (1.4) at 9F m c &U at Cou.' where Of/Otcou. represents the rate of change of the distribution function arising from particle collisions. For a collisionless plasma, one may set

8 f /8tcou. = 0.

Substitution of this collisionless Boltzmann, i.e., Liouville equation into equation (1.3) yields the result

= kB d rd3v lnf {. (f) + -[q ( + x)f] }. (1.5) Integration by parts leads to the following modified version of equation (1.5)

dS _f3 LOf q -.v- Of]
= -kBZ d rd v - + f + -E + j - 0. (1.6) Consequently, the entropy is a constant of the motion known as a Casimir invariant.

The total energy of the plasma system is given by

W= - d1 f 3r(2 + 2 - 2) + f d3rdv v2(f- fo), (1.7) =8ir 2J where al denotes a constant external magnetic field. The energy is normalized since W = 0 at equilibrium. The Maxwell equations are V. =41rp, cVxE a=
Ot'
V -f3 = 0, cV xBi=47J f(1.8 atr (1.8)

where the charge density p and current density J are given by p= EZe f d3vf, f = I Ze fd3v f. Using these equations and the Boltzmann equation (1.4) allows one to prove that W is a constant of the motion. Therefore, the combined system F = W + S is a constant of the motion.

One may expand the distribution function in terms of powers of a small initial departure from thermal equilibrium

f = fo ffl +f2 . -. (1.9) There are corresponding changes in the electric and magnetic fields E = 0 + El + 2.**,

= + + 2 + ... (1.10)

To second order the entropy has the following form

S= -kB E drd3vjfj (1 +A- m )+ f2(1+A- 7vT) + , (1.11) 2kBT 2kBT 2fo,

where the constant A = ln[N2/3m/2rkBT]. Due to particle conservation, the terms above in (1 + A) will vanish. Expressing equation (1.7) to second order allows us to write equation (1.11) as

S = -kB 8 d3r[2 + ]2] 1 d3rd3v- }. (1.12) We obtain a bracketed expression which is essentially a sum of positive terms. However, we know from equation (1.6) that S is a constant of the motion. Consequently, it is not possible for any of the quantities f1, B1 or f, to increase monotonically. For example, an exponential increase in time of fl would not be permitted.

This argument by Newcomb disproved the purported proof by Gordeyev (1952) that such an exponential growth of the distribution f could occur for a collisionless plasma in thermal equilibrium. Newcomb considered the conservation of only one constraint. However, in later work, Kruskal and Oberman (1958) considered the conservation of a general set of Casimir constraints in a paper dealing with plasma stability. Gardner (1963) provided additional insight with the recognition of the importance of phase space conservation in regard to electrostatic Vlasov stability theory. His arguments were as follows.

One may assume that f (t, 9, V) denotes the distribution function of one species of particle present in a plasma. Since the plasma is collisionless, the distribution will satisfy the Vlasov equation
8f
+ 6- Vdf + -. Vef = 0, (1.13) at m

where the force F(t, Y, i7) obeys the condition Vy - F= O. (1.14) The motion of one particle is dictated by the usual dynamical equations df/dt = 9, (1.15) mdV/dt = F. (1.16) Equations (1.15) and (1.16) define a flow in i, 6 phase space. By combining equations (1.14)(1.16) we see that the flow is incompressible. Also, equation (1.13) states that the value of f, at the phase space position of any fixed particle, will not change as the particle moves about in phase space. In other words, the distribution remains invariant when translated along any phase space trajectory that is consistent with the dynamical equations. One may interpret the phase space motion as that of an incompressible fluid flow of variable density

f. The initial value of f will be

f(0, Y, 6) = fo(, ). (1.17) The kinetic energy of the plasma will take the form W(t) = M v2f(t, Y, )did&. (1.18) One may integrate in Y over a finite box if the assumption of periodic & dependence of the distribution f is made.

With the initial value of fo, we wish to determine a lower bound for the kinetic energy of the plasma. Such a lower bound may be written as W = M v2f( )dd , (1.19) where f, (i), corresponding to a later value of the evolving distribution, represents a monotone decreasing function of v2. Also, for any number a > 0, the phase space volume of the region where f, (6) > a is equal to the phase space volume of the region where fo(F, v) > a. This condition holds because the phase space flow is incompressible with the value of f at a particle conserved during subsequent evolution of the system.

With these enforced conditions, one wishes to determine what the minimum possible value for W(t) will be. It is straightforward to see that the lowest-energy state corresponds to the case in which more massive particles (f large) are even closer to & = 0. For example, if we choose two particles f and f' with respective velocities v and v', then, if we are dealing with the lowest-energy state, the condition f > f' implies that v < v'. This state corresponds to fi (). Consequently, the amount of kinetic energy originally present in fo(i, 6), which can be given up to the field, is at most

W (0) - Wi.

(1.20)

Afterwards Lynden-Bell and Sanitt (1969) applied the method of conserved constraints to study the energy associated with a linearized perturbation of a galactic dynamical system. They referred to this as "a trick due to Newcomb." However, it was left to Bartholomew (1971) to finally deduce the mathematical form of the phase-preserving perturbation of the Boltzmann distribution which leaves general Casimir constraints invariant. Bartholomew's paper dealt with the problem of galactic stability.

Bartholomew treated a galaxy as a system of point masses moving under the influence of the collective gravitational potential of the entire system. One may define a phase space function F = F(qi,pi, t) over the space of generalized coordinates qi and conjugate momenta Pi. A phase space volume element may be defined as dr = dqldq2dq3dpidp2dp3. The continuity of flow through volume dr of phase space may be expressed as 8F a dq a dp
O + F + F =0. (1.21) 5T+ 5qi (dt pg di

Using Hamilton's canonical equations of motion (for the case of a unit mass) allows one to rewrite equation (1.21) as

8F 0 8H 8 49H + - F = 0. (1.22) t 8qi i Pi i

The Poisson bracket may be defined as OA BB OA OB
[A, B] - .p 0Bi a B(1.23) a9i aPi api a9i

This enables one to redefine equation (1.22) as OF
t+ [F, H] = 0. (1.24) The Hamiltonian H may be written as the sum of two terms. The first term, which corresponds to the kinetic energy, is a function of the generalized coordinates. The second

term corresponds to the gravitational potential T. Using Cartesian coordinates one may write the matter density as p= Fd3P, (1.25) which yields the gravitational term /Gf p'd r' =G F'dT'1 (1.26) IF ea ir - j -1

For the sake of convenience, one may give the gravitational term of the Hamiltonian a more general formulation

- K(qi,pi;q'i,P'i)F(qi, p'i)dr', (1.27) where K is a symmetric kernel. The second gravitational term has a linear dependence on the distribution F. One may rewrite equation (1.22) as OF 8qi OF Opi OF
+ + - 0, (1.28)
Bt Of 8qi at api

which corresponds to
dF(qi,pi, t) (1.29) dt -= 0. (1.29) dt

Assuming that the galactic system is perturbed from equilibrium by an infinitesimal amount

F + F (1.30) leads to a corresponding change in the Hamiltonian 6H = - K6F'dT'. (1.31) The Vlasov equation (1.31) will undergo a similar linearized perturbation t (F) + [6F, H] + [F, 6H] = 0. (1.32)

One wishes to determine whether, from the perspective of stability theory, the perturbation 5F will experience monotonic growth or will oscillate about equilibrium. As discussed in Bartholomew (1971) the relevant perturbations F are those which conserve angular momentum and total mass, and are generated by a canonical transformation. We insist on a canonical transformation because the Hamiltonian equations dqj dp'
dt = [qi, H] dt = [pi, H] (1.33) generate translations in phase space qi + dqi,

Pi + dpi, (1.34) by means of an infinitesimal canonical transformation. We are only interested in perturbations which are consistent with the equations of dynamics.

One wishes to move stars from the phase space point (qi,pi) to an infinitesimally close nearby point (qi + i, Pi + ti). The perturbation corresponds to an infinitesimal canonical transformation provided that there exists a generating function g such that Og
a- (1.35a) SPi,

i = . (1.35b) 89qi'

A more compact notation for equations (1.35) may be introduced by use of the Poisson bracket

fi = [qi,g], 7 i = [pi,g]. (1.36) The change in F may be found by writing the following equation F + a-(FOj) + -(Fqi) = 0, (1.37) 8qi 0Pi

which follows by analogy with equation (1.21). This leads to the proper form for the phasepreserving perturbation

F = -[F, g], (1.38) where 6F is defined in terms of the generating function g. Substitution of equation (1.38) into equation (1.32) yields the result F,] = [[F,g],H] + [F, 6H]. (1.39) Multiplication of this equation by ag/at and integration over all phase space leads to

[F, d = - JF, ] [H, g]dT - [F, g K[F', g']dr'dr. (1.40) The left-hand side of this equation vanishes by virtue of antisymmetry

The right-hand side of equation (1.40) can be proven symmetric in g and Og/at, allowing one to rewrite it as

i { [F, g][H, g]dr - [ F, g]K[F', g']drdrT'} = 0. (1.42) The functional in brackets corresponds to a second order energy perturbation.

Assuming a stationary reference frame, the Hamiltonian H is the energy per unit mass. The total energy E, for a fixed potential field, would be given by E = f FHdr. (1.43) However, if a perturbation of F alters the energy of other material by varying the potential, then the equation (1.43) can be written in the form E J FHdT + J FKF'drdr'. (1.44)

A first order perturbation of this energy can be calculated to be

6E = (F + 6F)(H + 6H)d" + (F + 6F)K(F' + SF')drdr' - E (1.45) = HFdv - 1 JFKbF'd-rdT'.
= f f f

Therefore, the first order change in energy is 6E = - f H[F, g]dT = f [F, H]gdr = 0. (1.46) Thus, there is no change in the energy of an equilibrium for a lowest order perturbation.

1.2 Constrained Hamiltonian Dynamics

It was Dirac (1950) who first studied constrained Hamiltonian dynamics. We may follow his treatment of the subject. Typically, one may consider a dynamical system of N degrees of freedom described by generalized coordinates qn (where n takes the values 1, 2,..., N) and velocities dqn/dt = qn. The dynamics of the system is described by a Lagrangian L L(qn, 4n), (1.47) which possesses the corresponding momenta BL
Pn = OL (1.48) 09qn

The variables qn, 4n, and Pn may be varied by a small amount Sqn, 4n, and bPn of the order of e. In our calculations, we only work to the order of accuracy of e. However, variation of equation (1.48) leads to difficulties because the left-hand side will differ from the right-hand side by a quantity of order E. Following Dirac (1950), we may class equations as to whether they remain accurate to order e under variation or not. The Lagrangian (1.47) remains valid under variation, since, by definition, the variation in L must equal the variation of L(qn, 4n). Equations of this type are referred to as strong equations, while equations, such

as (1.48), are referred to as weak equations. The following algebraic rules exist for the weak and strong equations; for general cases JA = 0 (1.49) (A 0 is a strong equation) and 6X 0 (1.50) (X = 0 is a weak equation), where the notations - for strong equivalence and = for weak equivalence have been employed. We can conclude that 6X2 = 2X6X = 0

where X = 0 is a weak equation. Consequently, we have deduced the strong equation X2 -= 0. A strong equation

XIX2 0,

can be constructed from two weak equations X1 = 0 and X2 = 0.

The standard case occurs when the N quantities OL/8q4n of equation (1.48) are all independent functions of the N velocities qn. For this case one may determine each velocity On as a function of the coordinates qn and momenta p.. However, if the BL/Bqn are not independent functions of the velocities n, then we may eliminate q's from equation (1.48) and derive one or more equations 0(q,p) = 0 (1.51) which are functions of coordinates q and momenta p. The equations (1.51) are weak equations. One may consider a complete set of independent equations (1.51)

m(q,p) = 0, m = 1,2,...,M.

(1.52)

A function of the q's and p's, which vanishes due to equation (1.48), may be written as a linear function of the cm with coefficients that are functions of the q's and p's. The strong and weak equations may be given a geometrical interpretation. Initially, one may visualize a 3N dimensional space with coordinates qn, n, and p,. This space will possess a 2N dimensional hypersurface where equation (1.48) holds. This hypersurface will be called R. Equations (1.51), which are deduced from equation (1.48), will also hold on this hypersurface. We may also define a 3N dimensional region of all points that are within a distance of order E from the hypersurface R. This region will be referred to as R. A weak equation will hold in the region 7, while a strong equation will hold in the region 1,. The Hamiltonian is defined in the usual manner as H = P.n - L, (1.53) where the standard summation convention is employed. The Hamiltonian (1.53) can be subjected to a variation as follows

H = 6(pnn - L)

= Prqn + nPn - OL/Oqn.6q - A1,9nn

= gndPn - OL/Oqn6qn. (1.54) Consequently, 6H does not depend on the 6q's. This result holds irregardless of whether or not the standard case applies. The definition (equation 1.53) of the Hamiltonian holds throughout the 3N dimensional space of q's, q's and p's. The result (equation 1.54) holds in the region R, to first order. Therefore, assuming that q and p remain constant, a first order variation in the velocities qn corresponds to a second order variation in H. Thus, for constant qn and Pn, the variation in H, corresponding to a finite variation in q., will be of

the first order provided that the variation is performed in R,. The variation in H will be zero if it is performed in the region R. It follows that, in the region R, the Hamiltonian is a function of q and p. For the region R one may define the weak Hamiltonian function H = h(q,p). (1.55) The function h is the ordinary Hamiltonian in the standard case. The following general variation

6(H - h) = (qn - p ) - (L + h )6qn (1.56) 9pn o9n a9n)

may be performed from a point in R. If the variation is within the region R, then it is obvious that 6(H - h) = 0. For such a variation of 6bqn and 6pn, the dynamical equations (1.48) are preserved by an appropriate choice of 6n. The 6q's and 6p's are restricted in the sense that 64m = 0 for all values of m. However, for arbitrary perturbations 6q, 6p which do not satisfy this condition, we compute

6(H - h) = Vmdc5m (1.57) for a suitable choice of coefficients. These coefficients vm are functions of the q's and p's.

Equation (1.57) can be reformulated as

(H - h - vmbm) = 6(H - h) - vmb0m - #m6bVm = 0, (1.58) where equation (1.52) has been used. One may use equation (1.58) to deduce the following strong equation

H - h + vmom (1.59) which holds to the first order in R,. A variation of equation (1.59) yields the result

6H = h + m6m

8h Oh 8mO(m 8 m Om nqn) (1.60) = 6Pn + 6qn + Vm ,-pn +a 96. (1.60) 19P. an,, ( P. O9n ) Comparison of equation (1.60) with equation (1.54) yields the following expressions On = h om, (1.61a) nPn + mPn
9L Oh Om (1.61b) Oqn - Oqn + Vm Oq

The usual Lagrangian equations of motion 9L
Pn- Oq,,

can be combined with equation (1.61b) to yield the Hamiltonian equation of motion Oh Om
p - - Vm (1.62) Oqn qn

The Hamiltonian equation corresponding to equation (1.62) is equation (1.61a).

It is convenient to formulate the Hamiltonian equations of motion in terms of the Poisson bracket notation. Any two functions ( and 7 of the q's and p's possess the Poisson bracket (, 09] 0- q r (1.63) [ -qn Opn Opn Oqn(

Poisson brackets are, by their definition, subject to the following rules

[, 9] = -[77, (], (1.64a) Of Of
[ (1, 7( 2, - [71, J +> [ 2] + ..., (1.64b) [(, [ (, ]] + [7, [(, (]] + [(, [ , ]] = 0. (1.64c) According to Dirac (1950) the notion of a Poisson bracket may be extended to include functions of q's where the velocities are not expressible as functions of q's and p's. These

generalized Poisson brackets are still subject to the properties (1.64a)-(1.64c). The following weak equations
OA 9A aA
- = 0 = 0 - 0 (1.65) Oqn can , Pn

can be derived from the strong equation A - 0. For any ( one may deduce that [(, A] = 0 (1.66) by using equation (1.64b). It is possible, although by no means necessary, that [, A] - 0. However, for a weak equation X = 0 one cannot generally infer that [(, X] = 0.

If g is a function of the q's and p's then the Hamiltonian equations (1.61a) and (1.62) allow us to write

g h 8+ m a9 ( A +v m) = [g,h] + vm[g, m]. (1.67) Using equation (1.52) allows one to rewrite this expression as S= [g,h] + vm[g, 0m] + [g, vm]ï¿½m = [g, H], (1.68) which is the generalized Hamiltonian equation of motion.

If the Lagrangian is homogeneous of the first degree in the velocities q4, then the momenta (1.48) will be homogeneous of degree zero in the q's. Consequently, they will depend only on the ratios of the q's. It follows that the p's cannot all be independent of the q's because there are only N - 1 independent ratios of the 4's corresponding to N p's. Therefore, at least one relation of type (1.51) must exist between the q's and the p's. The Euler theorem is applicable in this case and leads to L = q n A
n = o '9-4--q 1

L = qnPn. (1.69) This yields the following weak equation H=0 (1.70) which holds in the region R. Therefore, we can assume that h = 0 and that H = vmm. (1.71) It follows that the general equation of motion reduces to the form g = vm[g, 0m]. (1.72) As noted by Dirac (1950), the Lagrangian for any dynamical system can be made to satisfy the condition for homogeneous velocities by using an additional coordinate qo to represent the time t and the condition 4,, = 1 is then used to make the Lagrangian homogeneous of the first degree in all the velocities, including 4,. Therefore, without any loss of generality, one may consider only the homogeneous velocity theory.

The equations of motion must maintain the validity of the constraint equations (1.52). It follows that substitution of 0,n, for g in equation (1.72) must yield the set of equations Vm[0m, e'] = 0. (1.73) The system of equations (1.73) will be assumed to be reduced as much as possible with the set of equations (1.52). Each of the resulting equations must fall into one of the four categories:

Type 1. It involves some of the variables Vm.

Type 2. It involves the variables q and p and takes the form x(q,p) = 0.

However, it is independent of the variables Vm and independent of the

equations 0m (1.52).

Type 3. It reduces to the form 0 = 0.

Type 4. It reduces to the form 1 = 0. An equation of type 2 will lead us to another consistency condition that is analogous to equation (1.73), namely vm [0m, X] = 0, (1.74) for the constraint

x(q,p) = 0. (1.75a) The new consistency condition (1.74) may be reduced as far as possible by using eqs. (1.52) and (1.75a). Once again, eq. (1.74) will be one of the four types. If it is type 2, then we will obtain another consistency condition. Therefore, we must repeat the process until an equation of another type is obtained. If we obtain an equation of type 4, then the equations of dynamics are inconsistent. Therefore, we may ignore this case. Equations of type 3 are identically satisfied. It is necessary to examine the cases of type 1 and type 2.

A complete set of equations of type 2 may be written

xk(q,p) = 0, k = 1,2,...,K. (1.75b) The equations (1.75b) are chosen to exhibit order c variations such as the constraint equations (1.52); both sets are weak. We define a constraint mri' to be a first class constraint if its Poisson bracket with all other constraints 0 and x vanishes, i.e., if [ Om,m] = 0, m = 1,2,...,M

[m',Xk] = O, k = 1,2,...,K. (1.76) Equations (1.76) only have to hold as consequences of om = 0 and Xk = 0, i.e., they can be weak. A constraint Om not satisfying these conditions is termed second class.

A linear transformation may be performed on the constraints bm

em* = 7mm'm, '. (1.77) The 7-y matrix coefficients are functions of the coordinates q and momenta p such that their determinant will not vanish in the weak sense. It follows that the 0* constraints are completely equivalent to the 0 constraints. A transformation of this kind may be performed to bring as many constraints as possible into the first class. The first class constraints will be denoted 0,, and the second class constraints will be denoted 00, where 3 = 1, 2, ..., B and a = B + 1, B + 2,..., M. The consistency conditions will reduce to the format vy[0,3, ,] = 0, 3,0' = 1,2,...,B

v[ï¿½, Xk] = 0, k=1,2,...,K. (1.78) These are all of the type 1 equations. These equations prove that either the v#'s must all vanish or that the matrix is, in the weak sense, of rank less than B. Dirac (1950) proved that the first possibility is correct. Therefore, if the maximum number of O's possible have been placed in the first class, the v's corresponding to the second class constraints will all be zero. Therefore, the Hamiltonian H (1.71) can be written in terms of only the first class constraints as

H = v.0 (1.79) and the general equation of motion becomes

9 = v[g, 0]. (1.80) We may form a determinant from a set of functions Os (s = 1, 2, ... , S) of the q's and p's such that the determinant is nonvanishing in the weak sense. Let cs, correspond to the

cofactor of [0s, 0s,] divided by the determinant A. It follows that csS, E -Cs s

and

cS k ,[Os , Os] 6, s,,.t Dirac (1950) defined a new Poisson bracket [(, 4]* for any two quantities ( and 7 as [, 1* = [, 7] + [, Os]Css, [OS, ,?].

(1.81)

(1.82)

(1.83)

The new Poisson brackets satisfy the three conditions on Poisson brackets (1.64a)(1.64c). For any ( we deduce the following condition on the new Poisson brackets [, 0]* = [K,Os] + [, 0,]c,,s,,[Ov,0] =K[, O'] - K4, o ,], 6SI

=0.

(1.84)

We may choose the O's to consist entirely of O's and x's. The determinant A will vanish unless the O's are second class. It follows that [Os, H] = 0 for every value of s. Therefore, the new Poisson brackets give the Hamiltonian equations of motion

[g,H]* = [g,H] = g,

(1.85)

where g is any function of the q's and p's.

1.3 Covariant Poisson Brackets for Relativistic Field Theories

J.E. Marsden, R. Montgomery, P.J. Morrison, and W.B. Thompson (1986) were the first to write the equations of some specific relativistic field theories in covariant Poisson bracket form. They demonstrated that the field equations are equivalent to the following bracket

{F, S} = 0.

(1.86)

In this expression F is an arbitrary function of the fields and S is an action integral. The relativistic fields were grouped into two categories: (i) pure fields and (ii) media fields. The pure fields, corresponding typically to gauge fields, will be associated with functionals F of the basic field variables
For pure fields, the covariant bracket possesses a space-time vector field VA. If a 3+1 Dirac-ADM decomposition is implemented, then this vector field VA corresponds to a choice of foliation of the space-time manifold. In the case of media fields, the bracket is a covariant extension of the Lie-Poisson type discussed in section 1.2.

Marsden et al. (1986) deduced their results for relativistic field theories by first considering the case of elementary particle mechanics. The familiar canonical Hamiltonian equations
OH (1.87a)
aPi
OH
Pi q i, (1.87b) follow from an initial variation 6S = 0 of the action integral S f] = (piqi - H(q, p))dt. (1.88) The integral is treated as a functional on F, the phase space of trajectories (q(t),p(t)) associated with movement of a particle. Appropriate boundary conditions (cf. Arnold (1978,

p. 243)) are imposed on this phase space. The variational principle may be formulated in terms of a Poisson bracket defined on F. For phase space functionals F and G, one may define the bracket
f(I 6G 5G 6F\
F, G} (F) = 6G 6G dt, (1.89) f bqi JPi 6qi 6Pi

where the functional derivatives take the form d F tSF 6 F dF(F + s6r)|s=o - W )dt = 6q + & pi dt (1.90) and the variations 6r vanish at the endpoints of F. One may verify that 1 solves Hamilton's equation iff

{F, S}(F) = 0, (1.91) for arbitrary functionals F. Marsden et al. (1986) generalized this bracket operation to relativistic field theories. This covariant theory, however, does not single out a single spacetime direction, but treats space and time in an equivalent manner.

As well as the covariant bracket form, a 3+1 decomposition of equation (1.86) may be performed by assuming that F is of the form F = f n(t).F(p, q)dt, (1.92) where n(t) is an arbitrary function of time and Y is an arbitrary function of spatial coordinates q and spatial momenta p. Substitution of the functional F (1.92) and action S[F] (1.88) into the bracket (1.91) yields the result {F, S}(y) = n(t)(F - {F, H} (3))dt = 0, (1.93) where {F, H}(3) is the conventional Poisson bracket. The function n(t) is arbitrary, implying that

7 = {r, H} (3).

(1.94)

Among the examples considered by Marsden et al. (1986) were the Maxwell equations and the relativistic flat space Vlasov-Maxwell system. We will treat these examples in what follows. For the Maxwell equation system, one may denote the usual four-vector potential by A. In mathematical terminology, the potential is a one-form and we may construct the electromagnetic field tensor F (two-form) from it in the following manner F = dA,

i.e.,

Fp, = 8,A,, - 8,AP, (1.95) where ,, = /Ox8" and the index takes the values p = 0, 1, 2,3. The standard electromagnetic field Lagrangian with minimal coupling to an external current density J" is L[A] = Ldx = f (- FvF" - AJ")d4x, (1.96) where the Minkowski metric is used to raise and lower indices. One may derive the covariant momentum variables

HP"V- F"'. (1.97) The functional derivatives are defined by analogy with equation (1.90) except that there is a constraint on II'" due to its antisymmetry

- F(IIM" + sJII ") s=o =/ HF "d4 d--H-- + s6(")Io f FiF "d4x, (1.98) ds fI 6fl"

where 6HII" is an antisymmetric perturbation. A covariant Poisson bracket may be defined as

{F,G}v(A,II) G G IF) V"d4x, (1.99) 6A I bl>" bA , biH "

where V" is an arbitrary vector field on space-time and the functionals F and G are dependent on the field variables A. and H". An electromagnetic action S may be defined as a

covariant analogue of equation (1.88) S[A,H] = f[WA,, - H(A, I)]d4x,

(1.100)

where

1
H(A,II) = 4VWII1 +A JA

= IIV"Ay,v - L'. It may be shown that Maxwell's equations can be derived from {F,S}v(A, II) =0, for any choice of F and V. One may derive the conditions 6S
= 0 and 6S
= 0,

(1.101)

(1.102)

(1.103a)

(1.103b)

which imply

(1.104a)

II"" = -(9A' - 9'AO)

II"",, = - J.

(1.104b)

Taking into account the definition (1.95), the mathematical identities correspond to the Maxwell equations.

A 3+1 spacetime decomposition of this formalism may be performed (cf. Marsden and Weinstein (1982)). Coordinates are chosen so that the space-time vector field V is

V - 9 =-. (1.105) aX 0 ac

We may, for a closed system, choose J = 0 and rewrite the action S as S = { f [IioAi,o - 7]d3x}dt, (1.106) where the Hamiltonian density has the form 71 = (IIijHij + IioHio) (1.107) and Latin indices take the values i = 1, 2,3. We will choose a functional of the form F = n(t)F[Ai, HiO]dt, (1.108) where Ai and IiTo are the respective 3+1 field and conjugate momenta variables. A computation of the bracket (1.102) yields the result {F, S}afao = 6 (Ai,o +i Ho0 } d} n(t)dt
f I~- - Q(3
= { F - {FH} }n(t)dt, (1.109) where

h = ftd 3X (1.110) and {., .}(3) is the canonical Poisson bracket for the conjugate field variables Ai and IIO. Due to the arbitrariness of the function n(t), we may conclude that F = {F,H}(). (1.111) It is assumed that H is only a function of the conjugate field variables Ai and I1i = gio. This follows by substituting II' = -(,iA' - oJA') (1.112)

into equation (1.107).

We may now examine the case of the relativistic Vlasov-Maxwell equations. A special relativistic particle moves in an external electromagnetic field F = dA in accord with the Lorentz force law
dx" du' e
= u , F"'u,, (1.113) dr dT m

where e is the electric charge, m is the particle rest mass and T is the proper time of the particle. The momenta canonically conjugate to x" is

e
pA = mum + -A . (1.114)
c

One may construct the following particle Hamiltonian H = - upA (1.115)
2 2m c c with the corresponding Hamilton equations that are equivalent to dx" _ p O . dp, OH e uV OAA, dr Opa m' dr 80x c Ox'

The Boltzmann plasma distribution is constant along the particle world trajectories O f e Of _,OA=,
df fu" + u vA = 0. (1.116) dr 82x c Op, Ox"

This collisionless Boltzmann equation may be reformulated in terms of the following bracket {f, H}zp = 0, (1.117) where
fg 1Of Og Og Of {fg}P - Ox Op. Ox Op

The basic media field of the plasma is the Boltzmann distribution f. We define the bracket of two functionals F and G of the distribution to have the form {F,G}(f)= f {F, G dxd4P. (1.118) 6f 6f xp

The derivation of this bracket will be given in section 1.4. For now, we may accept it as given without further proof. One may write the plasma action S as S[f = f(x,p)H(x,p)d4xdp, (1.119) such that JS/6f = H. It is a straightforward exercise to prove that the covariant bracket equation

{F, S}(f) = 0 (1.120) is equivalent to the Vlasov equation (1.116). The relativistic Vlasov-Maxwell system may be analyzed in terms of the set (A, II "', f). One may construct the following covariant Poisson bracket

{F,G}v(A,II,f) = fVG 6G IF Vd4x 6A AIHPv JAA JH "1

/G d4xd4p. (1.121) I f 6 f Xp

One may define the action S for the Vlasov-Maxwell system to be S[A, H, f] = HP A -IIII P d4X + ff(x,p) (P ,) (~" - A) d4xd4p. (1.122) The field equations are formulated in terms of the relativistic Poisson bracket

{F, S}v(A, II, f) = 0, (1.123) for all choices of F and V. From equation (1.123) we deduce the following identities IS IS
5ii -0,5A -0,
6111- =0

and

f f IFIS d4xd4p = 0. (1.124) I f I f up

The mathematical identities (1.124) are quite useful. They yield the relativistic VlasovMaxwell equations
Of u+e Of U, OA,,
Ox1t c Opm O9X

8F"= u uf (x,p)d4p,
c

and

F, = O A, - Or,A,. (1.125)

1.4 Derivation of the Plasma Bracket The plasma bracket was first derived in its present form by Morrison (1980a). However, Gibbons (1980) independently derived it. Also, the work of Iwinski and Turski (1976) should be mentioned in this regard. In the previous section, we dealt primarily with Lie brackets that were constructed from field variables OA and their conjugate momenta IA. In this section, we wish to consider the derivation of the plasma bracket (1.118). Since f is noncanonical (without a corresponding conjugate momentum), this bracket will differ from the previous field-momentum brackets.

Following the treatment in Morrison (1980a), we will again deal with the Vlasov-Maxwell Hamiltonian system. Morrison (1980a) chose to formulate a Poisson bracket in terms of the noncanonical variables f,(1, '6, t), f(:, t) and B(ï¿½, t), where the Boltzmann distribution of species a is denoted as fa(Y, 6, t), the electric field intensity is denoted as E(ï¿½, t), and the magnetic field intensity is denoted as /(, t). In rationalized Gaussian units, with the speed of light set to unity, the Vlasov-Maxwell system can be written as Ot (:F, t) = -V x (, t), (1.126a) E(Y, t) = xB(Y, t) - ef f(z, t)P(zji)dz (1.126b)
a RI

and
Off~t B fa(z, t) Btfa(z, t) = -.9f.(z, t) e I [f(v, t) + 6x B(Y, t)] - P(zlf)di, (1.126c) Of ma R2 9f where the Boltzmann distribution fa is a function of the phase space variable z = (, 6). In the coupling terms of equations (1.126b) and (1.126c), the operator P(zlf) = J(ï¿½- r-) has been used. The regions of integration are defined to be R1 = AxR3 and R2 A, where R = (-o, +oo) and A C R3.

One wishes to formulate this system (1.126a)-(1.126c) as

89V /ot = [V, H], i = 0, 1, ..., 6 (1.127) where -P takes the values

( = f, i = 0 (1.128a) =E, i = 1,2,3, (1.128b) = B, i = 4,5,6, (1.128c) and the Hamiltonian functional is written as H{} = mov fadz + (E2 + B2)d&F. (1.129) The operator [.,.] corresponds to the Poisson bracket. One may assume that the solutions of the Vlasov-Maxwell system exist in a vector space w = w, x w2 that is defined over R. The subspace w, has elements that are functions of z, while the subspace w2 has elements that are functions of :i. The operator P(zl ), appearing in equations (1.126b) and (1.126c), maps elements of one subspace into another. The vector space w has an associated inner product

(glh) = J glhldz + f Rg2h2d, (1.130)

where gi e w, and g2 E w2 are the two components of g - (g, g2) E w = wl x w2. Use of an antisymmetric operator O'j on w and the inner product (1.130) allows one to define a Poisson bracket in the following manner [F, G] = (6F/6xiOibjG/6xj). (1.131) a,i,j

The quantities F and G are elements of Q, a vector space (over R) of Frichet differentiable functionals of the functions fa, E and j. Differentiability is defined with respect to the L2 norm (1.130). The elements in Q have the form F{X'} = F{x0} + F2{Xi}, i # 0, (1.132) where

Fp{X} = F(xk, k')dxp, 3= 1,2, and

X1 = z, X2 1 , il = 0, i2 = 1, 2, 3. The subscript k on Xk'O denotes partial derivatives of a general order k. The Hamiltonian functional (1.129) possesses the form (1.132) with k = 0. Arbitrary C functions of Xki (for all i) are also elements of Q. In equation (1.131) the quantity 6F/6xi is the functional derivative of F with respect to Xi.

One may define the functional derivative as

(d/dc)F{x0(z) + Ew(z)}IE=o = (6F/6xo w), (1.133) where, without loss of generality, one may set i = 0. It is important to note that both w(z) and 6F/3X0 E w, and for i 0 F/6xi E w2. For the purposes of computation, the relations 6x0(z)/6x0(z') = 6(z - z') and 6Xi(Y)/6Xi(f) = 6( - if') are useful. Functional differentiation of the k = 0 Hamiltonian (1.129) will involve only the following three quantities WH/6 fo = -mv2, (1.134a) 2 6H/6E = E, (1.134b) and WH/B = . (1.134c) The bracket (1.131) is (i) antisymmetric [F, G] = -[G, F] and (ii) satisfies the Jacobi identity [E, [F, G]] + [F, [G, E]] + [G, [E, F]] = 0, where E, F and G E Q. These are the standard properties associated with a Poisson bracket. A Lie algebra is defined as a Poisson bracket together with a vector space (Arnold (1978)). One may define a Hamiltonian system to be a set of partial differential equations with an integral invariant and a bracket satisfying properties (i) and (ii), such that the system may possess the form of equation (1.127), i.e., otF= [F, H]. The antisymmetric operator OIj will be written in the following cosymplectic form E 03 D 0--+2 O -+2 -D 03 where 03 is a 3 x 3 array of zeros and D = [Eijk(O/9Xj)] ('Eijk is the Levi-Civita tensor). The braces appearing in the upper left corner denote the usual Poisson bracket {g, h} = - 9h g h (1.135) W ag ag (9F which will map elements of wi into wl. The lower right-hand block (consisting of D's and 03's) will map elements of w2 into w2. The off-diagonal elements _-+, f (.) Pd, (1.136a) m0 R2 0v and 12R -( x PdY (1.136b) maR2 8 will, as seen by their notation, map elements of w2 into wl1. The other off-diagonal elements - 2 mo fR Pdz (1.137a) and PI-2 = e ()( x ) Pdz (1.137b) will map elements of wl into w2. One may rewrite equation (1.131) in the more explicit form [F, G] = [F, G]I + [F, G]2, (1.138) where [F, G] - 6fa ,fa dz - 6F 62-+ 1 G ) PEf dz _ 6F 6G I -B .-6 )dz (1.139a) and [F, G]2 -P G di a fR2 6E \6f +f 6PI-+2( G~ d24 R2 6B a/ (6F 6G 6G. 6F)d, . x-9x df. (1.139b) 2 \E 6B 6B 6E Using this bracket, one may derive the Vlasov-Maxwell system (1.126a)-(1.126c) from equation (1.127). The first term of equation (1.139a) will yield the Vlasov equation without the coupling term involving electromagnetic field variables. The final term in equation (1.139b) yields the vacuum Maxwell equations. The other terms in equation (1.139a) and (1.139b) correspond to field-distribution couplings. One may study the Vlasov-Poisson Hamiltonian system by neglecting terms in equation (1.138) and (1.129) that involve B. The resultant expression [F, G] = fa(z) 6 , dz (1.140) is obtained by writing the electric field E as a functional of the distribution fa. The corresponding Vlasov equation will be 8tfo = -{fa,Hpa}, (1.141) where HE{fae} = (R Hxa(z) fa(z)dz (1.142a) +-J1J fa(z)fo(z')H2a(ziz')dzdz'), 2 , n, 1 Hi1a = mv2 (1.142b) 2 and H2a = e _2/1F- i4 . (1.142c) The particle Hamiltonian takes the form Hpa = 6HE/6f,. CHAPTER 2 NEWTONIAN COSMOLOGY AND THE VLASOV-MAXWELL SYSTEM A Newtonian cosmology (Peebles (1980)) is typically viewed as the infinite radius limit of a homogeneous and isotropic sphere of matter that expands homologously with the expansion rate governed by a time-dependent scale factor a(t). We may formulate a collisionless Boltzmann, that is, gravitational Vlasov equation to describe the evolution of a one-particle distribution function f for such a scenario. We may consider so-called "equilibrium" solutions fo which correspond to expanding, steady state distributions (Peebles (1980); Bisnovatyi-Kogan and Zel'dovich (1970)). This is as straightforward as the construction of equilibria for isolated self-gravitating systems. However, for an isolated system, the equilibrium distributions fo lack explicit time-dependence. This facilitates the use of energy arguments to study the stability problem (Antonov (1961); Lynden-Bell and Sanitt (1969); Kandrup and Sygnet (1985)). Formulated in an inertial frame, cosmological so-called "equilibria" possess an explicit time-dependence which renders the standard energy arguments inadmissable. There is a way, however, to reformulate the problem so that cosmological "equilibria" possess no explicit time-dependence. This is done by reformulating the Vlasov description in terms of the noninertial average comoving frame. In this frame, the cosmological "equilibria" possess no explicit time-dependence. Under such a transformation, the Hamiltonian will acquire an explicit time-dependence. In spite of this, a cosymplectic structure may still be found and an energy criterion for linear instability can be given. It should be remembered that it is not necessary to use energy arguments to study the problem of stability for cosmolog34 ical equilibria. Instead, we may resort to a direct analysis of the linearized perturbation equations to determine the time-dependence of quantities such as the perturbed density 3p. However, because energy arguments are extremely useful in the study of the stability of isolated equilibria, they may be amenable to the study of Newtonian cosmological equilibria. The technique used here to extract a nontrivial stability criterion is quite general in the sense that, in the average co-moving frame, the Hamiltonian acquires an explicit time-dependence and the equilibrium corresponds to an energy extremum. Therefore, the general techniques may be applicable to other physical problems. 2.1 The Transformed Vlasov-Poisson Equation The distribution function f(Ra, Pa, m,T) associated with the inertial frame may be defined as the number density of particles of mass m with momentum Pa at the point Ra for the time T. The gravitational Vlasov equation 8f pa f I 4f -ï¿½+ P- a - m = 0 (2.1) OT m BRa oRa OPa determines the evolution of the distribution f. The gravitational potential 4I is selfconsistently determined by the expression /~m' f (k, ', m', T) 4(R,T) = -G dSR'd3P'dm' f(,Pm, T) (2.2) IR - R11 We wish to transform to a new set of variables {ra,pa, m, t} from the original variables {Ra, Pa, m, T} via a time-dependent canonical transformation of the form ra = a(T) 1Ra, Pa = a(t)[Pa - mH(t)6abRb] and dt = a(t)-2dT (2.3) where H - d(log a)/dT, and a is an arbitrary function of time. The transformed Vlasov-Poisson equation will be of the form Of pa.9f [O'I 1 .9fS+ - ma(t) - 02ra af = 0, (2.4) _5t m Bra Bra ap. where -m'" f ,J m', t) W(F,t) = -G d 3r'd 3p'dm' Ml V4, , 0 and 2 = -a2(d2 a/dT2). Equation (2.4) holds true for an arbitrary choice of a(T). However, an appropriate choice of a will cause the steady state equilibria to take a simple form. In this context, we may choose a(T) to satisfy the equation 2d2a 4rG 4rG a2 p(T)a 3 Po, (2.5) where p(T) is the average mass density of the Universe at time T. In the average co-moving frame, the quantity Po = a3p represents the constant, time-independent density of matter in the Universe. For this choice of a(T) the physical peculiar velocity pa Va - HRa m will decay as 1/a. It follows that the momentum Pa is conserved. We may construct a timeindependent solution fo(Pa) for the transformed Vlasov equation (2.4) from this constant of the motion. With the assumption that fo is independent of ra, we may deduce that d O 3 / 3 d m'fo(F,, n, t) (r, t) = -G drdp'dm' p po 47rG ora IF - P 3 = G a -r 72 l - 3r Pra" (2.6) Therefore, we can write o49/8ra = Q22ra. It follows that the O f/opa force term in the Vlasov equation (2.4) will vanish. The af/Ora term will vanish if spatial homogeneity is assumed. Consequently, we find that the distribution corresponds to a time-independent equilibrium af /Ot - 0. 2.2 The Vlasov-Poisson Hamiltonian Structure Hamiltonian formulations of the gravitational Vlasov-Poisson and Vlasov-Einstein equations, for isolated self-gravitating systems, have already been presented. By direct analogy, the Vlasov equation (2.4) for the Newtonian cosmological system can also be shown to admit a Hamiltonian formulation. It is necessary (Morrison (1980a); Morrison and Pfirsch (1990)) to identify a Lie bracket [A, B] and a Hamiltonian function H in order that the Vlasov-Poisson system may be written as 8f - [H, f]. (2.7) In accord with the ordinary Vlasov-Poisson system, we can define the bracket operation as [A, BI = dFf 6' , (2.8) f I6f ' f where {a, b} is the ordinary Poisson bracket, /6f is a functional derivative of the Boltzmann distribution and dP = d3axd3pdm is a differentiable phase space volume element. It is straightforward to verify that the bracket is antisymmetric and satisfies the Jacobi identity [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 (2.9) and therefore constitutes a bona fide Lie bracket. Consequently, the evolution generated by any Hamiltonian H and this Lie bracket will be symplectic. We may view the evolution as a generalized canonical transformation in an infinite-dimensional phase space of distribution functions. We shall select the "natural" Hamiltonian H = JdFp2 Ga(t) 3 rd3r ,[P(r) - P o][p(r') - Po] (2.10) H - d f -drdr(2.10) 1 2m 2 f F - P~ to generate the transformed Vlasov equation (2.4). In equation (2.10) we have p(F, t) = d pdmmf (F, , m,t) and Po satisfies equation (2.5). We calculate 6H/6f = E to have the form 2 E = + a(t)mW + Ga(t) d3r' PO (2.11) 2m ] IF - eP The Vlasov equation (2.4) is then generated from equation (2.7) through this choice of Hamiltonian H Of 6H S= I-, = {E, f }. (2.12) Other choices, besides equation (2.10), for the Hamiltonian are permitted. We may, for example, choose S P2 Ga(t) d , d'mf(r, p, m, t) m'f(r', p', m', t) H1 dr f- dr d H 2m 2 fF / mt2r -a(t) dr M22 f, (2.13) where Q2 = 4rGpo/3. This Hamiltonian differs from equation (2.10) by a function independent of f and only dependent on time. The choice of this Hamiltonian is disadvantageous, however, because it remains explicitly time-independent even for a homogeneous equilibrium. Also, for such an equilibrium, the energy E -= H' p2 + a( a(t)mQ2r2 (2.14) 6f 2m 2 is different from the "natural" energy E = p2/2m in that it possesses an overall function of time. It is generally known that, in accord with conservation of phase, the Vlasov equation possesses an infinite number of constraints. If the distribution function behaves properly "at infinity" then we may integrate by parts to prove that the numerical value of an arbitrary constraint C= f dFx(f) (2.15) is conserved. It follows that the time derivative of C vanishes dC d- = [C, H] - 0 (2.16) (To validate the integrations by parts it is sufficient to assume that fo is differentiable and, as r--+oo, independent of r with an isotropic velocity distribution. Although they are not physically intuitive, periodic boundary conditions could also be employed as well as other, less restrictive requirements as r--+oo ). It is not possible to define a unique Hamiltonian because of the existence of these Casimir constraints. The Hamiltonian H may be renormalized by adding a constraint C which will, however, leave the Vlasov equation (2.12) unaffected because [C, f] = f, = o0. (2.17) The Casimir constraints severely restrict the evolution of the Boltzmann distribution f as dictated by the Vlasov equation. These Casimirs (Morrison (1980a); Morrison and Pfirsch (1990); Kandrup (1990, 1991a); Kandrup and Morrison (1993)) constrain the evolution to an infinite-dimensional hypersurface of the infinite-dimensional phase space. The hypersurface is defined by the constancy of all the constraints. It follows that, when dealing with the problem of linear stability, we should only consider perturbations of the distribution 6f that preserve the constancy of the constraints. Only perturbations that propagate dynamically are relevant to the study of linear stability. We may, in a straightforward fashion, find the generic perturbation of the distribution that will conserve all of the constraints. It can be shown (Kandrup (1990); Morrison (1980a); Bartholomew (1971); Morrison and Pfirsch (1990)) that 6C will vanish if the perturbation 6f is generated from the equilibrium fo through a canonical transformation. Therefore, the perturbation must be of the form fo + f = exp({g,.})fo = fo + {g, fo} + (1/2!){g, {g, fo}} +..., (2.18) with a generating function g. It is simple, using the dynamically accessible perturbation 6if of equation (2.18), to prove that the first variation of the Hamiltonian 6H vanishes. This implies that all equilibria are extrema of the Hamiltonian (2.10). We can prove this by first calculating the expression J)H dF6(1)fP2 - Gma(t)( df,m'fo(X',p', ', t) d3 P_ )]O f 2m (Fli- fj jFr-P = fdN() ffE (2.19) and then substituting in the phase-preserving perturbation P) f = {g, fo}. The net result is J)H=f d{g, fo}E = d{E, fo}g = 0, where, for an equilibrium, Bfo S= {E, fo} = 0. We now consider the second variation of the Hamiltonian 6(2)H. We may calculate the following expression 6(2)H dF6(2) fE Ga(t) , mI(F) f(, p,m,t)m'(1) f (', p', m',t)2 2r- Substituting 6(1)f and (2)f, as given by equation (2.18), into this expression yields, after an integration by parts, the final expression 1 Ga(t) dm{g, fo}m'{g', fo'} (2)H = dF{g, fo}{g,E} - a() dFJm' f g fo}r f0' (2.21) 2 2 f F-Pfy This expression can be rewritten in a simpler form. Since the equilibrium fo is a function of Pa and m, we may write the Poisson bracket as Og Ofo {g, fo = 9 W " 1gJ xa ONa The overall homogeneity of the distribution makes it useful to utilize the Fourier transformed generating function 1 g(k a,Pf, ) /2 d3xe-i tg(xa,Pa, t). (2.22) g~k~pa t)--(27r)3/2 Using the Poisson bracket associated with overall homogeneity and the transform (2.22) will allow one to rewrite the second variation of the Hamiltonian as 6(2)H - d'k d3plg(k,p, m,t)12 (k. 16) k. a) -2rGa(t) f dk f dpfdp [mg(k, p, m,t) -f] E of']2.3 x [m'g(k,p',m',t) . (2.23) with u=ff/m. If we take fo to be an isotropic velocity distribution then this expression can be rewritten in a simpler format. For an isotropic velocity distribution, the equilibrium fo depends on the momentum Pa through the square of the magnitude p2. Consequently, one can write {g, fo} = FE - where, for an equilibrium, E = p2/2m and FE = o9fo/OE. Therefore, we write the second order perturbation of H as (2)H= f d3k d p(-FE)jg(k,p,mrn,t)kilU -27rGa(t) Jd 3k J d3p f d3p, [mg(k,p, m,t) k FE] x [m'g(k, p', m', t) F' . (2.24) 2.3 Stability of Collisionless Vlasov-Poisson Equilibria A simple stability criterion can be obtained from equation (2.23) or equation (2.24). The time derivative d6(2) H Gda ,m{g, f}m'{g', fo25 a = - a dF dF l (2.25) di 2 dt is negative semi-definite because, for an expanding universe, da/dt > 0. Therefore, in the average co-moving frame, any initial phase-preserving perturbation undergoes an intrinsically dissipative evolution. The perturbed energy 6(2)H will decrease after the initial perturbation. One may consider the case where 6(2)H(to) < 0, where to, is the time of the initial perturbation. Since the evolution is dissipative the quantity 6(2)H(t) can only become more negative. The system will continue to evolve away from equilibrium with the magnitude of the second order perturbation 16(2)HI increasing. Consequently, the system is linearly unstable. A perturbation of a certain minimum wavelength will cause the second negative contribution of equations (2.23) and (2.24) to have a magnitude greater than the first term which is of indefinite sign. This minimum wavelength will be of the order of magnitude of the Jeans length. An order of magnitude estimate indicates that the first and second terms of equation (2.24) will be comparable in magnitude for a characteristic perturbation wave number k2 ~ Ga(t)po/92, where f is a characteristic velocity and Po is the co-moving density. This estimate is performed by assuming that both the unperturbed distribution fo and the generating function g are well behaved, and that the derivative FE is everywhere negative. One can calculate the physical wavelength A ,- (k/a)-' ( V2/ Gp)1/2 which corresponds to the Jeans length, by utilizing the expressions for the physical density p = po/a3 and the physical peculiar velocity V = a-'v. The energy must be negative for wavelengths greater than the Jeans length. Specifying the form of fo enables us to obtain a better estimate of the physical wavelength. Therefore, the derivation of the Jeans instability via an energy argument given here is rigorous. By making certain assumptions, it can be shown that fo is unstable to perturbations with wavelength much shorter than the Jeans length. One example of this occurs for an equilibrium fo with an isotropic distribution of velocities and a derivative FE that is not everywhere negative. As a specific case of this, we consider the distribution fo = Eb exp(-fE), where b is a positive constant. Substituting into equation (2.24) a generating function g with nontrivial k dependence that is sharply peaked about the velocities where FE is positive leads to an instability. In the previous example, this occurs as v2-40. Therefore, for this example, both terms in equation (2.24) will be negative leading to a second order energy perturbation 6(2)H and a corresponding time derivative db(2)H/dt which are both negative. Another example involves any plane-symmetric equilibrium fo(fa) = fo(p~). Such an equilibria is unstable if there exists a range of velocities and a direction for which the derivative Of/Opa2 is positive. If we choose a k-vector, k = ka&, aligned in an appropriate direction such that 2 af Mt) 12 1a2UaIf -g(k,p,m,t) 2(k- " ) (" -) = -g(k,p,m, t)2kaP a = -2mg(k,p, m, t) 121ka l2 1U2 Of (2.26) apa2 ,(.6 then, for Of/Opa2 > 0, this expression will be intrinsically negative. However, if there does not exist a direction in velocity space for which Of /OPa2 is positive, then the integrand of the first term integral in equation (2.23) is positive and the equilibria will be stable when subjected to perturbations of sufficiently short wavelengths. For example, if we choose a k-vector k = kaa 2(k _) - = _j9 21k. 2U = -2mjgl2lkj21U.12 a2 > 0. (2.27) a# .9~Pa ~ It follows that the aforementioned equilibria will be stable for all perturbations of characteristic wavelength much shorter than the Jeans length provided that they are monotonically decreasing functions of speed for all directions of velocity space. Consequently, we do not allow for population inversions. This argument does not imply that a negative energy perturbation of characteristic length scale R will demonstrate instability associated with a natural time of tD = R/V. Instead, the magnitude of 5(2)H will increase with a time scale determined by the variation of a(t). Consequently, the Hubble time scale tH sets a lower bound on the growth time of the instability. Even if we consider the case of a time-independent energy 6(2)H, e.g., perturbations of static equilibria possessing compact support or, as an approximate case, a cosmological setting with time scale << tH, a dynamical instability with a time scale to(R) could be implied for a negative energy perturbation 6(2)H < 0. However, it is not certain that an instability will follow from a time-independent 6(2)H < 0 because none of the negative energy perturbations must necessarily be coupled to the positive energy perturbations. A sufficient, but not necessary (Kandrup (1991a)), condition of stability, for a time-independent energy with a static setting, occurs when 6(2)H > 0. We now consider a straightforward counterexample. The Hamiltonian of a two degree of freedom system 1 21 2 H !(P 12 q2) - 1(P2 W22q22) (2.28) 2 2 possesses an equilibrium solution qi = Pl = q2 = p2 = 0 which is stable if the frequencies are constant, yet becomes unstable if dw22/dt > 0. The instability resulting from dissipation, through the effects of gravitational radiation, has already been studied for rotating perfect fluid stars (Friedman and Schutz (1978)) and generic rotating, axisymmetric equilibrium solutions of the gravitational Vlasov-Poisson system for isolated systems. The energy argument dealing with instability that is given here is directly analogous to the previously studied cases. However, the Newtonian cosmology is intrinsically negative, while, in the previous cases, an additional source of dissipation, such as gravitational radiation, must be invoked to bring about energetic instability. From a theoretical viewpoint, this is due to the fact that the equilibria are static in the conformal-but not true-sense. 2.4 Introduction to the Curved Space Vlasov-Maxwell System At this point we shall consider the Hamiltonian formulation of the Vlasov-Maxwell system in a curved background space-time. This exercise will serve as a springboard to a similar treatment of the Vlasov-Einstein system which will be dealt with in chapter 3. The Vlasov-Maxwell system couples the Maxwell equations, governing the behavior of an electromagnetic field, together with the Boltzmann distribution f which serves as a source for the electromagnetic field. The Boltzmann distribution f is coupled to the electromagnetic field in the sense that it is influenced by it while serving as its source. The evolution of the distribution f is self-consistently determined by the collisionless Boltzmann (Liouville) equation. If we take the Boltzmann distribution f to be a function of the space-time coordinates xa and the physical (not canonical) momentum P', then it may be defined with respect to the tangent bundle corresponding to the background space-time. Therefore, the distribution f satisfies the following Liouville equation (cf. Israel (1972) and Stewart (1971)) -f + (eFoA P' - Pa ,vP"P) - = 0, (2.29) where r,, is the Christoffel symbol defined by the space-time metric g,, and the Faraday tensor F,, satisfies the Maxwell equations VFP" = J = e (_g)-1/2d4 p f J m and V ,*F"" = 0, (2.30) where * denotes a dual tensor. We recover the electrostatic Vlasov-Poisson system by taking the Newtonian limit c--+oo. For any dynamical system, there are a variety of methods for implementing a Hamiltonian formulation. We shall utilize the most general technique, which is to proceed at a formal algebraic level. Accordingly, we shall work with a phase space -y and a Hamiltonian H defined with respect to it. Also, a Lie bracket (F, G) which acts on functionals F and G of the phase space, will be used. It will be shown that, with suitable choices for the Hamiltonian H and Lie bracket, the Vlasov-Maxwell system may be formulated in terms of the constraint equations OF = (F, H), where F is an arbitrary functional. From classical mechanics, the standard Poisson bracket equation OfF = {F, H}, and its associated Hamiltonian H, demonstrate that the dynamics in the six-dimensional phase space (x',pi) is generated through a canonical transformation. Likewise, the Lie bracket equation A9F = (F, H) implies that the dynamics of F in the phase space -y is generated by a type of generalized canonical transformation. However, this cannot be a canonical transformation because -y does not have canonical coordinates. Initially, we might presume to take the phase space -y as being constructed from spatial coordinates xi and momenta pi, and electromagnetic coordinates, i.e., the vector potential Ai and conjugate momenta li. Unfortunately, this presumption is incorrect. It would be appropriate to select a six-dimensional phase space (xi,pi) for studying the dynamics of a Boltzmann distribution in a fixed electromagnetic field. If we take E as representing the canonical particle energy, then, in this case, the Liouville equation can be written as Otf = {e, f}. Conversely, the phase space (Ai, HI) would be employed for the examination of the dynamics of an electromagnetic field with respect to a fixed current source J. Taking the electromagnetic Hamiltonian (2.41), with an additional minimal coupling f (-g) -12d3 xJiAi, enables one to generate canonically the Maxwell equations corresponding to the problem. For the Vlasov-Maxwell system, however, one is not free to hold either the electromagnetic field or the Boltzmann distribution fixed. Our Hamiltonian description must take into account the reciprocity of the interaction between field and source, i.e., the electromagnetic field controls the dynamics of the Boltzmann distribution f which acts as the source of the electromagnetic field. In this case, we will take f to be a dynamical variable in the y phase space (Ai, Hi, f). However, the y phase space is not canonical in contrast to the six-dimensional (xi,Pi) phase space or the infinite dimensional (Ai, II') phase space. One can separate the electromagnetic variables into canonical pairs Ai(x) and Hi(xJ), but such a division of the distribution f into canonical pairs is not straightforward. If we attempt such a decomposition, then we must explicitly implement the infinite number of constraints associated with the VlasovMaxwell system because of conservation of phase. These constraints restrict the evolution to a still infinite-dimensional hypersurface of the infinite-dimensional -Y space. The distribution f may be decomposed into canonical pairs only in this hypersurface of the phase space. Since the cosymplectic structure is degenerate from the perspective of the full 7y phase space, this should be obvious. It remains valid, however, to use a noncanonical phase space Hamiltonian formulation, but, at some point in the analysis, the constraints corresponding to phase conservation must be explicitly implemented. Since the dynamics of the Vlasov-Maxwell system are generated by a generalized canonical transformation, it is possible to intuitively understand the evolution of the system. As for the Newtonian cosmology, a Hamiltonian formulation allows the problem of stability to be studied through the use of energy arguments. We wish to examine the energetic behavior of phase-preserving perturbations 6X = (bAi, 61P, 6f) of the equilibria Xo = (Aio, Ho', fo) associated with the Vlasov-Maxwell system. The interaction between electromagnetic field and Boltzmann distribution occurs on a fixed background space-time which possesses a timelike Killing field. It is straightforward to deduce a Hamiltonian formulation for the flat space electrostatic Vlasov-Poisson system. The electrostatic system has no radiative degrees of freedom and, with a choice of suitable boundary conditions, the electric field E (xl) at time t can be written as a functional of the Boltzmann distribution f at the same time instant. We must find a suitable Green function corresponding to the Poisson equation. Therefore, the only dynamical variable of the system is the Boltzmann distribution f . So, the 7-y phase space reduces to the phase space of distribution functions f. The Hamiltonian will have functional dependence H = H[f] and the Lie bracket (F, G) will act on functionals F[f] and G[f of the phase space. This Hamiltonian structure for the electrostatic Vlasov-Poisson system has been examined by Morrison (1980a, b) and Gibbons (1981). It is simple, for the Vlasov-Poisson system, to find phase-preserving perturbations (Gardner (1963); Bartholomew (1971); Morrison (1987); Morrison and Pfirsch (1989, 1990, 1992)) 6f that yield energy extrema 60)H = 0. We can deduce nontrivial energy stability criteria by calculating the second order variation 6(2)H. One example of a Vlasov-Poisson system would be a neutral plasma. Such a plasma consists of a fixed homogeneous, positive background of heavy ions interspersed with a gas of light electrons. On average, in a neutral plasma, the negative and positive charges cancel out and there is no external field present. A homogeneous and isotropic equilibrium electron distribution can be described by a function fo(E), where E = p2/2m is the particle energy. If the derivative 8fo/8E = FE is everywhere negative, then 6(2)H is positive for all phase-preserving perturbations bf which implies that the equilibrium is linearly stable. However, if, for some range of energies, FE exhibits a population inversion, then there exist perturbations 6f such that 6(2)H < 0, which, however, does not necessarily indicate a linear instability (Holm, Marsden, Ratiu, and Weinstein (1985); Bernstein (1958); Gardner (1963); Penrose (1963)). Similar reasoning can be employed to derive a Hamiltonian formulation for the gravitational Vlasov-Poisson system (Kandrup (1990)). For self-gravitating systems, there are no homogeneous, static equilibria and the sign of 3(2)H becomes more ambiguous because of the attractive nature of the gravitational interaction. For the gravitational Vlasov-Poisson system there exists (Sygnet, Des Forets, Lachize-Rey, and Pellat (1984); Kandrup and Sygnet (1985); Kandrup (1991a)) a stability theorem which is a direct analog of a similar theorem for isotropic electrostatic equilibria. For the gravitational Vlasov-Poisson system, any spherically symmetric equilibria f(E, m) that is a monotonically decreasing function of the particle energy E = p2/2m + mdo, for all values of the mass species m, is necessarily stable. The symbol 4Ino represents the gravitational potential which is a functional of the Boltzmann distribution fo. The flat space Vlasov-Maxwell system has had its corresponding Hamiltonian structure studied through a variety of methods (Iwinski and Turski (1976); Morrison (1980a); Marsden and Weinstein (1982)). The stability criteria for the flat space Vlasov-Maxwell system are directly analogous to those of the aforementioned Vlasov-Poisson system. Therefore, for a dynamically accessible perturbation, 6)H = 0, and the sign of the second variation 6(2) H provides a useful criterion for stability. The Hamiltonian structure of an electromagnetic fluid in special relativity, has, to a certain extent, been analyzed (Holm (1987)). We wish to derive the Hamiltonian structure for the general curved space Vlasov-Maxwell system. The Lie bracket utilized by Marsden and Weinstein (1982) for the case of a flat space Vlasov-Maxwell Hamiltonian system will be used here. However, unlike Marsden and Weinstein, the formulation will be given, with the exception of equations (2.59) and (2.60), in terms of the conjugate electromagnetic variables Ai and HI and the canonical particle momentum pi. It is possible (Marsden and Weinstein (1982)) to choose the electric and magnetic field densities Ei and B' and the contravariant physical momentum P' as alternate variables. However, this choice, for a curved space-time background, yields an unnecessarily complex formulation. For example, one can make a comparison between equation (2.45) and equations (2.59) and (2.60) to see this. The total Hamiltonian consists of an electromagnetic piece HEM and a matter contribution HM. The flat space electromagnetic Hamiltonian, corresponding to the canonical formulation, is generalized to the setting of a curved background space-time. The matter Hamiltonian used by Marsden and Weinstein (1982) is different from the one used here. If we write the canonical formalism in terms of an ADM 3+1 split into space and time, and select momentum variables corresponding to particle mass m and spatial momentum pi, rather than four-momenta p, (or Pa or P0), then, for simplicity of application, we should use the matter Hamiltonian HM discussed here, but not the form considered by Marsden and Weinstein (1982). With these choices for the Hamiltonian components it will become obvious that (i) the Vlasov equation is generated by the canonical particle energy E and (ii) the current J' is given by the functional derivative 6HM/6Aj. 2.5 The Vlasov-Maxwell Hamiltonian Formulation In analyzing the Hamiltonian structure of the curved space Vlasov-Maxwell system, it is convenient to work with the canonical coordinates (xv, p,). The canonical momentum p, can be written in terms of the physical momentum P" and vector potential AA as po = g,,P' + eA,. For a fixed particle mass m, the mass shell constraint g,~,PP' = gAVPttPV = -m2 allows one to fix the value of any four-momentum component in terms of the remaining three. The expression for the fundamental phase space element is ((-g) 12d4x) ( 1/2 dp) = -d4xd4p. ( -_g)12 ) The explicit 3+1 split into space and time, along with the mass shell constraint, makes it convenient to choose pi and m as momentum variables. This choice allows one to deduce the expression -mdm = gtPdP = PtdPt = (pt - eAt)dpt. One calculates that dpt = -mdm/Pt, which is used to rewrite the covariant volume element as 4 p__ t 3 3d 3 -d xd p-> - -drd3 xd p - = dTd3 xd pdm = dTdFdm, (2.31) in Pt/ where T represents the invariant proper time and dF represents the invariant 6-D phase space volume element corresponding to a fixed particle mass m and a constant time hypersurface. The number of particles with mass m in the infinitesimal six dimensional phase space volume element at time t can be defined as f(xi, t, pi, m)d 3xd3 pdm, (2.32) where f denotes the distribution function. The distribution f must be an (observer independent) invariant because, from equation (2.31), d3xd3pdm is also invariant. If the masses of the particles are restricted to a single value mo, the distribution function can be written as f (x, t, Pi, m) = f(x, t,pi, mo)6D(m - Mno) (2.33) The electromagnetic Hamiltonian can be calculated in the following manner. One takes the standard electromagnetic action SEM = - -1 (_g)1/2d4xFvF"v J dt d xL dt d 3X(g)2L, (2.34) where F,, = 8 ,A, - 89,A,, and imposes the electromagnetic gauge condition At = 0. One can decompose the spacetime metric g,,, according to the ADM formalism, into the lapse function N, the shift vector N', and the spatial three-metric hij. The line element (cf. Arnowitt, Deser and Misner (1962, p. 227)) ds2 = -gttdt2 + 2gtidtdxi + gijddxidj, (2.35) has metric components gtt = NiN' - N2, git = Ni and gij = hij. (2.36) The contravariant metric components are gtt 1 it Ni g = N21 9 N2 and NzNj gNj = hj N2 (2.37) One can raise and lower indices with the spatial three-metric. The inner product hijhjk 6,k furnishes an example of this. The "time" coordinate t is chosen so that an "equilibrium" corresponds to a space-time metric and electromagnetic field that have no t dependence. This "time" condition applies to static and stationary space-times, which possess timelike Killing vectors fields t', with an electromagnetic field seen as independent of time from a rest frame. The field momentum can be calculated as BL 8(OtAi) (-g)1/2Fit = hijtAj - h/2 (h - NN)Fkj (-g)' - fl~A3 h Nt"' N2 )Fk3 hi/21/2 N hiJtAJ - h N /2 h ikFjk (2.38) and inverted to yield the expression N htAi = hi/2hij + N Fji. (2.39) The electromagnetic Hamiltonian HEM(Hit, Ai) - d3x(IIOtAi - L) (2.40) can be written as dHEM 3XJ dx hijiiHJ + d3xiNjFji H E M " = 2- 1 2 + I d3xNhl/2 hik hJlFijFkl. (2.41) The matter Hamiltonian can be written as the minimal coupling between the distribution function f and the canonical particle energy e HM = dFf e. (2.42) The canonical particle energy e = -Pt satisfies the expression gttpt2 + 2gtiPt (Pi - eAi) + gij (Pi - eAi)(pj - eAj) =-m2, (2.43) where the gauge condition At = 0 has been imposed. The full electromagnetic Hamiltonian may be written as H = HEM +HM- d3xNhl/2jBiAi = HEM + HM + HB, (2.44) where, in analogy with a fixed background charge density, an additional coupling term HB, corresponding to a fixed, externally imposed charge current jB', has been added. The vector potential Ai dependence of the energy e, as made explicit in equation (2.43), implies that the matter Hamiltonian HM couples the electromagnetic field to the Boltzmann distribution f. The equations (2.42) and (2.44) have been written in terms of a single species of matter. For the case of multiple species, we must define Boltzmann distributions fa for each species a and replace (2.42) with a summation over separate distributions faThe Lie bracket operation (F, G) will act on pairs of functionals F[Ai, Hi, f] and G[A, Hi, f] which are defined on the infinite dimensional phase space (Ai, Hi, f). Consequently, the bracket may be written as (I a F bG SiG JF) / {F bG} (F,G)[Ai,IP,f] x = dG 6 ) + dFfI f , (2.45) f JAi 611 g li 6 f, 'j 6f 6f' where a Ob Ob a {a, b} = -- O - --c~ ai OPi 8xi pi represents the ordinary canonical Poisson bracket, and 6/6X represents the functional derivative corresponding to the variable X. It is straightforward to prove that the bracket is antisymmetric and satisfies the Jacobi identity (F, (G, H)) + (G, (H, F)) + (H, (F, G)) = 0, (2.46) which implies that it is a bona fide Lie bracket. If one considers restricted functionals F[f] and G[f], that possess only f dependence, the bracket (2.45) reduces to (F,G)[f] = drf 6, , (2.47) f f 6f which is the proper bracket of the Vlasov-Poisson (Morrison (1980a, b); Gibbons (1981); Kandrup (1980)) system. For functionals F[A,, I'] and G[Ai, '], whose dependence is constrained to Ai and H', the bracket becomes 3( 6F bG 6G 6F (F,G)[A,II] = d3x - - " (2.48) J ( 6A, 61 i A, 3fl ) For vacuum electrodynamics (cf. Wald (1984)), formulated in terms of a specified 3+1 decomposition, this is the proper bracket. One may obtain equation (2.48) from the covariant bracket ((F,G)) = dxV"( W G -fG 6F (2.49) f A,. M11- JA, 6H61 ' where II" = JS/6(8,A,) and VI is an arbitrary vector field corresponding to a suitable foliation. The electromagnetic bracket given by Marsden et al. (1986) reduces to equation (2.49) in the vacuum limit. For an appropriate choice of bracket and Hamiltonian, the constraint equation OtF = (F, H) (2.50) should reproduce the correct dynamical equations of the Vlasov-Maxwell system. To prove that this is the case, one must first calculate the following three identities 6H -W = E, (2.51a) 6H N Hi= h-/2hjj + NJFji, (2.51b) and JH 6A- J(NI - NHIJ) + k(Nhl/2hiJhklFLJ) - Nh1/2(Ji +JBi), (2.51c) where S= e (-g) -1/2d3pf (P eA . One can use the three identities to prove that the Vlasov-Maxwell system is obtainable from equation (2.50). It is straightforward to check that the phase space variables, Ag, H' and f, satisfy the appropriate dynamical equations JH Oit Ai= (Ai, H) = 6H, (2.52a) atHi= (H, H) = (2.52b) 6Aj and Otf = (f,H) = - f, = {ef}. (2.52c) The first two of these equations are the canonical equations for an electromagnetic field with a total current source tot = J + jB'. The equation for OtAi is equation (2.39) which defines the field momentum HI. Likewise, the second equation is the 3+1 decomposition of (-g)1/2VFi= (_g)l/2Jtoti, (2.53) which is the Maxwell equation -Otf + V x B = ,t generalized to curved space. The Vlasov equation can be written as Of Pi OOf P A Of 1 PyP, Ogi Of - = - - -+-(.4 Ot Pt xi Pt ai Opi 2 pt Oxi pi'2.54) where Pp = pt, - eA,, and P" = g"v(p, - eA,) are expressed in terms of the canonical variables xi and pi. This Vlasov equation is different from the usual one which is expressed in terms of the canonical momentum, with f(x'i, t,pi). However, a transformation to the physical coordinates (x', t, P) will demonstrate that f(x'i, t, P) satisfies the standard Vlasov equation (cf. eq. (2.29)) Of OA Of P [Of e OAj Of at Ot OP, Pt Ox8 Oxi Pj P3 DAj Of 1 PP,, Og " Of +e- + (2.55) P i o Pi 2 Pt Oxi opi( One of the curved space dynamical Maxwell equations is given by equation (2.52b). One may obtain the curved space generalization of the other dynamical Maxwell equation OtB + x = 0. For example, we may define the quantity Bi = iikjAk = h1/2e ijkO Ak. (2.56) Through the use of the alternating symbol Eijk or the alternating tensor e, it will satisfy the following equation of motion OtBi i= EjkojtAk = O [fijk (h2 hkl' + NIFkI) . (2.57) In an alternate formulation, OtB' can be written in terms of the electric Ei -lHi and magnetic B' field densities OtBi' = -j ( 2 ijkhkEl) + Oj (Bi Nj - Bj Ni). (2.58) Either equation (2.57) or (2.58) is the dynamical component of the dual Maxwell equation V,*F" =- 0. One can construct the curved space generalizations of the constraint equations V - = 0 and V- E = p. The first is a geometric identity resulting from the fact that Bi is constructed from an alternating symbol. The second constraint possesses a more complex character. The continuity equation VJP = 0 and the dynamical equation for VFAi = 0 ensure that, if the electric field constraint is initially enforced, the constraint will be maintained through time. Also, gauge invariance (cf. Misner, Thorne and Wheeler (1973)) can be used to demonstrate that the electric constraint will hold initially. It is possible to rewrite the bracket (2.45) in terms of variables which are physically more intuitive. According to Marsden and Weinstein (1982), the bracket may be written in terms of Ei = -Ii and Bi. These variables correspond to the true electromagnetic degrees of freedom and appear in equation (2.58). The bracket will take the form (dzhl/2 e ijk 6 a 6G id a i JEi Oxj Bk 5E-x 8 6Bk W, f (2.59) + f f" bf 'f with functionals F[Bi, Ei, f] and G[Bi, E, f]. Also, if one rewrites the distribution f(x', , pi) in terms of the physical momentum, instead of the canonical P, this will lead to a modified bracket form (F,\= d3zh/2eijk JP 8{ 6Bk 6Ei Oz 6Bk dl' J , if if + d(I 6 ajO 6aJ86F 6E'i Pi6f Epi i6f I 8e B (6F\ 6 6G\ ï¿½J drh- 1/2 eijk Bi -' ( G) 2.60) with functionals F[Bi, Ei, f] and G[Bi, Ei, 1]. It is clear that equations (2.59) and (2.60) correspond to covariant analogues of the equations (5.1) and (7.1) found in Kandrup and Morrison (1993). 2.6 Perturbations of Time-Independent Vlasov-Maxwell Equilibria If there is to be a meaningful definition of "equilibrium" then the curved space, which the Vlasov-Maxwell system is situated in, will be a static or stationary space-time that possesses a timelike Killing vector. An "equilibrium" will correspond to the case where both the electromagnetic field and matter distribution are independent of time. Consequently, a "natural" choice of time coordinate t will cause the metric gy, and electromagnetic field F,, to lack time dependence. This choice leads to a time derivative of the canonical momentum HI that is equal to zero. The "electric field" 1I must vanish if the derivative OtAi is to be independent of time. However, unless the second derivative Ot2A A 0, the F,, tensor will not be independent of time. When the canonical momentum pi = P + eAi is expressed in terms of the physical variables x' and P, it will, at "equilibrium", possess explicit time dependence because of the non-vanishing of OtAi. Therefore, for equilibria of this type, the time derivative Otfo, cannot be set to zero. Instead, the equilibrium distribution fo must obey the relation Bfo DA, Of0o OA, Ofo = e ---- --= - (2.61) at at 8P 8t p One wishes to know, for a given equilibrium Xo = (Ai, Hio', fo), what mathematical form the dynamically accessible perturbation 6X = (Ai, 6l, 6f ) will possess. It is straightforward to determine this because there exists an infinite number of conserved constraints, corresponding to conservation of phase, that are associated with evolution in the Vlasov-Maxwell system. Specifically, this implies that the numerical value of any functional constraint C[f()] = drx(f) (2.62) remains constant as the distribution f evolves, i.e., dC[f]/dt - 0. (2.63) Consequently, any dynamically accessible perturbation 6f must leave the numerical value of any such constraint unchanged. It follows that 6C[f] C[fo + 6f] - C[fo] 0, (2.64) for all orders of perturbation theory. A perturbation must obey this mathematical requirement to be dynamically accessible, i.e., in accord with the equations of dynamics. A solution to (2.64) will yield the form of the phase-preserving perturbation. It may be shown perturbatively that the deformation 6f, which serves as a solution (Bloch, Krishnaprasad, Marsden and Ratiu (1991)) to equation (2.64), must be 1 / = fo + 6f = exp({g,.})fo = fo + {g, fo} + g, {g, fo}} +..., (2.65) where g is a generating function of the variables x and pi, and {a, b} denotes the ordinary canonical Poisson bracket. So, a canonical transformation, with a generating function g, corresponds to a dynamical perturbation. Unlike the distribution fo, the canonical variables Ai and IP do not satisfy corresponding independent constraint equations similar to equation (2.61). Thus, the dynamically accessible perturbations 6Ai and 6Hi are only constrained by the field equations whose source is provided by the dynamically accessible perturbation 6f. One can demonstrate that the first variation 6P)H of the Hamiltonian (2.44) vanishes for a dynamically accessible perturbation 6() f (2.65). The first variation of the electromagnetic part of the Hamiltonian possesses the form 3()HEM = dxji( hN2hij13 + NJFji) - d3x6AiO(HiNJ - fIJNi) -f dx Aiaj (Nh 1/2hikhLFkl) =f d3x6bi(8tAi) + f d3xNhl/26AiVPF'i. (2.66) The last equality has been obtained from equation (2.39) for 8tAi and the fact that, for the case of a time-independent equilibrium with tHIli = 0, terms proportional to 6Ai may be combined to form VLFPi. Utilizing the unperturbed Maxwell equation VUFI" = J" + jBv leads to the expression 6()HEM = d3x(OtAi)6Hi + f d3xNhl/2 (ji +jBi)6Ai. (2.67) Also, the matter Hamiltonian has the first variation )Hm =J dF(f A6Ai + e6f = - d3xNhl/2 i6A + f dref. (2.68) The coupling Hamiltonian will have the variational form 6)H = - f d3xNhl/2jB 6Ai. (2.69) One may combine equations (2.67)-(2.69) to get the total variation of the Hamiltonian )H =f d3x(otAi)SHi + f drf f. (2.70) This last expression has been, of course, calculated for an arbitrary perturbation. With the special choice of a dynamically accessible perturbation, it is straightforward to show that this variation is vanishing. For an equilibrium, the magnetic field is static atBt = 0. From equation (2.57) one can deduce that 8tAi may be written as the gradient of an arbitrary scalar, so that OtAi = -Oi 4. (2.71) Substitution of equation (2.71) into the first term of equation (2.70) yields the expression J d3 x(tA)6Ii = - d3xNhl/2oi6Fit = f d3xT8i[Nhl/2 6Fit] = f d3xNh/2 Vi6Fit, (2.72) where an integration by parts has been performed and the expression Ii = Nhl/2Fit has been employed. Using the perturbed field equation and writing JJt in terms of 6f leads to Sd3x(OtAi)611' = f d3xNhl/ j6t = e f dPW'f. (2.73) By substitution of the dynamically accessible form of 6f (2.65) into this expression, one can obtain the final result e f dPf 6f =e dF1{g, fo} = -efdg{, fo}. (2.74) Also, substitution of the dynamically accessible 6f into the second term of equation (2.70) leads to /dI'E6f = dFe{g, fo} = -dFg{E, fo} = -f dgtfo = -e d gtAj . (2.75) The final expression of (2.75) may be rewritten in the following form ef dFg(Oi) f = e dFg{, fo}, (2.76) by replacing otAi with -OiT. Consequently, both the first (cf. 2.74) and second terms cancel out with each other. Therefore, the energy H has an extremal 61)H = 0 for a time-independent equilibrium in the curved space Vlasov-Maxwell system. In this setting, the space-time possesses a timelike Killing vector field and, at equilibrium, a static electromagnetic field F,, (the electric and magnetic fields are time-independent as viewed by a fixed observer). One now may wish to calculate the second variation 6(2)H. This variation may be conveniently reduced to the sum of three terms 6(2) H = 6(2) H + 6(2) H2 + 6(2) H3, (2.77) where 6(2) H f d3 X hijbllii + j d3x6biN 6Fji +1 d3xNhl/2hikhj6Fij6FkL (2.78) are the terms from HEM that possess products of first variations, 6(2)H2 = d3x6(2)Wi( h hijIJ + NJFji) -f d3x6(2)AjO(IiNJ - HJ Ni) - fd3x6(2)AiOj(Nhl/2h ikhjiFkl) (2.79) are the terms from HEM that have second variations of canonical variables, and 6(2)H3 -= d[Ej(2)f + 26e6f + 6(2)Ef0] - d3xNh/2jBi6(2)Ai (2.80) are terms corresponding to HM and HB. In these expressions, E represents the unperturbed particle energy and a variation 6X not possessing a superscript represents a first variation (1). One can readily calculate that the first and second variations of the particle energy E are Be i P -eAi3 E = = -e pt A (2.81) and 6(2)6_ 1 a2_ A j + (2)A 2 oAi9Aj OA i e- fP - eA)A Pe - eA' 6 Pi-eA JA - e pt' 6(2)Ai. (2.82) Therefore, combining the second and third terms in j(2)H3 leads to the result f d[6(2) Efo + 6E6f = eri'P e - e e~\ - f Nhl/2d3ji6(2)Ai -e f d [6(Pi'-eAi + (P'-eA i ) 6f] 3A, +2 f dFfo6( Pi - eAi = - d3xNhl/2 Ji6(2)Ai - d3xbJi6Ai -e2 f Nhl/2d3xNhl/2oijA 6Aj, (2.83) where (ij -J -1/2d3pf a2f0 AiAoAj Using the unperturbed field equation allows one to rewrite the second contribution 6(2) H2 as j(2) H2 'd3x(2)li(8tAi) + d3x(-g)1/2(2)AiV,F"'P = f d3x6(2) i(8(tAi) + d3xNhl/26(2) Ai(J' + jBi). (2.84) One can combine the second term of equation (2.84) with the last term of equation (2.80) to cancel out the first term of equation (2.83). Therefore, in the case of an arbitrary perturbation 6X, the second variation 6(2)H will be 6(2)H = d3XN hij6HiI6 + d x6IiN6Fys 2 e2fd1/2/OJ6iA + d3xNhl/2hikhjl6FJFkl - d3xNhl/2ij6AidAj - f d3xNhl/2jibAi + f d3x6(2)IIi(OtAi) + fdrE(2)f. (2.85) Restricting attention to perturbations that are dynamically accessible allows one to rewrite the last two terms of equation (2.85) in a simpler form. Using the scalar function %, one may write f d3x6(2) 'itAi = f d3xtqji6(2)fli f d3xNh/2 V6(2)Fit = fNhl/2d3xz6(2)Jt. (2.86) In the case of a dynamically accessible perturbation, one can rewrite equation (2.86) as e dPrT(2) f = e d' {g, {g,fo}} = - dif{g, fo}{g,e,}. (2.87) Likewise, the last term of equation (2.85) will take the form dre1(2) f d ,fo dF{gfo}{g, }. (2.88) 65 Therefore, for a dynamically accessible perturbation, the second order variation 8(2)H possesses the form 6(2)H d3xN hijHi6HJi + d3xNHi6Fji + d3xNhl/2hikh jlFij6Fkl - e d2xNh /2 Aj6Aj - d3xNhl/26j iA - J dF {g, fo }{g, e + eqI}. (2.89) This expression can be reformulated in terms of the electric and magnetic field densities ( E' and B' ) as 6(2)H = d3x hijh (EibE + 6B6BJ) - f d3x N /2eijkbEJ6Bk -2 1 d3 xNhl/2 i j6AJ6A - d3xNhl/2 6Ji'Ai - 1 d'{g, fo}{g, e + ef}, (2.90) where, as before, the alternating tensor is denoted as eijk. 2.7 Stability Criteria for Time-Independent Vlasov-Maxwell Equilibria It is straightforward to show that the Poisson bracket {g, e + e T}, which appears in equation (2.90), possesses the mathematical form Pi Og 8 09Bg {g,e +ef}= {g,E}= Pi + e--P Pt 8z i 89 Pi P3 BAy 8g 1 PyP,, 8g " Og +e P-. g 1 P1j', 49 -g (2.91) t Oxi pi 2 Pt 8xi api If the physical momenta P are considered fundamental, it may be rewritten as P' = g 8+ qOg PJ Og 1 PP,, g09" Og {g, E} = + ex - eF ptP 2 t Ox (2.92) Pt 8zi 8zi api ye~ ptSi 2 Pt 8zi 4pi The definition of % leads to the result {g, E} = -Dg, (2.93) where Pi . OAi 0 PJ oAi oAj 0 1PP, 8g " 8 - e +e +-- Pt Bxie-- +pi Pt 8 8zx 0xi ),-i 2 Pt 8xi aPi represents the unperturbed Liouville operator. The evolution of the distribution occurs along the characteristics corresponding to time-independent equilibria. This evolution is governed by the Liouville operator. The evolution equations do not imply conservation of the canonical energy e whether there exists a time-independent equilibrium or not. Contrary to expectation, one will find de c Pi 0Ai d- = -e P (2.94) dt at Pt at which, due to the time dependence of the vector potential, will not vanish. Since II is time-independent, one may compute - = P- (2.95) at pt t ' which implies that dE/dt - 0. It follows that E, not E, will be a conserved quantity for a time-independent equilibrium. One can construct a time-independent solution fo(E, m) to the Vlasov equation from these constants of the motion. Other equilibria are possible, however, this choice corresponds to an ensemble of particles with an isotropic velocity distribution. Such equilibria are ubiquitous. For this choice, equation (2.90) may be written as 6(2)H = dx hi (I'lIj B + B'6BJ) 2 Jh1/2 3 i k_ e 2 2i + d3x 1/2eijk6HIJ6Bk - f dxNh /2Oii6AjiAj - d 3xNh/26j 6A + I dF(-FE)l{g, E}2. (2.96) One may examine the various limits of (2.96) or (2.90). In the case of pure vacuum electromagnetism, the energy associated with a linearized vacuum fluctuation possesses the form 6( 2d3x__/2hij(ri6jï¿½ 6Bi6Bi) 6(2)H = da hyJIJ + daB) 2 Jh/2t d+ dx -2eijk6H6B, (2.97) which is obtained from equation (2.96) by turning off the matter distribution. As another example, one may examine spherically symmetric configurations in the non-relativistic approximation. Spherical symmetry implies that the shift vector N' vanishes and the slow motion approximation implies that the contributions 6J'iAi and 6Ai6Aj, which are of order (v/c)2, may be safely neglected. This yields the covariant analogue of the linearized energy associated with the flat space Vlasov-Poisson system, viz. 6(2) d3xH = N 2hij6HI6P - dF{g, fo}{g,E}. (2.98) 6()H=' d hjim One can deduce nontrivial stability criteria from the sign of the energy 6(2)H. If, for example, 6(2)H > 0, for all generating functions g, then any static equilibrium must be linearly stable. On the other hand, if, for some generating function g, the condition 6(2)H < 0 holds, then a linear instability is not necessarily guaranteed. Such a configuration, although it possesses linear stability, will probably be nonlinearly unstable or unstable if subjected to dissipative effects (Holm, Marsden, Ratiu and Weinstein (1985); Moser (1968); Morrison (1987)). Whether an equilibrium corresponds to an energy minimum is, generally speaking, difficult to determine. There is, however, one case where it is simple to show that the condition 6(2)H < 0 exists for some phase-preserving perturbation. For example, assume that the energy derivative FE is not everywhere negative. If one chooses a perturbation such that 6JU = d{g, fo} p = 0, (2.99) i.e., the overall charge density remains unaffected but the velocity profile is shuffled, then 68 one can assume 6Aj and 6l are zero. However, the dynamical perturbation must obey the relation 6(2)H = - df(-FE)I{g, E} 2. (2.100) By shuffling the velocities a significant amount where FE is positive in phase space, and by a minimal amount elsewhere, one can obtain a negative energy (2.100). This is the curved space analogue of the fact, first observed by Morrison and Pfirsch (1989), that "all interesting equilibria are either linearly unstable or possess negative energy modes." CHAPTER 3 THE VLASOV-EINSTEIN SYSTEM One may now consider the full covariant Vlasov-Einstein system. We attempt to give a Hamiltonian formulation of the collisionless Boltzmann equation of General Relativity (Vlasov-Einstein system). Energy stability criteria will be deduced for this Hamiltonian system. For this system (cf. Israel (1972, p. 201); Stewart (1971)), the Boltzmann distribution f will be defined with respect to the cotangent bundle corresponding to the space-time of choice. The characteristics, which correspond to geodesics of the space-time, are the trajectories along which the distribution f evolves. The system is complicated by nonlinearity because the space-time is determined by the distribution f which acts as a source of the mean field Einstein equation. Thus, the background space-time is not fixed and eternal as it was for the case of the Vlasov-Maxwell system in chapter 2. Therefore, in this scenario, the Boltzmann distribution has its evolution governed by the Vlasov equation pa 8f 1 8gbc PbPc Of m a 29xA - 0, (a,b,c,... 0,1,2,3) (3.1) m 8xa 2 Oxa m 8Pa where the space-time metric gab must self-consistently satisfy the mean field Einstein equation Gab = 87rTab = 8r f p paPb. (3.2) There have been earlier investigations of the linear stability of restricted cases of the Vlasov-Einstein system. One example corresponds to spherically symmetric equilibrium solutions whose perturbations are also spherically symmetric. For example, in the earliest investigation by Ipser and Thorne (1968), a variational principle is derived from the dy69 namical equations. Unstable modes can be detected by analyzing the variational principle. Ipser and Thorne (1968) write the perturbed distribution 6f as a sum 6f = bf_ + 6f+, where bf+ and 6f_ are even and odd, respectively, with respect to spatial momentum inversion. It was noted that, in certain circumstances, the odd perturbation component bf satisfies the operator equation 9t26f_ = T&f_. This equation is second order in time and the symmetric operator T, which is defined with respect to a suitable Hilbert space, is dependent only on the equilibrium fo. The pioneering work on this technique was carried out by Antonov (1961), albeit in a non-relativistic setting. When applied to a relativistic system, this method has been subjected to extensive numerical analysis by Ipser (1969a, b) and Fackerell (1970, 1971) among others. Ipser (1980) then went on to analyze spherically symmetric systems in terms of plasma physics techniques. An extension of the Newtonian "energy arguments" (cf. Ipser and Horwitz (1979)) was carried out with the use of an energy-Casimir argument to obtain linear stability criteria. This approach, due to Newcomb (cf. the summary in the appendix of Bernstein (1958)), had originally appeared in plasma physics. This approach is also discussed by Morrison and Pfirsch (1989) (and in internally cited references). The basic idea is as follows. For generic perturbations 6 f, equilibrium solutions fo of the VlasovEinstein system do not correspond to energy extremals because the constraints associated with phase conservation imply the existence of infinitely many conserved quantities, C[f]. It is possible, however, for equilibria to be extremal by considering only perturbations which ensure the conservation of one particular constraint, c[f]. Thus, the sign of the constrained second variation, j(2)He, yields a nontrivial stability criterion. For example, the condition 6(2)Hc > 0 ensures linear stability. Recent work has applied these plasma physics arguments (cf. the papers of Morrison and Greene (1980), Morrison (1980a) and Morrison and Pfirsch (1989), and internally cited references) to Newtonian galactic dynamics (cf. Kandrup (1990, 1991a, b)). Kandrup and Morrison (1993) have adapted these ideas in order to furnish a more systematic Hamiltonian formulation. In this work, the Hamiltonian character of the dynamical evolution, which was ignored in earlier research, is addressed. Also, all of the constraints (not just one) are implemented. Consequently, the earlier work of Ipser (1980) is clarified and extended. So far, only the stability of spherically symmetric equilibria, subjected to spherically symmetric perturbations, have been studied. Research prior to Kandrup and Morrison (1993) assumed that the Boltzmann distribution was restricted in form. For example, the distribution was supposed to be a monotonically decreasing function of the particle energy e. Lie algebraic techniques were employed by Kandrup and Morrison (1993) to show that the spherically symmetric Vlasov-Einstein system, subjected to a three-plus-one decomposition into space and time, is Hamiltonian. Consequently, it is necessary to identify a bracket [., .], i.e., a cosymplectic structure and a Hamiltonian function H, which are used to construct a Vlasov equation Ot f = [f, H] for the distribution function f, where at denotes a coordinate time derivative. Since the dynamical evolution is governed by a "generalized canonical transformation" in the infinite-dimensional phase space associated with distribution functions f, it is obvious that the Vlasov-Einstein system is Hamiltonian in character. However, the transformation is not truly canonical since f is a non-canonical variable. Due to constraints that are associated with conservation of phase, the Hamiltonian system is constrained. These phase-preserving constraints can be straightforwardly implemented in an explicit fashion. This enables one to prove that time-independent equilibria are energy extremals, such that 6P)H = 0 for all dynamically accessible perturbations bf. If the second variation 6(2)H is non-negative for all dynamically accessible perturbations, then the system is guaranteed to be linearly stable. However, as noted in Kandrup and Morrison (1993), a negative second variation 6(2)H < 0, corresponding to some dynamically accessible perturbation 6f, does not prove linear instability. It should, however, at least guarantee structural instability with regard to dissipation or any nonlinearities which are present (cf. Moser (1968); Morrison (1987); Bloch et al. (1991)). Unfortunately, although this approach has proven highly effective, it is still restricted to systems, whether perturbed or not, that are spherically symmetric. Most related research, with the notable exception of Ipser and Semenzato (1979), has been directly based on this assumption. Due to the Birkhoff theorem, the spherical symmetry of the system ensures that gravitational radiation will not be present, i.e., the radiative degrees of freedom will not be activated by the dynamics. Therefore, for appropriate boundary conditions at infinity or any horizons that may be present, the gravitational metric 9ab at time t will be uniquely determined by the stress-energy tensor Tab at that instant. This can be seen by implementing the standard Schwarzschild gauge, where only the pure radial gr and pure temporal gtt metric components remain free. For this particular gauge, the t - t component of the Einstein field equation fixes grr uniquely in terms of the energy density Ttt. Also, if grr and the radial "pressure" Trr are both known, then the r - r component of the field equation will fix gu. This situation is analogous to the externally imposed curved space Vlasov-Maxwell system considered in chapter 2. For the case of spherical symmetry, the magnetic divergence equation guarantees the vanishing of the magnetic field. The symmetry ensures that only the radial component of the electric field Er does not vanish. The radial component is fixed uniquely by solving the Poisson equation for a charge distribution p. It is possible to devise a Hamiltonian formulation which is applicable to general space-times, rather than only the spherically symmetric case considered by Kandrup and Morrison (1993). This will complicate the physics because the gravitational metric gab must be treated as a dynamical variable, as well as the Boltzmann distribution fo. Kandrup and Morrison employed only the distribution function fo as a dynamical variable. They worked in the context of an infinite-dimensional phase space of distribution functions fo at a given instant of time t. For a general space-time, we must employ three dynamical variables. These are (1) the spatial three-metric hab(Xi) for a t = constant hypersurface and (2) the conjugate momentum Hab(xi), along with (3) the Boltzmann distribution fo. One can define functionals F[hab, Hab, f] in terms of the infinite-dimensional phase space (hab, Iab, f). It is necessary to select a Lie bracket (.,.) and a Hamiltonian function H such that the correct dynamical equations governing the evolution of hab, nab, and f are generated by the constraint equation 8tF = (F, H). 3.1 The Vlasov-Einstein Hamiltonian Formulation It will be convenient for the dynamical analysis of a combined space-time-distribution system to invoke an ADM decomposition into space and time (cf. Wald (1984)). One may view the space-time, described by a metric gab, as a foliation into a series of spacelike hypersurfaces. The hypersurfaces are parametrized by a global time function t, and na will denote a unit normal vector to the hypersurface. The spatial three-metric hab(Xzi) for at = constant hypersurface is defined by the projection tensor, hab = gab + nanb (a, b, ... = 0, 1, 2,3). (3.3) A timelike vector field ta, which satisfies the condition taVat = 1, will be introduced. This vector field may be decomposed into a lapse and shift, respectively N and No, which are normal and tangential components with respect to the hypersurface, i.e., N = -tana and Na = habtb. The spatial three-metric hab and its conjugate momentum IIab correspond to dynamical degrees of freedom, while the lapse N and shift Na are gauge variables with respect to the Hamiltonian and momentum constraints. By construction, it is obvious that the time-time and space-time components of the spatial metric vanish, i.e., htt = hit = htt = hit = 0 (i, j,... = 1, 2, 3). Therefore, the ADM decomposition of the space-time metric gab is gtt = NiN' - N2, gt = N, and gij = hij, (3.4) and possesses contravariant components gtt 1 N' g = gi --N 2'1 N2 and 'i 0 h - NiNJ N2 (3.5) Analogous to the Vlasov-Maxwell system, the time coordinate is a "natural" coordinate corresponding to a timelike Killing field. An equilibrium solution of the Vlasov-Einstein system implies that the distribution function, the metric, and the conjugate momentum tensor are all independent of time t. A Lie bracket (F, G), acting on functional pairs F[hab, Igab, f] and G[hab, gab, f], will define the cosymplectic structure of the Hamiltonian formulation. For the fundamental dynamical variables hab, [ab, and f, we take (F, G) to possess the form (F,G)[hab, gab, f] = 167r d 3x G 6G hab Fab f Jhab 6fgab -6ab 6nab 6F bG + df f ,(3.6) 6f 5 where b9a Ob Ob Oa Oxi Opi 8xi OPi denotes the standard canonical Poisson bracket, and 6/6X denotes the functional derivative with respect to the variable X. For a t = constant hypersurface, the covariant phase space volume element can be written as dF = d axd3pdm. (3.7) If d3p refers to spatial momentum, as defined in the cotangent bundle, then d3xd3p is covariant. Therefore, the spatial coordinates x', the spatial momentum pi, and the mass m are held fixed with respect to variations carried out at some time t. Consequently, the volume elements dF and dx are not subject to variation, i.e., 1d = d 3x = 0. Kandrup and Morrison (1993) discuss other possible choices in greater detail (e.g., the original IpserThorne (1968) prescription). It is straightforward to show that the bracket is antisymmetric and satisfies the Jacobi identity (F, (G, H)) + (G, (H, F)) + (H, (F, G)) = 0. (3.8) Therefore, it is a bona fide Lie bracket. Consequently, the dynamics associated with this bracket and any Hamiltonian H will be symplectic with regard to the infinite-dimensional phase space (hab, 11ab, f). The bracket (3.6) is an obvious analog to the bracket for the curved space Vlasov-Maxwell system. In the absence of matter, the bracket (F, G) corresponds to the natural bracket of vacuum gravity (cf. appendix E in Wald (1984)). In the case of a spherically symmetric space-time, the metric can be written as a functional of the distribution f and eliminated as a dynamical variable. Consequently, the bracket (F, G) will reduce to the bracket [F, G] used by Kandrup and Morrison (1993). It is relatively straightforward to identify the Hamiltonian function H. The Hamiltonian will be the sum of a purely gravitational piece HG and a matter contribution HM. The Hamiltonian of vacuum gravity (cf. Arnowitt, Deser, and Misner (1962)) has the form HG d3X-G, where G h1/2 (N[_3)R +h-1 (abfl - 112)] 167r 2 -2Nb[Da(h-1/2Hab)] + 2Da(h-1/2NbHab)}. (39) In terms of notation, the covariant derivative operator is denoted Da and the scalar curvature of the spatial metric hab is denoted (3)R. The matter Hamiltonian is taken to be the minimal coupling of distribution to particle energy, viz. HM = f d3xz M = dFfE, (3.10) where the ordinary particle energy is denoted e = -tapa. For appropriate "time" coordinates, the particle energy is E = -Pt. The fundamental momentum variables, which appear in equation (3.10), are the spatial momentum components Pi and the particle mass m. The particle energy e is a function of the variables {xi,i, m, t} and may be defined by the mass shell condition, E2 2ENapa (hab NaNb 2 N2 N2 N2 )PaPb = - . (3.11) An alternative form of HM is HM = Nh 2d3xTtt, where tt= dp f tpt=f d3pdpt fptpt (3.12) (-g)1/2 m Nhl/2 P is the t-t component of the stress-energy tensor written as a momentum integral of the distribution. By variation of the gauge variables N and Na, one can demonstrate that the Hamiltonian and momentum constraints hold. Performing the calculation Oe (E + Napa)2 ON - -N3pt (3.13) 8N -N3 pt enables one to determine the functional derivative 6HM d3pdm (e + NaPa)2] N f N -N2pt 1/2 d pdpt (e+ NaPa )2 b/2 = h J- hh/2Taba, (3.14) fNhl/2 N2 where na is the t = constant hypersurface unit normal vector. The identity of the form na = (1/N)(ta - Na) has been used in the final equality. The condition 6H/6N = 0, i.e., that H is extremal with respect to variations of the lapse N, must hold. This may be calculated by combining equation (3.12) with 6HG/6N. The final result is the Hamiltonian constraint equation Gabanb = 87rTabna nb, (3.15) which may be rendered as -3R + h- (lab ab 1112)= 167r fd3pdpt (e + Napa)2 (3.16) 2 Nhl/2 I N2 M. Likewise, one may calculate the result 6HM h/2 d3pdPt hbapb(E NCpcp) f = -h /(3.17) 6Na f Nhl/2 [ N Im' which leads to the momentum constraint 6H/6Na = 0, habGbcnC = 87rhabTbcnc, (3.18) that may be written as 2Db(h-1/2fab) = -167r d3pdpt [hbapb(E + Npc) f (3.19) Nh1/2 N m The initial value problem demands the imposition of these constraints. However, this is auxilliary to the dynamical aspects of the Hamiltonian formulation. Assuming that the dynamical equations are generated by the Hamiltonian H and Lie bracket (., .), it is guaranteed that the constraint conditions will be enforced and propagated to later times. If the bracket and Hamiltonian correspond to a suitable Hamiltonian formulation, then the constraint equation atF = (F, H), (3.20) which holds for arbitrary functionals F, must generate the dynamical equations of the Vlasov-Einstein system. The dynamical equations of the gravitational field can be investigated first. From computation these take the form Othab = (hab, H) = 16r 6Hb (3.21a) and OtHab = (lab, H) = -16r JH. (3.21b) Jhab Along with the functional derivative 6HM 5hab INhl/2 d3pdpt bd NbNchad NbNdhac' 2ENahbdpd f 2 Nhl/n2 acb _ 2 PcPd -N2 J- (3.22) the momentum constraint (3.19) and the standard equations for 5HG/bhab and HG/ggab (cf. Arnowitt, Deser and Misner (1962)) can be combined to yield the dynamical Einstein equations, viz. 1-6H 1 Othab = 16r -H = 2h-1/2N(ab - 2habll) + 2D (aNb), (3.23a) and JH 0tilab = -167r 6hab = -Nh/2 ((3)Rab (3) Rh ab) + 1Nh /2 ab (Ilcd Ied 12) 2 2 2 -2Nh- 1/2 (facfcb _ Hab) + hl/2(DaDbN - habDcDcN) 2 +hl /2 Dc(h -1/2 Nclab) - 2HIc(aDcNb) +87rNhl/2 d3pdpt f hacPcPdhbd. (3.23b) fNhl/2 M The Vlasov equation possesses the form Otf = (f, H) = {e, f}. (3.23c) This is just the 3+1 decomposition of the covariant Vlasov equation (3.1). 3.2 All Vlasov-Einstein Equilibria are Energy Extremals A solution to the dynamical equations, {hoab(X i), IIoab( i), fo(i,pi,m)}, that is timeindependent, will be an equilibrium solution of the Vlasov-Einstein system. Consequently, one may conclude that o9thoab = Otoab = tfo = 0. The aforementioned dynamical equations, combined with the conditions of equilibrium, imply that 6H/hab = 6H/6Hab = {eo, fo} = 0. The particle energy at equilibrium is denoted eo,. Generic time-independent equilibria, {hoab, Hoab, fo}, only constitute energy extremals with regard to the restricted class of dynamically accessible perturbations. However, one cannot assume that the first variation 6P)H vanishes for perturbations of an arbitrary type. The first variation 6(1)H will take the general form 6 )H= Jd, W hab+ 6 1 ha(1)ab+ 67L(1)NN + (1)Na + 6 (1) f . (3.24) Employing the dynamical equations and the conditions of constraint leads to S)H = 1 f d3x(-tHabS(1)hab + athab(1)Hab) + dE 6(1)f. (3.25) 16f The equilibrium condition guarantees that atlo'b = athoab = 0. The last term involving 6(1)f will only vanish for a phase-preserving perturbation, i.e., /)H = dFeo{g, fo} = - / dfPg{e0o, fo} = 0, (3.26) where an integration by parts has been performed. The final equality follows from the equilibrium condition {o fo} = tfo = 0. Also, one should note that the condition (1)H = 0 can be used to define an equilibrium. 3.3 The Second Variation 6(2)H The set of variables {hab, Iab, N, Na, f} can be collectively denoted as Y'. The second variation 6(2)H can be written in terms of this "book-keeping" notation as 6(2)H = - 6(2)yl (2)H 6()y (1)YJ, (3.27) where the functional derivatives are computed with respect to the equilibrium distribution. The second variation 6(2)H provides a nontrivial criterion for linear stability. Nonlinear stability criteria can be obtained by analyzing arbitrarily high orders of variation. Employing the conditions of constraint, 8I/8N = 87/ONa = 0, and the dynamical equations evaluated at a time-independent equilibrium, 87"i/9hab = M/lab = 0, causes the first term of (3.27) to reduce to f dFEo (2)f. (3.28) Evaluating this term for a phase-preserving perturbation 6(2)f of the form (2.18) yields the result dFEo0(2)f d {g, {g, o}}= - f d{g,o}{g, fo}. (3.29) Therefore, the second variation can be written as 6(2)H= - fdr{g,eo}{g,fo}++ 1 dr 8f) 6(1)yl(i)yJ 6H)HYY (3.30) where the vacuum Hamiltonian is denoted HG. The second term in equation (3.30) can be rewritten as the following expression 1yf 6.Y .. + dr1I 6(l)yb6()f, (3.31) IJIJ y which undergoes further reduction to the following format: = dTf f M 1) +6,-f - f 6 ( ) + = dT6(1)f6()YI ' f 1 f 02e f 0e\ f d PI [f 6(l ) ( Y63()y Y.(3.32) 2" J Oy1yJ The sum over variables Y' in equation (3.32) does not include f or IIab because the particle energy e does not depend on these variables. The first term of equation (3.32) can be written as J d3x [6pN + 6Ja6Na + l (NSab - 2Najb)6hab], (3.33) where equation (3.13) for DE/8N, and analogous expressions for De/8Na and Be/Dhab, have been used in the derivation. Equation (3.33) employs the notation p = hl/2 naTabb, ja = hl/2habTb c and Sab = h 1/2hacTcdhbd. (3.34) First order variations 6p, 6Ja and 6Sab are expressed in terms of 5(1)f = {g, fo}. The second term in equation (3.32) may be evaluated by noting that ,92E 02c a2 0 -, (3.35a) ON2 ONONa ONaNb and that 82 2 N2hachbdpcPd, (3.35b) OhabON 2N(e + Napa) a2 hC(ahb)dpd, (3.35c) BhabO c and 2 1 N4 = hae bf hcgh dhpePfPgPh OhabOhcd 4 (- + NkPk)3 N2 + f (b ha)(ch d)ePPf - 2hf(b ha)(cNd)pf. (3.35d) (c + NkPk) Use of equations (3.35b)-(3.35d) yields the expressions - dFf a d3xSab6h bN, (3.36) OhabON 2 - drf ab6c =J dxJbhac hab6Nc (3.37) Shos89Nc and - dFf a~OE habhcd o8hab8 cd = d3x (NSacd - NhacSbd - 2hacjb Nd) bhab6hcd, (3.38a) where Sabcd = h1/2 hae hbf hcghdh dp f PeP!PgPh (3.38b) (_g)1/2 M (nkpk)2 Equations (3.36)-(3.38), together with equation (3.33), lead to the following expression for equation (3.32) d3x[6pbN + 6JabNa + 1 (NSab - 2NaJb)6hab - SabhabON I 2 2 +Jbhac6habbNc + (1NSabcd - lNhacSbd - hacjbNd) habbhcd]. (3.39) One may use the identities 6(NaJb) bhac6N + N aJb - hacjbNd 6hcd (3.40a) and 6(NSab) = Sab6N + NJSab (3.40b) to rewrite equation (3.39) in a more straightforward manner. The use of equation (3.30) leads to the final result 1/ 6(2)H =-- dl'{g, eo}{g, fo} + d3x [6pN + 6JabNa + NbSabbhab - Na Jb6hab 2 + (NSabcd aNhacSbd 6hab6hcd + 6(2)HG[6(1')Y], (3.41a) where 6(2) HG () = 62)HG (1)YI(1)YJ, (3.41b) 6()H[61)] +2 6yigyi I J corresponds to the second variation 6(2)HG associated with the first order perturbation 6(1)YI. The matter contributions to 6(2)H may be reformulated in terms of the distribution function. This alternate form can be written as 6(2)H= -1 dF{g,eo}{g, fo} + J dl{g, fo} ( + Napa)6N - habpb6Na + (2( NCpc) hachbdpcPd - NahbdPd 6hab + 2(E + N ï¿½P) + dlf hab cd Nd ae bf cg dh e I I~f~habhd 8(E+NkPk)3 PePfPgPh N2 hachdehbfpePPf + hachbf Ndpf 2(E + NkPk) e +6(2)HG[(6(1)Y. (3.42) A routine calculation of the gravitational contribution yields 16r6(2)HG[6(1)Y] h I2d3X{[((3)Rab _ 1hab(3)R) - 1hg'fab llcd - 2 f2 2 2 +2h-l(IacHeb- illHab)] 6Nhab + 2h-'(1Ia - habl)6N6Hab +hl~i~~ 2 2 - (DaDb N - habDcDe6N)6hab} + f d3x{h /2Dc[h-1/2(2Hac6Nb - Iab c)1ab + 26rHabDatNb} + 1 d3x(ND2H - NaD2 a), (3.43) where D2H and D2Aa correspond to the following expressions: D2H = h-1/2 {[IabHab- 112][ (6h)2 + 16hcd 6hcd] - 2flabicd (hbddhac hcdbhab)6hee {[a 2j 21J4'yc 2 2 -2(lab 6ab _ 1H6H)cc + 2HlabI d(6hacbhbd - 16habbhcd) 1 2 2 +26HIab cd (hachbd - habhcd) 2 +2(hbd6hac + hachbd - 1hcd6hab - hab6hcd) cdab + ab6flcd)l 2 2 -hl/2{(3)R[1(haa)2_ 16hab bhab] - (3)Rab6h ab6h cc +2(3)Rab hac6hcb - 6hab(DcDbdhac + DCDadhbc) + hCc(Db Da hab - DbDb shaa) +26hcd(DaDadhcd + DdDebha a - DdDa hac) + [Dcbhbb - Dbhbc] (2Da6hac - D haa) + Da6hdDa6hcd + Dd 6hac(Ddhca - Dchda)} (3.44a) 2 and D2Aa = 2(6Ibchad _ Ibc 6had)(D6hbd + DbShcd - Dd6hbc). (3.44b) These formulae are given in the appendix of Moncrief (1976). The perturbed constraint equations permit a further simplification of the second variation 6(2)HG. The exact Hamiltonian constraint can be written (3)R + h- 1 (f2 lablab) = 161rh-1/2p, (3.45) where p is the energy density. A linearized perturbation of this constraint will have the form -6hab (3) ab _ cDc6hdd + DaDb hab +h-hcdhed(flab Hab- l12) - 2h- llab( l ab- abI) 2 2 -2h- lhab (Hbcfac - 1Il =ab) = 16h- 126p - 8rh- 1/2phab6hab. (3.46) The constraint (3.45) may be used to eliminate the unperturbed p from this equation, giving the expression ((3)Rab + 1 hab(3)R)6hab - habDcDC6hab + DaDb 6hab 2 +l h-'hd hcd(Iab fab- f12) - 2h- 1 6IIab( ab habf) 2 2 2 -2h -1Shab(flbcHac - 1IIHab) = 16/rh- 112p. (3.47) 2 Likewise, for the exact momentum constraint Da(h-1/2lab) = 87rh-1/2Jb, (3.48) one may write its linearised perturbation as 2Da5ilab + flachbd(2DcShda - Dd6hac) = 167r6Jb. (3.49) The perturbed constraint equations (3.47) and (3.49) may be used to reduce equation (3.43) to the form =- f dax(6pSN + SJa6N0) 6(2)HG[6(1)yI] _ _ 3 1 fd 3X(ND2H - NaD2Aa). (3.50) 321r This expression may be substituted into equation (3.41a) to obtain the final form for J(2)H , viz. 1 f JdP{g,eo}{g, fo} 6(2)H = - dT{g, eo9fo} + f d3x[ N6Sabhab- N a Jb6hb + (NSabed _ NhacSbd) hab6hc] 1 3 + f d3x(ND2H - NaD2Aa). (3.51) 327r3.1 This formalism can be used to study the stability of spherical equilibria with respect to non-radial perturbations. With the notable exception of Ipser and Semenzato (1979), little work has been devoted to this problem. Consequently, the subject is not well understood. However, the energy functional (3.41b) should be, in principle, amenable to the testing of non-radial perturbations that cause 6(2)H < 0. Also, the formalism can be used to test more general spherical equilibria than in Ipser and Semenzato (1979). These authors, in their development of the symmetric operator formalism mentioned in the introduction, restricted the equilibrium to the form fo = fo(e, m), where - and m correspond, respectively, to the particle energy and mass. A generic spherical equilibrium may be written as fo = fo(E, J2, m) , where the squared angular momentum, associated with rotational symmetry, is denoted J2. Ipser and Semenzato (1979) also imposed the restriction that the equilibrium has a monotonically decreasing dependence on e. Therefore, the derivative afo/Oe is intrinsically negative with respect to all values of the particle energy E. The formalism developed here does not place constraints on the form of the equilibrium distribution. Another objective is the study of the stability of axisymmetric equilibria. Little research has been carried out on this subject. The motivation for this work comes from the fact that physical equilibria generally possess angular momentum. Consequently, there should be a flattening of the matter distribution due to the effects of rotation. Shapiro and Teukolsky (1993a,b) have succeeded in generating axisymmetric equilibria through a useful numerical algorithm which they have recently developed. It is plausible to conjecture that the stability of rotating, axisymmetric systems can be studied with the formalism developed here. One may attempt to prove that certain classes of equilibria always possess phase-preserving perturbations for which the energy decreases. Subsequently, further investigation may explain if, for relatively short time scales, these perturbations result in physical instabilities. Unfortunately, it is not straightforward to follow through with this program. Attacking the problem by means of a post-Newtonian expansion leads to inconclusive results. It appears that a generic rotating, axisymmetric solution of the Einstein equations must be found before stability results can be deduced from it. However, such a solution is not known. CHAPTER 4 THE VLASOV-BRANS-DICKE SYSTEM Carl Brans and Robert Dicke (1961), in an attempt to make Mach's principle compatible with General Relativity, formulated a scalar-tensor theory of gravity in which the Newtonian gravitational constant Go is represented by a scalar field. This is the celebrated Brans-Dicke gravitational theory. It was shown by Dicke (1962) that Brans-Dicke theory may be written in an equivalent representation in which the inertial masses of elementary particles vary as a function of the Brans-Dicke scalar field while the gravitational constant Go remains fixed. The Brans-Dicke theory may be formulated in either representation. Although the physical interpretation of gravitational phenomenon varies between the two representations, the actual physical predictions remain equivalent. For example, as pointed out by Dicke (1962), the gravitational red-shift in the variable mass representation is only a partially metric phenomenon. Since the particle masses are affected by the scalar field, part of the red-shift results from changes in the energy levels of the atoms. Consequently, the red-shift is not entirely metric induced. In this representation, the measures provided by rulers and clocks are not invariant with respect to position in space-time. Also, free falling matter does not follow geodesic trajectories in space-time. However, (massless) photons still do. The geodesic equation of the variable G representation is d 1 d (mgoiju) - mgjk,iuuk = 0, (4.1) where T denotes the proper time and ui is the relativistic four-velocity. One can switch to the variable mass representation through the transformation m = 01/2m (4.2) and gij = -1gij. (4.3) The speed of light c (units: [L][T]-) and Planck's constant h (units: [M][L]2 [T]-1) remain invariant under this transformation. Consequently, the geodesic equation for the variable mass representation has the form d 1 k d-r (hgij3j) - -2gkzUU + , = 0. (4.4) As shown by Toton (1970), the original Brans-Dicke variational principle d(g)12(R - w + 167rG LM) = 0 (4.5) f 0 C4 may be reformulated, by means of the conformal map (4.3), as 6f d4X(g)1/2(R-( + 167r - LM) = 0, (4.6) where LM is obtained from LM by replacing gy, in LM by g,#. The field equations associated with the original Brans-Dicke variational principle (4.5) are 8rG w 1 . GI, = -- To, + 4;,, - g,w;ao;a + [#;ud;v - 2 g" ,a;O] (4.7) and (2w + 3)0;a;a = 8rG"Ta (4.8) C4 where Gpv - Rpv - gvR. As noted by Bruckman and Velizquez (1993), the presence of second order derivatives of 4 in (4.7) make the variable G representation of Brans-Dicke theory unsuitable for application of the canonical ADM Hamiltonian formalism. Through examination of the field equations associated with the variational principle (4.6), viz. 87rGo, ,T 3 1 (49) C4 o 2 4.9) and (313 V\ V' 16irGo 1 8rGo 6LM 3+ w) V+V - ï¿½w) 16G, Lm + = 0, (4.10)
2 2 c4 c 4 where

4) = Ino, (4.11) one may see that equation (4.9) has the form of the standard Einstein field equation. Equation (4.9) can be written as

87rGo,tot
G,, - -C4 l, (4.12) where P is the sum of a matter stress-energy tensor and a scalar field pseudo stress-energy tensor. Consequently, because of the similarity to Einstein theory, the ADM formalism is easily applied to the variable mass representation of the theory.

Following Kandrup and O'Neill (1994), we may, while employing an ADM splitting into space and time, demonstrate that the collisionless Boltzmann, i.e., Liouville equation of the Brans-Dicke gravitational theory is Hamiltonian. By analogy with the Vlasov-Einstein case, criteria for the linear stability of time-independent equilibria, corresponding to relativistic matter configurations, will be derived.

As for notation, all physical quantities subjected to the transformations (4.2) and (4.3) will be denoted with a bar superscript. This notation is employed in the equations (4.2) and (4.3). Unbarred quantities, of course, have not been subjected to this transformation. In terms of physical variables, one works with a spatial three-metric hab and scalar field 0, as

well as their respective conjugate momenta IIab and H. These variables are sufficient for the vacuum metric-scalar Hamiltonian formalism. In the presence of matter, one must include the distribution function f as a fifth variable. Actually, as will be shown, it is necessary to work with the barred distribution f. Consequently, the Hamiltonian formalism is written in terms of a mixed system of variables. One may define functionals F = F[hab, 1-ab; 4, ; f] in terms of an infinite-dimensional phase space (hab, lab; 0, I; f).

It will be shown that the Vlasov-Brans-Dicke dynamical equations are equivalent to the constraint equations OtF = (F, H), where F denotes a phase space functional, Ot denotes a coordinate time derivative, H denotes a Hamiltonian and (.,.) denotes a bracket operation, to be later defined, which acts on the functional pairs F[hab, ab; 4, HI; f] and G[hab, fab; 4, H; fJ.

One may demonstrate that the first order variation is extremal,

6(')H[(1)X] = 0, (4.13) when subjected to a perturbation

(1)X = {J(1)hab, 6(1)nab; (1), 6(1)i; 6(1)f} (4.14) that is dynamically accessible. This perturbation will be consistent with both the Hamiltonian and momentum constraint equations and with matter field constraints due to conservation of phase. The expression 6(2)H may be computed, the sign of which is related to the stability of the time-independent equilibrium. A positive sign, J(2)H > 0, guarantees that the system is stable. A negative sign, 6(2)H < 0, does not guarantee linear instability, but should, at least, ensure nonlinear nonstability or nonstability with respect to dissipative effects.

4.1 The Vlasov-Brans-Dicke Hamiltonian Formulation

The ADM decomposition into space and time will be implemented in exactly the same way as in Kandrup and O'Neill (1994), which, in turn, followed the treatment given in Wald (1984). The metric line element will transform according to equation (4.2) as ds2 = 9,dx'dx = 0-1g,,dxdxv = 4-Id2. (4.15) Therefore, the transformed metric line element is d. = -1/2ds. (4.16) The expression (4.16) is the same as Dicke's (1962) A-1/2dg = ds, where the arbitrary well-behaved function A is set equal to 0-1. Consequently, the relativistic four-velocity transforms as

fA = u1/2u". (4.17) The expression (4.2) enables one to derive an expression for the transformed fourmomentum, viz.

P, = #. (4.18) The covariant form for the transformed four-momentum is given by PA = P., (4.19) which implies that the covariant four-momentum is the same with respect to both representations of the Brans-Dicke theory. In the Vlasov-Einstein case (Kandrup and O'Neill (1994)), one worked with a distribution function of the form f = f(xo,pa, m, t), where xa denotes the spatial coordinates, Pac, denotes the covariant spatial momenta, m denotes the particle mass and t is the time. For the variable mass representation of Brans-Dicke theory,

the distribution will be

f = f(7,, f(x",po, i, t). (4.20) It should be noted that only the mass variable m is altered by the transformation equations (4.2) and (4.3). The scalar field 0 cannot be factored out of the distribution (4.20), implying that the barred distribution f must be used in the Hamiltonian formulation.

One can postulate the following bracket

[ ax F 6G bF ,bG
(F ,G )[hab, ab;4 , I; f] 161r d 3 W h - b b J b hab 611ab 6IIab 6'-b

+167r dx( 6F 6G F )G
6 II 6Hl&

+161r df f{ , (4.21) where 6/16X denotes the usual functional derivative with respect to the variable X, and BA OB OB BA
{A,B} = BI 9oB O9 (4.22) '9XtapVa e9aa9pa

is the ordinary canonical Poisson bracket written in terms of the variable mass representation. Other notations in equations (4.21) and (4.22) are as in Kandrup and O'Neill (1994). It is interesting to note that the bracket is of a mixed composition, with the matter component written in terms of the transformed variables.

The bracket (4.21) is a bona fide Lie bracket because (a) it is antisymmetric and (b) it satisfies the Jacobi identity

(F, (G, H)) + (G, (H, F)) + (H, (F, G)) = 0. (4.23) We may separate the Hamiltonian function into two parts: one part, HG, which is purely gravitational and another part, HM, which governs the matter coupling to the gravitational field. Variation of the action given by Brans and Dicke (1961), leads to the Brans-Dicke

field equations written in the variable G representation. Subjecting these equations to the transformations (4.2) and (4.3) yields an altered form of them. Due to the fact that the scalar field is determined by the field equations, the quantitative physical predictions of the theory remain the same. The variable mass representation of the Brans-Dicke action (4.6) does not couple the Ricci scalar invariant R to the scalar field 0. Therefore, when the ADM formalism is implemented, one can use partial integration to rid the action of unwanted boundary terms in a straightforward manner. Thus, it is convenient to work within the variable mass representation.

Following the treatment given by Toton (1970), one can write HG as

HG= d3xXJG, (4.24) where the Hamiltonian density is

HG = 4 /2 _(3) -(1 ab ab _ 1 2 h- 2f2 167rGo 2 (6 + 4w) +(3 -1 oh a -aRb 1h(Ia ab - 2) +
+ + haba'b] - 2Nb[Da(h-1/2 ab -1/2 ,b]}, (4.25) where Da denotes the covariant derivative associated with the spatial three-metric hab and

(3)R denotes the three-curvature scalar corresponding to hab. The matter Hamiltonian, written with respect to the variable mass representation, takes the form It, = fdrff. (4.26) If it is demanded that the Hamiltonian H be extremal with respect to variations in the lapse N and shift Na, then the Hamiltonian and momentum constraints are enforced. Following the treatment given in Kandrup and O'Neill (1994), one may derive the expression e (e + Napa)2
ON = -N3pt (4.27)

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HAMILTONIAN STRUCTURE AND STABILITY OF RELATIVISTIC GRAVITATIONAL THEORIES By ERIC O'NEILL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000

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ACKNOWLEDGEMENTS I wish to express my sincere gratitude and thanks to my doctoral supervisor, Professor Henry E. Kandrup, for his guidance and input into my thesis project during the years that I worked on it. He initiated me into the study of stabihty problems in relativistic systems by means of phase-preserving perturbations. During later stages of the project his commentary and text editing were a great source of help to me. I have learned a lot from him in terms of his approach to theoretical physics research. I also wish to express my thanks towards my family for financial support and encouragement during the yecirs spent working on my degree. Also, I would like to express my thanks to Dr. Samuel Mikaelian for providing me with thesis formatting tex macros. ii

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TABLE OF CONTENTS page ACKNOWLEDGEMENTS ii ABSTRACT v CHAPTERS 1 INTRODUCTION 1 LI Theory of Casimir Invariants 2 1.2 Constrained Hamiltonian Dynamics 11 1.3 Covariant Poisson Brackets for Relativistic Field Theories 20 1.4 Derivation of the Plasma Bracket 28 2 NEWTONIAN COSMOLOGY AND VLASOV-MAXWELL SYSTEM 34 2.1 The Transformed Vlasov-Poisson Equation 35 2.2 The Vlasov-Poisson Hamiltonian Structure 37 2.3 Stability of Collisionless Vlasov-Poisson Equilibria 42 2.4 Introduction to the Curved Space Vlasov-Maxwell System 45 2.5 The Vlasov-Maxwell Hamiltonian formulation 51 2.6 Perturbations of Time-Independent Vlasov-Maxwell Equilibria 58 2.7 Stability Criteria for Time-Independent Vlasov-Majcwell Equilibria 65 3 VLASOV-EINSTEIN SYSTEM 69 3.1 The Vlasov-Einstein Hamiltonian Formulation 73 3.2 All Vlasov-Einstein Equilibria are Energy Extremals 79 3.3 The Second Hamiltonian Variation S^'^^H 80 4 VLASOV-BRANS-DICKE SYSTEM 88 4.1 The Vlasov-Brans-Dicke Hamiltonian Formulation 92 4.2 The Second Hamiltonian Variation (J^^'/f 102 4.3 The Question of Stability HO iii

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5 VLASOV-KIBBLE SYSTEM 114 5.1 Kibble Gauge Invariant Theory of Gravity 115 5.2 Vlasov-Kibble Hamiltonian Structure and Formulation 117 5.3 Initial Variation of the Vlasov-Kibble Hamiltonian 129 5.4 Second Hamiltonian variation S^"^^!! 132 SUMMARY 141 REFERENCES 145 BIOGRAPHICAL SKETCH 149 iv

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy HAMILTONIAN STRUCTURE AND STABILITY OF RELATIVISTIC GRAVITATIONAL THEORIES By Eric O'Neill May 2000 Chairman: Henry E. Kandrup Major Department: Physics The research contained within this thesis is concerned with the Hamiltonian structure and stability of several relativistic physical systems. Its primary focus involves a comparison of the Hamiltonian structure and stability properties of two alternative gravitational theories with General Relativity. Specifically, the Brans-Dicke and Kibble theories of gravity are the subject of study. Working in the context of an Arnowitt-Deser-Misner splitting into space plus time, it may be demonstrated that the Vlasov-Einstein system, i.e., the collisionless Boltzmann equation of General Relativity, is Hamiltonian, and then this Hamiltonian character may be used to derive nontrivial criteria for linear and nonlinear stability of time-independent equilibria. Unlike all earlier work on the problem of stability, the formulation provided here is completely general,

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incorporating no assumptions regarding spatial symmetries or the form of the equihbrium. The two alternative gravitational theories are studied by using the same methods. The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function /, the spatial metric hab, and the conjugate momentum 11"''. The Hamiltonian formulation entails the identification of a Lie bracket {F,G), defined for pairs of functionals F[/iab, IT"'', /] and G[/ia6, 11"'', /], and a Hamiltonian function /f[/ia6, n"'', /], so chosen that the equations of motion dtF = {F,H) for arbitrary F, with dt a coordinate time derivative, are equivalent to the Vlasov-Einstein system. An analogous mathematical structure is deduced for the Vlasov-Brans-Dicke and Vlasov-Kibble Hamiltonian systems, where the gravitational field variables differ from General Relativity. An explicit expression is derived for the most general dynamically accessible perturbation SX = {Sf, 6hab, 611"''} which satisfies the Hamiltonian and momentum field constraints and the matter constraints associated with conservation of phase, and it is shown that all equilibria are energy extremals with respect to such 6X, i.e., 6^^^H[SX] = 0. The sign of the second variation, 6^^^H, which is also computed, is thus related directly to the problem of Uneeir stabiUty. US^'^^H > 0 for all dynamically accessible perturbations SX, the equilibrium is guaranteed to be linearly stable. The existence of some perturbation 6X for which S^^^H[SX] < 0 does not necessarily signal a lineaj instability. vi

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CHAPTER 1 INTRODUCTION Throughout this dissertation the Hamiltonian structure, and energetic stability, of several relativistic coUisionless plasma systems will be studied. In chapter 2, the collisionless Vlasov-Newtonian cosmology and the fixed curved background spacetime coUisionless Vlasov-Maxwell system will be examined. In the remaining three chapters, the Hamiltonian structure of three relativistic theories of gravitation will be analyzed in detail. Chapter 3 consists of a study of the Hamiltonian structure of the Vlasov-Einstein system (General Relativity), while chapter 4 is a corresponding study of the Vlasov-Brans-Dicke Hamiltonian system. Finally, chapter 5 deals with the Vlasov-Kibble Hamiltonian system. The Hamiltonian of each gravitational theory will be written in terms of the ADM formalism (Arnowitt, Deser, and Misner (1962)). The use of this formalism facilitates a convenient 3+1 split into space and time variables. From a dynamical standpoint, this makes the analysis of static equilibrium matter configurations particularly straightforward. For each theory, there are field variables and corresponding conjugate momenta. Following Dirac (1950), one can construct a bracket out of these canonically conjugate dynamical variable pairs. The equations of dynamics for the conjugate variables may be derived from the bracket. This applies to the field variables of the gravitation theories. A Boltzmann distribution / will be used to describe the matter distribution. This distribution corresponds to the case of a coUisionless matter distribution and it will satisfy the Liouville, i.e., coUisionless Boltzmann equation. This variable / does not possess a conjugate variable. Therefore, it is not straightforward to construct a Dirac bracket for it. 1

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2 1.1 Theory of Casimir Invariants Early work with the theory of Casimir invariants dates back to the late 1950's with the research of W. Newcomb (cf. the appendix in Bernstein (1958)). Newcomb considered a perturbation of a plasma Hamiltonian for which the entropy, serving as a Casimir invariant, remains fixed. His idea, which went as follows, was developed to show that general small motions of a coUisionless plasma about thermal equilibrium cannot exhibit exponential growth in time. Newcomb utilized the fact that entropy is a constant of the motion for collisionless systems. denote the Maxwell distribution. In equation (1.1) the particle density is denoted by N and Boltzmann's constant is denoted by fc^The mass and velocity of the plasma particles are denoted, respectively, as m and v. The temperature at thermal equilibrium is denoted by T. Newcomb worked with the following expression for the entropy where the summation is taken over all species of particle present in the plcisma. Since 5 = 0 for thermal equilibrium, the expression for entropy (1.2) is normalized. For a departure from thermal equilibrium, that is spatially bounded, the integral (1.2) is well defined. For spatial disturbances of the form e"'"'*^''', the integration may be taken We will let (1.1) (1.2) over a cube of side length 2n/k in which one face is perpendiculcir to the wave vector it. One may compute the time rate of change of S to be (1.3)

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The term df /dt will vanish due to the conservation of the total number of particles (factor the partial time derivative out of the integral). For the rest of the integral, it is necessary to utilize the Boltzmann equation (Chapman and Cowhng (1939)) 5/ , . 5/ , q , v^^^\ df _ df dt or m dv dt co/i.' (1.4) where df /dt^^n represents the rate of change of the distribution fimction arising from particle collisions. For a coUisionless plasma, one may set Substitution of this coUisionless Boltzmann, i.e., Liouville equation into equation (1.3) yields the result Integration by parts leads to the following modified version of equation (1.5) (1.5) 0. (1.6) Consequently, the entropy is a constant of the motion known as a Casimir invariant. The total energy of the plasma system is given by W = d'r{E' + Bl) + d'rd'v--fvV fo), (1.7) where Bg denotes a constant external magnetic field. The energy is normalized since W = 0 at equilibrium. The Maxwell equations are V-E = 4Trp, cVxE = dB^ dt ' dE VB = 0, cVxB = 4nJ+ Â— , dt (1.8)

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where the charge density p and current density J are given by p = Y,Ze J d^vf, J = Y,Zejd\vf. Using these equations and the Boltzmann equation (1.4) allows one to prove that is a constant of the motion. Therefore, the combined system F = W + S is a constant of the motion. One may expand the distribution function in terms of powers of a small initial departure from thermal equilibrium / = /o + /l+/2 + ... . (1.9) There are corresponding changes in the electric and magnetic fields B = Bo + Bi + B2 + ... . (1.10) To second order the entropy has the following form where the constant A = | ln[N^^^m/2'KkBT]. Due to particle conservation, the terms above in (1 + A) will vanish. Expressing equation (1.7) to second order allows us to write equation (1.11) as We obtain a bracketed expression which is essentially a sum of positive terms. However, we know from equation (1.6) that 5 is a constant of the motion. Consequently, it is not possible for any of the quantities Ei, By or /i to increase monotonically. For example, an exponential increase in time of /i would not be permitted.

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5 This argument by Newcomb disproved the purported proof by Gordeyev (1952) that such an exponential growth of the distribution / could occur for a collisionless plasma in thermal equilibrium. Newcomb considered the conservation of only one constraint. However, in later work, Kruskal and Oberman (1958) considered the conservation of a general set of Casimir constraints in a paper dealing with plasma stability. Gardner (1963) provided additional insight with the recognition of the importance of phase space conservation in regard to electrostatic Vlasov stability theory. His arguments were as follows. One may assume that f{t, x, v) denotes the distribution function of one species of particle present in a plasma. Since the plasma is collisionless, the distribution will satisfy the Vlasov equation * . % + v-Vxf+--^vf = 0, (1.13) at m where the force F{t, x, v) obeys the condition Vff-F = 0. (1.14) The motion of one particle is dictated by the usual dynamical equations dx/dt = v, (1.15) mdv/dt = F. (1.16) Equations (1.15) and (1.16) define a flow in x, v phase space. By combining equations (1.14)(1.16) we see that the flow is incompressible. Also, equation (1.13) states that the value of /, at the phase space position of any fixed particle, will not change as the particle moves about in phase space. In other words, the distribution remains invariant when translated along any phase space trajectory that is consistent with the dynamical equations. One may interpret the phase space motion as that of an incompressible fluid flow of variable density

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6 /. The initial value of / will be f{0,x,v) = fo{x,v). (1.17) The kinetic energy of the plasma will take the form (1.18) One may integrate in x over a finite box if the assumption of periodic x dependence of the distribution / is made. With the initial value of fo, we wish to determine a lower bound for the kinetic energy of the plasma. Such a lower bound may be written as where /i(v), corresponding to a later value of the evolving distribution, represents a monoregion where fi{v) > a is equal to the phase space volume of the region where fo{x, v) > a. This condition holds because the phase space flow is incompressible with the value of / at a particle conserved during subsequent evolution of the system. With these enforced conditions, one wishes to determine what the minimum possible value for W{t) will be. It is straightforward to see that the lowest-energy state corresponds to the case in which more massive particles (/ large) are even closer to v = 0. For example, if we choose two particles / and /' with respective velocities v and v', then, if we are dealing with the lowest-energy state, the condition f > f implies that v < v'. This state corresponds to fi{v). Consequently, the amount of kinetic energy originally present in fo{x,v), which can be given up to the field, is at most (1.19) tone decreasing function of u^. Also, for any number a > 0, the phase space volume of the WiO)-Wi. (1.20)

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Afterwards Lynden-Bell and Sanitt (1969) applied the method of conserved constraints to study the energy associated with a linearized perturbation of a galactic dynamical system. They referred to this as "a trick due to Newcomb." However, it was left to Bartholomew (1971) to finally deduce the mathematical form of the phase-preserving perturbation of the Boltzmann distribution which leaves general Casimir constraints invariant. Bartholomew's paper dealt with the problem of galactic stability. Bartholomew treated a galaxy as a system of point masses moving under the influence of the collective gravitational potential of the entire system. One may define a phase space function F = F{qi,pi,t) over the space of generalized coordinates Qi and conjugate momenta p,. A phase space volume element may be defined as dr = dqidq2dq3dpidp2dp3. The continuity of flow through volume dr of phase space may be expressed as Using Hamilton's canonical equations of motion (for the case of a unit mass) allows one to rewrite equation (1.21) as dF d ( ^dH\ d f ^dH\ + ^ FÂ— Â— FÂ— =0. (1.22) dt dqi V dpi J dpi \ dqi J The Poisson bracket may be defined as dAdB dAdB , ^ [A,B] = Â—Â—-Â— Â— . 1.23 dqi dpi dpi dqi This enables one to redefine equation (1.22) as dF dt + [F,H] = 0. (1.24) The Hamiltonian H may be written as the sum of two terms. The first term, which corresponds to the kinetic energy, is a function of the generalized coordinates. The second

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term corresponds to the gravitational potential ^. Using Cartesian coordinates one may write the matter density as p = I Fd'p, (1.25) which yields the gravitational term J \r Â— r\ J \r Â— r\ For the sake of convenience, one may give the gravitational term of the Hamiltonian a more general formulation / K{qi,pi;q\,p',)F{q\,p',)dT', (1.27) where K is a symmetric kernel. The second gravitational term has a linear dependence on the distribution F. One may rewrite equation (1.22) as ^ + ^^ + ^5^=0, (1.28) dt dt dqi dt dpi which corresponds to dF{q^,pi,t) = 0. (1.29) dt Assuming that the galactic system is perturbed from equiUbrium by an infinitesimal amount F + 6F (1.30) leads to a corresponding change in the Hamiltonian = j KSF'dr'. (1.31) The Vlasov equation (1.31) will undergo a similar linearized perturbation -{6F) + [dF, H] + [F, 6H] 0. (1.32)

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9 One wishes to determine whether, from the perspective of stability theory, the perturbation 6F will experience monotonic growth or will oscillate about equihbrium. As discussed in Bartholomew (1971) the relevant perturbations 6F are those which conserve angular momentum and total mass, and are generated by a canonical transformation. We insist on a canonical transformation because the Hamiltonian equations f = fe.Â«l f = 1P.Â«1 (1.33) generate translations in phase space Qi + dqi, Pi + dpi, (1.34) by means of an infinitesimal canonical transformation. We are only interested in perturbations which are consistent with the equations of dynamics. One wishes to move stars from the phase space point {qi,Pi) to an infinitesimally close nearby point {qi + ^i,Pi + rn). The perturbation corresponds to an infinitesimal canonical transformation provided that there exists a generating function g such that 6 = 1^, (l-35a) dpi m = -^(1.356) dqi A more compact notation for equations (1.35) may be introduced by use of the Poisson bracket 6 = [gi,5], m = \Px.9](1.36) The change in F may be found by writing the following equation

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10 which follows by analogy with equation (1.21). This leads to the proper form for the phasepreserving perturbation SF = -[F,g], (1.38) where 5F is defined in terms of the generating function g. Substitution of equation (1.38) into equation (1.32) yields the result F ^ ' dt -[[F,g],H] + [F,6H]. (1.39) Multiplication of this equation by dg/dt and integration over all phase space leads to ' dt K[F',g']dT'dT. (1.40) The left-hand side of this equation vanishes by virtue of antisymmetry dg2 dt d92 [p dgi dr. (1.41) The right-hand side of equation (1.40) can be proven symmetric in g and dg/dt, allowing one to rewrite it as {-\ I [F,g][H,g]dT -\j j [F,g]K[F\g']drdT'^ = 0. (1.42) d^ r 1 dt The functional in brackets corresponds to a second order energy perturbation. Assuming a stationary reference frame, the Hamiltonian H is the energy per unit mass. The total energy E, for a fixed potential field, would be given by -I E = / FHdT. (1.43) However, if a perturbation of F alters the energy of other material by varying the potential, then the equation (1.43) can be written in the form FKF'dTdr'. (1.44)

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11 A first order perturbation of this energy can be calculated to be Therefore, the first order change in energy is 5E = (1.46) Thus, there is no change in the energy of an equilibrium for a lowest order perturbation. It was Dirac (1950) who first studied constrained Hamiltonian dynamics. We may follow his treatment of the subject. Typically, one may consider a dynamical system of N degrees of freedom described by generalized coordinates gÂ„ (where n takes the values 1, 2,..., N) and velocities dq-n/dt = 9Â„. The dynamics of the system is described by a Lagrangian which possesses the corresponding momenta Pn = (1.48) dqn The variables qn, gm and pn may be varied by a small amount <5gÂ„, JgÂ„, and 6pn of the order of e. In our calculations, we only work to the order of accuracy of e. However, vajiation of equation (1.48) leads to difficulties because the left-hand side will differ from the right-hand side by a quantity of order e. Following Dirac (1950), we may class equations as to whether they remain accurate to order e under variation or not. The Lagrangian (1.47) remains valid under variation, since, by definition, the variation in L must equal the variation of HlnjQn)Equations of this type are referred to as strong equations, while equations, such 1.2 Constrained Hamiltonian Dynamics L = L{qn,q, (1.47)

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12 as (1.48), are referred to as weak equations. The following algebraic rules exist for the weak and strong equations; for general cases 6A = 0 (1.49) (yl = 0 is a strong equation) and 6X^0 (1.50) (X = 0 is a weak equation), where the notations = for strong equivalence and = for weak equivalence have been employed. We can conclude that SX"^ = 2X6X = 0 where X = 0 is a weak equation. Consequently, we have deduced the strong equation X^ = 0. A strong equation X1X2 = 0, can be constructed from two weak equations Xi = 0 and X2 Â— 0. The standard case occurs when the N quantities dL/dqn of equation (1.48) are all independent functions of the N velocities gÂ„. For this case one may determine each velocity 9Â„ as a function of the coordinates 9Â„ and momenta pÂ„. However, if the dL/dqn are not independent functions of the velocities then we may eliminate g's from equation (1.48) and derive one or more equations . . ... 4>{q,p)=Q . , , (1.51) which are functions of coordinates q and momenta p. The equations (1.51) are weak equations. One may consider a complete set of independent equations (1.51) <^m(g,p)=0, m = l,2,...,M. (1.52)

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13 A function of the g's and p's, which vanishes due to equation (1.48), may be written as a linear function of the (pm with coefficients that are functions of the g's and p's. The strong and weak equations may be given a geometrical interpretation. Initially, one may visualize a 3N dimensional space with coordinates qmQn^ and pÂ„. This space will possess a 2N dimensional hypersurface where equation (1.48) holds. This hypersurface will be called TZ. Equations (1.51), which are deduced from equation (1.48), will also hold on this hypersurface. We may also define a 3N dimensional region of all points that are within a distance of order e from the hypersurface TZ. This region will be referred to as 7^(. A weak equation will hold in the region TZ, while a strong equation will hold in the region TZ^ . The Hamiltonian is defined in the usual manner as H = pnQn L, (1.53) where the standard summation convention is employed. The Hamiltonian (1.53) can be subjected to a variation as follows SH = SipnQn L) = Pn^n + qn^Pn dL/dQnSqn dL/dqnSqn = qnSpn dL/dqnSqn. (1.54) Consequently, 6H does not depend on the 6q's. This result holds irregardless of whether or not the standard case applies. The definition (equation 1.53) of the Hamiltonian holds throughout the 3N dimensional space of q's, q's and p's. The result (equation 1.54) holds in the region TZ^ to first order. Therefore, assuming that q and p remain constant, a first order variation in the velocities corresponds to a second order variation in H. Thus, for constant gÂ„ and pÂ„, the variation in H, corresponding to a finite variation in 9Â„, will be of

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14 the first order provided that the variation is performed in TZ^. The variation in H will be zero if it is performed in the region 11. It follows that, in the region TZ, the Hcimiltonian is a function of q and p. For the region TZ one may define the weak Hamiltonian function H = h{q,p). (1.55) The function h is the ordinary Hamiltonian in the stand?ird case. The following general variation 5{H -h) = (1.56) may be performed from a point in TZ. If the variation is within the region TZ, then it is obvious that 5{H Â— /i) = 0. For such a variation of Sqn and Spn, the dynamical equations (1.48) are preserved by an appropriate choice of Sq^. The 5g's and (5p's are restricted in the sense that 6(f>m = 0 for all values of m. However, for arbitrary pertmrbations Sq, 6p which do not satisfy this condition, we compute S{H -h)= Vm6m (1.57) for a suitable choice of coeflficients. These coefficients Vm are functions of the g's and p's. Equation (1.57) can be reformulated as S{H -hVm(f>m) 6{H -h)VmS(pm " ^rn^Vm = 0, (1.58) where equation (1.52) has been used. One may use equation (1.58) to deduce the following strong equation H = h + Vm(t>m ; (1.59) which holds to the first order in TZ^. A variation of equation (1.59) yields the result 6H = Sh + VmS(l>m

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15 = ^Spn + ^Oqn + Vm ^ dpn + ^ dqn . (1.60) OPn Oqn \ OPn Oqn J Comparison of equation (1.60) with equation (1.54) yields the following expressions dh d(f)m ,^ . qn = -^+Vm'^ , (1.61a) OPn OPn dL dh d(f>m = + Â— Â• (1.616) oqn oqn dqn The usual Lagrangian equations of motion dL > * '. Pn = OQn . . * can be combined with equation (1.61b) to yield the Hamiltonian equation of motion dh d(f>m ,^ Pn = Wm^; Â— Â• (1.62) oqn oqn The Hamiltonian equation corresponding to equation (1.62) is equation (1.61a). It is convenient to formulate the Hamiltonian equations of motion in terms of the Poisson bracket notation. Any two functions ^ and r? of the g's and p's possess the Poisson bracket dqn dpn dpn dqn Poisson brackets are, by their definition, subject to the following rules \iA^-Wi\. (1.64a) [e,/(m,%,-)] = ^\iM + ^\iM + (1-646) [^,[r,,C]] + [r?,[C,e]] + [C,[e,^]] = 0. (1.64c) According to Dirac (1950) the notion of a Poisson bracket may be extended to include functions of q's where the velocities are not expressible as functions of q's and p's. These

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16 generalized Poisson brackets are still subject to the properties (1.64a)-(1.64c). The following weak equations |A=Â„, |A.o. |A=0 (1.65) dqn dqn dpn can be derived from the strong equation ^ = 0. For any ^ one may deduce that [e,A] = 0 , (1.66) by using equation (1.64b). It is possible, although by no means necessary, that [^,A] = 0. However, for a weak equation X = 0 one cannot generally infer that [^, X] = 0. If 9 is a function of the q's and p's then the Hamiltonian equations (1.61a) and (1.62) allow us to write . ^ / _^ ^ 9m A dg f dh ^ ^ d(f>m \ dqn\dpn ""dpnJ dpn\dqn ""dqn J ^[9,h]+Vm[9,(f)m](1.67) Using equation (1.52) allows one to rewrite this expression as g = [g, h] + Vm[g, m] + b, Vm](pm = [q, H], (1.68) which is the generalized Hamiltonian equation of motion. If the Lagrangian is homogeneous of the first degree in the velocities gÂ„, then the momenta (1.48) will be homogeneous of degree zero in the g's. Consequently, they will depend only on the ratios of the g's. It follows that the p's cannot all be independent of the g's because there are only N 1 independent ratios of the g's corresponding to N p's. Therefore, at least one relation of type (1.51) must exist between the g's and the p's. The Euler theorem is apphcable in this case and leads to

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17 L = qnPn(1.69) This yields the following weak equation (1.70) which holds in the region TZ. Therefore, we can assume that /i = 0 and that (1.71) It follows that the general equation of motion reduces to the form 9 = VmlgAm](1.72) As noted by Dirac (1950), the Lagrangian for any dynamical system can be made to satisfy the condition for homogeneous velocities by using an additional coordinate qo to represent the time t and the condition go = 1 is then used to make the Lagrangian homogeneous of the first degree in all the velocities, including qoTherefore, without any loss of generality, one may consider only the homogeneous velocity theory. The equations of motion must maintain the validity of the constraint equations (1.52). It follows that substitution of for g in equation (1.72) must yield the set of equations The system of equations (1.73) will be assumed to be reduced as much cis possible with the set of equations (1.52). Each of the resulting equations must fall into one of the four categories: Type 1. It involves some of the variables VmType 2. It involves the variables q and p and takes the form xiq^P) = 0However, it is independent of the variables Vm and independent of the (1.73)

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18 equations (1.52). Type 3. It reduces to the form 0 = 0. Type 4. It reduces to the form 1=0. An equation of type 2 will lead us to another consistency condition that is analogous to equation (1.73), namely Vm[m,X]=0, (1-74) for the constraint X(g,p) = 0. (1.75a) The new consistency condition (1.74) may be reduced as far as possible by using eqs. (1.52) and (1.75a). Once again, eq. (1.74) will be one of the four types. If it is type 2, then we will obtain another consistency condition. Therefore, we must repeat the process until an equation of another type is obtained. If we obtain an equation of type 4, then the equations of dynamics are inconsistent. Therefore, we may ignore this case. Equations of type 3 are identically satisfied. It is necessary to excimine the cases of type 1 and type 2. A complete set of equations of type 2 may be written ^ j Xkiq,p) = 0, k = l,2,...,K. (1.756) The equations (1.75b) are chosen to exhibit order e variations such as the constraint equations (1.52); both sets are weak. We define a constraint to be a first class constraint if its Poisson bracket with all other constraints (f) and x vanishes, i.e., if [0m',0m] = 0, m=l,2, ...,M [(f>m',Xk] = 0, k = l,2,...,K. (1.76) Equations (1.76) only have to hold as consequences of = 0 and Xk = 0, i.e., they can be weak. A constraint 4>m not satisfying these conditions is termed second class.

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19 A linear transformation may be performed on the constraints (1)^ 4>m* = 7mm' (l>m'(1-77) The 7 matrix coefficients are functions of the coordinates q and momenta p such that their determinant will not vanish in the weak sense. It follows that the cf)* constraints are completely equivalent to the 0 constraints. A transformation of this kind may be performed to bring as many constraints cis possible into the first class. The first class constraints will be denoted 4>a and the second class constraints will be denoted (f)p, where /3 = 1,2, ...,B and a = B + 1, B + 2, M. The consistency conditions will reduce to the format M0M = ^, P,I3' = 1,2,...,B V0[p,Xk] = O, k = \,2,...,K. (1.78) These are all of the type 1 equations. These equations prove that either the v^'s must all vanish or that the matrix is, in the weak sense, of rank less than B. Dirac (1950) proved that the first possibility is correct. Therefore, if the maximum number of ^'s possible have been placed in the first class, the u's corresponding to the second class constraints will all be zero. Therefore, the Hamiltonian (1.71) can be written in terms of only the first class constraints as H = Va(t>a (1.79) and the general equation of motion becomes P = Wa[ff,<^a](1.80) We may form a determinant from a set of functions 9s {s = I, 2, ... , S) of the q's and p's such that the determinant is nonvanishing in the weak sense. Let Css' correspond to the

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20 cofactor of [^si^s'] divided by the determinant A. It follows that Css' = -Cs's (1-81) and Css'[Gs,Os"] ^ ^s's"(1-82) Dirac (1950) defined a new Poisson bracket [^,r?]* for any two quantities ^ and t] as [^,T,Y = [i,T]] + [(,e,Me,,,Tj]. (i.ss) The new Poisson brackets satisfy the three conditions on Poisson brackets (1.64a)(1.64c). For any ^ we deduce the following condition on the new Poisson brackets [^,9sY = [te^] + [^,9,>]c,,s"[es",0s] = 0. (1.84) We may choose the ^'s to consist entirely of 0's and x'sThe determinant A will vanish unless the 's are second class. It follows that [9s, H] Â— 0 for every value of s. Therefore, the new Poisson brackets give the Hamiltonian equations of motion [g,H]* = \g,H]=g, (1.85) where g is any function of the g's and p's. 1.3 Covariant Poisson Brackets for Relativistic Field Theories J.E. Marsden, R. Montgomery, P.J. Morrison, and W.B. Thompson (1986) were the first to write the equations of some specific relativistic field theories in covariant Poisson bracket form. They demonstrated that the field equations are equivalent to the following bracket {i^,5} = 0. (1.86)

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21 In this expression F is an arbitrary function of the fields and S is an action integral. The relativistic fields were grouped into two categories: (i) pure fields and (ii) media fields. The pure fields, corresponding typically to gauge fields, will be associated with functionals F of the bcisic field variables (f)^ and their conjugate momenta H^'^. The media fields, which describe plasmas and relativistic fluids, are described by functionals F of only the field variables (without the conjugate momenta). An example of a media field would be a Boltzmann distribution / which possesses no corresponding conjugate momenta. It was noted in Marsden et al. (1986) that equation (1.86) satisfies Jacobi's Identity and the standard properties of Poisson brackets. Consequently, the space of fields is a Poisson Manifold (cf. Dirac (1950)). For pure fields, the covariant bracket possesses a space-time vector field V^. If a 3+1 Dirac-ADM decomposition is implemented, then this vector field corresponds to a choice of foliation of the space-time manifold. In the case of media fields, the bracket is a covciriant extension of the Lie-Poisson type discussed in section 1.2. Marsden et al. (1986) deduced their results for relativistic field theories by first considering the case of elementary particle mechanics. The familiar canonical Hamiltonian equations q' = ^, {1.87a) dpi P. = (1-876) follow from an initial variation J5 = 0 of the action integral S[T] = j{p4-H{q,p))dt. . (1.88) The integral is treated as a functional on P, the phase space of trajectories {q{t),p{t)) associated with movement of a particle. Appropriate boundary conditions (cf Arnold (1978,

PAGE 28

5 22 p. 243)) are imposed on this phase space. The variational principle may be formulated in terms of a Poisson bracket defined on F. For phase space functionals F and G, one may define the bracket where the fimctional derivatives take the form ds F{T + s6T)U=o = j {^-^^^)dt = / (^V + ^<^Pi)rf< (1-90) and the variations 5T vanish at the endpoints of V. One may verify that F solves Hamilton's equation iff {F,5}(F)=0, (1.91) for arbitrary functionals F. Marsden et al. (1986) generalized this bracket operation to relativistic field theories. This covariant theory, however, does not single out a single spacetime direction, but treats space and time in an equivalent manner. As well as the covariant bracket form, a 3+1 decomposition of equation (1.86) may be performed by assuming that F is of the form F = j n{t)T{p,q)dt, (1.92) where n{t) is an arbitrary function of time and T is an eirbitrary function of spatial coordinates q and spatial momenta p. Substitution of the functional F (1.92) and action 5[F] (1.88) into the bracket (1.91) yields the result ^ {F,5}(7) = I n{t){T-{T,H}^^^)dt = 0, (1.93) where {T, H}^^^ is the conventional Poisson bracket. The function n{t) is arbitrary, implying that :r={jr,i/}(3). (194)

PAGE 29

23 Among the examples considered by Marsden et al. (1986) were the Maxwell equations and the relativistic flat space Vlasov-Maxwell system. We will treat these examples in what follows. For the Maxwell equation system, one may denote the usual four-vector potential by A. In mathematical terminology, the potential is a one-form and we may construct the electromagnetic field tensor F (two-form) from it in the following manner F = dA, i.e., Ff^u = d^A, d^A^, (1.95) where 5^ = d/dx^ and the index takes the values /i = 0, 1,2,3. The standard electromagnetic field Lagrangian with minimal coupling to an external current density is L[A] = I Cd^x = j (-^F^^F"" A^J'')d^x, (1.96) where the Minkowski metric is used to raise and lower indices. One may derive the covariant momentum variables The functional derivatives are defined by analogy with equation (1.90) except that there is a constraint on 11'"' due to its antisymmetry -Fin^^'^ + s6Yin\s=, = / J^J^'^'d'^^ (1-98) where SI{^^'^ is an antisymmetric perturbation. A covariant Poisson bracket may be defined as where V is an arbitrary vector field on space-time and the functionals F and G are dependent on the field variables A,, and 11''''. An electromagnetic action S may be defined as a

PAGE 30

24 covariant analogue of equation (1.88) S[A, U] = jiW-'A^,^ H{A, U)]d:^x, (1.100) where H{A,U) = ^U^.U^^" + A^J*" = n'^M^,^ c. (1.101) It may be shown that Maxwell's equations can be derived from {F,S}v{AU)=0, (1.102) for any choice of F and V. One may derive the conditions = 0 ' (1.103a) and which imply and lÂ§-^ 0, (1.1036) n"" = -{d^A" d^A^") (1.104a) n'"',, = -7". (1.1046) Taking into account the definition (1.95), the mathematical identities correspond to the Maxwell equations. A 3+1 spacetime decomposition of this formalism may be performed (cf. Marsden and Weinstein (1982)). Coordinates are chosen so that the space-time vector field V is

PAGE 31

25 We may, for a closed system, choose J = 0 and rewrite the action S as S = j^j [^"^,,0 n]d^x^dt, (1.106) where the Hamiltonian density has the form ^ = ^(nijn'J+niotfÂ°) (i.ior) and Latin indices take the values i = 1, 2, 3. We will choose a functional of the form F ^ j n{t)F[Ai,ir^]dt, . ^ (1.108) where Ai and 11'Â° are the respective 3+1 field and conjugate momenta variables. A computation of the bracket (1.102) yields the result = {F,H}^^^}n(t)dt, (1.109) where H = jm^x (1.110) and {., is the canonical Poisson bracket for the conjugate field variables Ai and 11'Â°. Due to the arbitrariness of the function n{t), we may conclude that > = (1.111) It is assumed that H is only a function of the conjugate field variables Ai and IT' = 11'Â°. This follows by substituting n'J = -(aM^ -^A') (1.112) into equation (1.107).

PAGE 32

26 We may now examine the case of the relativistic Vlasov-Maxwell equations. A special relativistic particle moves in an external electromagnetic field F Â— dA in accord with the Lorentz force law ^ = Â„.; ^ = ^F'"'Â„Â„, (1.113) or ar m where e is the electric charge, m is the particle rest mass and r is the proper time of the particle. The momenta canonically conjugate to x*^ is 6 = mUfj, + -A^. (1-114) " . .. ' > 'Â• One may construct the following particle Hamiltonian ' with the corresponding Hamilton equations that are equivalent to dx^^_^H__p^^ dp^ _ dH _ e ^dA^ dr dpfj, m ' dr dx>^ c dx^^ The Boltzmann plasma distribution is constant along the pajrticle world trajectories Â— -Â— --^u''Â— -0 dr dxt^ cdpfj, dx** This collisionless Boltzmann equation may be reformulated in terms of the following bracket {f,H}^p^O, (1.117) where {fn\ g/ dg dg df \J^y}xp Q^^,Qp^ dxf'dp^' The basic media field of the plasma is the Boltzmann distribution /. We define the bracket of two functionals F and G of the distribution to have the form

PAGE 33

27 The derivation of this bracket will be given in section 1.4. For now, we may accept it as given without further proof. One may write the plasma action S as S[f] = j f{x,p)H{x,p)d''xd^p, (1.119) such that dS/6f = H. It is a straightforward exercise to prove that the covariant bracket equation {F,5}(/)=0 .. (1.120) is equivalent to the Vlasov equation (1.116). The relativistic Vlasov-Maxwell system may be analyzed in terms of the set (v4^, 11'^'^, /). One may construct the following covariant Poisson bracket One may define the action S for the Vlasov-Maxwell system to be + 1 fi^^P)^{pt^ l\) {P^ lA^)d''xdV (1.122) The field equations are formulated in terms of the relativistic Poisson bracket {F,S}v{A,U,f) = 0, ' (1.123) for all choices of F and V. Prom equation (1.123) we deduce the following identities ss ^ = 0, ^ = 0, and

PAGE 34

28 The mathematical identities (1.124) are quite useful. They yield the relativistic VlasovMaxwell equations df edf ,dA, H t; Â— U, dxf^ c dp^ ax'' d^F^"' = ^-lu-f{x,p)d'p, and F^, = d^A, d,A^. (1.125) 1.4 Derivation of the Plasma Bracket The plasma bracket was first derived in its present form by Morrison (1980a). However, Gibbons (1980) independently derived it. Also, the work of Iwinski and Turski (1976) should be mentioned in this regard. In the previous section, we dealt primarily with Lie brackets that were constructed from field variables and their conjugate momenta H^'*. In this section, we wish to consider the derivation of the plasma bracket (1.118). Since / is noncanonical (without a corresponding conjugate momentum), this bracket will differ from the previous field-momentum brackets. Following the treatment in Morrison (1980a), we will again deal with the Vlasov-Maxwell Hamiltonian system. Morrison (1980a) chose to formulate a Poisson bracket in terms of the noncanonical variables fa{x,v,t), E{x,t) and B{x,t), where the Boltzmann distribution of species a is denoted as faix,v,t), the electric field intensity is denoted as E{x,t), and the magnetic field intensity is denoted as B{x, t). In rationalized Gaussian units, with the speed of light set to unity, the Vlasov-Maxwell system can be written as dtBix, t) = ~VxE{x, t), (1.126a) dtE{x,t) = WxB{x,t) 5Zea / vfa{z,t)P{z\x)dz (1.1266)

PAGE 35

29 and dtfaiz,t) = -v-^^^^-^[ [Eix,t)+vxB{x,t)].^^^^P{z\x)dx, (1.126c) or maJ R2 where the Boltzmann distribution fa is a function of the phase space variable z = (f, v). In the coupHng terms of equations (1.126b) and (1.126c), the operator P{z\x) Â— S{x Â— r) has been used. The regions of integration are defined to he Ri = AxR^ and R2 = A, where R = (-00, +00) and A C R^. One wishes to formulate this system (1.126a)-(1.126c) as d^ydt = [^\H], i-0,l,...,6 (1.127) where takes the values ^fa, i = 0 (1.128a) = E, 2 = 1,2,3, (1.1286) = B, i = 4,5,6, (1.128c) and the Hamiltonian functional is written as Hm = E / l^av'^fadz + I liE"" + B^)dx. (1.129) The operator [., .] corresponds to the Poisson bracket. One may assume that the solutions of the Vlasov-Maxwell system exist in a vector space cu = uji x uj2 that is defined over R. The subspace cji has elements that are functions of z, while the subspace U2 has elements that are functions off. The operator Piz\x), appearing in equations (1.126b) and (1.126c), maps elements of one subspace into another. The vector space lj has an associated inner product {9\h) = J ^ gihidz + J ^ g2h2dx, (1.130)

PAGE 36

30 where gi G uJi and 32 Â€ 0^2 are the two components of g = (31,52) G u} = ujy x ui2Use of an antisymmetric operator O^^ on u; and the inner product (1.130) allows one to define a Poisson bracket in the following manner [F, G]=J2 {SF/5x'\0'^6G/Sx^). (1.131) The quantities F and G are elements of fi, a vector space (over R) of Frechet differentiable functional of the functions fa, E and B. Differentiability is defined with respect to the L? norm (1.130). The elements in have the form ^ Â• Â• Â• F{x'} = i^{xÂ°} + F2{x'}, i / 0, . f (1.132) where Mx'}= f F0{x0,XkVdx0, /3=1,2, and Xi = Z, X2 = X, n = 0, 12 = 1,2,3. The subscript k on xjt^" denotes partial derivatives of a general order A;. The Hamiltonian functional (1.129) possesses the form (1.132) with k = 0. Arbitrary CÂ°Â° functions of xa;' (for all i) are also elements of Q. In equation (1.131) the quantity 6F/Sx'is the functional derivative of F with respect to x'One may define the functional derivative as {d/de)F{x''iz) + ew{z)}\,=o = (SF/Sx^^lw), (1.133) where, without loss of generality, one may set i = 0. It is important to note that both wiz) and 6F/SxÂ° 6 uji and for i / 0 6F/Sx' G 1V2. For the purposes of computation, the relations 6x^iz)/SxÂ°{z') = 6{z z') and 5x'{x)l5x\x') = 5{x ~ x") are useful. Functional differentiation of the A; = 0 Hamiltonian (1.129) will involve only the following three

PAGE 37

31 quantities SH/dfa = ^TUav'^, SH/6E = E, and SH/6B = B. (1.134a) (1.1346) (1.134c) The bracket (1.131) is (i) antisymmetric [F, G] = Â— [G, F] and (ii) satisfies the Jacobi identity [E, [F,G]] + [F, [G,E]] + [G, [E,F]] = 0, where E, F and G Â€ fi. These are the standard properties associated with a Poisson bracket. A Lie algebra is defined as a Poisson bracket together with a vector space (Arnold (1978)). One may define a Hamiltonian system to be a set of partial difTerential equations with an integral invariant and a bracket satisfying properties (i) and (ii), such that the system may possess the form of equation (1.127), i.e., dtF = [F, H]. The antisymmetric operator O'^ will be written in the following cosymplectic form -{fa,-} p2->l p2^1 '^B pl->2 D -D 03 where O3 is a 3 x 3 array of zeros and D = [eijk{d/dxj)] {eijk is the Levi-Civita tensor). The braces appearing in the upper left corner denote the usual Poisson bracket ^ _ dg dh dg dh ' dr dv dv df (1.135) which will map elements of uji into uJi. The lower right-hand block (consisting of D's and 03's) will map elements of U2 into uj2The off-diagonal elements .dfa ma JR2 ov (1.136a)

PAGE 38

32 and Pr^^^f (.)(^xv)pdx (1.1366) will, as seen by their notation, map elements of u}2 into wi . The other off-diagonal elements ^ f {)^Pdz (1.137a) and Ph-^'^ Â— f {)(^^v)pdz (1.1376) will map elements of wi into a;2. One may rewrite equation (1.131) in the more explicit form [F,G] = [F,G]i + [F,G]2, (1.138) where and + / 7^-^X7^ =VxÂ— df. 1.1396) Jr2\6E 6B SB SeJ ' Using this bracket, one may derive the Vlasov-Maxwell system (1.126a)-(1.126c) from equation (1.127). The first term of equation (1.139a) will yield the Vlasov equation without the coupling term involving electromagnetic field variables. The final term in equation (1.139b) yields the vacuum Maxwell equations. The other terms in equation (1.139a) and (1.139b) correspond to field-distribution couplings.

PAGE 39

33 One may study the Vlasov-Poisson Hamiltonian system by neglecting terms in equation (1.138) and (1.129) that involve B. The resultant expression is obtained by writing the electric field ^ as a functional of the distribution /Â„. The corresponding Vlasov equation will be 5t/a = -{/a,^fp"}, (1.141) where HE{fa} = J2(f Hi''iz)fa{z)dz (1.142a) \ ^ /r^ faiz)faiz')H2''{z\z')dzdz'y + jRi JRi 1 Hi'^ = (1.1426) and = e^^l\r-i^\. (1.142c) The particle Hamiltonian takes the form = SHE/^fa-

PAGE 40

CHAPTER 2 NEWTONIAN COSMOLOGY AND THE VLASOV-MAXWELL SYSTEM A Newtonian cosmology (Peebles (1980)) is typically viewed as the infinite radius limit of a homogeneous and isotropic sphere of matter that expands homologously with the expansion rate governed by a time-dependent scale factor a{t). We may formulate a coUisionless Boltzmann, that is, gravitational Vlasov equation to describe the evolution of a one-particle distribution function / for such a scenario. We may consider so-called "equilibrium" solutions fo which correspond to expanding, steady state distributions (Peebles (1980); Bisnovatyi-Kogan and Zel'dovich (1970)). This is as straightforward as the construction of equilibria for isolated self-gravitating systems. However, for an isolated system, the equilibrium distributions fo lack explicit time-dependence. This facilitates the use of energy arguments to study the stability problem (Antonov (1961); Lynden-Bell and Sanitt (1969); Kandrup and Sygnet (1985)). Formulated in an inertial frame, cosmological so-called "equilibria" possess an explicit time-dependence which renders the standard energy arguments inadmissable. There is a way, however, to reformulate the problem so that cosmological "equilibria" possess no explicit time-dependence. This is done by reformulating the Vlasov description in terms of the noninertial average comoving frame. In this frame, the cosmological "equilibria" possess no expUcit time-dependence. Under such a transformation, the Hamiltonicin will acquire an expHcit time-dependence. In spite of this, a cosymplectic structure may still be found and an energy criterion for linear instability can be given. It should be remembered that it is not necessary to use energy arguments to study the problem of stability for cosmolog34

PAGE 41

35 ical equilibria. Instead, we may resort to a direct analysis of the linearized perturbation equations to determine the time-dependence of quantities such as the pertiurbed density 8p. However, because energy arguments are extremely useful in the study of the stability of isolated equilibria, they may be amenable to the study of Newtonian cosmological equilibria. The technique used here to extract a nontrivial stability criterion is quite general in the sense that, in the average co-moving frame, the Hamiltonian acquires an explicit time-dependence and the equilibrium corresponds to an energy extremum. Therefore, the general techniques may be applicable to other physical problems. 2.1 The Transformed Vlasov-Poisson Equation The distribution function J{R'^,Pa.,m,T) associated with the inertial frame may be defined as the number density of particles of mass m with momentum Pa at the point Ra for the time T. The gravitational Vlasov equation determines the evolution of the distribution /. The gravitational potential $is selfconsistently determined by the expression$(^,T) = -G / d'R'd'P'dm' "''^^^yi'^'''^K (2.2) J \R-R'\ ^ ' We wish to transform to a new set of variables {r",pa,m, <} from the original variables {/?", Pa, m,T} via a time-dependent canonical transformation of the form r" = a{T)-'R^ pÂ„ = a{t)[Pa mH{t)6abR'] and dt = a{t)~^dT (2.3)

PAGE 42

36 where H = d{losa)/dT, and o is an arbitrary function of time. The transformed Vlasov-Poisson equation will be of the form p7 + ma t) dr" J dpa ^^=0, (2.4) where = -G j d^r'cfp' and Equation (2.4) holds true for an arbitrary choice of a{T). However, an appropriate choice of a will cause the steady state equilibria to take a simple form. In this context, we may choose a(T) to satisfy the equation -'^2=~^PiTW = -^Po, (2.5) where p{T) is the average mass density of the Universe at time T. In the average co-moving frame, the quantity po = a^p represents the constant, time-independent density of matter in the Universe. For this choice of a(T) the physical peculiar velocity pa = ifi?" m will decay as 1/a. It follows that the momentum pa is conserved. We may construct a timeindependent solution fo{pa) for the transformed Vlasov equation (2.4) from this constant of the motion. With the assumption that is independent of r", we may deduce that dr"^ ' ' dr" J ^ f-P

PAGE 43

37 or"' J r Â— r (2.6) Therefore, we can write jdr"^ = fi-^r". It follows that the df/dpa force term in the Vlasov equation (2.4) will vanish. The df/dr"' term will vanish if spatial homogeneity is assumed. Consequently, we find that the distribution corresponds to a time-independent equilibrium df/dt = 0. Hamiltonian formulations of the gravitational Vlasov-Poisson and VlasovEinstein equations, for isolated self-gravitating systems, have already been presented. By direct analogy, the Vlasov equation (2.4) for the Newtonian cosmological system can also be shown to admit a Hamiltonian formulation. It is necessary (Morrison (1980a); Morrison and Pfirsch (1990)) to identify a Lie bracket [A,B] and a Hamiltonian function H in order that the Vlasov-Poisson system may be written as In accord with the ordinary Vlasov-Poisson system, we can define the bracket operation as where {a, b} is the ordinary Poisson bracket, S/Sf is a functional derivative of the Boltzmann distribution and dT = (fixd^pdm is a differentiable phase space volume element. It is straightforward to verify that the bracket is antisymmetric and satisfies the Jacobi identity and therefore constitutes a bona fide Lie bracket. Consequently, the evolution generated by any Hamiltonian H and this Lie bracket will be symplectic. We may view the evolution as a 2.2 The Vlcisov-Poisson Hamiltonian Structure (2.7) (2.8) [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 (2.9)

PAGE 44

38 generalized canonical transformation in an infinite-dimensional phase space of distribution functions. We shall select the "natural" Hamiltonian H= [dr^f-^[d\[ d\' ^^(^^ , (2.10) J 2m 2 J J \r-r\ to generate the transformed Vlasov equation (2.4). In equation (2.10) we have p{r,t) = J d^pdTnmf{f,p,m,t) and po satisfies equation (2.5). We calculate SH/6f = E to have the form E=-^+ a{t)m
PAGE 45

39 is different from the "natural" energy E = p^/2m in that it possesses an overall function of time. It is generally known that, in accord with conservation of phase, the Vlasov equation possesses an infinite number of constraints. If the distribution function behaves properly "at infinity" then we may integrate by parts to prove that the numerical value of an arbitrary constraint C^jdTxU) (2.15) is conserved. It follows that the time derivative of C vanishes ^ = [C,H]=0 (2.16) (To validate the integrations by parts it is sufficient to assume that fo is difFerentiable and, as r->oo, independent of r with an isotropic velocity distribution. Although they are not physically intuitive, periodic boundary conditions could also be employed as well as other, less restrictive requirements a& r-^oo ). It is not possible to define a unique Hamiltonian because of the existence of these Casimir constraints. The Hamiltonian H may be renormalized by adding a constraint C which will, however, leave the Vlasov equation (2.12) unaffected because [^^'/] = {/.|7}=0. (2.17) The Casimir constraints severely restrict the evolution of the Boltzmann distribution / as dictated by the Vlasov equation. These Casimirs (Morrison (1980a); Morrison and Pfirsch (1990); Kandrup (1990, 1991a); Kandrup and Morrison (1993)) constrain the evolution to an infinite-dimensional hypersurface of the infinite-dimensional phase space. The hypersurface is defined by the constancy of all the constraints. It follows that, when dealing with the problem of linear stability, we should only consider perturbations of the distribution 8f that preserve the constancy of the constraints. Only perturbations that propagate dynamically

PAGE 46

40 are relevant to the study of linear stability. We may, in a straightforward fashion, find the generic perturbation of the distribution that will conserve all of the constraints. It can be shown (Kandrup (1990); Morrison (1980a); Bartholomew (1971); Morrison and Pfirsch (1990)) that 6C will vanish if the perturbation Sf is generated from the equilibrium /Â„ through a canonical transformation. Therefore, the perturbation must be of the form fo + 6f = exp{{g, .})fo = fo + {g, fo} + (l/2!){5, {g, fo}} + (2.18) with a generating function g. It is simple, using the dynamically accessible pertiurbation df of equation (2.18), to prove that the first variation of the Hamiltonian 5H vanishes. This implies that all equilibria are extrema of the Hamiltonian (2.10). We can prove this by first calculating the expression J L2m \J |r-r| J \r~'r'\J. = J dFS^^^fE (2.19) and then substituting in the phase-preserving perturbation ^(^)/ = {g,fo}The net result is = I dr{g, f,}E = 1 dr{E, fo}g = 0, where, for an equilibrium. We now consider the second variation of the Hamiltonian We may calculate the following expression j{2)// = i|drj(2)/Â£; 2 J J IfPI ^ '

PAGE 47

41 Substituting and as given by equation (2.18), into this expression yields, after an integration by parts, the final expression This expression can be rewritten in a simpler form. Since the equilibrium /Â„ is a function of Pa and m, we may write the Poisson bracket as The overall homogeneity of the distribution makes it useful to utilize the Fourier transformed generating function 9{k\pa,t) ^^-^ I d^xe-^^-^g{x\pa,t). (2.22) Using the Poisson bracket associated with overall homogeneity and the transform (2.22) will allow one to rewrite the second variation of the Hamiltonian as S^'^H = -yd'k I d'p\g{k,p,m,t)\\k-u)(^k-^^ -2nGa{t) J d'k J d'p J d'p'^mg{k,p,m,t)j^ ^ x\m'g{k,p',m',t)-^-%], (2.23) with u = p/m. If we take fo to be an isotropic velocity distribution then this expression can be rewritten in a simpler format. For an isotropic velocity distribution, the equilibrium /Â„ depends on the momentum pa through the square of the magnitude p^. Consequently, one can write

PAGE 48

42 where, for an equilibrium, E = p^/2m and Fe Â— dfo/dE. Therefore, we write the second order perturbation of H as <5(2)if = iy d'k j d^p{-FEMk,p,m,t)k-u\' -2irGa{t) j d^k j d^p J d^p \^g{k,p,m,t)^^^FE [k v! m'g{k,p\m',t)-^F'E 2.3 Stability of Collisionless Vlasov-Poisson Equilibria (2.24) A simple stability criterion can be obtained from equation (2.23) or equation (2.24). The time derivative is negative semi-definite because, for an expanding universe, da/dt > 0. Therefore, in the average co-moving frame, any initial phase-preserving perturbation undergoes an intrinsically dissipative evolution. The perturbed energy will decrease after the initial perturbation. One may consider the case where 6^'^^H{to) < 0, where to is the time of the initial perturbation. Since the evolution is dissipative the quantity 5^'^^H{t) can only become more negative. The system will continue to evolve away from equilibrium with the magnitude of the second order perturbation increasing. Consequently, the system is hnearly unstable. A perturbation of a certain minimum wavelength will cause the second negative contribution of equations (2.23) and (2.24) to have a magnitude greater than the first term which is of indefinite sign. This minimum wavelength will be of the order of magnitude of the Jeans length. An order of magnitude estimate indicates that the first and second terms of equation (2.24) will be comparable in magnitude for a characteristic perturbation wave number k^ ~ Ga{t)po/v^, where u is a characteristic velocity and po is

PAGE 49

43 the co-moving density. This estimate is performed by assuming that both the unperturbed distribution Jo and the generating function g are well behaved, and that the derivative Fe is everywhere negative. One can calculate the physical wavelength A ~ {k/a)~^ {V^/Gp)^^^, which corresponds to the Jeans length, by utilizing the expressions for the physical density p = Po/oi^ and the physical peculiar velocity V = a~^v. The energy must be negative for wavelengths greater than the Jeans length. Specifying the form of fo enables us to obtain a better estimate of the physical wavelength. Therefore, the derivation of the Jeans instability via an energy argument given here is rigorous. By mciking certain assumptions, it can be shown that /Â„ is unstable to pertmbations with wavelength much shorter than the Jeans length. One example of this occurs for an equilibrium /Â„ with an isotropic distribution of velocities cind a derivative Fe that is not everywhere negative. As a specific case of this, we consider the distribution fo = E''exp{Â—pE), where 6 is a positive constant. Substituting into equation (2.24) a generating function g with nontrivial k dependence that is sharply peaked about the velocities where Fe is positive leads to an instability. In the previous example, this occurs as v^-^O. Therefore, for this example, both terms in equation (2.24) will be negative leading to a second order energy perturbation and a corresponding time derivative d6^'^^H/dt which are both negative. Another example involves any plane-symmetric equihbrium foipa) = foipi)Such an equilibria is unstable if there exists a range of velocities and a direction for which the derivative df/ dp J is positive. If we choose a A;vector, k = kaO, aligned in an appropriate direction such that -\9ik,p, m, t)\'{k .u)(^k.^^= -lgik,p, m, t)\'\k,fu,K = -2m\g{k,p,m,t)\'^\ka\^\ua\'^-^, (2.26)

PAGE 50

44 then, for df Idpa^ > 0, this expression will be intrinsically negative. However, if there does not exist a direction in velocity space for which df/dpa^ is positive, then the integrand of the first term integral in equation (2.23) is positive and the equilibria will be stable when subjected to perturbations of sufficiently short wavelengths. For example, if we choose a fc-vector k = kaO. -\gf{k-u)(k-^'j = _|5|2|/,^|2Â„^^ = _2Â„l|g|2|fcj2|^^|2^ > q (2.27) It follows that the aforementioned equilibria will be stable for all perturbations of charcicteristic wavelength much shorter than the Jeans length provided that they are monotonically decreasing functions of speed for all directions of velocity space. Consequently, we do not allow for population inversions. This argument does not imply that a negative energy perturbation of characteristic length scale R will demonstrate instability associated with a natural time of = R/V. Instead, the magnitude of will increase with a time scale determined by the variation of a{t). Consequently, the Hubble time scale tn sets a lower bound on the growth time of the instabihty. Even if we consider the case of a time-independent energy 6^"^^!!, e.g., perturbations of static equilibria possessing compact support or, as an approximate case, a cosmological setting with time scale < i//, a dynamical instabihty with a time scale toiR) could be implied for a negative energy perturbation S^'^'>H < 0. However, it is not certain that an instability will follow from a time-independent S^'^'^H < 0 because none of the negative energy perturbations must necessarily be coupled to the positive energy perturbations. A sufficient, but not necessary (Kandrup (1991a)), condition of stability, for a time-independent energy with a static setting, occurs when J^^^/f > 0.

PAGE 51

45 We now consider a straightforward counterexample. The Hamiltonian of a two degree of freedom system H = \{p^ + u;rW) 1{P2' + u>2W) (2.28) possesses an equilibrium solution gi = pi = 92 = P2 = 0 which is stable if the frequencies are constant, yet becomes unstable if du2^/dt > 0. The instability resulting from dissipation, through the effects of gravitational radiation, has already been studied for rotating perfect fluid stars (Friedman and Schutz (1978)) and generic rotating, axisymmetric equihbrium solutions of the gravitational Vlasov-Poisson system for isolated systems. The energy argument dealing with instability that is given here is directly analogous to the previously studied cases. However, the Newtonian cosmology is intrinsically negative, while, in the previous cases, an additional source of dissipation, such as gravitational radiation, must be invoked to bring about energetic instability. From a theoretical viewpoint, this is due to the fact that the equilibria are static in the conformal-but not true-sense. 2.4 Introduction to the Curved Space Vlasov-Maxwell System At this point we shall consider the Hamiltonian formulation of the Vlasov-Maxwell system in a curved backgroimd space-time. This exercise will serve as a springboard to a similar treatment of the Vlasov-Einstein system which will be dealt with in chapter 3. The Vlasov-Maxwell system couples the Maxwell equations, governing the behavior of an electromagnetic field, together with the Boltzmann distribution / which serves as a source for the electromagnetic field. The Boltzmann distribution / is coupled to the electromagnetic field in the sense that it is influenced by it while serving as its source. The evolution of the distribution / is self-consistently determined by the collisionless Boltzmann (Liouville) equation. If we take the Boltzmann distribution / to be a function of the space-time coordinates xÂ° and the physical {not canonical) momentum P", then it may be defined with respect to the

PAGE 52

46 tangent bundle corresponding to the background space-time. Therefore, the distribution / satisfies the following Liouville equation (cf. Israel (1972) and Stewart (1971)) + (-^%^' r^.P'^P'')^ = 0, (2.29) where F"^^ is the Christoffel symbol defined by the space-time metric and the Fciraday tensor F^j, satisfies the Maxwell equations J m and V^*F'"' = 0, (2.30) where * denotes a dual tensor. We recover the electrostatic Vlasov-Poisson system by taking the Newtonian limit c->oo. For any dynamical system, there are a variety of methods for implementing a Hamiltonian formulation. We shall utilize the most general technique, which is to proceed at a formal algebraic level. Accordingly, we shall work with a phase space 7 and a Hamiltonian H defined with respect to it. Also, a Lie bracket (F, G) which acts on functionals F and G of the phase space, will be used. It will be shown that, with suitable choices for the Hamiltonian H and Lie bracket, the VlasovMaxwell system may be formulated in terms of the constraint equations dtF = {F,H), where F is an arbitrary functional. From classical mechanics, the standard Poisson bracket equation dtF = {F,H}, and its associated Hamiltonian H, demonstrate that the dynamics in the six-dimensional phase space (x'.p.) is generated through a canonical transformation. Likewise, the Lie bracket equation dtF = {F,H) implies that the dynamics of F in the phase space 7 is generated by a type of generalized canonical transformation. However, this cannot be a canonical transformation because 7 does not have canonical coordinates.

PAGE 53

47 Initially, we might presume to take the phase space 7 as being constructed from spatial coordinates and momenta pi, and electromagnetic coordinates, i.e., the vector potential Ai and conjugate momenta 11^. Unfortunately, this presumption is incorrect. It would be appropriate to select a six-dimensional phase space (x\pi) for studying the dynamics of a Boltzmann distribution in a fixed electromagnetic field. If we take e as representing the canonical particle energy, then, in this case, the Liouville equation can be written as dtf = {Â£)/}Â• Conversely, the phase space {Ai,W) would be employed for the examination of the dynamics of an electromagnetic field with respect to a fixed current source J'. Taking the electromagnetic Hamiltonian (2.41), with an additional minimal coupUng / {-g)~^^^(fxJiA\ enables one to generate canonically the Maxwell equations corresponding to the problem. For the Vlasov-Majcwell system, however, one is not free to hold either the electromagnetic field or the Boltzmann distribution fixed. Our Hamiltonian description must taJce into account the reciprocity of the interaction between field and source, i.e., the electromagnetic field controls the dynamics of the Boltzmann distribution / which acts as the source of the electromagnetic field. In this case, we will take / to be a dynamical variable in the 7 phase space {Ai, 11', /). However, the 7 phase space is not canonical in contrast to the six-dimensional {x\pi) phase space or the infinite dimensional {Ai, IV) phase space. One can separate the electromagnetic variables into canonical pairs Ai{x^) and W{x^), but such a division of the distribution / into canonical pairs is not straightforward. If we attempt such a decomposition, then we must explicitly implement the infinite number of constraints associated with the VlasovMaxwell system because of conservation of phase. These constraints restrict the evolution to a still infinite-dimensional hypersurface of the infinite-dimensional 7 space. The distribution / may be decomposed into canonical pairs only in this hypersurface of the phase

PAGE 54

48 space. Since the cosymplectic structure is degenerate from the perspective of the full 7 phase space, this should be obvious. It remains valid, however, to use a noncanonical phase space Hamiltonian formulation, but, at some point in the analysis, the constraints corresponding to phase conservation must be explicitly implemented. Since the dynamics of the Vlasov-Maxwell system are generated by a generalized canonical transformation, it is possible to intuitively understand the evolution of the system. As for the Newtonian cosmology, a Hamiltonian formulation allows the problem of stability to be studied through the use of energy arguments. We wish to examine the energetic behavior of phase-preserving perturbations 6X = {6Ai,SU\6f) of the equilibria Xg = iAiÂ°,U.o\fo) associated with the Vlasov-Maxwell system. The interaction between electromagnetic field and Boltzmann distribution occurs on a fixed background space-time which possesses a timelike Killing field. It is straightforward to deduce a Hamiltonian formulation for the flat space electrostatic Vlasov-Poisson system. Â•' ' The electrostatic system has no radiative degrees of freedom and, with a choice of suitable boundary conditions, the electric field E^{x^) at time t can be written as a functional of the Boltzmann distribution / at the same time instant. We must find a suitable Green function corresponding to the Poisson equation. Therefore, the only dynamical variable of the system is the Boltzmann distribution /. So, the 7 phase space reduces to the phase space of distribution functions /. The Hamiltonian will have functional dependence H = H[f] and the Lie bracket (F, G) will act on functional F[f] and G[f] of the phase space. This Hamiltonian structure for the electrostatic Vlasov-Poisson system has been examined by Morrison (1980a, b) and Gibbons (1981). It is simple, for the Vlasov-Poisson system, to find phase-preserving perturbations (Gardner (1963); Bartholomew (1971); Morrison (1987); Morrison and Pfirsch (1989, 1990, 1992)) Sf that yield energy extrema 5^^'>H = 0.

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49 We can deduce nontrivial energy stability criteria by calculating the second order variation S^^^H. One example of a Vlasov-Poisson system would be a neutral plasma. Such a plasma consists of a fixed homogeneous, positive background of heavy ions interspersed with a gas of light electrons. On average, in a neutral plasma, the negative and positive charges cancel out and there is no external field present. A homogeneous and isotropic equilibrium electron distribution can be described by a function /o(-E), where E Â— p^l2m is the particle energy. If the derivative dfo/dE = Fe is everywhere negative, then (J^^^if is positive for all phase-preserving perturbations 5f which implies that the equilibrium is linearly stable. However, if, for some range of energies, Fe exhibits a population inversion, then there exist perturbations 5f such that < 0, which, however, does not necessarily indicate a linear instabihty (Holm, Marsden, Ratiu, and Weinstein (1985); Bernstein (1958); Gardner (1963); Penrose (1963)). Similar reasoning can be employed to derive a Hamiltonian formulation for the gravitational Vlasov-Poisson system (Kandrup (1990)). For self-gravitating systems, there are no homogeneous, static equilibria and the sign of 6^'^'>H becomes more ambiguous because of the attractive nature of the gravitational interaction. For the gravitational Vlasov-Poisson system there exists (Sygnet, Des Forets, Lachize-Rey, and Pellat (1984); Kandrup and Sygnet (1985); Kandrup (1991a)) a stability theorem which is a direct analog of a similar theorem for isotropic electrostatic equilibria. For the gravitational Vlasov-Poisson system, any spherically symmetric equilibria f{E,m) that is a monotonically decreasing function of the particle energy E = p'^l2m + m$o, for all values of the mass species m, is necessarily stable. The symbol represents the gravitational potential which is a functional of the Boltzmann distribution /Â„. The flat space VlasovMaxwell system has had its corresponding Hamiltonian structure studied through a variety of methods (Iwinski and Turski PAGE 56 50 (1976); Morrison (1980a); Marsden and Weinstein (1982)). The stability criteria for the flat space Vlasov-Maxwell system are directly analogous to those of the aforementioned Vlasov-Poisson system. Therefore, for a dynamically accessible perturbation, S^^'^H Â— 0, and the sign of the second variation (5'^' H provides a useful criterion for stability. The Hamiltonian structure of an electromagnetic fluid in special relativity, has, to a certain extent, been analyzed (Holm (1987)). We wish to derive the Hamiltonian structure for the general curved space Vlasov-Maxwell system. The Lie bracket utilized by Marsden and Weinstein (1982) for the case of a flat space Vlasov-Maxwell Hamiltonian system will be used here. However, unlike Marsden and Weinstein, the formulation will be given, with the exception of equations (2.59) and (2.60), in terms of the conjugate electromagnetic variables Ai and 11* and the canonical particle momentum pi. It is possible (Marsden and Weinstein (1982)) to choose the electric and magnetic field densities and and the contravariant physical momentum as alternate variables. However, this choice, for a curved space-time background, yields an unnecessarily complex formulation. For example, one can make a comparison between equation (2.45) and equations (2.59) and (2.60) to see this. The total Hamiltonian consists of an electromagnetic piece Hem and a matter contribution HmThe flat space electromagnetic Hamiltonian, corresponding to the canonical formulation, is generalized to the setting of a curved background space-time. The matter Hamiltonian used by Marsden and Weinstein (1982) is different from the one used here. If we write the canonical formalism in terms of an ADM 3-1-1 split into space and time, and select momentum variables corresponding to particle mass m and spatial momentum pi, rather than four-momenta (or or P'*), then, for simplicity of apphcation, we should use the matter Hamiltonian discussed here, but not the form considered by Marsden PAGE 57 51 and Weinstein (1982). With these choices for the Hamiltonian components it will become obvious that (i) the Vlasov equation is generated by the canonical particle energy e and (ii) the current J' is given by the functional derivative SHM/^Ai. 2.5 The Vlasov-Maxwell Hamiltonian Formulation In analyzing the Hamiltonian structure of the curved spa^e Vlasov-Maxwell system, it is convenient to work with the canonical coordinates The canonical momentum Pfi can be written in terms of the physical momentum P^^ and vector potential as Pfj. = gfiuP" + eAf^. For a fixed particle mass m, the mass shell constraint gfiuP^P" Â— g^^P^Pv = Â—m^ allows one to fix the value of any four-momentum component in terms of the remaining three. The expression for the fundamental phase space element is The explicit 3-1-1 split into space and time, along with the mass shell constraint, makes it convenient to choose pi and m as momentum variables. This choice allows one to deduce the expression -mdm = g^i'Pf.dPt = P^dPt = {p^ eA*)dpt. One calculates that dpt = Â—mdm/P^, which is used to rewrite the covariant volume element as -d'^xd'^p^ ^drd^xd^p^-^^^j = drd^xd^pdm = drdrdm, (2.31) where t represents the invariant proper time and dT represents the invariant 6-D phase space volume element corresponding to a fixed particle mass m and a constant time hypersurface. The number of particles with mass m in the infinitesimal six dimensional phase space volume element at time t can be defined as f{x\t,pÂ„m)d^xd^pdm, (2.32) PAGE 58 52 where / denotes the distribution function. The distribution / must be an (observer independent) invariant because, from equation (2.31), (Px(fipdm is also invariant. If the masses of the particles are restricted to a single value tUo, the distribution function can be written as f{x\t,pi,m) = f{x\t,pi,mo)SD{m rrio). (2.33) The electromagnetic Hamiltonian can be calculated in the following manner. One takes the standard electromagnetic action Sem = -\j {-gf'd^xF^.F^'' = J dt J d^xL = J dt J (fix{-g)^^^C, (2.34) where F^u = dfiAi, Â— di/A^, and imposes the electromagnetic gauge condition At = 0. One can decompose the spacetime metric g^j,, according to the ADM formalism, into the lapse function N, the shift vector N\ and the spatial three-metric hij. The line element (cf. Arnowitt, Deser and Misner (1962, p. 227)) ds'^ = -gttdt^ + 2gtidtdx' + gtjdx'dx^, : (2.35) has metric components gtt^NiN'-N\ gu = N, and 9ij = hij. (2.36) The contravariant metric components are PAGE 59 53 and 9'' = (2.37) One can raise and lower indices with the spatial three-metric. The inner product hijh^'^ = furnishes an example of this. The "time" coordinate t is chosen so that an "equilibrium" corresponds to a space-time metric and electromagnetic field that have no t dependence. This "time" condition applies to static and stationary space-times, which possess timelike Killing vectors fields with an electromagnetic field seen as independent of time from a rest frame. The field momentum can be calculated as dL d{dtA,) /,l/2 ATj = ^W^dtA,-h}l^Â—h''^F,, . . (2.38) and inverted to yield the expression , . dtAi = ^KjW + WFji. (2.39) The electromagnetic Hamiltonian HEM{n'\Ai) = I (fx{U^dtA-L) (2.40) can be written as Hem = ^ J d^x-^hijU'W + J d^xU'N^Fji +\j d\Nh'l^h}^hi'F,,Fki. (2.41) The matter Hamiltonian can be written as the minimal coupling between the distribution function / and the canonical particle energy e Hm = I dTfe. (2.42) PAGE 60 54 The canonical particle energy e = Â—pt satisfies the expression gV + ^g^'PtiPi -eAi)+ g'^pi eA^){pj eAj) = -m^, (2.43) where the gauge condition At = 0 has been imposed. The full electromagnetic Hamiltonian may be written as H = Hem + HmJ d?xNh}l'^jB'A^ = Hem + Hm + Hg, (2.44) where, in analogy with a fixed background charge density, an additional coupling term Hb, corresponding to a fixed, externally imposed charge current jb', has been added. The vector potential Ai dependence of the energy e, as made explicit in equation (2.43), implies that the matter Hamiltonian Hm couples the electromagnetic field to the Boltzmann distribution /. The equations (2.42) and (2.44) have been written in terms of a single species of matter. For the case of multiple species, we must define Boltzmann distributions fa for each species a and replace (2.42) with a summation over separate distributions /q. The Lie bracket operation (F, G) will act on pairs of functional F[Ai, II', /] and G[AÂ„ 11', /] which are defined on the infinite dimensional phase space {Ai,U.\f). Consequently, the bracket may be written as /jpr^\iA rri f ^3 fSFSG SG SF\ /Â• _ (6F 6G) where . ,^_da db db da ^ ~ dx'dpi ~ dx'dpi , represents the ordinary canonical Poisson bracket, and 6/SX represents the functional derivative corresponding to the variable X. It is straightforward to prove that the bracket is antisymmetric and satisfies the Jacobi identity {F, {G, H)) + {G, {H, F)) + {H, {F, G)} = 0, (2.46) PAGE 61 55 which implies that it is a bona fide Lie bracket. If one considers restricted functionals F[f] and G[f], that possess only / dependence, the bracket (2.45) reduces to (f.G)[/l = /dr/{^^,^}, (2.47) which is the proper bracket of the Vlasov-Poisson (Morrison (1980a, b); Gibbons (1981); Kandrup (1980)) system. For functionals F[Aj,n*] and G[Aj,n'], whose dependence is constrained to Ai and W, the bracket becomes {F.G)[A,n-l=yA(--^--^). (2.48) For vacuum electrodynamics (cf. Wald (1984)), formulated in terms of a specified 3+1 decomposition, this is the proper bracket. One may obtain equation (2.48) from the covariant bracket r^^rufSF 6G 6G SF \ , , ((F.G))=yAF'(^Â— -Â— (2.49) where 11'^'' = SS/6{dfiA^) and is an arbitrary vector field corresponding to a suitable foHation. The electromagnetic bracket given by Marsden et al. (1986) reduces to equation (2.49) in the vacuum limit. For an appropriate choice of bracket and Hamiltonian, the constraint equation dtF=^{F,H) (2.50) should reproduce the correct dynamical equations of the Vlasov-Maxwell system. To prove that this is the case, one must first calculate the following three identities 6H JJ=^^ (2.51a) ^ = ^'*Â»^n^ + ^'^:/(2.516) and 8H Â— = d,{WW Nm^] + dkiNh'/^h^^h'^Fi,) Nh^l^f + jB% (2.51c) PAGE 62 56 where ^=e/(-.)-/Vp/(?l^). One can use the three identities to prove that the Vlasov-Maxwell system is obtainable from equation (2.50). It is straightforward to check that the phase space variables, Ai, W and /, satisfy the appropriate dynamical equations 5H dtAi = {A^,H) = -^, :. (2.52a) dtn' = {U\H) = -Â— (2.526) oAi and dtf = {f,H) = |/, ^1 = {e,f}. (2.52c) The first two of these equations are the canonical equations for an electromagnetic field with a total current source = +jB^The equation for dtAi is equation (2.39) which defines the field momentum 11'. Likewise, the second equation is the 3+1 decomposition of (-5)1/2V^F^Â« = {-g)'^^Jtot\ (2.53) which is the Maxwell equation -dtE + 'V x B Jtot generalized to curved space. The Vlasov equation can be written as 5/ ^ P^dl _ P^dA^dl IPf^dgf^dl (0 r,A\ dt Ptdx^ dx^ dpi 2 F' dxi dp^' ^ ' where P^, = p^, ~ eA^, and P^' = g'"'{p^ eA^) are expressed in terms of the canonical variables and pi. This Vlasov equation is different from the usual one which is expressed in terms of the canonical momentum, with f{x\t,pi). However, a transformation to the physical coordinates {x\ t, Pi) will demonstrate that f{x\ t, Pi) satisfies the standard Vlasov equation (cf. eq. (2.29)) dl_ dAjdf ^ P\ dt ^ dt dP^ ~ pi df OA, df e Â— dx' dx' dPj J PAGE 63 57 PÂ« dx^ dpi 2 pi dx' dpi' ^ ' One of the curved space dynamical Maxwell equations is given by equation (2.52b). One may obtain the curved space generalization of the other dynamical Maxwell equation dtB + V X .B = 0. For example, we may define the quantity B' = e'^'^djAk = h}l'^e'^''djAk. (2.56) Through the use of the alternating symbol e'-''^ or the alternating tensor e'^*^, it will satisfy the following equation of motion (2.57) In an alternate formulation, dtB^ can be written in terms of the electric = -11' and magnetic B' field densities dtB^ = -dj (^^e^^'^hkiE^^ + dj (b^N^ B^N^^ . (2.58) Either equation (2.57) or (2.58) is the dynamical component of the dual Maxwell equation y^*piii' Â— 0. One can construct the curved space generalizations of the constraint equations V Â• B = 0 and V E Â— p. The first is a geometric identity resulting firom the fact that is constructed from an alternating symbol. The second constraint possesses a more complex character. The continuity equation V^J'' = 0 and the dynamical equation for V^F''' 0 ensure that, if the electric field constraint is initially enforced, the constraint will be maintained through time. Also, gauge invariance (cf. Misner, Thome and Wheeler (1973)) can be used to demonstrate that the electric constraint will hold initially. It is possible to rewrite the bracket (2.45) in terms of variables which are physically more intuitive. According to Marsden and Weinstein (1982), the bracket may be written in terms of E^ = PAGE 64 58 Â—11' and B'. These variables correspond to the true electromagnetic degrees of freedom and appear in equation (2.58). The bracket will take the form (F G) = [d\h'l'e^^A^^(^\ ^^(^\ ' J [sEidxAsB'^) 6&dxAsB'^) + with functional F[B',E\ f] and G[B\ E\f]. Also, if one rewrites the distribution f{x\ t, Pi) in terms of the physical momentum, instead of the canonical pi, this will lead to a modified bracket form 'J [SE' dxi\SB'' ) SE'dx'\SB>=). + Sf Sf 6F df SG 6G df SF (2.60) J dp,\5f)dpk\df)^' with functional F[B\F?J] and G[B^,E\f]. It is clear that equations (2.59) and (2.60) correspond to covariant analogues of the equations (5.1) and (7.1) found in Kandrup and Morrison (1993). 2-6 Perturbations of Time-Independent Vlasov-Maxwell Equilibria If there is to be a meaningful definition of "equilibrium" then the curved space, which the Vlasov-Maxwell system is situated in, will be a static or stationary space-time that possesses a timelike Killing vector. An "equilibrium" will correspond to the case where both the electromagnetic field and matter distribution are independent of time. Consequently, a "natural" choice of time coordinate t will cause the metric g^.^ and electromagnetic field F^j, to lack time dependence. This choice leads to a time derivative of the canonical momentum PAGE 65 59 that is equal to zero. The "electric field" IT* must vanish if the derivative dtAi is to be independent of time. However, unless the second derivative dt^Ai = 0, the F^^ tensor will not be independent of time. When the canonical momentum pi = Pi + eAi is expressed in terms of the physical variables and P,, it will, at "equilibrium", possess explicit time dependence because of the nonvanishing of dtAi. Therefore, for equihbria of this type, the time derivative dtfo cannot be set to zero. Instead, the equilibrium distribution must obey the relation dfo _ ^dAdfo _ ^dAdfo dt dt dP, ~ dt dpi ^"^-^^^ One wishes to know, for a given equilibrium Xg = {AiÂ°,Wo,fo), what mathematical form the dynamically accessible perturbation 6X = {SAt,6W,6f) will possess. It is straightforward to determine this because there exists an infinite number of conserved constraints, corresponding to conservation of phase, that are associated with evolution in the Vlasov-Maxwell system. Specifically, this implies that the numerical value of any functional constraint C[f{t)] = I dTxif) (2.62) remains constant as the distribution / evolves, i.e., dC[f]/dt = 0. (2.63) Consequently, any dynamically accessible perturbation Sf must leave the numerical value of any such constraint unchanged. It follows that 6C[f] = C[fo + 6f]-C[fo]^0, (2.64) for all orders of perturbation theory. A perturbation must obey this mathematical requirement to be dynamically accessible, i.e., in accord with the equations of dynamics. A PAGE 66 60 solution to (2.64) will yield the form of the phase-preserving perturbation. It may be shown perturbatively that the deformation which serves as a solution (Bloch, Krishnaprasad, Marsden and Ratiu (1991)) to equation (2.64), must be f = fo + Sf = exp{{g, .})/Â„ = fo + {g, fo} + ^{g, {g, fo}} + (2.65) where 5 is a generating function of the variables and pi, and (a, 6} denotes the ordinary canonical Poisson bracket. So, a canonical transformation, with a generating function g, corresponds to a dynamical pertiurbation. Unlike the distribution /Â„, the canonical variables Ai and 11' do not satisfy corresponding independent constraint equations similar to equation (2.61). Thus, the dynamically accessible perturbations SAi and 61V axe only constrained by the field equations whose source is provided by the dynamically accessible perturbation 6f. One can demonstrate that the first variation 6^^^H of the Hamiltonian (2.44) vanishes for a dynamically accessible perturbation (2.65). The first variation of the electromagnetic part of the Hamiltonian possesses the form j d^x5A,dj{Nh}l'^h'^hi^Fki) = j d^x5U\dtA,) + j d^xNh^/^SAiV^F"'. (2.66) The last equality has been obtained from equation (2.39) for dtAi and the fact that, for the case of a time-independent equilibrium with dtW = 0, terms proportional to 6Ai may be combined to form Vf,F^'\ Utilizing the unperturbed Maxwell equation V^.F'^" = J" + Jb" leads to the expression S^'^Hem = I d^x{dtAi)6U' + I d^xNh'^^iJ' + jB')5Ai. (2.67) PAGE 67 61 Also, the matter Hamiltonian has the first variation + j dVeSf. (2.68) The coupling Hamiltonian will have the variational form 6^^^Hb = I d^xNh^^^js'SAi. (2.69) One may combine equations (2.67)-(2.69) to get the total variation of the Hamiltonian S^^^H = I d^x{dtAi)SU' + I dVeSf. (2.70) This last expression has been, of course, calculated for an arbitrary perturbation. With the special choice of a dynamically accessible perturbation, it is straightforward to show that this variation is vanishing. For an equihbrium, the magnetic field is static dtB^ = 0. From equation (2.57) one can deduce that dtAi may be written as the gradient of an arbitrary scalar, so that dtAi = -di^. (2.71) Substitution of equation (2.71) into the first term of equation (2.70) yields the expression j d^x{dtAi)5Yi' ^j d^xNh^/^di^SF'* = J d^x^di[Nh^/^6F'^] = J d^xNh^^^^ViSF'\ (2.72) where an integration by parts has been performed and the expression H' = Nh^/^ F'* has been employed. Using the perturbed field equation and writing J J* in terms of 6f leads to J d^xidtA^)SW = I d^xNh}''^^5J' = e j dT^Sf. (2.73) PAGE 68 62 By substitution of the dynamically accessible form of 6f (2.65) into this expression, one can obtain the final result e j dr PAGE 69 63 j (fx6^^^A,dj{U'N^ WN') j d^x5^'^'^A^dj{Nh}l'^h}^y^Fki) (2.79) are the terms from Hem that have second variations of canonical variables, and j (fxNh^/^je'S^^^A, (2.80) are terms corresponding to Hm and HbIn these expressions, e represents the unperturbed particle energy and a variation SX not possessing a superscript represents a first variation 5(1). One can readily calculate that the first and second variations of the particle energy e axe de _ /P' eA 6e and 2 Therefore, combining the second and third terms in S^'^^Hs leads to the result I dr[S^^hfo + 6eSf] = 5Ai j Nh'/^d^xNh'/^e'^6A6Aj, (2.83) ,2 where dA^dAj ' PAGE 70 64 Using the unperturbed field equation allows one to rewrite the second contribution (J(2)if2 as = j d^x(5(2)n'(aiA) + / d\Nh^l^6^'^'^Ai{f + jfi'). (2.84) One can combine the second term of equation (2.84) with the last term of equation (2.80) to cancel out the first term of equation (2.83). Therefore, in the case of an arbitrary perturbation 5X, the second variation 5^^^H will be +^ j d^Nh'/^h^'h^'dF^jSFki y I d^xNh'/^'&^SAMj J d^xNh}''^5f5Ai + j d\6^'''^IV{dtAi) + j dre(5(2) /. (2.85) Restricting attention to perturbations that are dynamically accessible allows one to rewrite the last two terms of equation (2.85) in a simpler form. Using the scalar function one may write j d^x5^'^^IVdtAi = j d^x^diS^'^^Ii^ = J d^xNh'/^^Vi6(^^F^' = j iV/ii/2d3^*j(2) jÂ«. (2.86) In the case of a dynamically accessible perturbation, one can rewrite equation (2.86) as e I ^ g I dri{ff,{p,/Â„}}^ = dr{9jo}{9,e^}. (2.87) Likewise, the last term of equation (2.85) will take the form I dre6^')f = \j dr{g, {ff, fo]]e = -\j dr{g, e). (2.88) PAGE 71 65 Therefore, for a dynamically accessible perturbation, the second order variation S^^^H possesses the form + -J d^xNh^l^h'''h^^6Fij8Fki -J J d^xNh}l'^Q'HAi6Aj j d\Nh'IHr5A^ -~l dr{g, fo}{g, e + e^}. (2.89) This expression can be reformulated in terms of the electric and magnetic field densities ( Â£;'andB')as . Â• S(^^H =y d\^h,,{5E'5E^ + 6B'6B^) j d^x-^e^.^SE^B' -jI d^xNh^l^Q^HAi&Aj j d^xNh}l'^6f6Ai -lldT{9,fo}{g,e + e'^}, (2.90) where, as before, the alternating tensor is denoted as eijk. 2-7 Stability Criteria for Time-Independent VlasovMaxwell Equilibria It is straightforward to show that the Poisson bracket {g,Â£ + e*}, which appears in equation (2.90), possesses the mathematical form {g.e + e^} = {g,E} = --+e-^^^ ^^P^dAj^dg_ _ IP^dg^dg^ dx' ^p^ 2 PÂ« dx^ dpi ' If the physical momenta Pj are considered fundamental, it may be rewritten as (2.91) {g,E} = ^^+e^^-eF^^-l^^9"' ^3 ^ P'dx^^^dx^dP, ^^''ptdPi 2 pt 'd^dPi' ^^^^> The definition of ^' leads to the result {g,E} = -Vg, (2.93) PAGE 72 66 where ~ P* dx' ^ dt dPi Pt Kdxi dx' ) ^P^^ 2 dx' dPi represents the unperturbed Liouville operator. The evolution of the distribution occurs along the characteristics corresponding to time-independent equilibria. This evolution is governed by the Liouville operator. The evolution equations do not imply conservation of the canonical energy e whether there exists a time-independent equilibrium or not. Contrary to expectation, one will find de de P' dAi dt = dt=-'p^^^ (2.94) which, due to the time dependence of the vector potential, will not vanish. Since 11 is time-independent, one may compute _ P' dAj dt ~pt~df' (2-^^^ which implies that dE/dt = 0. It follows that E, not e, will be a conserved quantity for a time-independent equilibrium. One can construct a time-independent solution fo{E,m) to the Vlasov equation from these constants of the motion. Other equilibria are possible, however, this choice corresponds to an ensemble of particles with an isotropic velocity distribution. Such equilibria are ubiquitous. For this choice, equation (2.90) may be written as + 1 d^x^e^.kSU^dB'^ y I d^xNh'/^&^6A^SAJ -J d\Nh"Hr5A, + y dTi-FE)\{9,E}\\ (2.96) One may examine the various limits of (2.96) or (2.90). In the case of pure vacuum electromagnetism, the energy associated with a linearized vacuum fluctuation possesses the PAGE 73 67 form 2 J h^l^ + j d\^eijk6UUB'', (2.97) which is obtained from equation (2.96) by turning off the matter distribution. As another example, one may examine spherically symmetric configurations in the non-relativistic approximation. Spherical symmetry implies that the shift vector vanishes and the slow motion approximation implies that the contributions 6J^6Ai and SAi6Aj, which are of order (v/c)^, may be safely neglected. This yields the covariant analogue of the Unearized energy associated with the flat space Vlasov-Poisson system, viz. H. If, for example, > 0, for all generating functions g, then any static equilibrium must be linearly stable. On the other hand, if, for some generating function g, the condition < 0 holds, then a linear instability is not necessarily guaranteed. Such a configuration, although it possesses linear stability, will probably be nonlinearly unstable or unstable if subjected to dissipative efiects (Holm, Marsden, Ratiu and Weinstein (1985); Moser (1968); Morrison (1987)). Whether an equilibrium corresponds to an energy minimum is, generally speaking, diflicult to determine. There is, however, one case where it is simple to show that the condition < 0 exists for some phase-preserving perturbation. For example, assume that the energy derivative Fe is not everywhere negative. If one chooses a perturbation such that SJ'^ = J dr{gJo}p^^O, (2.99) i.e., the overall charge density remains unaff'ected but the velocity profile is shuffled, then PAGE 74 68 one can assume 5Ai and ^IT' are zero. However, the dynamical perturbation must obey the relation 6^'')H=\jdT{-FE)\{g,E)\\ (2.100) By shuffling the velocities a significant amount where Fe is positive in phase space, and by a minimed amount elsewhere, one can obtain a negative energy (2.100). This is the curved space analogue of the fact, first observed by Morrison and Pfirsch (1989), that "all interesting equilibria are either linearly unstable or possess negative energy modes." PAGE 75 CHAPTER 3 THE VLASOV-EINSTEIN SYSTEM One may now consider the full covariant Vlasov-Einstein system. We attempt to give a Hamiltonian formulation of the collisionless Boltzmann equation of General Relativity (Vlasov-Einstein system). Energy stability criteria will be deduced for this Hamiltonian system. For this system (cf. Israel (1972, p. 201); Stewart (1971)), the Boltzmann distribution / will be defined with respect to the cotangent bundle corresponding to the space-time of choice. The characteristics, which correspond to geodesies of the space-time, are the trajectories along which the distribution / evolves. The system is complicated by nonlinearity because the space-time is determined by the distribution / which acts as a source of the mean field Einstein equation. Thus, the background space-time is not fixed and eternal as it was for the case of the Vlasov-Maxwell system in chapter 2. Therefore, in this scenario, the Boltzmann distribution has its evolution governed by the Vlasov equation -^-0^ Â— ^ = 0, a, 6, c,... =0,1,2,3 3.1 m ox" 2 Sx" m opa where the space-time metric gab must self-consistently satisfy the mean field Einstein equation G\ = 87rT\ = Stt / -^^Ip-p,. (3.2) J (-3)1/2 m There have been earlier investigations of the linear stability of restricted cases of the Vlasov-Einstein system. One example corresponds to spherically symmetric equilibrium solutions whose pertmbations are also spherically symmetric. For example, in the earliest investigation by Ipser and Thorne (1968), a variational principle is derived from the dy69 PAGE 76 70 namical equations. Unstable modes can be detected by analyzing the variational principle. Ipser and Thome (1968) write the perturbed distribution Sf as a. sum 6f = Sf+ Sf+, where 6f+ and 6fare even and odd, respectively, with respect to spatial momentum inversion. It was noted that, in certain circumstances, the odd perturbation component satisfies the operator equation dt^Sf= Tdf-. This equation is second order in time and the symmetric operator T, which is defined with respect to a suitable Hilbert space, is dependent only on the equilibrium /Â„. The pioneering work on this technique was carried out by Antonov (1961), albeit in a non-relativistic setting. When applied to a relativistic system, this method has been subjected to extensive numerical analysis by Ipser (1969a, b) and Fackerell (1970, 1971) among others. Ipser (1980) then went on to analyze spherically symmetric systems in terms of plasma physics techniques. An extension of the Newtonian "energy arguments" (cf. Ipser and Horwitz (1979)) was carried out with the use of an energy-Casimir argument to obtain linear stability criteria. This approach, due to Newcomb (cf. the summary in the appendix of Bernstein (1958)), had originally appeared in plasma physics. This approach is also discussed by Morrison and Pfirsch (1989) (and in internally cited references). The basic idea is as follows. For generic perturbations 6f, equilibrium solutions /Â„ of the VlasovEinstein system do not correspond to energy extremals because the constraints associated with phase conservation imply the existence of infinitely many conserved quantities, C[f]. It is possible, however, for equilibria to be extremal by considering only perturbations which ensure the conservation of one particular constraint, c[f]. Thus, the sign of the constrained second variation, yields a nontrivial stabihty criterion. For example, the condition > 0 ensures linear stability. PAGE 77 71 Recent work has applied these plasma physics arguments (of. the papers of Morrison and Greene (1980), Morrison (1980a) and Morrison and Pfirsch (1989), and internally cited references) to Newtonian galactic dynamics (cf. Kandrup (1990, 1991a, b)). Kandrup and Morrison (1993) have adapted these ideas in order to furnish a more systematic Hamiltonian formulation. In this work, the Hamiltonian character of the dynamical evolution, which was ignored in earlier research, is addressed. Also, all of the constraints (not just one) are implemented. Consequently, the earlier work of Ipser (1980) is clarified and extended. So far, only the stability of spherically symmetric equilibria, subjected to spherically symmetric perturbations, have been studied. Research prior to Kandrup and Morrison (1993) assumed that the Boltzmann distribution was restricted in form. For example, the distribution was supposed to be a monotonically decreasing function of the pcirticle energy e. Lie algebraic techniques were employed by Kandrup and Morrison (1993) to show that the spherically symmetric Vlasov-Einstein system, subjected to a three-plus-one decomposition into space and time, is Hamiltonian. Consequently, it is necessary to identify a bracket [., .], i.e., a cosymplectic structure and a Hamiltonian fvmction H, which are used to construct a Vlasov equation dtf = [/, H] for the distribution function /, where dt denotes a coordinate time derivative. Since the dynamical evolution is governed by a "generalized canonical transformation" in the infinite-dimensional phase space associated with distribution functions /, it is obvious that the Vlasov-Einstein system is Hamiltonian in character. However, the transformation is not truly canonical since / is a non-canonical variable. Due to constraints that are associated with conservation of phase, the Hamiltonian system is constrained. These phase-preserving constraints can be straightforwardly implemented in an explicit fashion. This enables one to prove that time-independent equilibria are energy extremals, such that S^^^H = 0 for all dynamically accessible perturbations 6f. PAGE 78 72 If the second variation S^^^H is non-negative for all dynamically accessible perturbations, then the system is guaranteed to be linearly stable. However, as noted in Kandrup and Morrison (1993), a negative second variation S'^'^^H < 0, corresponding to some dynamically accessible perturbation 6f, does not prove hnear instability. It should, however, at least guarantee structural instability with regard to dissipation or any nonlinear it ies which are present (cf. Moser (1968); Morrison (1987); Bloch et al. (1991)). Unfortunately, although this approach has proven highly effective, it is still restricted to systems, whether perturbed or not, that are spherically symmetric. Most related research, with the notable exception of Ipser and Semenzato (1979), has been directly based on this assumption. Due to the Birkhoff theorem, the spherical symmetry of the system ensures that gravitational radiation will not be present, i.e., the radiative degrees of freedom will not be activated by the dynamics. Therefore, for appropriate boundary conditions at infinity or any horizons that may be present, the gravitational metric gab at time t will be uniquely determined by the stress-energy tensor Tab at that instant. This can be seen by implementing the standard Schwarzschild gauge, where only the pure radial grr and pure temporal gu metric components remain free. For this particular gauge, the t-t component of the Einstein field equation fixes grr uniquely in terms of the energy density T^f Also, if grr and the radial "pressure" T^r are both known, then the r r component of the field equation will &x guThis situation is analogous to the externally imposed curved space Vlasov-Maxwell system considered in chapter 2. For the case of spherical symmetry, the magnetic divergence equation guarantees the vanishing of the magnetic field. The symmetry ensures that only the radial component of the electric field Er does not vanish. The radial component is fixed uniquely by solving the Poisson equation for a charge distribution p. It is possible to PAGE 79 73 devise a Hamiltonian formulation which is applicable to general space-times, rather than only the spherically symmetric case considered by Kandrup and Morrison (1993). This will complicate the physics because the gravitational metric gab must be treated as a dynamical variable, as well as the Boltzmann distribution foKandrup and Morrison employed only the distribution function fo as a dynamical variable. They worked in the context of an infinite-dimensional phase space of distribution functions fo at a given instant of time t. For a general space-time, we must employ three dynamical variables. These are (1) the spatial threemetric habix^) for a f = constant hypersurface and (2) the conjugate momentum W^{x^), along with (3) the Boltzmann distribution fg. One can define fimctionals F[/ia(,,n"'',/] in terms of the infinite-dimensional phase space (/ia6, IT"'', /). It is necessary to select a Lie bracket (.,.} and a Hamiltonian function H such that the correct dynamical equations governing the evolution of hab, 11"'', and / are generated by the constraint equation dtF = {F,H). 3.1 The Vlasov-Einstein Hamiltonian Formulation It will be convenient for the dynamical analysis of a combined space-time-distribution system to invoke an ADM decomposition into space and time (of. Wald (1984)). One may view the space-time, described by a metric gab, as a foliation into a series of spacelike hypersurfaces. The hypersurfaces are parametrized by a global time function t, and n" will denote a unit normal vector to the hypersurface. The spatial three-metric hab{x') for a < = constant hypersurface is defined by the projection tensor, hab = gab + riaTib (a, 6, ... = 0, 1, 2, 3). (3.3) A timelike vector field t", which satisfies the condition fVat = 1, will be introduced. This vector field may be decomposed into a lapse and shift, respectively N and TV", which are PAGE 80 74 normal and tangential components with respect to the hypersurface, i.e., N = Â—f^Ua and Na = habt^. The spatial three-metric hab and its conjugate momentum 11"'' correspond to dynamical degrees of freedom, while the lapse N and shift Na are gauge variables with respect to the Hamiltonian and momentum constraints. By construction, it is obvious that the time-time and space-time components of the spatial metric vanish, i.e., htt = = /i" Â— /i'* = 0 ... = 1,2, 3). Therefore, the ADM decomposition of the space-time metric gab is gu = NiN'-N^, git = N^, and 9ii = hij, (3.4) and possesses contravariant components ^ iV2' ^ ~ iV2 and 9'' = (3.5) Analogous to the Vlasov-Maxwell system, the time coordinate is a "natural" coordinate corresponding to a timelike Killing field. An equilibrium solution of the Vlasov-Einstein system implies that the distribution function, the metric, and the conjugate momentum tensor are all independent of time t. A Lie bracket acting on functional pairs F[hab,W'',f] and G[hab,W'', f], will define the cosymplectic structure of the Hamiltonian formulation. For the fundamental dynamical variables hab, ^Â°^ and /, we take {F, G) to possess the form {F,G)[hab,U^\f] = Wn I d$$^^^ - PAGE 81 75 f ^JSF 6G] where dx^ dpi dx'^ dpi denotes the standard canonical Poisson bracket, aind S/6X denotes the functional derivative with respect to the variable X. For a i = constant hypersurface, the covariant phase space volume element can be written as dT = (fxcfpdm. (3.7) If d^p refers to spatial momentum, as defined in the cotangent bundle, then d^xd^p is covariant. Therefore, the spatial coordinates x\ the spatial momentum pi, and the mass m are held fixed with respect to variations carried out at some time t. Consequently, the volume elements dT and d^x are not subject to variation, i.e., SdT = 6d^x = 0. Kandrup and Morrison (1993) discuss other possible choices in greater detail (e.g., the original IpserThorne (1968) prescription). It is straightforward to show that the bracket is antisymmetric and satisfies the Jacobi identity * (F, {G, H)) + (G, (if, F)) + {H, {F, G)) = 0. (3.8) Therefore, it is a bona fide Lie bracket. Consequently, the dynamics associated with this bracket and any Hamiltonian H will be symplectic with regard to the infinite-dimensional phase space {hab, /). The bracket (3.6) is an obvious analog to the bracket for the curved space Vlasov-Maxwell system. In the absence of matter, the bracket {F, G) corresponds to the natural bracket of vacuum gravity (cf. appendix E in Wald (1984)). In the case of a spherically symmetric space-time, the metric can be written as a functional of the distribution / and eliminated as a dynamical variable. Consequently, the bracket (F,G) PAGE 82 76 will reduce to the bracket [F,G] used by Kandrup and Morrison (1993). It is relatively straightforward to identify the Hamiltonian function H. The Hamiltonian will be the sum of a purely gravitational piece He and a matter contribution HmThe Hamiltonian of vacuum gravity (cf Arnowitt, Deser, and Misner (1962)) has the form Hg where nc = ^h'^^{N[-^^^R + h-^u'^'iiab ^n2)] -2iV(,[Z)a(/i-V2nÂ°Â«')] + 2Da{h-'^^Nt,U'''>)}. (3.9) In terms of notation, the covariant derivative operator is denoted Da and the scalar curvature of the spatial metric hab is denoted ^^^R. The matter Hamiltonian is taken to be the minimal coupling of distribution to particle energy, viz. Hm = j d^xHM = J dVfe, .' (3.10) where the ordinary particle energy is denoted e = fpaFor appropriate "time" coordinates, the particle energy is e = -pt. The fundamental momentum variables, which appear in equation (3.10), are the spatial momentum components pi and the particle mass m. The particle energy e is a function of the variables {x',pÂ„ m, t} and may be defined by the mass shell condition, + [h"' -j^jPaPb -m2. (3.11) N2 ^2 An alternative form of Hm is Hm = J Nh}l'^d\Tt\ where rpt f d^P f t f d^pdpt f t PAGE 83 77 is the t-t component of the stress-energy tensor written as a momentum integral of the distribution. By vajriation of the gauge variables N and Na, one can demonstrate that the Hamiltonian and momentum constraints hold. Performing the calculation de (e + N'^paf ON -N^pt enables one to determine the functional derivative (3.13) 6H M "V 6N I PAGE 84 78 The initial value problem demands the imposition of these constraints. However, this is auxilliary to the dynamical aspects of the Hamiltonian formulation. Assuming that the dynamical equations are generated by the Hamiltonian H and Lie bracket {.,.), it is guaranteed that the constraint conditions will be enforced and propagated to later times. If the bracket and Hamiltonian correspond to a suitable Hamiltonian formulation, then the constraint equation Â• dtF = {F,H), (3.20) which holds for arbitrary functionals F, must generate the dynamical equations of the Vlasov-Einstein system. The dynamical equations of the gravitational field can be investigated first. Prom computation these take the form dthab = {hab, H) = IGtt^ (3.21a) and Q^^ab ^ H) = -167r^. (3.216) Along with the functional derivative 5Hm Â— , (3.22) m ~2^^ Jm^iV ^ ^h^' the momentum constraint (3.19) and the standard equations for SHc/Shab and SHg/SU"'' (cf. Arnowitt, Deser and Misner (1962)) can be combined to yield the dynamical Einstein equations, viz. ^i^-" 1^^^ = 2/i-'/'A^(nÂ„6 IhabU) + 2D^,N^), (3.23a) and ajH"'' = -167r 6h ab PAGE 85 79 -2iV/i-i/2(n"^nc'' ^nn"*) + h}l'^{D''D''N h^^D^DcN) +SnNh'/' [ ^ U'^^PcPdh"'. (3.236) The Vlasov equation possesses the form dtf = {f,H) = {e,f}. (3.23c) This is just the 3+1 decomposition of the covariant Vlasov equation (3.1). 3.2 All Vlasov-Einstein Equilibria are Energy Extremals A solution to the dynamical equations, {hÂ°abix^),^o''''{^')Jo{x\pi,m)}, that is timeindependent, will be an equilibrium solution of the Vlasov-Einstein system. Consequently, one may conclude that dthÂ°ab = dt^o'"'' = dtfo = 0. The aforementioned dynamical equations, combined with the conditions of equilibrium, imply that 6H/6hab = 6H/6W'' = {^o, fo} = 0. The particle energy at equilibrium is denoted Eq. Generic time-independent equilibria, {/i^aft, IIo"*, /o}, only constitute energy extremals with regard to the restricted class of dynamically accessible perturbations. However, one cannot assume that the first variation (J^^^if vanishes for perturbations of an arbitrary type. The first variation S^^^H will take the general form Employing the dynamical equations and the conditions of constraint leads to ^"Lj <^M-dtTi'''5^')ha, + dtK,6^')Yl-'') + j dTeoSW f. (3.25) PAGE 86 80 The equilibrium condition guarantees that dtTlo"''' Â— dthÂ°ab Â— 0. The last term involving S^^^f will only vanish for a phase-preserving perturbation, i.e., S^'^H = I dTsoig, fo}--/ dTgi^o, fo} = 0, . . (3.26) where an integration by parts has been performed. The final equality follows from the equilibrium condition {Â£o,/o} = dtfo = 0. Also, one should note that the condition S^^^H = 0 can be used to define an equilibrium. 3.3 The Second Variation S^^^H The set of variables {hab, H"*, N, Na, /} can be collectively denoted as . The second variation S^^^H can be written in terms of this "book-keeping" notation as = E Â§,s'''y' + ^ E E "v'^wy^ (3.27) where the functional derivatives are computed with respect to the equilibrium distribution. The second variation provides a nontrivial criterion for linear stability. Nonlinear stability criteria can be obtained by analyzing arbitrarily high orders of variation. Employing the conditions of constraint, dU/dN = dU/dNa = 0, and the dynamical equations evaluated at a time-independent equilibrium, dH/dhab = dUjdW^ = 0, causes the first term of (3.27) to reduce to / dTsoS^^^f. (3.28) Evaluating this term for a phase-preserving perturbation S^'^^f of the form (2.18) yields the result I dTsoS^'^f = 1| dTsoig, {gJo}} = dT{g,eo}{9,fo}. (3.29) Therefore, the second variation can be written as PAGE 87 81 I J +5EE|^'"">"*"'>'^ (3.30) where the vacuum Hamiltonian is denoted Hq. The second term in equation (3.30) can be rewritten as the following expression ^ E E / ^fQ^s^'^y's^'^y' + E / dr^si^^r^s^^)/, (3.31) which undergoes further reduction to the following format: de dY' de \ . de -lZEl<^f^si^^y^si^^y^ = E / <^s^'^y's^'ifSh) -\i:El^fJ^s'''y's^''y'(3.32) The sum over variables Y' in equation (3.32) does not include / or IT"* because the particle energy e does not depend on these variables. The first term of equation (3.32) can be written as I d^x ^6p6N + SrSNa + i(J(iV5"'' 2N''J^)5Kb , (3.33) where equation (3.13) for de/dN, and analogous expressions for de/dNa and de/dhab, have been used in the derivation. Equation (3.33) employs the notation and S"' = h'/'^h'^-T.^h^. (3.34) PAGE 88 82 First order variations 8p, (57Â° and ^5Â°* are expressed in terms of 6^^^ f Â— {g,fo}The second term in equation (3.32) may be evaluated by noting that dN^ dNdNa dNadNb = 0, (3.35a) and that ^ N^h'"'h'"^PcPd, (3.356) dhabdN 2N{e + N^pa) and dhabdN, d'^e 1 7V4 = h'^''h^^%, (3.35c) :y'^h'fh'^h'"^p,PfPgPf, dhabdhcd 4 (e + AT'^pfc)' + '^h^^'fi'^^^'N'^^pf. (3.35d) Use of equations (3.35b)-(3.35d) yields the expressions I dhabdN ^^"''^^ ^^''^'^''^habSN, (3.36) / '^^ dhbdN ^^''^^^^ = I d^^j''h'"'5Kb5N^ (3.37) and where J OhabOhcd = J (fx QiV5"'"^'' Nh^'S*"^ 2/i"V''iV''j 5hab5h^, (3.38o) gabcd ^ h'/^haef^bff^cg^dk f _jhp IPePjPgPh Equations (3.36)-(3.38), together with equation (3.33), lead to the following expression for equation (3.32) / d\[8p5N + SrSNa + \5{NS''^ 2N''J^)5Kb -S^^dhabdN PAGE 89 One may use the identities 83 ^J^h^-^ShabSNc + {^NS"-^^'^ ^Nh^'^S^'^ h"-^J^N'^^ ShabShcd]. (3.39) ax jb uac jb \Tdi (3.40a) and SiNS"'') = S''''SN + NSS' Â•ab i to rewrite equation (3.39) in a more straightforward manner. The use of equation (3.30) leads to the final result + Sp6N + 5r5Na + \N6S''HKb N^SJ^Shab 1 ^Nh'^'S'^^jShabShcd where (3.406) (3.41a) (3.416) corresponds to the second variation 6^'^'>Hg associated with the first order perturbation rpj^^ matter contributions to S^'^^H may be reformulated in terms of the distribution function. This alternate form can be written as ^(2) H = -\ldr{9,eo}{9Jo} + 1 dr{g, /4 + iVV) PAGE 90 84 iV2 2(Â£ + Ni^pk) +5(2)i/c[ Sh'^>'{D'Db6hac + DWaShbc) + 5h\{D'D''5Kb D''DbSh$$ +2Sh"'(D''DaSh,d + DdD,Sh\ DdD'^SKc) "1 + ~D'^5h\ Db6h!"^ {2D''5hac Dc5h\) PAGE 91 85 + lDaShcdD''Sh'='^ + D'^Sha'^iDaShc'' Dc6h/)} (3.44a) and P^A" = 2{6n''^h'"^ U''^6h'"^){DcShbd + D^Shcd DdShu). (3.446) These formulae are given in the appendix of Moncrief (1976). The perturbed constraint equations permit a further simplification of the second variation S^^^HqThe exact Hamiltonian constraint can be written ^^^R + h-\^U^ Hatn"'') = 167r/i-i/V, (3.45) where p is the energy density. A linearized perturbation of this constraint will have the form -dhJ^^R'''' DcD^Sh/ + D^DHhab -2h-^ShabiU\Yl'"' ^nn"*) = Wirh-^^Hp Sixh-^''^ ph'^Hhab. (3.46) The constraint (3.45) may be used to eliminate the unperturbed p from this equation, giving the expression +\h-'h'^''6Ka{ii'''nab \u') 2h-'5n-\nab \KbYi) -2h-HKb{n\U.''^ ^-im'^^) = 167r/i-i/2^p. (3.47) Likewise, for the exact momentum constraint one may write its linearised perturbation as 2^,^"" + U'^^h>"'{2D,Shda DdShac) = WnSj". (3.49) PAGE 92 86 The perturbed constraint equations (3.47) and (3.49) may be used to reduce equation (3.43) to the form 6^^^Hg[6''^^Y^] = Jd^x{SpSN + SrSNa) + [ d^x{ND^H NaD^A"). (3.50) SZTT J This expression may be substituted into equation (3.41a) to obtain the final form for , viz. 6^^^H = -^ldr{9,eo}{g,fo} + j d^x ^^N6S"-''dhab N''6J''Shab + ^ / d^'xiND^H NaD^A''). . (3.51) This formaUsm can be used to study the stability of spherical equilibria with respect to non-radial perturbations. With the notable exception of Ipser and Semenzato (1979), Uttle work has been devoted to this problem. Consequently, the subject is not well understood. However, the energy functional (3.41b) should be, in principle, amenable to the testing of nonradial perturbations that cause S^^^H < 0. Also, the formalism can be used to test more general spherical equihbria than in Ipser and Semenzato (1979). These authors, in their development of the symmetric operator formahsm mentioned in the introduction, restricted the equihbrium to the form /Â„ = /Â„(Â£:, m), where e and m correspond, respectively, to the particle energy and mass. A generic spherical equihbrium may be written as fo = fo{Â£,J'^,m) , where the squared angular momentum, associated with rotational symmetry, is denoted J^. Ipser and Semenzato (1979) also imposed the restriction that the equilibrium has a monotonically decreasing dependence on e. Therefore, the derivative dfo/ de is intrinsically negative with respect to all values of the particle energy e. The PAGE 93 87 formalism developed here does not place constraints on the form of the equilibrium distribution. Another objective is the study of the stability of axisymmetric equilibria. Little research has been carried out on this subject. The motivation for this work comes from the fact that physical equilibria generally possess angular momentum. Consequently, there should be a flattening of the matter distribution due to the effects of rotation. Shapiro and Teukolsky (1993a,b) have succeeded in generating axisymmetric equilibria through a useful numerical algorithm which they have recently developed. It is plausible to conjecture that the stability of rotating, axisymmetric systems can be studied with the formalism developed here. One may attempt to prove that certain classes of equilibria always possess phase-preserving perturbations for which the energy decreases. Subsequently, further investigation may explain if, for relatively short time scales, these perturbations result in physical instabilities. Unfortunately, it is not straightforward to follow through with this program. Attacking the problem by means of a post-Newtonian expansion leads to inconclusive results. It appears that a generic rotating, axisymmetric solution of the Einstein equations must be found before stability results can be deduced from it. However, such a solution is not known. PAGE 94 CHAPTER 4 THE VLASOV-BRANS-DICKE SYSTEM Carl Brans and Robert Dicke (1961), in an attempt to make Mach's principle compatible with General Relativity, formulated a scalartensor theory of gravity in which the Newtonian gravitational constant Go is represented by a scalar field. This is the celebrated Brans-Dicke gravitational theory. It was shown by Dicke (1962) that Brans-Dicke theory may be written in an equivalent representation in which the inertial masses of elementary particles vary as a function of the Brans-Dicke scalar field while the gravitational constant Go remains fixed. The Brans-Dicke theory may be formulated in either representation. Although the physical interpretation of gravitational phenomenon varies between the two representations, the actual physical predictions remain equivalent. For example, as pointed out by Dicke (1962), the gravitational red-shift in the variable mass representation is only a partially metric phenomenon. Since the particle masses are affected by the scalar field, part of the red-shift results from changes in the energy levels of the atoms. Consequently, the red-shift is not entirely metric induced. In this representation, the measures provided by rulers and clocks are not invariant with respect to position in space-time. Also, firee falling matter does not follow geodesic trajectories in space-time. However, (massless) photons still do. The geodesic equation of the variable G representation is ^{mgiju^) ^mgjk,iU^u'' = 0, (4.1) where r denotes the proper time and u' is the relativistic four-velocity. One can switch to 88 PAGE 95 89 the variable mass representation through the transformation m = ^l'^m (4.2) and Qij = ~^9ij(4.3) The speed of Hght c (units: [L][T]~^) and Planck's constant h (units: [M][L]^ remain invariant under this transformation. Consequently, the geodesic equation for the variable mass representation has the form d . 1 .. Â— {mgiju') -mgjk,iu'u'' + m^i = 0. (4.4) As shown by Toton (1970), the original Brans-Dicke variational principle 6 1 d^x{-g)"\R cj^^ + IStt^Lm) = 0 (4.5) may be reformulated, by means of the conformal map (4.3), as S I M-<,?'\R (I + ^) 4^ + le^f^iiM) = 0. (4.6) where Lm is obtained from Lm by replacing in Lm by g^^/(/>. The field equations associated with the original Brans-Dicke variational principle (4.5) are SttG uj 1 (t>Gnu = Â—^T^>^ + (l>;piu 9txi^'";a + '^[;,i;u ^9nu(l>''^(t>;a] (4.7) and i2u; + 3)'-.^, = ^T\, (4.8) where G^^ = ^g^^i?. As noted by Bruckman and Velazquez (1993), the presence of second order derivatives of in (4.7) make the variable G representation of Brans-Dicke theory unsuitable for application of the canonical ADM Hamiltonian formalism. PAGE 96 90 Through examination of the field equations associated with the variational principle (4.6), viz. G,^=^^-^^ + {^-+uy^,,^,^ ^,,.$-Â°$,.) (4.9) and where$ = ln(/), (4.11) one may see that equation (4.9) has the form of the standard Einstein field equation. Equation (4.9) can be written as G,u = ^t*Â°,^ (4.12) where T^Â°' is the sum of a matter stress-energy tensor and a scalar field pseudo stress-energy tensor. Consequently, because of the similarity to Einstein theory, the ADM formalism is easily applied to the variable mass representation of the theory. Following Kandrup and O'Neill (1994), we may, while employing an ADM splitting into space and time, demonstrate that the coUisionless Boltzmann, i.e., Liouville equation of the Brans-Dicke gravitational theory is Hamiltonian. By analogy with the Vlasov-Einstein case, criteria for the linear stability of time-independent equilibria, corresponding to relativistic matter configurations, will be derived. As for notation, all physical quantities subjected to the transformations (4.2) and (4.3) will be denoted with a bar superscript. This notation is employed in the equations (4.2) and (4.3). Unbarred quantities, of course, have not been subjected to this transformation. In terms of physical variables, one works with a spatial three-metric hab and scalar field 0, as

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91 well as their respective conjugate momenta 11Â°'' and 11. These variables are sufficient for the vacuum metric-scalar Hamiltonian formalism. In the presence of matter, one must include the distribution function / as a fifth variable. Actually, as will be shown, it is necessary to work with the barred distribution /. Consequently, the Hamiltonian formalism is written in terms of a mixed system of variables. One may define functionals F = F[hab, 11"''; (}>, 11; /] in terms of an infinite-dimensional phase space {hab, H"''; 11; /). It will be shown that the Vlasov-Brans-Dicke dynamical equations are equivalent to the constraint equations dtF = {F,H), where F denotes a phase space functional, dt denotes a coordinate time derivative, H denotes a Hamiltonian and (., .) denotes a bracket operation, to be later defined, which acts on the functional pairs F[/ia6, n"**; H; /] and G[hab,n-b;,Il;f]. ' ^'1'^ One may demonstrate that the first order variation is extremal, = 0, (4.13) when subjected to a perturbation = H > 0, guarantees that the system is stable. A negative sign, (J^^^if < 0, does not guarantee linear instabihty, but should, at least, ensure nonhnear nonstability or nonstability with respect to dissipative effects.

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92 4.1 The Vlasov-Brans-Dicke Hamiltonian Formulation The ADM decomposition into space and time will be implemented in exactly the same way as in Kandrup and O'Neill (1994), which, in turn, followed the treatment given in Wald (1984). The metric line element will transform according to equation (4.2) as = g^^dx^'dx" = (f>-^gf,^dx^'dx'' = (l>-^ds'^. (4.15) Therefore, the transformed metric line element is ds = 4>'^''^ds. (4.16) The expression (4.16) is the same as Dicke's (1962) X'^^'^ds = ds, where the arbitrary well-behaved function A is set equal to "^ Consequently, the relativistic fourvelocity transforms as u'' =
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93 the distribution will be f = f{x'',pa,fh,t) ^ f{xÂ°,pcÂ„m,t). (4.20) It should be noted that only the mass variable m is altered by the transformation equations (4.2) and (4.3). The scalar field (j) cannot be factored out of the distribution (4.20), implying that the barred distribution / must be used in the Hamiltonian formulation. One can postulate the following bracket . Â•/r ^ Â• SF SG 6F SG \ Jhab6U^>> SUabSh'''>J 3 f6FSG SF6G\ (F,G)[/iÂ„6,n"*;0,n;/] = IGtt J d^x^where d/SX denotes the usual functional derivative with respect to the variable X, and , . dA dB dB OA , , is the ordinary canonical Poisson bracket written in terms of the variable mass representation. Other notations in equations (4.21) and (4.22) are as in Kandrup and O'Neill (1994). It is interesting to note that the bracket is of a mixed composition, with the matter component written in terms of the transformed variables. The bracket (4.21) is a bona fide Lie bracket because (a) it is antisymmetric and (b) it satisfies the Jacobi identity (F, (G, H)) + (G, {H, F)) + {H, {F, G)) = 0. (4.23) We may separate the Hamiltonian function into two parts: one part, He, which is purely gravitational and another part, Hm, which governs the matter coupling to the gravitational field. Variation of the action given by Brans and Dicke (1961), leads to the Brans-Dicke

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94 field equations written in the variable G representation. Subjecting these equations to the transformations (4.2) and (4.3) yields an altered form of them. Due to the fact that the scalar field is determined by the field equations, the quantitative physical predictions of the theory remain the same. The variable mass representation of the Brans-Dicke action (4.6) does not couple the Ricci scalar invariant R to the scalar field (f). Therefore, when the ADM formalism is implemented, one can use partial integration to rid the action of unwanted boundary terms in a straightforward manner. Thus, it is convenient to work within the variable mass representation. Following the treatment given by Toton (1970), one can write Hq as Hg^ j d^x-Ha, (4.24) where the Hamiltonian density is + (' + ^^^1 2JV,[D.(ft-"^n"') '-h-"m4'% (4.25) where Da denotes the covariant derivative associated with the spatial three-metric hab and ^^^R denotes the three-ciurvature scalar corresponding to habThe matter Hamiltonian, written with respect to the variable mass representation, takes the form Hm = J dTfe. (4.26) If it is demanded that the Hamiltonian H be extremal with respect to variations in the lapse N and shift Na, then the Hamiltonian and momentum constraints are enforced. Following the treatment given in Kandrup and O'Neill (1994), one may derive the expression de {e + N^paf dN -N^pt (4.27)

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95 which implies that 6Hm f d^pdfh 5N -I N 2n -NY ^ (f>y^J Ml/2 01/2 ie + N'^Pa) 2, iV2 fh (4.28) where n" is a unit normal vector orthogonal to a t = constant hypersurface. The final equality follows from the identity n" = [t^ Â— NÂ°')IN . It should be noted that the derivation of equation (4.28) has used the relation e = e. Likewise, one may derive an expression for SHg/SN, viz. 6Hr 6N IGttGo /ii/2 (6 + 4a;) V2 (4.29) The Hamiltonian constraint, 6H/SN = 0, is obtained by combining equations (4.28) and (4.29), which yields the expression +
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where m = AT'' N N ' one may rewrite equation (4.32) as . _ Â— . LÂ— n^rÂ» c4 ,/)2 Substitution of the following relation n^n'' = 0. into equation (4.35) yields the resultant expression -^n'^n = 0. C4 Use of the equation 2 (6 + 4a;) V2 J 2 (f>'^ will further modify equation (4.38) into the form 2' (/)2 c4 (/.2

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97 Finally, use of the transformed determinant (^3/2 (4.40) will allow one to derive the Hamiltonian constraint ab (t>,a,b 01/2 (4.41) This expression is in agreement with equation (4.30). The momentum constraint can be derived in a similar, and straightforward, manner. First of all, one may calculate the functional derivative 6Hm/6N(1, viz. 6Hm _ f d^pdpt 6Nd ~ J Nh^/^ N f m (4.42) The functional derivative of the gravitational contribution will be 6Hg _ 6Na ~ IGnGo {-2/1^/2 ^Â„(/i-i/2n'"') + n^-''}. (4.43) Combining equations (4.42) and (4.43) yields the momentum constraint -2/ii/2i)^(/i-i/2n"d) + n<^.rf 167rGo/ii/2 fd^pdpt f d'^pdpt â€¢"Pa{i+N'Pc) N I rh (4.44) The momentum constraint (4.44) may be checked by taking the inner product of the Brans-Dicke field equation with the spatial metric /i"'' and the unit normal vector n^. This will yield the following expression -/i"''%n'^ = 0. 9 (4.45)

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98 The relations (4.46) and enable one to rewrite equation (4.45) in the form ^ ^* ' , (4.47) (4.48) Finally, one can calculate ) m (4.49) which, when substituted into equation (4.48), yields the desired momentum constraint -2/ii/2z);5(/i-i/2n^") + n<^'" 167rGo/ii/2 r d^pdpt r d^pd I Ml /2 N ) I rfi (4.50) The requirement that the correct dynamical equations be generated by the constraint equation dtF = {F,H), (4.51) which holds for arbitrary functionals F, implies that the Lie bracket and Hamiltonian combine to generate the Vlasov-Brans-Dicke system. Consequently, one may construct the appropriate dynamical equations dthab {hab,H) = IGtt dt = ((/>, H) = 167r 6H dH a^^Â°'' = (^Â«^//) = -167^ dH ^hab (4.52a) (4.526) (4.52c)

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99 and dtU = (n, H) = -167r^. (4.52d) Explicit forms for equations (4.52a) to (4.52d) may be derived. These are dthab = lGTr-^^=2Nh-'/^{Uab-\habU'c) + 2D^aNb), (4.53a) +/."'DÂ„(W"A-'/2n'.) + 32^~^fl 16Â»^^ (4.53c) and 6h ab 2 2 2 -^n-'-A^" + 8.^ / i0|A-p*S(4.53., Equation (4.53b) may be verified by computing the conjugate momentum 11 = dL/d^. Since we are working in the pseudo-Einstein representation of the Brans-Dicke theory, the equation (4.53a) for dthab is what we would expect. To prove equation (4.53c) we must rewrite equation (4.10) as (Z^2.)NUy^^-^^'^^L,.'^'i!^% ,4.54)

-^ (f)^ 6(j) ^ '

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100 where \$ = ln, (4.59)

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101 where equation (4.53b) has been used in the derivation. Consequently, one can write equation (4.59) as de df de df df or as de df de dj df where e = e, = and pa = PaSubstitution of de ^ dg^"" Pf,p^ dx"' dx"2p* (4.62a) and de _ p*^ dpc (4.626) into equation (4.61) yields the result a fPÂ° I ^9^>mP.df df pi dx^ dx'^ 2p* dpa d(f>"^' (4.63) From Bartholomew (1971), we know that any phase-preserving perturbation can be written in the form Sf^exp{{g,.})fo = {gJo} + ^{9,{g,fo}} + ... . (4.64) We know that generic time-independent equilibria, {hÂ°ab, Ho"*; ,6Y[-5f), (4.65) which satisfy C[f + 8f]-C[f]=Q, (4.66) where C[f] = I dTxif). (4.67)

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102 For a phase-preserving perturbation S^^^f, we can prove that the first variation S^^^H vanishes identically. We calculate the first order vajiation of the Hamiltonian which reduces to + 1 dTeoS^'^f. (4.69) For a time-independent equilibrium this expression reduces to = I dfÂ£Â„(5(i)/= I dTsoigj'o} = / dTgisoJo} = 0, (4.70) because {Â£oJo} = dtf-^dt
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103 at an unperturbed equilibrium, reduce the first term of equation (4.73) to the single contribution E^jS^'^y' = I dreo6^'^f. (4.74) Therefore, one may write ^(^^^f as ' (4.75) where the second order phase-preserving perturbation 6^'^^ f = ^{g, {g, /o}}, given in equation (4.64), has been used in the derivation. Following the calculation given in chapter 3 for the Vlasov-Einstein system (Kandrup and O'Neill (1994)), we find that + ^Ei:^*"'V"-"">'^(4.76) Due to the fact that e Â— e, one can rewrite the expression E/Â«<')yV.)(/^) (4.77a) as From Kandrup and O'Neill (1994), we know that S^'^dT = 0, (4.78) and, since df = ^^/"^dT = (p^^'^d^xd^pdm, one can calculate the variation of the transformed phase space volume element dWdv = h-'^'6('UdT + 0i/2j(i)rfr = i^df . (4.79) ^ 1 (j)

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104 Expanding the summation of equation (4.77b) yields the result /*^<.,^Â„.,(;^) ,|^,a,^Â„Â„Â„(/|L) ,Â„Â„, This can be written as where J ]_ 2(f) 2(p 2 4

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105 Qjy Rafted _ ^Nh'^'^S'"^ h'"'j''N'^y^^^habS^^^hcd where abed ^ ^^laej^bfj^cgj^dh f ^'^P f PePfPgPh gaoca / A combination of equations (4.81) and (4.85) yields the final result J i 2(f) 2^ +^<5(i)(iV5"'' 2iVj'')j(i)/iÂ„ft \{NS''^ 2iVV'')^(i)/iÂ„6^ ^ 4 ^(^_j^Sabcd _ ^J^f^acgbd _ h^^Jf'N'^y^^^habS^^^hcd Â• This expression may be further reduced through the use of the identities (4.85) (4.86) (4.87) (4.88a) and SPINS'''') = S''''d''^^N + iV(5(l)5"^ We can write the expression for 6^^^H as ab 2(p 2 4 (j) l^^afccd _ iiV/j-5'"')^(i)/iÂ„fe5(i)/i,^ o Z (4.886) (4.89)

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106 where S^'^Ha[6^'^Y] = +\j:T.^^^^'^y'S^'^y' (4-90) represents the second order energy cissociated with a first order perturbation S^^^Y. A straightforward calculation of the gravitational contribution yields +2(^+0; + 2 (6 + 4a;) (6 + 4w) (6 + 4a;) + ^ld\{ND^H-NaD^A''). (4.91) We may compute D^H and D^a". One may write D'^H as the sum of two terms D'^Hi and D'^H2. The first contribution to D^H (depending solely on metric terms) is given by -2nÂ°''n^''(Awrf(i)/,Â„, \h^5^')Kb)8(')h\ 2(nÂ„6j(i)nÂ°* \m^')ii)6^')h\ ^ 2

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107 +2(5(l)n"''5(l)n'^'^(/lac/lM \Kbhcd) -h'/^{i^)R\h6Wh\f Is^'^h'^'S^'^hab] ^'^RabS^'^h'^'sWh^ -5^^h''\D''Db5'^^^Kc + D'^DaS^^^hbc) + S^^^h^ciD^D^S^^^hab D^Db5^^^h\) +26^^h"'{D''DaS^^^hcd + DdDcd'^^h^'a + DdD^'S^^^Kc) + ]^DJ^^^h,dD''6^^^h'"^ + DH^^^ha'iDdS^^^K" DcS^^hd")}. (4.92a) The second contribution to D'^H (with dependence on scalar terms) is given by: (6 + 4u}) +2/i-i5(i'/i^ (6 + 4u;) (6 + 4a;) . -2/1-1 '1 2((5(i)2((?(i)n)^ 80n(^(i).^j(i)n . (6 + 4a;) ^ (6 + 4a;) (6 + 4a;) -2 ^(
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108 -4<5(i'nj(i'/iÂ°V,6 4(j(i)n/i"'' + (4.92c) The energy functional may be reduced to a simpler format by using the perturbed constraint equation. The Hamiltonian constraint can be written as where P=^TÂ„6nV. (4.936) One can compute the perturbed form of equation (4.93a) to be +/i-i/i^''j(i)/ied(n"''nÂ„6 ln2) -2/i-i(j(i)n"''(nab \habn) 2/i-ij(i)/iÂ„,(n''eninn"'') ^ 2 ""(6 + 40,) (6 + 4u) " (6 + 4w) The use of the unperturbed constraint (4.93a) allows one to derive the expression +^'^^^^/ic
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109 -[l+.y/'h^'^h^'6^'^K, = (4.95) One may carry out the same procedure for the momentum constraint 2 which has a perturbed form -SWu''' Ud^'^h'^^cf,,, US^'U'" ^^(J(i) J^ (4.96) The utilization of equations (4.95) and (4.96) allows one to rewrite equation (4.91) in the form 6^^^Hg[S^'^Y] = -|d3^(
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110 4.3 The Question of Stability It has been shown by Reddy (1973) that a BirkhofF theorem is not generally satisfied by a spherically symmetric matter configuration in the Brans-Dicke (1961) gravitational theory. Therefore, it is possible that gravitational radiation may escape from a pulsating spherically symmetric source. If, in the case of a spherical matter configuration, a negative second order perturbation of the Hamiltonian < 0 is permitted, will the system tend towards equilibrium or will it possess a run-away instability? We may answer this question by determining the sign of the time derivative of the second order Hamiltonian perturbation dt (4.99) To determine the sign of equation (4.99) one may exploit a Hamiltonian series expansion about equilibrium of the type given by Bruckman and Velazquez (1993), viz. H = Ho + S^'^H + ^6^^^H + ... . (4.100) The condition of equilibrium, S^^^H = 0, allows one to rewrite the equation (4.100) in the format H = Ho + l:6^^^H + ... . (4.101) The following expression 2 dE 1 d(J(2)^ IS may be deduced since Hg is a constant and H = E. Determining the sign of dE/dt equivalent to finding the sign of equation (4.99). We know firom Wagoner (1970) that the simplest situation in which pure scalar radiation is produced remains the radial pulsation of a spherically symmetric matter configuration.

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Ill Following Wagoner (1970) one may deduce that the scalar-energy loss from such a configuration will be where the s subscript denotes scalar, Ro denotes a suitable far-field radial coordinate, c represents the speed of light, and T," corresponds to the time-time component of the stress-energy tensor associated with the scalar field. The equation (4.103) holds for the variable G representation. Intuitively, this expression is intrinsically negative because scalar monopole radiation, from a radially pulsating spherical source, will carry away energy, rendering the right side of equation (4.103) positive definite. Equation (4.103) can be written in terms of the variable mass representation as The variable mass form for the stress-energy tensor associated with the scalar field is found, by inspection of equation (4.9), to be (4.103) (4.104) SttG* (4.105a) where G* = Go (4.1056) Choosing a spherically symmetric metric element ds2 ^ _ (4.106) permits one to compute the following expression for T^", viz.

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112 The scalar field may be reabsorbed into the metric coefficients yielding the result rjil. which, as can be demonstrated, is equal to the time-time component of the scalar field stress-energy tensor in the variable G representation. Consequently, one can conclude that fdE\ fdE\ This result implies that the physics of the system is invariant with respect to choice of representation. From Brans and Dicke (1961) we have the contravariant metric components Ji ,-2. _ (1 + {MI2c^ot)[{2u: + 4)/(2a; + 3)]^/^)'^^ 9 -e Â— -Â— (4.110a) (1 (M/2c2<^or)[(2a; + 4)/(2a; + and (1 + (M/2c2 0, (4.112)

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113 which implies the following conditions: (f),<0 Â±f^H<0. (4.113) One may deduce the following theorem, namely: the negative energy extremal of a spherical matter configuration can manifest run-away energetic behavior with increasingly negative energies and a gradual slide away from equilibrium. However, this run-away instability hinges upon the fact that a negative energy extremal does indeed exist for a spherical matter configuration. An interesting point made by Wagoner (1970) and Nutku (1969) is that there is no scalar field energy loss at the order of the post-Newtonian approximation. Scalar energy is only lost at higher than the post-Newtonian approximation.

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CHAPTER 5 THE VLASOV-KIBBLE SYSTEM In 1956, Utiyama (1956) tried to show that General Relativity could be formulated as a Yang-Mills theory (Yang and Mills (1954)) of the homogeneous Lorentz group. Corresponding to the six space-time dependent parameters e'^^x) = Â— e-''(x) of the homogeneous Lorentz group are six vector gauge fields A^^^^. Utiyama, on an a priori basis, introduced curvilinear coordinates and a tetrad of vectors hk^ corresponding to a vierbein system of gauge fields. Therefore, the attempt to show that General Relativity is a Yang-Mills theory was only partially successful because no explanation for the a priori introduction of the curvilinear coordinate system and vierbein tetrad hk^ was given. Furthermore, it was not possible to relate the Yang-Mills gauge fields A'^^ to the Christoffel symbols T"^^^ in a unique manner except by making an ad hoc assumption that the quantities T'^^^ calculated from A'^f^ are symmetric. In 1961, Kibble (1961) demonstrated that the vierbein fields h^^ could naturally appear in a gravitational Yang-Mills theory through an extension of the gauge group to the inhomogeneous lO-parameter Lorentz (Poincare) group. The tetrad of gauge fields hk'' would then be associated with the four additional translation parameters of the (extended) Lorentz group. Kibble was able to prove that a gravitational theory, which is remarkably similar to General Relativity, can be derived as a Yang-Mills theory of the Poincare group associated with a Minkowskian space-time. We will attempt, here, to provide a Hamiltonian formulation of the Vlasov-Kibble gauge invariant gravity system. In Kandrup and O'Neill (1994), a Hamiltonian function was utilized in which the gravitational field is treated on the same 114

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115 footing as the Boltzmann distribution /. For this theory, an analogous formulation will be employed in which a vierbein b^f, (inverse -.h}.^ ,b'' fj.hk' = 5/) and a local affine connection {A^^ = Â—A^\) function as dynamical variables. In this chapter, then, there will be five different dynamical variables, namely: (1) the tetrad field b'^^ and its (2) conjugate momentum Iljt'', as well as (3) the local affine connection A'^fj, and its (4) conjugate momentum Uij'^, and (5) the Boltzmann distribution function /. As in Kandrup and O'Neill (1994) one may identify a Hamiltonian function H and an extended bracket {A,B). These are chosen so that the equations of constraint dtF = {F, H), for all functional F = F[b''^Â„ Rfc'^; A'^ ^, n^/; /], are equivalent to the correct dynamical equations of motion for 6*^^, 11*;^, A'^, Uij^ and /. 5.1 Kibble Gauge Invariant Theory of Gravity One may consider an arbitrary matter field Lagrangian which is invariant with respect to the global Poincare group Â£^ = Â£^(x,a,x), (5.1) with dkX = dxl dx'' and x a column vector that is transformation invariant with respect to some representation of the Lorentz group. This theory will remain invariant with respect to the local Poincare group if we introduce a tetrad field 6^^ and a local afiine connection A'^f, which corresponds to the six Lorentz parameters e'^ (e'J = -e^') of the same group. The covariant derivative of the matter field has the following form (cf. Kibble (1961)) DkX = hk'^V.x = hk^{d^ + \a'^^S,,)x, (5.2) with hk" denoting the inverse tetrad field {b^hk" = 6/) and Sij denoting the Lorentz group generators. The gauge invariant matter Lagrangian will be C'= bC^ix,D,x) = bC'^ix,X;k), (5.3)

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116 where b = det 6*^^ and D^x = X;kProm the two gauge fields one may construct two different types of gauge field strengths, viz. (1) the torsion TV. 2V[,6V] = 2(6' [^,,] + A',[J^]) (5.4) and (2) the curvature i?V = 2(^'^[^,.]+^V[.^"^H)(5-5) The notation -^[^^,/] denotes the condition of antisymmetry, viz. One may take the gravitational Lagrangian density to possess the general form (Nikolic (1984)) ' ' CO = bC^{T,,k,R^Jkl). (5.7) In this expression we have converted Greek indices to Latin indices by contraction with the inverse tetrad field h^^, e.g., T'^mu = hm^T''^^. Latin indices correspond to local Lorentz (anholonomic) indices, while Greek indices correspond to coordinate indices (holonomic). The first letters of these alphabets (a, 6, c, a, 0, 7,...) run over values 1, 2, 3, but other letters take on the values 0, 1, 2, 3. The diagonal of the Minkowski metric rnj corresponds to (+, -, -, -). The antisymmetric tensors e'-?*^' and e"*"^ possess the values e''^^^ = e^^^ = 1. The Ricci tensor can be defined as Rij = R'^i^j. Making the rather typical assumption of a matter field that minimally couples to gravity leads to a Lagrangian of the form + (5.8) Prom Kibble (1961), one may define the currents T\ = dC^ldK^ = hh^i^dC"^ ldx;k)X-,i ~ (5.9a)

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117 S^, = -2{dC^ldA'^^) = -hh^'^idC'^ |^x,k)S^jX. (5.96) where, from eqs. (5.2) and (5.3), we have = Â£'"{x,X,^,/ifc^ A^^,} = bC''{x,X;k}. (5.10) For a gravitational Lagrangian of the form a = bR'hj = hR, (5.11) the conservation laws (Kibble (1961)) possess the form {TKhk^\^ + T\hk^\, = \i:\jR'^^., (5.12a) S%|^ = 7-^V-7;A''. (5.126) From Kibble (1961), the equations of motion are &(i2^fc \5'jR) = -r^/i/, (5.13a) -[6(/i,^/i/ h.^^hn^u = (5.136) where R'hi = ht,^hi''Wi^,. (5.14) 5-2 Vlasov-Kibble Hamiltonian Structure and Formulation The dynamical variables that we deal with primarily are the tetrad field 6*^^ and affine connection A'J along with their respective momenta Efc" and Hj/. Torsion and curvature are defined in terms of antisymmetric derivatives of b''^ and A'J^. Consequently, the velocities 6*^0,0 and A^^q^ do not appear and we have the corresponding primary constraints, VIZ. nfcÂ° = 0, (5.15a)

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118 UijÂ° = 0. (5.156) First of all, one must consider how to pass from the inverse tetrad field hk^ to the following set of variables nk,h^Â°',N,NÂ°'. With respect to a locai Lorentz basis, the components of the unit normal vector n to the xÂ° = const, hypersurface are given by nk = hk''/\fg^, : (5.16) where g'^" = hk^h!"'. (5.17) Following Nikolic (1984) it is possible to decompose the inverse tetrad hk*^ into components orthogonal and parallel with respect to the local Lorentz indices: hk^ = hj." + nkh^f", (5.18a) /lit" = S-k'nif" = {6k' nkn')hi'', (5.186) h^f" = n'^hk^ = n''. (5.18c) One may see that h-,Â° = 0, V = (V^6,;j, (5.19) where t^)^"/? corresponds to the three-dimensional contravariant metric. We may infer that rifc and hk" do not depend on 6*^oIntroduction of the lapse N = l/^ = nkh\ (5.20) and shift functions TV= ^ ^_a^^k^ (5 21) allows one to write equation (5.18c) as = l/AT, /i^" = -m/N. (5.22)

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Therefore, we can pass from hk^ to {nk,hij.Â°',N,N'^} without difficulty. A similar decomposition of b'^n leads to the following results = b'^o = 6% + n'^b^o = iV*^ + n*iV (5.23a) and b''a = b^a + n%^c.^b^cc' (5.236) The purely gravitational part of the canonical Hamiltonian can be written as (Nikolic (1984)) n'^can = nfc"6^,o + ^n,/A'^a,o hcP. (5.24) It is a straightforward exercise to derive the following velocities (Nikolic (1984)) fc^,o = iVb'aT^-^ + Nl^T^f, r^o(O), (5.25a) A'^afi = Nb'aR''lÂ± + N<^R''a0 R^'aoiO), (5.256) where T*ao(0) = -6^,^ + 6'Â«^^o (5.26a) R''a0{0) -A'^o,a + ASo^'^a (5.266) These velocities contain spatial derivatives of the unphysical variables 6'=o and A^^o (these unphysical variables correspond to the momentum constraints (5.15a, b)). A simple calculation (Nikolic (1984)) yields the following canonical Hamiltonian n^'can = N'H''^ + N'"H'^^ ^A'hW'ij + I>Â«Â°,Â„, (5.27) where ^""^ a = (6'=on,+ i^'^on,/) , . (5.28a) ^ .a

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120 n''^j = 2n[i"6,]Â„ + v^n^/, (5.286) n^a = n/T'=;3a b'aV^n/ + ^n,/i?'^';jÂ«, (5.28c) -n^^VaHfe". (5.28d) In equation (5.28d) the notations 11^' = Uk^'b'^a and Hi/ = Uij^'b^a have been employed. The unphysical variables make their appearance in the divergence term (5.28a). The bracketed expression in equation (5.28d) can be written as a function of T'^j^, ^k/J, R^-'kh Uij'^/J and n*^ , i.e., velocities T''j_^ and -R'-^^x can be eliminated from it. This will be shown later. Therefore, it is independent of unphysical variables and corresponds to the only dynamical part of the gravitational canonical Hamiltonian. Using equations (5.20) and (5.21) we may rewrite the canonical Hamiltonian in the form = b''on''k \A'^on''^, + D^",,, (5.29) so, apart from the divergence term, VPcan has linear dependence on the unphysical variables 6*^0 and A'^q. One can revert to the ADM form by using the transformations = n'"Hk, (5.30a) y-a = b'^a'Hk. (5.306) One must, also, examine the role which the primary constraints (5.15a, b) play in the Hamiltonian structure of the theory. The total gravitational Hamiltonian may be initially written as -Htot = n,''6%,o + lu,j'^A'^^,o bC"". (5.31)

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121 Since 6*^0 and A^^q are unphysical variables, this expression may be rewritten as = Nn^ + N'^Ua \a'^qH^, + LÂ»Â°,a + nfeOft^.o + ^n./A'^o,o, (5.32) where 11^Â° and rTij" are the primary constraints (5.15a, b). Following Nikolic (1984) we will consider 6*^0,0 and A'^ g to be arbitrary multipliers, denoted respectively as u'^q and u^^q. Starting with the basic dynamical variables b^a-, njtÂ°, A^^a, Hij" and /, one can define the cosymplectic structure as a bracket {F, G) which acts on functional pairs such as F[b''a, Hfc"; yl'^a, n,/; /] and H^"; A'^ a, n,/; /]. One can postulate a bracket of the form (F,G,K.,n.";.Â«..n,Â«;/l = mÂ§-^^. + 16 ( rfi ( \ '^J 6A'^a. SUij'' OT,/ 6 Ail a J J \6AJ\SU^J'' SU^j''6A3'a) f ^,JF 6G, where S/SX denotes the usual functional derivative with respect to the variables X and , . dA dB dB dA , ^ is the ordinary canonical Poisson bracket. The form of equation (5.33) is chosen to satisfy the Poisson bracket (Nikolic (1984)) {A'^^, n,,^} = 26\k6^i^5^^6{x f ). (5.35) It is straightforward to demonstrate that equation (5.33) constitutes a bona fide Lie bracket because it is antisymmetric and satisfies the Jacobi Identity (F, (G, H)) + {G, {H, F)) + {H, (F, G)} = 0. (5.36)

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122 The requirement that the equations of constraint dtF = {F,H), (5.37) which hold for arbitrary functionals F, generate the correct dynamical equations implies that the Lie bracket and Hamiltonian combine to generate the Vlasov-Kibble gauge invariant gravity system. Through simple construction, we derive the expressions dtb'a = {b'a, H) = 167r-^, (5.38a) dtTlk'' = (n,Â°, H) = -167r^, (5.386) SH dtA'^c. = {A'^a, H) = 327r^j^ (5.38c) and a^n^/ = (n,/, i?) = -327r;r^. (5.38d) Explicit forms for expressions (5.38a) to (5.38d) may be derived. Before doing this, however, one must analyze the explicit form of the matter Hamiltonian. Following Nikolic (1984) we will assume minimal coupling between the matter and gauge fields so that the Hamiltonian is a sum of matter and gravitational parts. The covariant derivative DkX contains the time derivative x,o (this covariant derivative is present in the gauge invariant matter field Lagrangian (5.3)). One may decompose DkX into orthogonal and parallel components DkX = rikD^x + DkX = rikh^^V^x + /^it^VaX(5.39) The covariant derivative D^x does not depend on velocities or unphysical variables. The matter field Lagrangian (5.3) may be reexpressed in terms of these new variables as Â£^ = Â£^(x,i?xX,^fcX;n*). (5.40)

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123 All the dependence of equation (5.40) on velocities and unphysical variables is through Dixit is a straightforward exercise to derive the following velocity of the matter field (Nikolic (1984)) Xfi = ND^X + N''VaX-lA''oS^JX (5.41) The determinant b may be factorized as b = det 6'=^ = Â— det 6"q = NJ (5.42) no which will enable us to calculate the corresponding field momentum n = = J^zÂ—-. 5.43 The canonical matter Hamiltonian has the form n^can = Ux,o bC^, (5.44) which, upon substitution of equation (5.41), yields the following ADM components n^ij = nSijX, (5.45a) n^o. = nVaX, (5.456) dC M IdD^X D^X-t M (5.45c) and D^",a = 0. (5.45d) It should be noted that U'^ ^ is the Legendre transformation of with respect to Dx.XThus, it can be expressed as a function of x, Â£>jtX. "'^ and II/J.. Since we are employing a statistical mechanics formulation (involving a Boltzmann distribution function) to describe

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124 the matter Hamiltonian, we are immediately presented with a problem. If we use the matter Hamiltonian (cf. Kandrup and O'Neill (1994)) Hm^ I dTfe, (5.46) where e = Â—pt, the expression for e is typically derived from the mass shell constraint g'^'^P^Pu = -m2, (5.47) which, in our vierbein formulation, can be written as hk^h'"'p^p^ = -m^. (5.48) This expression may be subjected to the standard ADM decomposition Hm^ I d^x{Np + N^'J^}, (5.49) where = p, (5.50a) n^a = Ja. (5.506) Clearly, it seems that we are lacking an extra term Ti^ ij in the ADM decomposition; with this term the ADM decomposition would possess the form Hm = I d^x{Np + N'^J^ i^'^oT^A^ . .}. (5.51) To make an appropriate modification of the matter Hamiltonian, we may consider the representation of a perfect fluid in terms of a set of six potentials (Schutz (1971)) Til pi^ = mu^ = g^^"~{<^^, + + eS^^y (5.52) Using the covariant derivative prescription (5.2), one may modify this expression for the contravariant four-momentum to P' = r + a/3,, + 9S,.) + ^^'^,^(5,,0 + aS,,0 + eS^.S). (5.53)

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125 The Lorentz group generators Sij may be formulated in a coordinate representation as Sij Xidj Xjdi. (5.54) Therefore, the expression (5.53) may be written as pf^ = gf^-(p^ + ^A'j^L,j), (5.55) where Lij corresponds to the orbital angular momentum. This leads to a mass shell constraint g'^'iP^ + lA'^^Lij){p, + ^A'^'M = -m\ (5.56) This constraint is an analog to the mass shell constraint of electromagnetism g'^'iPii + eA^)(jpy + eAu) = -m^. (5.57) In our vierbein formulation, the mass shell constraint may be written as h'Vip^ + \a'^ ^,Lij){j>, + ^A'^KLki) = -m\ (5.58) The analogy with electromagnetism is of importance. Electromagnetism can be viewed (theoretically) as a gauge theory of the group U{\). However, there is no a priori reason for the prescription Pfi^Pfi + eA^ in the context of classical physics. The prescription is due to gauge invariance in quantum mechanics. Likewise, there is no basis in classical physics for the prescription The derivation of this prescription from the perfect fluid representation should be seen only as a heuristic derivation. The charges Lij may or may not correspond to orbital angular momentum. Only experimentation can detect them and prove their existence.

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126 It follows that the energy of a single massive particle will be of the form E = -pt^A'^Lij = e ^A'^Ly. (5.59) Therefore, we may write the matter Hamiltonian Hm (5.46) as Hm = I dFfE = I dTfe -^j dTfA^^oLij. (5.60) This yields the desired form for Hm (5.46), viz. Hm = J d^xd^pdm f^-^ J d^xA'^o J d^pdmfLij j d\{Np + Ar% ]^A'\V.^,j). ^ (5.61) One may derive explicit expressions for the dynamical equations (5.38). For the physical variables b'-y and A''"^ we have dtb^^ = Nb"'^T^rnÂ± + N'^T^^^ + b^on + b^oA^k-y A'^objy (5.62a) and 1 +A<"-o,^ A'^joA^^ + A^'oA^^y. (5.626) These expressions were previously derived. It is important to note the link between these dynamical equations and their conjugate momenta. The momentum expressions for b^^ and A^^fj, are I _ d{bc^_) ^ dcP_ and , d{bC^_ d-cp

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127 where Uk = Uk^b^a and Uij'' = Ilij^b'^Q. The Lagrangian is the same as that in equation (5.7), except that the torsion and curvature have been decomposed as T^^ = 2T\jj_n^] + T^-^ (5.64a) and R'^ki = 2R'\j,^ni^ + W^j,l. (5.646) The Lagrangian (5.11) has no torsion dependence and so ITjt" = 0. It follows that dtb'^ = 0 by setting 11^Â° equal to zero in the ADM Hamiltonian decomposition (5.28). The corresponding equations for the conjugate momenta Ylk^ and II jj'^ are dtUk" = 0 (5.65a) and (5.656) Without torsion the momentum variable Iljt" vanishes identically. The dynamical equation for Ilij" can be derived from the field equation (5.13b) by using expression (5.63b). We calculate the Vlasov equation to be dtfo = {fo,H) = {E,fo}. (5.66) This will take the form Pt dx^ dx2Pt dpc, ^ P' 2 dx''Wa -{L^JJo}, (5.67) where P^=Pfj, + ^A'^f^Lij. This is, in fact, the Vlasov equation in the gauge A^^o = 0. One may write {EJo} = {e,fo}-{lA^'oL,,Jo}

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128 f\ l^^o, dfo 1 ij . . 2 dx^ "^dpa 2' (5.68) and obtain the full Vlasov equation dtfo = {e, fo] = 2PÂ« " ''dpc 2Pi ax" '^dpa + 0 Dt ^''/'{^iji/o} Â— (5.69) We now wish to derive the Hamiltonian initial constraint equations. A variation of the unphysical variables 6*^0 and A'^o in equation (5.29) yields nknÂ± + h-k^'-Ha = [ d^p-PkP J m (5.70a) and 2Yl^^bj^^ + VaHij" = 0. To derive these constraints, the following expressions have been utilized (5.706) dE = -Pk (5.71a) and dE r= 0. (5.716) Alternatively, one may derive the Hamiltonian initial constraint equations by variation of the lapse and shift variables {N,N^} present in equation (5.23a). The vierbein 6*0 has four independent components and these correspond to the lapse N and three (independent) components of the shift vector N'' (remember that n^iV*^ = 0). The constraints are 1 J\-Uk'T''r:+~nu'R'^ 1 J = 16nJ 2J / NJ {E + iVÂ°(p, + '^A'^MY iV2 I m' (5.72a)

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129 (fipdpt NJ N m (5.726) and where the expressions 2n[i"6j]<, + = 0, dE ON'' and dE = 0 (5.72c) (5.73a) (5.736) (5.73c) have been used in the above derivations. In Kibble gauge invariant gravity, the stress-energy tensor T^^ is generally nonsymmetric because hk'^ does not always appear in the matter Hamiltonian density Hm through the symmetric combination g^"" (Kibble (1961)). 5.3 Initial Variation of the Hamiltonian We may now consider the first variation of the Vlasov-Kibble Hamiltonian. This procedure will be analogous to the computations of chapters three and four. One may write the ADM Hamiltonian in the following format j d^x{NJ H = Hq + Hm + 1 d^x(6*onfc" + ^A'^oHi/) ^ + 1 dTfe

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130 ^jd^A'^ofL^J. (5.74) 2 The divergence term in equation (5.74) is readily converted into a surface integral ^ dhaib'oUk" + ^A'^on./). (5.75) This surface integral is analogous to the surface integral that appears in the ADM Hamiltonian formulation of General Relativity (cf. Wald (1984)). For the choice of Lagrangian (5.11) considered here, this surface integral will fall oflF exponentially to zero at infinity. Equation (5.74) is the canonical Hamiltonian. The total (not the canonical) Hamiltonian is H = j d?x{NJ -^-j dTA'^ofLij, (5.76) where the surface integral (5.75) has been dropped (notice the presence of terms relating to the momentum constraints (5.15a) and (5.15b)). A first variation of the total Hamiltonian (5.76) yields the result +iWnÂ»<>6*Â„,Â„ + i('i(6*o,,)nÂ»Â» + i4("nÂ„Â»(/iÂ«Â„,Â„) + in./i<')(yi"Â„,Â„)} + 1
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131 In equation (5.77), we have the initial constraint equations (5.72), viz. SH/SN, 5H/SN'' and 6H/SA^^o. These constraints may be dropped out of equation (5.77). The dynamical equations (5.38) a^nfe" = -167r = 16.^, 6H dtA'^a = 327r and also will vanish because we are deahng with an isolated matter configuration in static equilibrium. Utihzing the primary constraints (5.15) allows one to reduce S^^^H still further. In Nikolic (1984) the multipliers A^^o^ and b'^o Q are arbitrary functions of time. Perturbation of the (enforced) primary constraints (5.15) leads to the following expressions S^'^Uk"" = 0, (5.78a) ^^''n,/ = 0. (5.786) Therefore, the variation reduces to two terms (j(i)H = I drS^'^fe -~l dTd^'^fL^jA'^o. (5.79) In analogy with electromagnetic theory (cf. chapter 2), one may impose the gauge condition A^^o = 0. Finally, we have the following result I dTeS^'^f = I dVeigJo} = -j drg{e,fo} = 0, (5.80) because the equihbrium configuration is static. We may conclude that the total Hamiltonian has a vanishing first variation 6^'^H = Q. (5.81)

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132 5.4 Second Hamiltonian Variation 5^'^'>H In analogy with the General Relativity and Brans-Dicke cases, one may let denote the set of variables {6^, n^", A'\,Yi,j'', N, iV^ /}. (5.82) The second variation of the Hamiltonian, S^"^^!!, can be written abstractly as = E mi'^'y' + ^ E E ^Smy'smY\ (5.83) where the functional derivatives are evaluated for the unperturbed equiUbrium. The first term of equation (5.83) has only one non-vanishing contribution 5H I where i:^Si'^y' = ldrEj')f, (5.84) E = e^A^'oUj. Therefore, one may write equation (5.83) as 4EE^Â«<"i"f"^^. (5.85) where the second order phase-preserving perturbation J^^) / = {g, /o}}, which appears in the perturbation expansion (Bartholomew (1971)) f = fo + 6f = fo + {g, fo} + ^{g, {g, /Â„}} + ~{g, {g, {g, /Â„}}} + (5.86) has been used. Following the calculation given in Kandrup and O'Neill (1994), we deduce ^^'^H =-\j dT{g, E^]{g, fo] + Yf dr^(^)y^5(i) [f^^

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133 The formulae for dE/dN, dE/dN'^ and dE/db'a can be used to calculate the following second term of equation (5.87) Y.j<^^'''y'^''\fÂ§j)" ' (5-88) We have the following dE/dY^ formulae: dE (E + N'^ip^ + U'^^Mf dE and we have: and (5.89a) = -P-k -h-k"Pa, (5.896) dE ^ /Â»fc"(Pp + ^^Vfef)ff'-^(p^ + ^A*^'^L,f) dE _ l g^''{p, + \A'i^.Lij)Lmn This yields the form of the second term of equation (5.87) = I d\[5p5N + SJkSN'' + SiNSnSb'^ ^^^"Â•"a^E^^n], (5.90) where, taking into account the fact that = / Ij^'^ l^''f^Lij)ip, + \A'',Lki)^, (5.91)

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134 Also, the expression for the angular momentum tensor has been employed SÂ°mn = -NJ I Â—g^'^iV, + \A''^L,i)Lmn^. (5.92) One writes the first order variations 6p, SJk and 6Si^ in terms of a perturbed distribution function = {3, fg}. It is straightforward to calculate the third term of equation (5.87). One must compute the following expressions d^E d^E d^E dN^ dNON'' dNi'dN^ d^E = 0, (5.93a) = Pih'', (5.936) as well as and where and db^ad^p pt pt +9 pt PlP0po P^P<^p0 pp0ppa = 9'"'{p. + \a'KU,) = g^^'^P, (5.94) Pi = hi^ip^ + U'^^Lij). (5.95) 2 In addition, these expressions must also be computed d'^E 1 L f^iN-^P' + P^}, (5.96a) d'^E _ 1 ^ dj^mn Qjsfk ~ ~2^k ^mm (5.966)

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135 8PE 1 P^LklP^Lmn iVÂ° and jV/J pa 1 J,/'" ~4Ar2 (pf)2'^'='-^"'" + l-pT^klLmn {5.96c) ^2^ 1 tPfPÂ« 1 mn Use of equations (5.93b)-(5.93d) leads to the corresponding terms I dTf^g-^Sb^JN'' = J (fxNh-k''SN''db^aSi\ (5.97a) / "^^W^N^^'"^^ " " / ^^^'^^'^^'"(^"'S'/' + Sn (5.976) and 2' ^ AT'^ 1 where := Jhi'^T/ =^J f i^PiP^^, (5.98a) J Nj m C0a_ T f SpdptPiPf'Pf and iVJm (pÂ«)2 Using equations (5.96a)(5.96d) leads to the additional terms (5.99a)

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136 " / '^^^ dAmn^dN^ ^^"'"^^^' = \ I d'xT,'mn5A^''c.hf5N\ (5.996) -\ I d\^E'\imnSA^^p6A'', 1 r nI^ +^ j c?^a:/"ArE"fc,^Â„^^-";5<5^'='Â„ (5.99c) and + 1 1 d\h,^{^Q;^^^ g'^^NQ,^^ + ^Q.'-'mn + ^Q.'\n}^b>^a6A--0, (5.99d) where S'^mn = -NJ I Â—g^'ip. + ^A'^'.L.O^mn A (5.100a) klmn-J j -, J NJ (Pt) ^'""m (5.1006) (5.100c) and r, i^P 1 f d^P Pt^P^P" . f ^t^ mn--^ J j^j ^pt^2 (5.100d) Using equations (5.90), (5.97a)-(5.97c) and (5.99a)-(5.99d) allows one to write the second and third terms of equation (5.87) as j d'x{6pSN + dJk6N>' ^6A^^J{E-^,) + 5(iV5,Â°)56'Â« +iV/ijt"<5iV^<56'Â„5/ 57VJ6'Â«(iV"5/ + 5,")

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137 1 iV* 1 a 1 + + ^Q,'\r. + TfQ^^'mn " 5"^ArQ,Â«_}<56^M-",}. (5.101) This allows us to write equation (5.87) as J(2) ff = -\l ^{9^ Eo}{g, fo} + I d^x{6p5N + 6Jk6N'' ^^"Â•^^E^^n + SiNSn^b^a +Nh-k''6N''6b^aSi^ SNdb^aiN^Si* + Si") ATT -^i:''^tmnSA^-06A'^Â„ ^E^\imnSA^"pSA'^, + l{nkJ^''mn hk^Nj'''mnWjA"'''p 1 N" 1 where represents the second order energy corresponding to the first order perturbation S'^^^Y. A standard calculation of the gravitational contribution then yields the result 167r(5(2)^^[5(i)y] =

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138 +2SN'{6JR'^j-k + J^h^R'^-ia + JShj^R'^-ffk + Jh-hf5R''^^n,] + j d\{N5(''^n''x + NH^^^n^'k ^A'^<^''):Â«%}, (5.103) where 5(2)-^G^^ ^^^^ (5(2) -^G.^. ^^j^g ^j^^ following forms: -2JShm''hn^6R^\0 ~h^^h/S^'^R"^\f, -JS^^'^h^R-^^^ J6hn"ShjR-^^0 (5.104a) and +4SJR^'-^phf5h-^^ + JS^^^h-k^R^\ +2J6hj,Â°hj^6R'^fS, + ^JShj^Sh-,''R^^0^ +J6(')h^R^^^-, + 2Jhi<^5hf6R'^^^ + Jh-,-hj0S('^R'^0, (5.1046) and

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139 In the above expressions the following notations have been employed: 6 J = Jhk^Sb^ (5.105a) 6^^h = Jhrh^hj^^'SU^^Sb^a + J6h-k''Sb''a + J/ijtÂ°<5(2)6*^ (5.1056) -M'Â„^A"^\. A\^6A^J^ (5.105c) <5(2)i2Â«i^, = 6(')A^^^^^ S^'^A^K^, + 6(')A\.A-^^ +2dA\JA''^^ + A\J^^^A''^^ -
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140 1 /V" 1 / d'ar{iV5(2)-^G^ ^ NH^'^^n'^k ^A'^o^^^'^H^i,} (5.106) loTT y 2

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SUMMARY In chapter two, we explored the Hamiltonian structures of a coUisionless Newtonian cosmology and the Vlasov-Maxwell system in a curved backgroimd space-time. In the case of the Newtonian cosmology, one found a cosymplectic structure for the coUisionless Boltzmann equation in the context of an expanding Newtonian cosmology, working in the average comoving frame. It is observed that all equilibrium solutions fo corresponding to homogeneous distributions of matter are energy extrema with respect to dynamically accessible perturbations that preserve all the constraints associated with conservation of phase, i.e., d^^'^H = 0. A simple expression is derived for the second Hamiltonian variation, S'^'^^H, associated with an arbitrary dynamically accessible perturbation, and it is shown that the time derivative dS^'^^H/dt is intrinsically negative. It follows that any such perturbation with 5^'^'>H < 0 necessarily triggers a linear instability. This provides a simple proof of the linear instability of all equilibria towards perturbations of characteristic wavelength longer than the Jeans length A J, and of the linear instability of equilibria exhibiting population inversions towards shorter wavelength perturbations as well. The basic approach used here to extract a nontrivial stability criterion is quite general, relying only on the facts that, in the average comoving frame, the evolution appears dissipative and the equilibrium is an energy extremum. One might , therefore, expect that it may find applications to other physical problems as well. It is important to note that the dissipative effect is intrinsic to the system itself. 141

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142 The Vlasov-Maxwell analysis in a fixed curved background space-time serves as an interesting extension of the well understood Hamiltonian structure of the fiat space VlasovMaxwell system. The formalism may be of use in studying electromagnetic effects in the neighbourhood of (for example) black holes. Chapter three gives a completely general Hamiltonian formulation of the Vlasov-Einstein system working within the context of an ADM decomposition into space plus time. This approach incorporates explicitly the radiative degrees of freedom of the gravitational field and, as such, requires no assumptions about any underlying symmetries. This formulation is effected using lie algebraic techniques, treating the distribution function /, the spatial metric hab, and the conjugate momentum 11Â°'' as dynamical variables. Although hab and 11"'' form a conjugate pair of canonical variables, / is not canonical, so that there exists no immediate analog of the canonical Hamiltonian equations and no immediate Lagrangian formulation, despite the fact that, in principle, a canonical formulation must exist (at least locally). There are at least two concrete problems of interest for which this new Hamiltonian formulation has immediate applications. The first of these involves an examination of the stability of spherically symmetric equihbria toward nonspherical perturbations. The energy functional can be used to address the stability of more general spherical equilibria than have hitherto been studied. Another, yet more interesting, problem involves an examination of the stability of axisymmetric equilibria, a general subject which has hitherto received absolutely no attention at all. One anticipates that realistic physical equilibria will be characterized by a nonvanishing total angular momentum but, typically, such equilibria would be expected to manifest deviations from spherical symmetry as a result of rotational flattening.

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143 A general examination of the problem of stability for axisymmetric equilibria was attempted. Unfortunately, to obtain specific information on the stability of such matter configurations, it is necessary to have quantitative information on the behavior of a given axisymmetric solution, i.e., one must have an axisymmetric solution of the Einstein field equations (corresponding exterior and interior solutions for an axisymmetric matter configuration). Chapter four dealt with the Brans-Dicke scalar-tensor theory. This theory has two representations: (i) the variable G representation and (ii) the variable mass representation. Only the variable mass representation has an ADM formulation, whereas the variable G representation does not possess one. The Vlasov-Brans-Dicke system does not satisfy a Birkhoff theorem. For example, a pulsating spherically symmetric matter configuration will emit scalar monopole radiation. If the system possesses a negative energy mode, then one may have a run-away instability. The amount of energy emitted is the same for both representations. Since the variable mass representation involves a statistical mechanical description of a gas of particles with masses that have space-time dependence, the Vlasov-Brans-Dicke formalism may have a wider range of apphcability to variable mass Boltzmann distributions. This dissertation gives the full form of the ADM Hamiltonian for the Vlasov-Brans-Dicke system. The gravitational part was derived by Toton (1970). However, the matter part of the ADM Hamiltonian was explicitly derived by the author. This lead to a derivation of the dynamical equations in the presence of matter. Also, a calculation of the second order Hamiltonian variation was provided. Chapter five dealt with the Vlasov-Kibble system. The ADM form of the matter Hamiltonian was identified. It was found that the angular momentum couples to the energy via

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144 the affine connection gravitational field variable. The theory possesses a peculiar mass shell constraint analogous to electromagnetism. The boost-angulaj: momentum tensor Lki takes the place of electrical charge e in electromagnetism. For a torsionless gravitational Lagrangian, the vierbein and its conjugate momentum are static. The static nature of the vierbein is a key element in fixing the time independence of the spherically symmetric (Schwarzschild) solution of the theory. Thus, for a torsionless Lagrangian, a Birkhoff theorem is satisfied. However, this may not be the case for a Lagrangian with torsion. The Vlasov-Kibble system has many features in common with the Vlasov-Maxwell system. For example, the Vlasov equation of this theory is analogous to that of electromagnetism. The specific ADM variables used in variational calculations of the Hamiltonian have also been identified. The ADM Hamiltonian can be written with respect to these variables or with respect to vierbein variables. In either representation the results are equivalent. Finally, it should be noted that the formalism is applicable to any generic gravitational theory that possesses an ADM formalism.

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REFERENCES Antonov, V.A., Astr. Zh. 4, 859 (1960) [translated in Sov. Astron. AJ 4, 859 (1961)] Arnold, V., Mathematical Methods of Classical Mechanics (SpringerVerlag, Berlin/New York/Heidelberg, 1978) Arnowitt, R., Deser, S., and Misner, C.W., in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962) Bao, D., Marsden, J.E., and Walton, R., Comm. Math. Phys. 99, 319 (1985) Bartholomew, R, Mon. Not. R. Astr. Soc. 151, 333 (1971) Bernstein, I., Phys. Rev. 109, 10 (1958) Bisnovatyi-Kogan, G.S., and Zel'dovich, Ya. B., Astr. Zh. 47, 758 (1970) [translated in Sov. Astron. AJ 14, 758 (1971)] Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., and Ratiu, T.S., in 1991 American Control Conference (IEEE, Evanston, 1991) Brans, C, and Dicke, R.H., Phys. Rev. 124, 925 (1961) Bruckman, W.F., and Velazquez, E.S., G.R.G. 25, 901 (1993) Chapman, S. and Cowling, T.G., The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1939) Dicke, R.H., Phys. Rev. 125, 2163 (1962) Dirac, P.A.M., Can. J. Math. 2, 129 (1950) Dirac, P.A.M., Recent Developments in General Relativity and Gravitation (PWN Polish Scientific Pubhcations, Warsaw, 1962) Fackerell, E.D., Astrophys. J. 160, 859 (1970) Fackerell, E.D., Astrophys. J. 165, 489 (1971) Friedman, J.L., and Schutz, B.F., Astrophys. J. 222, 281 (1978) 145

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147 Kruskal, M.D., and Oberman, C, Phys. Fluids 1, 275 (1958) Lynden-Bell, D., and Sanitt, N., Mon. Not. R. Astr. Soc. 143, 167 (1969) Marsden, J.E., Montgomery, R., Morrison, RJ., and Thompson, W.B., Ann. Phys. (N.Y.) 169, 29 (1986) ' i. \ Marsden, J.E., and Weinstein, A., Physica D 4, 394 (1982) Misner, C.W., Thorne, K.S., and Wheeler, J. A., Gravitation (W.H. Freeman, San Francisco, 1973) Moncrief, V., J. Math. Phys. 17, 1893 (1976) Morrison, P.J., Phys. Lett. A 80, 383 (1980a) Morrison, P.J., Phys. Rev. 45, 790 (1980b) Morrison, P.J., Z. Naturforsch. Teil A 42, 1115 (1987) Morrison, P.J., and Greene, J.M., Phys. Rev. Lett. 45, 790 (1980) Morrison, P.J., and Pfirsch, D., Phys. Rev. A 40, 3898 (1989) Morrison, P.J., and Pfirsch, D., Phys. Fluids B 2, 1105 (1990) Morrison, P.J., and Pfirsch, D., Phys. Fluids B 4, 3038 (1992) Moser, J.K., Mem. Am. Math. Soc. 81, 1 (1968) Newcomb, W.A., Ann. Phys. 3, 347 (1958) Nikolic, LA., Phys. Rev. D 30, 2508 (1984) Nutku, Y., Astrophys. J. 155, 999 (1969) Peebles, P.J.E., The Large Scale Structure of the Universe (Princeton University Press, 1980) t Penrose, O., Phys. Fluids 3, 258 (1963) Reddy, D.R.K., J. Phys. A: Math., Nucl. Gen. 6, 1867 (1973) Shapiro, S.L., and Teukolsky, S.A., Astrophys. J. 419, 622 (1993a) Shapiro, S.L., and Teukolsky, S.A., Astrophys. J. 419, 635 (1993b)

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148 Schutz, B.F., Phys. Rev. D 2, 2762 (1970) Stewart, J.M., Nonequilibrium Relativistic Kinetic Theory (SpringerVerlag, Berlin, 1971) Sygnet, J.F., Des Forets, G., Lachize-Rey, M., and Pellat, R., Astrophys. J. 246, 737 (1984) Toton, E.T., J. Math. Phys. 11, 1713 (1970) Utiyama, R., Phys. Rev. 101, 1597 (1956) Wagoner, R.V., Phys. Rev. D 1, 3209 (1970) Wald, R.M., General Relativity (University of Chicago Press, Chicago, 1984) Yang, C.N., and Mills, R.L., Phys. Rev. 96, 191 (1954)

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BIOGRAPHICAL SKETCH Eric O'Neill was born on November 17, 1964, in Sydney, Nova Scotia, Canada. His parents are Ronald Joseph O'Neill and Eugenie Elizabeth (nee Morrison) O'Neill. During the years 1966-67, he lived in Kingston, Ontario, and his sister, Rhonda, was born there in 1966. His family moved back to Nova Scotia in 1968. He started school in September 1970. He became interested in science at a fairly early age. His parents gave him a telescope as a Christmas gift. A total eclipse of the sun occurred in Nova Scotia in March 1970, and this helped propel his early interest in astronomy. The Apollo Moon missions were also a source of endless fascination. He started high school in September 1980. After difficulties with mathematics, he became interested in mathematical logic after reading a science fiction novel. Soon he started studying mathematics on his own as an interest. After finishing an honours B.Sc. at Dalhousie University, he worked with Professor Alan Coley of the mathematics department for an M.Sc. In September 1989 he arrived at the University of Florida. He passed the physics comprehensive exams in August 1990. He then worked with Professor Henry Emil Kandrup towards the Ph.D. in physics. 149

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