Group Title: spiralling binary system of black holes
Title: The spiralling binary system of black holes
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Title: The spiralling binary system of black holes
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Language: English
Creator: Blackburn, James Kent, 1957-
Copyright Date: 1990
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THE SPIRALLING BINARY
SYSTEM OF BLACK HOLES





By

JAMES KENT BLACKBURN


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


1990


























To Space Ship Earth














ACKNOWLEDGEMENTS


Many people have contributed to my education and made this disserta-

tion a reality. The individual making the most significant contribution towards

this end has been my research advisor, Professor Steven Detweiler. Through-

out our collaboration he has bestowed much encouragement and confidence.

He possesses a rare talent for communicating the complex in terms of the well

understood. He has been more than a mentor guiding his pupil on to greater

challenges; he has also been a good friend and I thank him for these qualities.

I am very grateful to Professors Jim Ipser, Jim Fry, Rick Field, and Steve

Gottesman for their participation as members of my supervisory committee. I

wish to express special thanks to the faculty, postdocs, and graduate students

of the astrophysics group in the physics department. Though small in number,

they have provided a wealth of dialog and comprehension. I would also like

to thank everyone else that has contributed to my understanding in physics,

especially my fellow graduate students.

I am endlessly grateful to my parents, my four sisters, and the rest of

my relatives for their encouragement, love, and care, especially my two grand-

mothers who passed away before the completion of my stay in Gainesville.

Finally, I give special thanks to my wife Laddawan Ruamsuwan for her

never-ending cheerful disposition while struggling herself with the questions of

physics, nature, and the cosmos.

















TABLE OF CONTENTS


page

ACKNOWLEDGEMENTS ......................................... iii

LIST OF TABLES .................................................. vi

LIST OF FIGURES ................................................viii

N O TAT IO N ..................................................... ... ix

ABSTRACT ........................................................ xii

CHAPTERS

1 INTRODUCTION .............................................. 1

2 INITIAL VALUE FORMULATION ............................ 7

2.1 Foliation of Space-Time ..................................... .7

2.2 Steady State Equations ............................... ..... 19

2.3 Boundary Conditions ...................................... 23

3 VARIATIONAL PRINCIPLE FOR TOTAL MASS............. 28

4 TRIAL GEOMETRIES ........................................32

5 NUMERICAL RESULTS ....................................... 49

5.1 Test Particle Geometry ...................................... 49

5.2 Equal Mass Geometry ..................................... .. 56

5.3 Arbitrary Mass Ratios ..................................... 70

5.4 Gravitational Radiation.................................... 79

5.5 Apparent Horizons.........................................89

6 SUMMARY AND CONCLUSIONS ............................96

iv









APPENDIX COMPUTER PROGRAM .......................... 101

REFERENCES .................................... ...............154

BIOGRAPHICAL SKETCH........................................157

















LIST OF TABLES


TABLE page

1 Stable Circular Analytical Test Particle Orbits .....................52

2 Unstable Circular Analytical Test Particle Orbits ................. 52

3 Numerical Test Particle Masses .................................... 55

4 Numerical Stable Circular Test Particle Orbits .....................55

5 Numerical Stable Circular Orbits for Equal Masses................ 65

6 Comparison of Stable Circular Orbits for Equal Masses ............. 69

7 Numerical Masses for Mass Ratio 0.005 ............................71

8 Numerical Orbits for Mass Ratio 0.005.............................71

9 Numerical Masses for Mass Ratio 0.02 .............................72

10 Numerical Orbits for Mass Ratio 0.02..............................72

11 Numerical Masses for Mass Ratio 0.03 .............................73

12 Numerical Orbits for Mass Ratio 0.03..............................73

13 Numerical Masses for Mass Ratio 0.05 .............................74

14 Numerical Orbits for Mass Ratio 0.05..............................74

15 Numerical Masses for Mass Ratio 0.10 .............................75

16 Numerical Orbits for Mass Ratio 0.10..............................75

17 Numerical Masses for Mass Ratio 0.25 .............................76

18 Numerical Orbits for Mass Ratio 0.25..............................76

19 Numerical Masses for Mass Ratio 0.50 .............................77

20 Numerical Orbits for Mass Ratio 0.50..............................77

vi









21 Numerical Masses for Mass Ratio 0.75 .............................78

22 Numerical Orbits for Mass Ratio 0.75..............................78

23 Gravitational Quadrupole Radiation for Equal Masses .............. 84


















LIST OF FIGURES


FIGURE page

1 Two Neighboring Spatial Slices of a Foliation ...................... 12

2 Hypersurface Topology for Binary System of Black Holes............25

3 Analytical Effective Potential for Test Particle .....................53

4 Numerical Effective Potential for Test Particle ..................... 57

5 Binding Energy of Numerical Test Particle.........................58

6 Single Parameter Newtonian Circular Orbits .......................60

7 Post-Newtonian Effective Potential for Equal Masses ..............63

8 Numerical Effective Potential for Equal Mass Black Holes ...........66

9 Numerical Binding Energy of Equal Mass Black Holes .............67

10 Quadrupole Radiation from Equal Mass Black Holes ..............85

11 Numerical Binding Energy versus Separation.......................87

12 Numerical Components of Velocity ................................ 88

















NOTATION


The conventions used in this dissertation are intended to be consistent

with most of the modern references to classical general relativity. In almost

all cases, the meaning will be clear from the context. When the possibility

for confusion exits, the actual meaning is presented. Geometrized units are

employed throughout the text, that is, units having the gravitational constant

and the speed of light both set equal to one. The choice of the metric signature

is (- + + +). The following is a descriptive list of symbols and their meanings

appearing within this dissertation:


gpv Metric of four dimensional space-time geometry

VV Covariant derivative associated with space-time metric

Lt Lie derivative in the direction of the four-vector t

(M, gi,) Space-time manifold with metric gpv

E, Space-like hypersurface embedded in space-time manifold

r Time parameter labeling each hypersurface

7ytU Metric of three dimensional spatial geometry

D, Covariant derivative associated with spatial metric

Khv Extrinsic curvature tensor

7tV Momentum conjugate to spatial metric

FP V Christoffel Symbols










Rapy Riemann Tensor of four dimensional spacetime geometry

Ra/py Riemann Tensor of three dimensional spatial geometry

Rjv Ricci Tensor of four dimensional spacetime geometry

RPv Ricci Tensor of three dimensional spatial geometry

R Ricci Scalar of four dimensional spacetime geometry

R Ricci Scalar of three dimensional spatial geometry

Guv Einstein tensor

Tiv Stress energy tensor

a Lapse function

/ Shift vector

41 Reduces to conformal factor in low velocity limit

,CG Lagrangian density of general relativity

7'IG Hamiltonian density of general relativity

HG Hamiltonian of general relativity

s Separation between the two black holes

m Schwarzschild test particle mass

E Energy of Schwarzschild test particle mass

m1 Mass of first black hole as measured on common sheet of
hypersurface

m2 Mass of second black hole as measured on common sheet of
hypersurface

p Reduced mass: mlm2/(ml + m2)

Mtotai Mass of both black holes as measured on common sheet

M1 Irreducible mass of first black hole as measured on first
isolated sheet of hypersurface, also called Mlirreducble

M2 Irreducible mass of second black hole as measured on second
isolated sheet of hypersurface, also called M2irreduable

Mo, Eo Extreme values of the mass determined by Hamiltonian

x










f Angular frequency of rotation of binary black holes

o0, Position of first black hole along x-axis in corotating frame

Xo2 Position of second black hole along x-axis in corotating frame
u Tangential velocity of first black hole

v Tangential velocity of second black hole

Jtota, Total angular momentum

Jtota, Scaled total angular momentum: Jtotai/(M1M2)

Zji Reduced quadrupole moment tensor
LGW Dimensionless luminosity of gravitational radiation

t Rate angular momentum lost to gravitational radiation

Yem Spherical harmonics
Y* Complex conjugate to spherical harmonics
















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


THE SPIRALLING BINARY
SYSTEM OF BLACK HOLES


By

James Kent Blackburn

May 1990
Chairman: Steven Detweiler
Major Department: Physics

A novel approach to the numerical analysis of the dynamic system of

two spiralling black holes is developed using the initial value formalism of gen-

eral relativity. As the two black holes spiral in on each other they will lose

energy and angular momentum in the form of gravitational radiation. When

the amplitude of this gravitational radiation is small enough, the space-time

for the two orbiting black holes can be approximated by a geometry which is

unchanging as seen by an observer in a frame of reference corotating with the

two black holes. Then a time-like Killing vector field is assumed to exist over

a finite region of the space-time geometry. A variational principle is found for

the total mass of the binary system based on the Hamiltonian of general rela-

tivity and is used to study the dynamics and stability of the close orbits. The

emission of gravitational waves within the context of the quadrupole moment

approximation is used to determine the secular evolution of the system. For

xii








black holes of equal mass, approximately 3% of the their initial mass, as deter-

mined when the two black holes are at rest at large separations, is emitted as

gravitational radiation with frequencies less than the quadrupole normal mode

frequency of the final coalesced Kerr black hole.















CHAPTER 1
INTRODUCTION


The concept of black holes has been on the minds of men for at least

two centuries. In 1783 John Michell, in a letter to Henry Cavendish, discussed

the gravitational confinement of light emitted from the surface of a star with

the same density as our sun but having a radius 500 times greater.1 Laplace

predicted in 1795 that a spherical object having a large mass of sufficiently

small radius would possess a gravitational field so great that even light could

not escape from its surface in accord with the Newtonian theory of gravity.2

Receiving little attention, Laplace removed his early consideration of a black

hole from later editions of his work. The consequences of gravitational fields of

this strength brought up in these early discussions lie dormant for more than

a century, until Einstein's theory of general relativity in 1915.3

Schwarzschild was able to formulate an exact solution to the gravitational

field surrounding a spherical mass within a month of Einstein's publication of

the theory of general relativity.4 Schwarzschild's solution contains the complete

description of the gravitational field associated with a spherically symmetric,

nonrotating, electrically neutral black hole. Einstein, not surprisingly, did not

expect an exact solution to the field equation of general relativity to exist and

commended Schwarzschild on his achievement.

During the next half century black holes would once again be disregarded

as an idea with enough physical significance to justify their study. Theoretical







2

studies of white dwarfs by Chandrasekhar in 1930 lead to the prediction of

an upper mass limit for white dwarfs of approximately 1.4 M0.5 Above this

maximum mass, known as the Chandrasekhar limit, a white dwarf would not

be saved from gravitational collapse by the degenerate relativistic electron

pressure of Fermi-Dirac statistics. The final state of such a stellar object was

now a matter of speculation. Eddington, unhappy with the prospects of black

holes, modified the equation of state of a degenerate relativistic gases such that

finite equilibrium states existed for stars of any mass.6 Landau also rejected the

idea of black holes as the final resting state of stars with masses greater than

the Chandrasekhar Limit. In his elementary explanation of the Chandrasekhar

Limit, Landau concludes,"Such masses exist quietly as stars and do not show

any such ridiculous tendencies we must conclude that all stars greater than

1.5 Me certainly possess regions in which the laws of quantum mechanics are

violated."7

Chadwick's discovery of the neutron lead to a new class of stellar object,

the neutron star.8 These objects are supported against gravitational collapse

by the pressure of degenerate neutrons and the strong interactions between

nucleons. The first detailed calculations of neutron star structure were carried

out by Oppenheimer and Volkoff in 1939.9 They also reached the conclusion

that a critical mass existed above which neutron stars could not reach a final

equilibrium state. Later Oppenheimer and Snyder made a detailed study of

the collapse of pressureless neutron stars and concluded that it was unlikely

that any law of physics would come to the rescue to prevent the gravitational

collapse for some stellar masses.10

Lacking observational support, the study of black holes yielded to the

opinions of the great astronomers and physicists of this era. In retrospect







3

Chandrasekhar stated, "Eddington's supreme authority in those years effec-

tively delayed the development of fruitful ideas along these lines for some thirty

years."11 Even with the critical mass limits derived from theory, serious consid-

erations of black holes did not surface again until the later half of this century.

The golden age of black holes began in the 1960s with contributions from

many physicists.12,13,14 Much of the excitement in black holes was sparked by

the observations of exotic stellar objects such as pulsars, quasars, and com-

pact X-ray sources.15 In 1963 Kerr discovered a family of exact solutions to

Einstein's vacuum field equations characterizing a chargeless, rotating black

hole.16 Later Newman constructed a charged generalization of this solution for

the Einstein-Maxwell field equations.17 This is the most general exact analytic

solution of a black hole known. It is parametrized by just three observable

quantities, total mass M, total angular momentum J, and total charge Q. Such

a simple parametrization of a black hole led Wheeler to the adage, "A black

hole has no hair."18

At roughly the same period in time new techniques were being developed

to handle the increasing number of global issues surrounding black hole physics.

Questions concerned with asymptotic flatness, event horizons, and singulari-

ties were resolved using these new global techniques.19,20,21,22 An extremely

useful conformal transformation of space-time developed by Penrose, designed

to bring infinity into a finite distance, illustrates the global features of a black

hole in what is known as a Penrose diagram.23 This diagram describes all the

causal relationships of past and future singularities, event horizons, space-like,

time-like, and null infinities, and the orientation of light cones at all events for

a black hole's space-time manifold. It plays a crucial role in formulating the

topic of this dissertation.







4

By the 1970s the studies of a black hole had reached maturity. Yet, the

universe as we know it can not be simply modeled by the space-time geometry

associated with a single black hole. If Einstein's field equations are to be

considered a viable mathematical model of nature, then they must contain the

descriptions of any physical system observed. Efforts to find new solutions have

been restricted to studies of space-time geometries with so much symmetry

that their applications to astrophysics become very limited.24 Perturbative

methods are also applied to problems of significance to astrophysics, but they

too are plagued by a limited range of application from perturbation theory's

very nature.25

Classical theories of physics such as mechanics and electromagnetism

have the feature that when initial data is specified in such a way as to satisfy

any constraints in the theory, then the initial data can be dynamically evolved

in a manner completely and uniquely determined by the specification of initial

data. A theory which can be represented by such a description is said to possess

an initial value formulation. The general theory of relativity also has an initial

value formulation;26 much of the pioneering development was carried out in

the 1960s by Arnowitt, Deser, and Misner.27

These efforts laid down the foundation for studying new classes of prob-

lems using the initial value formulation of the general theory of relativity. The

task of finding solutions is no less involved in this formulation. However, the

initial value formulation is well suited for numerical analysis. With the in-

creasing speed and accuracy of the digital computer finding applications in

many areas of research, the computer age inevitably was to influence general

relativity. Initial data ranging from gravitational radiation to colliding black

holes have been studied by these methods.28,29







5

One important unsolved classical problem in general relativity is the bi-

nary system of two orbiting black holes. Such a system of black holes will spiral

in on each other as a result of the energy and angular momentum losses asso-

ciated with gravitational radiation emitted by the system. The amplitude of

the radiation increases as the system becomes more tightly bound. Eventually,

the secular time scale becomes comparable to the orbital period. In the end

the black holes coalesce with a highly nonlinear burst of gravitational radia-

tion then settle down into a Kerr black hole. The details of this binary black

hole system poses an important challenge to the classical theory of general

relativity.30

The energy and angular momentum carried off by gravitational radia-

tion will be small throughout most of the evolution of the black holes. While

the time scale for secular effects caused by the radiation reaction are large in

comparison to the dynamical time scale of the orbit, an observer in a frame

of reference rotating with the black holes will see the space-time geometry as

approximately time independent. Making the time independence into a global

feature of the space-time geometry introduces an important simplification to

general relativity. This time independent approximation to the geometry ap-

plied to the spiralling binary black hole system is the subject of this disserta-

tion.

The geometry for a finite but sufficiently large region of the space-time

manifold is considered. Initial data consistent with the binary system of or-

biting black holes and satisfying the necessary constraints is placed on a par-

tial Cauchy surface of the manifold. This initial data evolves inside the future

Cauchy development consistently with the time symmetry of the rotating frame

of reference. The time translational vector field of this frame will be a Killing







6

vector field for the geometry. This Killing vector field results in a simplifica-

tion of the initial value formulation to a set of equations we call the steady

state equations. These equations are used in a variational principle for the

total mass, which when extremized provides the total mass of circular orbits.

The results of this novel technique are compared with calculations from earlier

methods when the parametrization of the binary system of black holes overlaps

these earlier techniques.

The quadrupole gravitational radiation is used to benchmark the validity

of the approximation of a time translational Killing vector for the space-time

geometry.31 As long as the amplitude of the gravitational radiation is small

on the boundaries of Cauchy surface approximate asymptotic flatness can be

imposed as a boundary condition. Under these circumstances the geometry

will be a reasonable approximation to the realistic geometry for this system.















CHAPTER 2
INITIAL VALUE FORMULATION


2.1 Foliation of Space-Time


The search for a mathematical representation of the physical universe in

classical general relativity begins with Einstein's field equations

1
Gap R,3 2g9R = 87rTa (2.1)

a set of coupled, second order, nonlinear, partial differential equations which

relate the inverse of curvature, through second order derivatives of the metric

coefficients on the right to sources of curvature, the distribution of mass and

energy on the left, for a space-time manifold. The elegant statement of Ein-

stein's equations, that given the mass energy distribution then solve for the

geometry of space-time, is misleading. This results from the coupling of the

definition of the mass energy distribution to the geometry. Without knowledge

of the geometry, the distribution of mass and energy cannot be specified.

In actuality, one can only describe equation (2.1) in terms of sources

and expressions for the geometry correctly after separating the initial value

data from the future evolution.32 Of the ten components to Einstein's field

equations, four equations relate the distribution of mass energy on a space-

like hypersurface to the geometry of the hypersurface, and the remaining six

equations determine how the geometry of the hypersurface and the mass energy

will evolve. The true statement of Einstein's field equations is that given a

7







8

space-like 3-geometry and its time rate of change, and given the fields of mass

energy and their time rates of change all at a particular time, solve the space-

time 4-geometry at that one time, and then evolve the mass energy fields and

4-geometry to all times. The details of this procedure are at the center of the

initial value formulation of general relativity.

In order to apply the initial value formalism to general relativity the

space-time manifold (M, gp), where gap are the metric coefficients, must be

well posed. That is, the gravitational field must be the time history of the

geometry of a space-like hypersurface viewed as a Cauchy problem.33 That

the space-time manifold (M, ga) possesses a Cauchy description requires the

manifold to be globally hyperbolic with the following features:

There are no closed causal paths.

M can be foliated by Cauchy surfaces Er.

The Cauchy surface is parametrized by a global time function r whose

gradient is time-like everywhere.

M has topology E x R where E is the topology of the Cauchy surface.

On such a manifold, (M, gap), an initial value formulation is well posed.34 This

is shown by using harmonic coordinates Xzic which satisfy


H -= VVy"Hc = 0 (2.2)


They are used to write Einstein's field equations as a quasilinear, diagonal,

second order hyperbolic system


-g aavg, + Fpy(g, Og) = 87rT, (2.3)

where the Fi,(g, ag) are the nonlinear terms that depend on the metric and

its first derivative.35







9

The harmonic coordinates xzH of equation (2.2) are not useful in the

initial value formalism. However, the coordinates to be used are related by a

coordinate transformation to the harmonic coordinates

axv
x1 = xc (2.4)


The coordinate transformation does not change the geometry of the space-time,

or the physics it contains. It does mean that we are free to work with a set

of coordinates which more closely characterize the problem at hand and still

be assured that the development of the Cauchy Surface will exist. The initial

data on the Cauchy surface has a maximal development; this is the manifold

(M, gap). The extent of the Cauchy development typically is large enough to

prove physically interesting.

Having demonstrated that a well posed initial value formalism is con-

sistent with treating the task of finding solutions in general relativity to the

Cauchy problem, the initial value formulation can now be described. The

space-time manifold (M, gap) is foliated by Cauchy surfaces, space-like hyper-

surfaces Er, parametrized by a universal time function r. Each Cauchy surface

will have a time-like normal vector field

n na = n# (2.5)


satisfying the normalization condition


(n,n)= -1 (2.6)

and having components na in the coordinate basis aO/xa or components n3 in

the general basis 6p. The selection of basis is also arbitrary though the choice

can lead to a simplification of calculations. As an example, a natural choice for







10

the initial value formulation, choose C0 = n, the normal to the hypersurface

surface, and 6i where i = (1,2,3), any three tangent vectors spanning this

hypersurface Er. Though not necessary, the three tangent vectors are generally

assumed to be the tangents to the coordinates

'i = (2.7)

providing a coordinate basis for the discussion of the three geometry of the

hypersurface.

The normal to each hypersurface of the foliation is also described through

the dual cotangent basis of vectors "a by the closed one-form

w = wa,' = wadxa (2.8)

where "' = dxa in a coordinate basis, and w satisfies the relation

(n,w) = 1 (2.9)

The one-form w describes the foliation of space-time and is related to the

universal time function r by the gradient

wO = aVer (2.10)

where V is the covariant derivative of the space-time manifold and a is the

lapse function. The lapse function a is a measure of the local rates of clock

on the hypersurface. Imposing these choices insures that n points in the di-

rection of increasing r, and gives the four-velocity vector field of observers

instantaneously at rest in the slice Er.

An important point to make about n is that, for a particular foliation of

the space-time manifold, it is not the most natural choice of orthogonal vector

fields for connecting each slice. Reconsidering equation (2.9)


(n, w) = (n,aVr) = (an, Vr)


(2.11)







11

reveals that the orthogonal vector field with components N3 = an13 is the

natural choose since the proper time interval between two nearby hypersurfaces

parametrized by r is a6r. However, N is not the most general vector to

measure this proper time interval, see Figure 1. Removing the restriction of

orthogonality, a general vector field t of the form


t = t"bo = (an" + # )6o (2.12)


with #'wa = 3an" = 0, will also satisfy the normalization condition


(t,Vr) = 1 (2.13)


The arbitrary spatial vector P/6, is the shift vector and represents the remain-

ing kinematic freedom associated with describing a particular space-time foli-

ation. The vector field t is the tangent to a congruence of curves parametrized

by 7 connecting the hypersurfaces Er of the foliation on (M, ga p).

The space-time metric gap of M induces the space-like metric 7ya on the

hypersurface Er given by


Ta# = gag + UwaWz (2.14)

with the inverse space-like metric given by

7ap = gap + nan (2.15)


That the metric yp is a spatial tensor results from


7 n = 0 (2.16)

The mixed form 7a5, one covariant index and one contravariant index, is the

unit operator of projection onto the hypersurface.

















































FIGURE 1: Two Neighboring Spatial Slices of a Foliation. Two neigh-
boring hypersurfaces of the space-time foliation illustrating the relationship
between the "time" vector t, the lapse function a, the shift vector /, and the
normal n in the expression t = (an" + 3 )6a.







13

The covariant derivative V of the space-time manifold also induces a

covariant derivative D on the hypersurface though the action of the projection

operation. Consider a scalar function $ defined on the hypersurface. The

covariant derivative on Er of D is given by

Dp y = 7aVaI (2.17)

Next consider a spatial vector existing in the hypersurface with contravariant

components va where va'w = 0. The covariant derivative on Er of v" is given

by

Dpyv =E 7'y'Vav (2.18)

As a last example consider the second rank mixed spatial tensor tap which

satisfies tapwa = tpn# = 0. The covariant derivative on E, of ta is given by

D -t 7y~7A VtA (2.19)

When the covariant derivative D on the hypersurface acts on the spatial

metric 0(Yf the following result


D =7ap = 7y-7 a7 pV(gp~ + nunv) = 0 (2.20)

using V7,gv = 0, and 7apn = 0, shows D to be the unique covariant deriva-

tive operator on the hypersurface ar associated with the metric cop of the

hypersurface.

Having established a metric -op and covariant derivative D on the hy-

persurface Er, the usual constructs of differential geometry can be defined on

the 3-geometry of the hypersurface. The Riemannian curvature tensor R(7) is

given by requiring that all spatial cotangent vectors wvdxv satisfy

1 1
D[D wv = 2(DaD# D#Da)wv = 2R.pa,3 (2.21)






14
To complete the initial data for the Cauchy surface also requires the
extrinsic curvature tensor K0p. The following are equivalent expressions for
the extrinsic curvature:
Kap = y-v'pV(,aVv)r



1 1 v
= 70.7 p(Vwy v + VVw.)

= y p(Vpn + Vvnp)
= -~ y7p(gyVpn + g7VEnY) (2.22)

=- y Vp (nV^gpv gVn + g gVvnL )
1 I -
= ^7 v7 pn9v
1 YU
= -2-7 7 pAnp
1
= 2- n7o7p

where is the Lie derivative. The Lie derivative Cn in the direction of the time-
like normal n to the Cauchy surface acts as a time derivative. The equation

Anv^L = 0 was used in the last two qualities. The final expression provides
an interpretation of the extrinsic curvature as the time rate of evolution, or
velocity, of the spatial metric. The extrinsic curvature's role in the initial data
formulation is to express the bending of the embedded hypersurface rE in the
space-time manifold (M, gaS). The extrinsic curvature, as a spatial tensor,

depends on the vector field n only on the Cauchy surface. Thus, one slice of
the foliation of space-time is characterized by (Er, ya, Kag), and provides the
initial data for the space-time manifold (M, ga,).
As previously mentioned, the 3-geometry is found first, followed by the
determination of the 4-geometry for that one value of the universal time r.







15

The geometry of the hypersurface is related to the space-time geometry by the

Gauss-Codazzi equations


Sa^ A-I Rpopv = RapA6 + Ka\Kpa KaKy (2.23a)

7Pa^yU7~Rppvn" = DaKgs DeK a (2.23b)

where the roman Rpcpv are the components of the Riemannian tensor of the

space-time geometry, and the italic RaPA6 are the components of the Rie-

mannian tensor of the hypersurface, which results from the projection onto

the Cauchy surface of the commutator of the space-time covariant derivatives,

V[aV#], applied to any spatial vector, and applied to the normal vector. Using

the symmetries of the Riemannian curvature tensor, and the equivalence of this

tensor to the Ricci tensor in three dimensions
Rappv = R[pv][(p]
(2.24)
R[upp]v = 0

and

Rapyiv = 27-,Rv]p + 27p[R ], + y7a[vLy]pR (2.25)

the Gauss-Codazzi equations relate 14 of the 20 components of the space-

time Riemannian curvature tensor to the initial data on each hypersurface Er.

These equations are consistent for any three dimensional geometry embedded

in a four dimensional geometry.

Requiring that the space-time geometry (M, gap) satisfy Einstein's field

equations (2.1) will impose several constraints on the initial data for the Cauchy

surface. Einstein's equations involve the Ricci tensor Rpv, and Ricci scalar R of

the space-time geometry. The Gauss-Codazzi equations relate the Riemannian

tensor of the space-time geometry to the initial data of the hypersurface. In

order to impose Einstein's equations, the Gauss-Codazzi equations must be






16

contracted into a form involving the Ricci tensor and Ricci scalar. Contracting

the indices a and A in equation (2.23a ) and using equation (2.14) gives

7yiygRav + nPnPRpp 6 nPn'n^ wRpR,p6 + nPn n, n"Vwpw6Rpgopv

= R6 + K\Kob K K# .

Using the symmetries of the Riemannian tensor this reduces to

7Yly^ R + nPnPRp#,, = R6 + KA\K6 KKA (2.26)

Contracting this equation over the indices / and 6 gives

R + 2nPn Rpp = R + K\K6, K6KAb (2.27)

This expression relating the Ricci scalar and Ricci tensor of the space-time

geometry to the Ricci scalar and extrinsic curvature of the hypersurface. Simi-

larly, by contracting the indices a and 6 in equation (2.23b ) and using equation

(2.14) gives


7Ranv + -ynPnPnVRp,,v = DaKa DK .

Again using the symmetries of the Riemannian tensor this reduces to

y7pRavnv = DaKa D6K% (2.28)

which relates the Ricci tensor of the space-time geometry to the covariant

derivative of the extrinsic curvature in the hypersurface.

Before invoking Einstein's field equations on equations (2.27), and (2.28),

consider the trace of equation (2.1)

1
Ga = R g R = 87rTa

which reduces to


R = -87rT .


(2.29)






17
This can now be substituted back into equation (2.1) giving


Ra( = 87r(Ta g pTao) (2.30)

which is an equivalent expression to Einstein's field equations. Substituting

equations (2.29), and (2.30) into equation (2.27) gives

167rp = 16Irn'nWTo = R + K K' KX6K6 (2.31)

where p = n'nPTI is the energy density measured by an observer that is

moving with the hypersurface Er. Making these same substitutions in equation

(2.28) gives

8srjf = 87rypTovnv = DaKa DK"a (2.32)

where jp = 7y~pTon is the momentum density vector measured by the same

observer moving with the hypersurface Er. Equations (2.31), and (2.32) are

constraints on the initial data of the Cauchy surface imposed by Einstein's field

equations. Equation (2.31) is known as the Hamiltonian constraint equation,

and equation (2.32) is known as the momentum constraint equation.

Having satisfied these equations, the initial data of the hypersurface can

be evolved forward in time along the congruence of curves parametrized by

r, having t = (an" + #3)6, as tangents. In order to evolve the initial data,

consider the Lie derivative of the extrinsic curvature CnKap. This will provide

the remaining 6 components of the space-time Riemannian curvature tensor.

The 6 components of the space-time curvature tensor involving second order

derivatives in a time-like direction are given by

nnKap -= n(7" ra(plv)) = n (7i7 UpV)pW2)

= 677 R6- JK.6 ^DDp& (2.33)
= Pn n Rbv K,6K DoDpa







18

In order to impose Einstein's field equations, the space-time Riemannian tensor

is replaced by expressions involving the space-time Ricci tensor and the initial

data on the hypersurface by substituting equation (2.26) into equation (2.33)

giving

CnKap = -7y 6pR6A + Rp + K Kap 2KK - DaDpa (2.34)

Next the Einstein's field equations are imposed through the substitution of

equation (2.30) into equation (2.34) giving

fCnKap = -87y 66 ,T6 + 47r-ypTAA
D(2.35)
+ Rap + KKapK\ 2KA, 1-DDpa 35)

This can now be put into a form representing the Lie derivative of the ex-

trinsic curvature in the direction of the tangents to the congruence of curves

parametrized by 7 using the linearity of the Lie derivative

CtKap = aCnKap + CpKap (2.36)

Applying equation (2.36) to equation (2.35) and defining the projection of the

spatial stress tensor Sap = 7y pT6A gives

LtKap = -87raSap + 47racopTA
(2.37)
+ aRp + aK Kap 2aKAKp DDpa + CfLKap

which is the evolution equation for the initial data of the Cauchy surface Er.

Conservation of energy and momentum,VaTa/ = 0, for a space-time

manifold (M, gap) is a consequence of the contracted Bianchi identity

VGP = 0 (2.38)

This is shown by taking the divergence of Einstein's field equations

VG/" = VpR"V gP"VVPR = 87rVT~' (2.39)
2







19

Observers moving with the hypersurface measure the stress energy tensor using

the previously defined quantities p, jA, and SPV to be


TPv = pn"nv" + 2j(Pnv) + Sl" (2.40)


Projecting the contracted Bianchi identities onto the normal n to the hyper-

surface and imposing Einstein's equations in terms of equation (2.40) gives the

continuity equation


tp = caSPKyv + apK" aDIjJ 2jPDpa + Cp (2.41)


Similarly, by projecting the contracted Bianchi identities onto the hypersurface

and imposing Einstein's equations in terms of equation (2.40) gives the Euler

equation


CtjC = 2aKP"jv + cajK", aDvSC"V + Sl"VDva + py"VDva + Cpj3 (2.42)


An important consequence of the contracted Bianchi identities (2.38) is that

initial data which satisfies the constraint equations (2.31), and (2.32), will con-

tinue to satisfy these constraint equations as it is evolved along the congruence

of curves parametrized by r from one slice to the next by the evolution equation

(2.37).


2.2 Steady State Equations


When specializing classical general relativity to the study of black holes,

the stress energy tensor, Tap, is set equal to the classical vacuum value of

zero. The gravitational field itself becomes the source of curvature through

the nonlinearity of Einstein's field equations (2.1). Equation (2.29) for zero







20

stress energy restricts the Ricci scalar of the space-time geometry to zero.

Substituting these vacuum values into Einstein's field equations gives

GvT""m) = Rp, = 0. (2.43)

The Ricci tensor for the vacuum space-time geometry must be zero. The initial

value formalism applied to vacuum space-times will also have zero stress energy.

The related quantities of energy density, momentum density vector, and spatial

stress tensor measured by an observer comoving with the hypersurface are also

zero
p=0

j = 0 (2.44)

S/ = 0 .
Applying these vacuum restrictions to the constraint equations (2.31), and

(2.32) of the initial value formalism gives the Hamiltonian constraint for the

vacuum

R + K\K'6 KI6KA~ = 0, (2.45)

and the momentum constraints for vacuum

DaK D = Ka = 0 (2.46)

Finally, applying the vacuum restrictions to the evolution equation (2.37) gives

LtKap = aRap + aK\Kap 2aKA KA DaDpo + CKrpP (2.47)

The continuity equation (2.41), and the Euler equation (2.42) are trivially

satisfied in a vacuum space-time geometry.

The initial value formalism for vacuum simplifies further when the tan-

gent vector t' is a Killing vector field. A vector field (O on any space-time

manifold (M, ga,) satisfying


Lgvy = Vgv + Vgip = 0


(2.48)







21

is known as a Killing vector field. Such a vector field results from the presence

of a symmetry on the space-time manifold. This symmetry gives rise to the

conserved quantity (au" along geodesics having ul for a tangent, as is shown

by evaluation of the expression

uPV(,u") = u+u"VP6, + GuAVPu"
(2.49)
=0

which is useful for integrating the geodesic equation uVvuP = 0, when sym-

metries are present.

As discussed in the introduction, the spiralling binary black hole prob-

lem possesses no symmetries. This is a consequence of the gravitation radiation

the system emits associated with the acceleration each black hole experiences.

Energy and angular momentum are carried off to null infinity by the gravita-

tional radiation causing the binary system of black holes to spiral in on itself.

However, when the holes are reasonably far apart and orbiting in near cir-

cular orbits, the secular time scale for decay of the orbit will be large when

compared with the dynamical time scale of the orbital motion. Then as an

approximation, the circular orbits will be stable. An observer corotating with

the black holes will see a space-time geometry which is constant in time. This

approximation will then possess a time symmetry in the corotating observers

frame. The corotating observer's Killing vector field is then


t'= (2.50)


where the prime denotes a quantity observed in the corotating frame. In the

inertial frame of reference, the black holes rotate at the angular frequency 0 as

viewed from spatial infinity. The shift vector connecting the coordinates of the







22

inertial frame to the coordinates of the corotating frame in the asymptotically

flat region is given by

S (2.51)

The natural choice of orthogonal vector field connecting the hypersurfaces in

the asymptotically flat regions of space-time for the inertial frame is


at
N = an u = .-. (2.52)

The Killing vector field for the asymptotically flat region of the space-time

geometry in the inertial frame of reference is given by equation (2.12). Substi-

tuting the asymptotically flat results of equations (2.51), and (2.52) gives

0 0 a
t = to- = + Q (2.53)
Oxu at Qt

as a coordinate transformation of equation (2.50) also reveals. The Killing vec-

tor field for all other regions requires that the space-time geometry be known.

Equation (2.53) is an asymptotic boundary condition on the Killing vector field

in the inertial frame of reference.

Having chosen the appropriate lapse function a, and shift vector /3 to

make the tangent t = (an" + /')6, a Killing vector field, then Ct, the Lie

derivatives in the direction of t, of various geometric quantities of the space-

time manifold (M, ga ) will be zero. In particular

CtTpV = 0 (2.54)

and

CtK, = 0 (2.55)

will be satisfied. In addition equations (2.22) and (2.54) allow the extrinsic

curvature to be written as


aKp = D(a p) .


(2.56)






23
The vacuum constraint equations, (2.45) and (2.46) are unchanged by this

symmetry. However, the vacuum evolution equation reduces to

aRp + acgKap 2aKhAKAp DaDpa + CKp = 0 (2.57)

The last term in equation (2.57) is equivalent to

ICfKap = P3DpKo.p + 2KP(,Dp)I, (2.58)

Substituting into equation (2.57) gives

aRop+oaK\Kap-2aKKXp -DODpa+OlDKOp+2K, Dp = 0. (2.59)

Under the symmetry of this Killing vector field, the initial value formal-

ism simplifies to a set of equations referred to as the steady state equations36

aKup = D(,Op) (2.60a)

R + KK6 K K' = 0 (2.60b)

DaK' D =pK 0 = 0 (2.60c)

DDpa- DK, p-2Kh,(Dp),-aRap-aK a Kop+2aKAK = 0 (2.60d)

which are collected here in summary. Equations (2.60a) through (2.60d ) are

the fundamental vacuum initial value formulae for the approximate symmetry

associated with the Killing vector field t used in the analysis of spiralling binary

black holes discussed in this dissertation.


2.3 Boundary Conditions


The boundary conditions for the space-time geometry select the particu-
lar solution to equation (2.43). In the spiralling binary black holes approxima-

tion, these boundary conditions are specified on the Cauchy surface. This is







24

due to the simplification introduced by the steady state symmetry. The spatial

geometry of the Cauchy surface chosen for the binary black holes consists of

three separate asymptotically flat regions each possessing its own spatial in-

finity as shown in Figure 2. The top sheet of the Cauchy surface contains two

black holes connected to the two lower isolated sheets by bridges similar to the

Einstein-Rosen bridge of the Schwarzschild black hole.37 This differs from the

inversion symmetric spatial geometry used in earlier studies which consisted of

just two asymptotically flat regions.38 Unlike the inversion symmetric geome-

try which requires that boundary conditions be placed at the two throats of

the black holes as well as at the two spatial infinities, the Cauchy surface hav-

ing three asymptotically flat regions has the advantage of requiring boundary

conditions only at the three spatial infinities.

The conditions on initial data for the Cauchy surface which guarantee

asymptotic flatness were found by Ashtekar and Hansen.39 In a coordinate

component manner when asymptotic Euclidean coordinates are used, these

conditions on the initial data require that the spatial metric of the hypersurface

approach the flat Euclidean spatial metric in the following limit


lim 7Yp = Y + O(-) (2.61)
r--oo

where fp, is the flat spatial metric of Euclidean geometry and r is the ra-

dial component of flat spherical coordinates. The components of the extrinsic

curvature in these coordinates must approach zero at least as fast as


lim Ku = 0(2) (2.62)
r-oo r rf r r

and the Ricci curvature tensor of the spatial hypersurface must approach zero

















































FIGURE 2: Hypersurface Topology for Binary System of Black Holes.
A space-like slice with three asymptotically flat regions, one on each sheet,
connected by two wormholes.








at the rate

lim Rv = O() (2.63)
r +oo r"

or faster in this Euclidean coordinate system.27

For a space-time manifold which is asymptotically flat the mass and

angular momentum of the geometry is found using two dimensional spherical

surface integrals near space-like infinity.40 The mass is given by the integral


16nrMi = lim f Vv(hY f/"h~ )dE (2.64)
r-+oo, j

where hP, = 7^, fyv, and Vy is the flat derivative operator. The linear

momentum is given by the surface integral

87rPi = lim -(KP" 7P" )dE, (2.65)


and the angular momentum is given by the surface integral


87rJ = lim e6"iV (KP 7PK')dEp (2.66)
"r--+ooi J

where (P is the axial Killing vector field at each spatial infinity and ec'6 is the

total anti-symmetric tensor. The subscript i is used to distinguish each of the

three asymptotically flat regions which the radial coordinate r is approaching

in the above limits.

Finally, the Killing vector field generated by imposing the time symme-

try on the space-time geometry is inconsistent with the exact geometry as a

result of the gravitational radiation, as already pointed out. In the frame of

the corotating observers, the gravitational radiation must have standing wave-

forms at large radial distances from the binary black holes. In a Euclidean

background geometry gravitational radiation falls off as 1/r and has an energy

density which falls off as 1/r2. Thus the contribution to the total energy from







27

large radial distances is divergent. However, if the amplitude is sufficiently

small, then a large radial value rmax can be found such that the contribution

to the total energy from the gravitational radiation is much less than the mass

of the black holes and rmax is sufficiently distant to make the assumptions of

asymptotic flatness valid in the Euclidean sense. This does mean that the ini-

tial data is being satisfied on a partial Cauchy surface which will have a finite

future Cauchy development consistent with causality and the Killing vector

field. Outside this future Cauchy development the geometry is unspecified and

not determined by the initial data. Only under these conditions will the time

symmetric approximation to the spiralling binary black holes be sufficiently

adequate.














CHAPTER 3
VARIATIONAL PRINCIPLE FOR TOTAL MASS


Determination of the variational principle for the mass begins with the

Hamiltonian density of general relativity27

WG = x ray V CG (3.1)

where the dot over the spatial metric denotes Lie derivative in the time direc-

tion

7"/v -= ty = -2aKpy + D1p3 + D,3p (3.2)

The conjugate momentum tensor density is related to the extrinsic curvature

by
711V = = -y(K yj( 'a) (3.3)

and the Lagrangian density G of general relativity is given by

CG = -Rg (3.4)

The Lagrangian density can be rewritten in terms of the initial value formalism

and the initial data of the hypersurface by using the following relationships:

J(- = ad (3.5)

R = 2(Gy, R,,)nn" (3.6)

Using the vacuum Hamiltonian constraint equation (2.45), the first term in

(3.6) becomes

2Gvninv = R Kp1vKPV + KPKv (3.7)

28






29
From the definition of the Riemann Tensor the second term in equation (3.6)

can be evaluated,

Rvn'n" = R 'n,,,"n

= nP(VV, VpV.)n" (3.8)

= KPK KpoaKa Vp(nVPVan) + V,(nVpna) .

The last two terms in equation (3.8) are total divergences and will be dropped

temporarily since they result in surface terms which will be described later.

Combining equations (3.5), (3.6), (3.7), and (3.8) in equation (3.4) gives the

Lagrangian density in terms of the initial value data

LG = atv/ (R + KvKP" K1K",) + neglected divergences (3.9)

Finally substituting this expression for the Lagrangian density (3.9) into the

Hamiltonian density (3.1) along with equations (3.2) and (3.3) gives

-G = -V [a(R K.vKP + K1Kv,) 2/uD,(K1' 7yK~ ,)]
(3.10)
neglected divergences

where all divergent terms have been suppressed until later discussion.

The Hamiltonian is given by the integration of the Hamiltonian density

(3.10) over the three dimensional space-like hypersurface Er


HG = J -HG (3.11)

This Hamiltonian, using the Hamiltonian density of equation (3.10), is known

to be correct for a closed universe having the closed hypersurface Er.26 How-

ever, when the space-time geometry is asymptotically flat the variation 6HE of

this Hamiltonian does not lead to Einstein's field equations. In order to correct

this shortcoming the Hamiltonian in equation (3.10) must be supplemented by

the addition of surface integrals which result from the careful treatment of the







30
previously neglected divergences. In a geometry with asymptotic flatness this

leads to 41

HE = HG +E (3.12)

where E is given by


E = lim aV,(h fP"vh)d (3.13)
r--+oo f

Variations 6HE which preserve asymptotic flatness do lead to Einstein's field

equations, and the value of HE for solutions to Einstein's field equations is E.

From equation (2.64) this value is shown to be


{HE}(6HEO) = {E}(6HE=0) = 16rMo (3.14)

Solving equation (3.14) for Mo gives the mass of the system

1 1
Mo = 16i HE(6HE=O) = 1 {HG + EI(6HE=0) (3.15)

Substituting the integrals (3.13) and (3.11) into equation (3.15) produces the

equation to be evaluated in a variational method for the mass

Mo = lim aV,(h" fPh')dE
167r r-oo, J

1- 6 7[a(R KpIK K + KP~ ) 2,pDv(K'" 7PvK(,)]dV
(3.16)
where V, is again the flat derivative operator and the surface integral is eval-

uated at the shared infinity of the hypersurface.

As demonstrated by equation (3.15), SHE = 0 must be satisfied in order

for Mo to be the mass of the system in equation (3.16). To study the variation

of HE consider a test geometry for the binary black holes system possessing

the time symmetry in a corotating frame as described by the steady state

approximation and characterized by the initial value data, a, P K ', and






31

7yp with their respective infinitesimal variations 6b, 6b 6KV', and 6ya,.
The quantity 6HE is given by30
6HE = 167rbMo

= J{bab] + 6/3cj], + SKI"jaj + bya, b, c, I ld]}dV (3.17)

+ Qob6Jo a6MI1 + Q1SJ1 a2SM2 + i226J2 .
The I] denotes linear combinations of the equation numbers (2.60a ) through

(2.60d ) enclosed, which will vanish when the steady state equations are sat-

isfied. The 6Ji's represent variations in the angular momentum at each of the

three asymptotically flat regions of the hypersurface Er. Similarly, the ,i's are

the angular frequencies, and the 6Mi's are the variations in masses measured

at each of the three asymptotically flat regions.

The variational principle for Mo can now be described. Given a trial

geometry on the hypersurface E, characterized by a, #P, KPL, and y then

any arbitrary infinitesimal variations, ba, Sbp, S6K', and S67p which do not

change the total angular momentum at each of the three asymptotically flat

regions of Er and also do not change the mass at the two asymptotically flat

regions associated with the interiors of the binary black holes, will satisfy the

condition SHE = 0 for trial geometries which also satisfy the steady state

equations. The total mass Mo is an extremum since it satisfies SMo = 0. This

signifies that an accurate determination of the total mass Mo can be obtained

when only a rough estimate of the geometry is available.

Application of the variational principle follows as such, given the trial

geometry for the hypersurface, evaluate equation (3.16) throughout the pa-

rameter space of the classes of variations preserving the restrictions described

above for the extremum. The mass of the system is given by this extremum

value. An error of order O(e) in the trial geometry will produce an error of

order O(e2) in the extremum value of the total mass Mo for the geometry.














CHAPTER 4
TRIAL GEOMETRIES


The trial geometry to be used in the variational principle for the mass

must be a physically reasonable approximation to the binary black hole system.

In addition the meaning of asymptotically flat must be made precise in order

that the surface integration appearing in the variation of the Hamiltonian of

equation (3.11) vanish. The detailed variation of -G is

-HG = {H6a + HSP + APv&yt, + BV67rpV}dV

lim SG,""P{a(Db&ypv) (Da)by7, }dEp
-+00 (4.1)
Elim VP{2 37ap + (2/P7rP PrV)bya}dE
i-..0i J


where the two surface integrals are evaluated at each of the three asymptotically

flat regions of the hypersurface Er. The terms appearing in the integrand are


H = V(7rpvr/"_ ( 1)2 R) (4.2)

which is proportional to the Hamiltonian constraint,

H, = -2Dy,7r (4.3)

which is proportional to the momentum constraint,

APU = a- (27r -"Iy7rv,) + 7yPDpP + 7,yPDpD (4.4)







33

which is the functional derivative of the Hamiltonian with respect to the spatial

metric tensor 7ps,

1 1 1
Bj, = --u(Rv 2- R) + 2-- a(7rOP7rop 2 )2)

-a( pr, prP) + fy(DpDva ypIPDDpa) (4.5)
V P
+ Du(irP1 ( )<- (D<^)ir

which is the functional derivative of the Hamiltonian with respect to the con-

jugate momentum tensor density rpv,, and the super-metric


G"vP = 12 V(Y( "P + 7yPy"v 27~L"aP) (4.6)


The particular asymptotic behavior for the initial data necessary to apply

the Hamiltonian (3.12) and the variational principle for the mass (3.16) are

given for the spatial metric by41

1
7pY fv oc (4.7)
r

and the gradient of the metric by the behavior

1
7 (4.8)

The conjugate momentum tensor density has the asymptotic behavior given

by
1
7, = (Kv 7,vK") OC r2 (4.9)

The lapse function has the asymptotic behavior given by


a 1 oc (4.10)
r

and the gradient of the lapse function has the asymptotic behavior

1
N .- .(4.11)







34
The shift vector has the asymptotic behavior given by


1" c (4.12)

and the gradient of the shift vector has the asymptotic behavior

1, oc (4.13)


Where the r for each behaves as the Schwarzschild radial coordinate in the

asymptotic limit

r = .x2 + 2 (4.14)

These are sufficient specifications on the asymptotic behavior of the geometry

to use equation (3.16) as a variational principle for the mass.

The trial geometry should also approximate several limiting cases where

analytical results can be used as a bases for comparison. These limiting cases

consist of the following list of properties:

At each of the three distinct infinities of the Cauchy surface the spa-

tial geometry must be asymptotically flat as outlined above in order to

guarantee well defined masses, linear moment, and angular moment.

When the separation between the two black holes is large, the geom-

etry must approach the superposition of two boosted black holes with

arbitrary velocities.

For very small black hole velocities the Hamiltonian constraint (2.60b )

and the momentum constraint (2.60c ) must be satisfied by the geometry

to order O(v2) in the velocities.

In the limit that one of the masses is much less than the other, the test

particle limit, the analytic results for geodesics of the Schwarzschild black

hole are reproduced.







35

The first requirement is necessary to establish well defined physical quan-

tities. The second and third requirements are consistent with special relativity

and Newtonian gravity. The last requirement allows for comparison to analytic

results of general relativity. This list of requirements is not necessarily com-

plete, however it does provide a reasonable basis of test for the trial geometry.

The infinitesimal line element for the four dimensional space-time geom-

etry is given by

ds2 = gp dxdx" (4.15)

Once the four dimensional space-time has been foliated into constant 7 space-

like hypersurfaces Er as previously outlined the infinitesimal line element in

terms of the lapse function a, the shift vector /?l, and the three dimensional

spatial metric 7Y,, is given by


ds2 = 2dt2 + 7yv(dxu + P#dt)(dx" + Vdt) (4.16)


The form of the spatial metric for the trial geometry in a coordinate basis

which approaches Cartesian coordinates in the asymptotic limit is given by


-pvdxPdx" = T4(dx2 + dy2 + dz2) 2(a12 V12) 2 (22 2) dy2
S(1- u2) (1 -v2)
(4.17)

The functions 9, 1i, 2, a1, and a2 are chosen to make the spatial metric on

the hypersurface Er model the geometry of two black holes in circular orbits,

with asymptotic boundary conditions, as close as possible. They are defined

in the trial geometry by
ml m2
T = 1 + + (4.18)
2pi 2P2
where ml and m2 are the masses of the two black holes as measured on the up-

per (shared) sheet of the hypersurface Er, also Pl and P2 are radial measures of







36

distance from each black hole given by the relationships to the quasi-Cartesian

coordinates
2
p12 = (x xo12 + 2+ z2 (4.19)

and
2
22 = (x o2)2 + + z2 (4.20)

where xo, and u are the position along the x-direction and 3-velocity of the

black hole of mass mi, and the x02 and v are the position along the x-direction

and 3-velocity of the black hole of mass m2. The functions 01 and 02 are

chosen to be
2
ml m2m2
1 = 1 + + m (4.21)
2pi 2s(m12 + 4p 4)
and
2
m2 mlm2
02 = 1 + (4.22)
2P2 2s(m22 + 4p24)
where s is the separation between the two black holes and is given by


s = Zol Xo2 (4.23)


The functions al and a2 are chosen to be


1- +=) m2m
(P + (1 Tfl21 4 (4.24)
S2p(+2)) 2s(m2 4p

and
(2p + 21 2 ) (4.25)

(1 + M2 2s(m22 + 4p24
2p,
In addition the shift vector 3 and the lapse function a are needed in order

to specify the four dimensional space-time metric gpv. The lapse function a






37
provides a measure for the rate of proper time over the hypersurface Er and
is chosen to be

M1 2p2( + ) 2p(1 )
p (l-, )(l ^2p,

(1 + 21(1 ) ( + 2p ))


( ^ 2
rn1 2m'))


p4 u'- 2(a2 -_ 14)_ 22 4))

x(1 + m2i + 2 2 m 12 p m 2 2 (4.26)

+ 1+ 2 (1 + m 1 ) m 222
2s 2s(+ m2212 + 4p24
+ 1 m12 22 m22P12 P 2
m12p22 + 4pi4 m22p + 4p2 4 )
The arbitrary spatial shift vector 3 is chosen to co-rotate each point on the hy-
persurface Er with the orbiting black holes. In this quasi-Cartesian coordinate
system the shift vector has components given by
( u2 22 2 22,12
3 1+( )( ) ) ))y (4.27a)
1 m2 12p22 +4p14 1 -2 22 12+ 4p24 4


(@4 UT (a2 ) -2 24))

3z = 0. (4.27c)

The quantity f appearing in the components of the shift vector 3 is the orbital
angular frequency of the black holes and is given by
u v
= (4.28)

The Christoffel symbols are needed in order that the covariant derivative
and the scalar curvature for the spatial geometry can be calculated. The
covariant Christoffel symbols are derived from the spatial metric using

I'6 = -2 9x6 + x Oxt) (4.29)
2 9xb 8 V a8







38

By having a diagonal spatial metric, these take on a particularly simple form:

rx 1 7xx (4.30)
2 8ox

1zxy = zyz = (4.31)
y xy 2ay

rz F 1 7= rz (4.32)
2 =z

Y 1 0yy (4.33)
2 ax

rzyz = rzzy = 0 (4.34)

rzz = z (4.35)
2 Ox
ry ax (4.36)
2 dy

1y = yyr yy (4.37)
=yxy Lyyx -- ^ 2 x

Tyxz = Fyzx = 0 (4.38)

S1 yy (4.39)
yyy-= 2 (y

yyz = yzyy (4.40)
2 dz
1 87zz
=yzz- y (4.41)

rzx = 1 (4.42)
2 Oz

Fzy = rzyz = 0 (4.43)

rFzz = zz = 7zz (4.44)

1 a/yy (4.45)
zyy 2 Dz
1 87zz
rzyz =rzz (4.46)

-1 (y.4
Fzzz 1 z (4.47)
2 az








The contravariant Christoffel symbols are given by

S0 = 17 0r+ ^ a v6 f (4.48)
b 27 89x6 xL' 9x

which also are rather simple expressions when the spatial metric is diagonal:

=z 1 87zz (4.49)
xz 27z- Ox

z =_ Z 1 97Y X (4.50)
xy yx 2x7zz Oy

I p= z 1 =X (4.51)
Sx zx ~2,y az

x y1 17 (4.52)
YY 2yxz Ox

rZz = ry = 0 (4.53)

r 1 O'yzz (4.54)
zz 27,xx Ox

Fy = -1 O- y (4.55)
2yyy Oy

rY = ry 1 OY (4.6
1 y r = (4.56)
Yx 27yy Ox

rYz = rYz = 0 (4.57)

p = 1 &Yyy (4.58)
=y 2-yy Oy

Syz (4.59)
Fz= FY 2yyy Oz

rzz 1 OYzz (4.60)
27yy ay

r 1 7Ozz (4.61)
Fz 27zz az
rz = rzy = 0 (4.62)

z = z 1 Oyzz (4.63)
z x 27zz Ox
rz1 0yy (4.64)
FZy 2zz Oz








Fyz = rz y1 yzz
y ~ zy~ 27z,, ay
1 87zz
zz 27zz az


(4.65)

(4.66)


The covariant derivative DP is defined using these Christoffel symbols.
As an example, consider the gradient of a second rank mixed tensor T3


DTP- = 9TP + T6, +1 aTf
p a 6 ~x4 a ap^


(4.67)


Other applications of the covariant derivative follow from this example with
the exception that the gradient of a scalar function is merely


(4.68)


All of which is in agreement with the standard definitions of differential geom-
etry.
The scalar curvature R also is derivable from the spatial metric of the hy-
persurface E, and will be required for the variational principle. For a diagonal
metric of the form
(7 0 0
7, = 0 yy 0 (4.69)
0 0 7zz
such as the spatial metric for the hypersurface E, of equation (4.17), the scalar


curvature becomes the relatively modest expression


R= TY
(7xxztyy)


(27Yz7yyxyy)

S(27zyW z)
(27^7x27zzzz)


(2)7xxTyyzyy)


(27yxxyyyyzz)


('Y7tzz) (7zX7yy)


(TyyYzz)


+ 91--19) + (Yi )

+ + (
(27yy7zz7zz) (2xyzz Yxxyy)
(28) ( Y )
(27yxxYyyTzz) (27'y7zyyYzz)

(7z ) ( (yY)
+ (+yy+zzyzz)
(2-yyuyy~zz) (27yyTzzTz)


(Yzzxzz) (Yyyyzz)

+
(27yy7yyyzz)


(27yzzx7 7zz)

+(20zzzz)
(27,yx7zz7yzz)


(4.70)


D, xP
_z a







41

which results from a computation similar to that of the Dingle formula.42

Actually, a simplification exist for the spatial metric of equation (4.17) since

there the y7z and 7zz components are equal. However, this simplification was

not used to its full advantage in the application of the variational principle and

will be left out here.

The complete set of initial data for the hypersurface Er requires in ad-

dition to the spatial metric ypv, the lapse function a, and the shift vector #",

that the extrinsic curvature K1, be defined as given by equation (2.22). How-

ever, this is a rather complicated expression when applied to the trial geometry

above. In the variational method outlined, only a reasonable approximation to

the extrinsic curvature is needed. Consider the linear superposition of the ex-

trinsic curvatures associated with two separate boosted black holes.The space-

time geometry for a single black hole in isotropic spherical coordinates is given

by the infinitesimal line element31

ds2 = -aisdt2 + 1s(dPso + P~od2 sod2) (4.71)


where
(1 M
aOiso = (4.72)
(1 + )
and
M
iso (1 + ). (4.73)
2Piso
The isotropic radial coordinate Piso is related to the Schwarzschild radial co-

ordinate rSch by

rsch = ( + ) piso. (4.74)
2PisoM







42

By making a Lorentz transformation on this metric the infinitesimal line

element for a single boosted black hole is obtained. For the quasi-Cartesian

coordinates it is found to be


= \ 1-- / d 1 -2 v2
V2 (4.75)
(24 2 2
( iso -v isody2 +4(dx2 +dz2

with the position of the boosted black hole given by

yBBH(t) = yoi + vt (4.76)

where v is the boost velocity of the black hole and yo, is the coordinate offset

of the black hole from the origin along the y axis. In order to put a spatial

slice of this geometry into the moving frame of reference, introduce the new

set of coordinates

t = t' (4.77a)

x = x' (4.77c)

y = y' + vt (4.77b)

z =z' (4.77d)

Then the infinitesimal line element is given by

ds2 (1 v2)a_ Isso dt'2
iso -v iso )
ISO ISO
S(' v2a(1 ,- v2)a? 2 (4.78)
is1 2 so Idy l+ 2 80 dt
ISO ISO
+ io(ddx'2 + dz'2)
Comparison of this infinitesimal line element to the general form for infinitesi-

mal line elements given by equation (4.15) leads to the definition of the square

of the lapse function A2BH for the single boosted black hole
A2 (1 -v2)a 2 4
ABH~ 0 (4.79)
ISO iSO







43

The definition of the shift vector is found to be

( v(1 v2 2 )
'BH P v2 (4.80)
3SO tso
And the spatial metric yBVBH, given by the three dimensional infinitesimal line

element becomes
_BBH- / 4 (d"'2+ dz'2)+ .is 2 V dy2
B1 v2a2
7BdxldxLv = %40(dx2 + dzy2) s 2so)dyt2 (4.81)


which agrees with the spatial slice of the boosted black hole geometry given in

equation (4.75). This boosted black hole geometry will be used to construct an

adequate extrinsic curvature tensor for the binary black holes' geometry. The

extrinsic curvature for this boosted black hole geometry is given by


2ABBHKIH -- /BBH + ,BBH. (4.82)


In the comoving frame the first term vanishes


Lt H = 0 (4.83)


Thus the extrinsic curvature for the boosted black hole geometry becomes
2ABBH KBBH =_ BBH
2ABBHMV = _l'?7.V
S+ BBH Qa (4.84)
= 0BH #^BBH | ,BBH p + o BBH BH
BBH als "7' BBH 710 XV fPBBH

In order to generate the extrinsic curvature for the binary black holes

geometry consider the two separate geometries for two independent boosted

black holes, one with boost velocity u offset from the origin by xol along the

x axis and the second with boost velocity v offset from the origin by xo2 along

the x axis. The extrinsic curvature is then approximated by


(4.85)






44
where Ki., and K2v, are constructed by comparison with the boosted black
hole's extrinsic curvature given in equation (4.84). The difference is that in
the place of the isotropic lapse function and conformal factor aio and 4iso
appearing in the expression for the boosted black hole's extrinsic curvature
are placed the functions al, a2 and k1, 02 of the binary black holes given by
equations (4.24), (4.25), (4.21), and (4.22). In addition the shift vector of the
boosted black hole JBBH is replaced by the shift vector of the binary black
holes with components given by equations (4.27a ) through (4.27c ). Once this
analogy is complete, the extrinsic curvature for the binary black holes geometry
has the following components:


Kzx = Klzz + K2xz
u_ 1 Y_ _2 (4.86)
2VT4yy 2 '-yy



Kyy = Klyy + K2yy
"(-Ba + 4yyF )
2 y (4.87)

v(-a2 + 4yy & )
+ 2
2 'I'4fyyy



Kzz = Kzz + K2zz
Su v2 +t (4.88)
2V4yyy 2 VT4yy









Kxy = Kyx = Kxly + K2 xy

Tyy0P a l al-1
SU 4 ay pI 27yy (4.89)
( ai 2,yyy
Styy P2 /(a2 a 2
+ V a x P2 27Y)



Kyz = Kzy = Klyz + K2yz

u tyy (P i a al
7 '4 Oz 8pI 27yy ) (4.90)
Hyy~Op2(9a2 8a2
+V Oz \P2 27 y



Kzz = Kzx = Klxz + K2xz
(4.91)
=0.

The masses associated with this trial geometry for each of the three
asymptotically flat regions of the spatial hypersurface Er are given by evalu-
ating the integral of equation (2.64). On the top sheet which contains both
black holes, the value of the total mass is found to be


Mtotal = + (4.92)

When solving for the masses at each of the bottom sheets of the hypersurface,
the trial geometry must be evaluated in the limit that Pl and P2 approach
zero. The value of the masses associated with each of these bottom sheets are
referred to as irreducible masses, and are found to be


b ( m2(1- u2)) (4.93)
Mliarredue = m1 1 + 21 (4.93)







46

for the asymptotically flat region containing only the one black hole located at

x01, and

M2irreduib = m2 1 + ( v2) (4.94)

for the asymptotically flat region containing only the one black hole located at

x02. The variable s is the separation between the two holes. It is the irreducible

masses which must be fixed in value for the purpose of comparing neighboring

geometries in the variations of the Hamiltonian of equation (3.16).

The linear momentum for each sheet of the hypersurface is defined by

equation (2.65). Evaluated on the top sheet, the magnitude of the total linear

momentum for this trial geometry is given by

mlu m2v (4.95)
Ptota- = + (4.95)

The total linear momentum can freely be chosen to have the value


Ptotal = 0 (4.96)

restricting the frame of reference to the center of mass frame. Then the value

of the second velocity v is restricted by the value of the first velocity u though

the constraint
mlu m2v
= (4.97)
N/1 u2 'v
The values of the linear moment as measured on each of the bottom two sheets

of the hypersurface are both zero.

The Angular momentum for each sheet of the hypersurface is defined by

equation (2.66). Again evaluated on the top sheet, the magnitude of the total

angular momentum for this trial geometry is given by


total + m 2 (4.98)
Vtoto = +







47

The values of the angular moment as measured on each of the bottom two

sheets of the hypersurface are both zero.

These equations taken in combination provide a single parameter for the

variational principle as follows. Select a choice of irreducible masses for the two

black holes Mlirredueible and M2irreducible along with a choice of the total angular

momentum Jtotal for the geometry, the above equations (4.93), (4.94), (4.95),

and (4.98) provide self consistent values for the velocities u and v, the masses

of the two black holes on the top sheet of the hypersurface ml and m2, and

the positions along the x-axis xo0 and xo2, given any value for the separation

s between the two black holes. The correct value of the separation for these

values of irreducible masses and total angular momentum is found by numerical

application of the variational method for the Hamiltonian of equation (3.16)

on a computer.

A concise algorithmic description of the numerical code applied to a uni-

versal Turing machine consists of the following:43

Choose an interesting set of values for the irreducible masses and total

angular momentum along with a reasonable estimate for the separation

between the two black holes. This estimate may be found through ap-

plication of Newtonian mechanics, post-Newtonian analysis, or the test

particle limit of the Schwarzschild geometry.

Evaluate equation (3.16) over a sufficiently large region of three dimen-

sional hypersurface E, to cover all three sheets of the slice out to their

respective asymptotically flat regions. Note that the surface integral in

equation (3.16) can actually be evaluated analytically for this trial ge-

ometry and has the value of (-2Jtotal).







48

Use the results of the evaluation of equation (3.16) in an extremizing

algorithm which efficiently alters the value of the separation between the

two black holes to a value corresponding to the minimum or maximum

value of the Hamiltonian of equation (3.16). This value of the Hamil-

tonian is the mass of the binary black hole system and the separation

provides the best trial geometry to the system.

In practice the procedure is much more complex as a result of the limitations of

modern computer programming languages. The actual numerical procedure is

in the form of a C program, a programming language not often used in physics

but one of the least limited in terms of flexibility. The C program has been

extensively tested where comparisons to analytically known results apply and


performs admirably.














CHAPTER 5
NUMERICAL RESULTS


5.1 Test Particle Geometry


In the limit that one of the black hole's mass approaches zero, the trial

geometry used in the variational principle should reproduce the results obtain-
able from the test particle geodesics of the Schwarzschild geometry. In this test

particle limit, the trial geometry does not satisfy the steady state equations,
(2.60a ) through (2.60d ), nevertheless the numerical results in this limit are

remarkably accurate.
The test particle geodesics are analytically derivable, and can easily be
expressed in a concise functional form.44 Since the constraints on the trial ge-
ometry of equation (4.17) restrict the motion of the two black holes to circular

orbits of fixed radial separation, only the analytic results for the test particle

in circular orbits about a Schwarzschild black hole will be applicable for com-

parison in this limit. The analytic expressions for orbital equations of motion
about a Schwarzschild black hole are found using the following set of coupled

differential equations in the usual Schwarzschild coordinates of (r, 0, 0):

(- Veff(r) (5.1)

where the effective potential Veff(r) is given by


Ve f(r) = ) + J )2 (5.2)
r r2








also
(J)a2
dr r2
d7 r2 (5.3)

and
d(E)
dr M (5.4)

The equation (5.1) though (5.4) together can be solved for motion, r(r), O(r)

and t(r) for a test particle of mass m orbiting a Schwarzschild black hole of

mass M with particle energy E and total angular momentum Jtot,,. In general,

the equation for r(r) involves an elliptic integral. However, by restricting

considerations to the circular orbits which satisfy


( =0, (5.5)

the solution for r(r) becomes simple. The circular orbits are found by setting

dV,ff (r)
=d -0. (5.6)
dr

Thus the square of the energy per test particle mass equals the effective po-

tential

(E) = Ve(r) (5.7)

Figure 3 illustrates (Ve/)2 for several values of Jto,,./mM. Differentiating the

effective potential with respect to the radius leads to the expression for the

radius of the circular orbits

Mr2 t- t )r + r 3M (I =0 (5.8)


Equation (5.8) has two solutions for the radii of the circular orbits. The outer

radial solutions, shown in Table 1, correspond to stable circular orbits, while







51

the inner radial solutions, shown in Table 2, correspond to unstable circular or-

bits. Finally, there are no circular orbits, stable or unstable, for a test particle

with Jtotl < 2v3mM, a result which does not appear in Newtonian considera-

tions. Combining equations (5.7) and (5.8), the following relationships for the

total angular momentum and energy are derived

M (5.9)
Jtota = mr 3M (5.9)

E m(r 2M) (5.10)
/r(r 2M)
r(r 3M)
The equations of motion (5.1) through (5.4) also provide the following expres-

sions for the tangential velocity

Jlosl 11 2M
= a 1 2M (5.11)
rE r

and the angular frequency

M =V (5.12)

as measured by an observer at infinity. Circular orbits exist down to radii of

3M. But stable circular orbits exist only for radii greater than 6M in these

Schwarzschild coordinates.

The binding energy per unit mass of a test particle in the innermost

stable orbit at r = 6M is given by

EB ing m -E = 1- 8 5.7191% (5.13)
m m 9

This percentage of the rest-mass energy of the particle is available for emission

in the form of gravitational radiation as the particle slowly spirals in from an

initial state of rest at infinity inward to the innermost stable circular orbit at

r = 6M.










TABLE 1: Stable Circular Analytical Test Particle Orbits

jtota, rseh/M Piso/M E/m OM1 v
3.46T 6.0000 4.9495 0.94280904 0.0680414 0.5000
3.50 7.0000 5.9580 0.94491118 0.0539949 0.4472
3 10.6453 9.6193 0.95830523 0.027914 0.3401
4.00 T12.0000 10.9772 0.96225045 0.0240563 0.3162
4.50 16.5876 15.5716 0.97167430 0.0148210 0.2618
5.00 21.5139 20.5017 0.97776736 0.0100213 0.2264
5.50 26.8730 25.8633 0.98201155 0.0071784 0.2005
6-00 32.6969 31.6891 0.98511216 0.0053486 0.1805
6.50 39.0000 37.9934 0.98745695 0.0041059 0.1644
7.00 45.7897 44.7841 0.98927822 0.0032274 0.1511
7.50 53.0703 52.0655 0.99072367 0.0025866 0.1399
8.00 60.84444 59.8402 0.99189151 0.0021070 0.1304
8.50 69.1138 68.1102 0.99284942 0.0017404 0.1221
9.00 77.8798 76.8766 0.99364540 0.0014550G 0.1148
9.50 87.1430 86.1401 0.99431433 0.0012293 0.1084
10.00 96.9042 95.9016 O0.99488211 0.0010483 0.1026
Note: Units with G = c = 1.
tActual value of j,,,., J ,,il/mM is 2/3.



TABLE 2: Unstable Circular Analytical Test Particle Orbits

total rsch/M Piso/M Elm QM1 v0
3.46 6.0000 4.9495 0.94280904 0.0680414 0.5000
3O50 5.2500 4.1903 0.94561086 0.0831306 0.5547
3.85 4.1772 3.0965 0.98181765 0.1171314 0.6777
4.00 4.0000 2.9142 1.00000000 0.1250000 0.7071
4.50 3.6624 2.5649 1.06732488 0.1426781 0.7756
5.00 3.4861 2.3811 1.14159110 0.1536336 0.8203
5.50 3.3770 2.2667 1.22039216 0.1611404 .8522
6.00 3.3036 2.1889 1.30239038 0.1665808 0.8760
6.50 3.2500 2.1328 1.38675049 0.1706770 0.8944
7.00 3.2103 2.0908 1.47291437 0.1738502 0.9090
7.50 3.1798 2.0583 1.56049136 0.1763645 0.9207
8.00 3.1556 2.0326 1.64919827 0.1783936 0.9302
8.50 3.1361 2.0119 1.73882411 0.1800567 0.9381
9.00 3.1202 1.9949 1.82920807 0.1814379 0.9448
9.50 3.1070 1.9808 1.92022523 0.1825981 0.9505
10.00 3.0958 1.9690 2.01177698 0.1835827 0.9553
Note: Units with G = c = 1.
tActual value of jtal E Jtoaa/mM is 2V3.

















1.05




1.00


0.95


0.90 -
j=0


0.85
0 5 10 15 20
Rlsotropic/M









FIGURE 3: Analytical Effective Potential for Test Particle. Stable cir-
cular orbits are located at the minima. Unstable circular orbits are located
at the maxima. No orbits exist for Jt,toa < 2V/mM, where Jot,, is the total
angular momentum, m is the mass of the test particle and M is the mass of
the black hole. The dotted line locates the positions of the numerically derived
stable circular orbits.







54

This is one of the most efficient mechanisms in nature for converting

mass into energy. The burning of nuclear fuel in the cores of stars is only 0.9%

efficient as hydrogen is combined to form iron.44 Such efficiency is a major

reason for studying black holes as a source of gravitational radiation and as

candidates for high energy compact objects.

By taking the black hole of mass ml to be the more massive black hole

and the black hole of mass m2 to be the less massive black hole in the trial

metric, then a relationship between the analytical results, for a test particle

in circular orbit about a Schwarzschild black hole, to the numerical results for

the variation method, requires the following identifications:


M = irreducible (5.14)

m = M2irreducible (5.15)

In addition the coordinates used in the analytical analysis are Schwarzschild

coordinates, (r, 0, 4). The trial metric is expressed in isotropic coordinates

(p, 0, 0). The two radial coordinates are related through equation (4.74). Tak-

ing these factors into account, the numerical results for black hole masses of

Mlirreducible = 1.00, and M2irredcible = 0.01, can be used to benchmark the trial
geometry with a test particle orbiting a Schwarzschild black hole. The values

of m1 and m2 appearing in equation (4.92) are summarized in Table 3 for

these values of irreducible masses along with the total mass Mot,, defined by

equation (4.92) and the extreme value of the mass returned by the variational

principle.

Table 4 provides a list of numerical results which can be compared to the

analytic results of Table 1. In Table 4 the quantity j,,,,i is the total angular

momentum divided by the mass of each black hole, s is the separation between











TABLE 3: Numerical Test Particle Masses

Total mi m2 Mto tal Eo
3.85 0.99940862 0.00940862 1.01000214 1.009565079
4.00 0.9992198 0.00952198 1.00986509 1.009612864
4.10 0.99957026 0.00957026 1.00983177 1.009638584
4.20 0.99960644 0.00960644 1.00981706 1.009661131
4.33 0.99963959 0.00963959 1.00981405 1.009686699
4.40 0.99965838 0.00965838 1.00981109 1.009699110
4.50 0.999687 0.00968070 1.00981099 1.009715259
4.60 0.99969985 0.00969985 1.00981313 1.009729954
4.80 0.99973405 0.00973405 1.00981890 1.009755968
5.o0 0.99976203 0.00976203 1.00982694 1.009777992
6.00 0.99984429 0.00984429 1.00987175 1.009852113
7.00 0.99988955 0.00988955 1.00990319 1.009893937
8.00 0.99991788 0.00991788 1.00992474 1.009919656
9. 0.99993595 0.00993595 1.00994011 14F 1.009937049
10.00 0.99994865 0.00994865 1.00995125 1.009949368
Note: Units with G = c = 1, and M2 = 0.01M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.




TABLE 4: Numerical Stable Circular Test Particle Orbits

Total s/M1 (Eo M)/M2 MI U0 v_
3.85 7.950 0.9565079 0.0581753 0.00485 0.4576
4.00 9.955 0.9612864 0.0394575 0.00402 0.3888
4.10 11.130 0.9638584 0.0326053 0.00369 0.3592
4.20 12.200 0.9661131 0.0279352 0.00344 0.3374
4.33 13.368 0.9686699 0.0240683 0.00324 0.3185
4.40 14.132 0.9699110 0.0219325 0.00311 0.3068
4.50 15.155 0.9715259 0.0195468 0.00297 0.2933
4.60 16.153 0.9729954 0.0176149 0.00285 0.2817
4.80 18.296 0.9755968 0.0143674 0.00262 0.2602
5.00 20.506 0.9777992 0.0119365 0.00244 0.2423
6.T00 31.606 0.9852113 0.0060513 0.00190 0.1894
7.00 44.764 0.9893937 0.0035240 0.00156 0.1562
8.00 60.380 0.9919656 0.0022150 0.00133 0.1324
9.00 77.556 0.9937049 0.0015107 0.00116 0.1160
10.00 96.875 0.9949368 0.0010760 0.00103 0.1032
Note: Units with G = c = 1, and M2 = 0.01M1.
M1 and M2 are irreducible masses.
Eo is the extremized mass.









the black holes, M *- is the energy per mass of the less massive black hole, Q

is the angular frequency, uO is the tangential velocity of the more massive black

hole and vO is the tangential velocity of the less massive black hole. The most

obvious difference in the test particle tables is the limiting value of Jlotai/mM.

The numerical results have no stable circular orbits for Jtota, < 3.85mM which

is roughly 11% larger than the analytic limit. A comparison of the energy

per mass of the particle with angular momentum Jto,,, = 3.85mM reveals a

difference of 0.2% between the analytical and numerical results. However, the

binding energy per mass for the numerical result is 4.35%, while the analytic

result is 4.17%. This translates into a 4.3% difference in the percentage of

rest-mass energy available for the emission of gravitational radiation between

the two results.

Using the Hamiltonian of equation (3.16) as the effective potential energy,

a series of curves can be generated which locate the stable circular orbits as

their minima. The numerically derived effective potentials for several values of

Jtotla/mM are illustrated in Figure 4. For a fixed value of angular momentum

the variational principle provides the extremum of mass to second order in

accuracy, while providing the position of the extremum only to first order. Thus

the numerical results for the extremized mass are most accurately presented

as functions of the angular momentum as shown in Figure 5.


5.2 Equal Mass Geometry


When the two black holes are of comparable masses, no analytic solutions

exist for circular orbits. The form chosen for the trial geometry does reduce




















1.0105-







E 1.0100 4.40

4.20

4.00

3.85
1.0095

J=3.50


0 5 10 15 20 25
S/M,








FIGURE 4: Numerical Effective Potential for Test Particle. Stable cir-
cular orbits are located at the minima. Maxima were not found with this trial
geometry. No orbits exist for Jo,,i < 3.85M1M2, where Jta, is the total angu-
lar momentum and M1 and M2 are the irreducible masses of the black holes.
The position of the analytic stable circular orbits for a test particle orbiting a
Schwarzschild black hole are represented by the dotted line.







58









0.00 1I 1



-0.01



-0.02
CM1

-0.03



-0.04



-0.05



3.5 4 4.5 5 5.5 6
Total








FIGURE 5: Binding Energy of Numerical Test Particle. The binding
energy per mass of the smaller black hole (Eo M1 M2)/M2 for the sta-
ble circular orbits is plotted against j,it. = Jtoal1/(M1M2) for the case of
M2 = 0.01M1 where M1 and M2 are the irreducible masses. The dotted line
represents the stable circular orbits for the analytic test particle.







59

to the time symmetric solution for any separation when the speeds of the two

black holes approach zero.45 In addition, when the separation is very large

the trial geometry reduce to two boosted black holes having arbitrarily large

speeds. The trial geometry will not be exact in any other limits. Still, the

numerical solutions can be compared to Newtonian calculations, as well as,

post-Newtonian calculations.

A Newtonian binary system of two point masses, ml, and m2, in circular

orbits, move about their common center of mass under the influence of the

Newtonian gravitational force

F= l2 1 (5.16)
12
This is in units with the universal gravitational constant G = 1, and the speed

of light c = 1. Next introduce two new mass variables, the total mass

Mtotl = mi + m2 (5.17)

and the reduced mass
mlm2 (5.18)
(mi + m2)
Then the Newtonian expression for the separation between the two point

masses in circular orbits is given by
j2
SN total = Jtotal (5.19)
PMSN -total
where Jtotai is the total angular momentum of the binary system. The scaled

angular momentum jtota is related to the total angular momentum by

Jtotai = jtotal(fiMtotal) = jtotal(mlm2) (5.20)

The total energy of the binary system of point masses, including the rest mass

energy of the two masses, is given by
3M2
EN = MAotal P3 Mta = Mtotal (5.21)
2 Jta, "2j 5toat
total total















0.05



0.00


-0.05



-0.10


-0.15



-0.20


20 40 60 80


100


Sewton/(M +M2)







FIGURE 6: Single Parameter Newtonian Circular Orbits. The binding
energy per reduced mass as a function of separation per total mass. A single
curve describes all possible Newtonian orbits in terms of one parameter, the
scaled total angular momentum jota.







61

An interesting result occurs when the following rescaling is performed:

(EN Meoto,) 1
(EN 1t (5.22)

SN j.2 (5.23)
Total
When equation (5.22) is plotted as a function of equation (5.23), a single curve

results which represents all possible stable circular orbits of a binary system in

Newtonian mechanics, as shown in Figure 6.

The post-Newtonian approximation developed out of a study of the equa-

tions of motion in a gravitational system described by general relativity.46 The

equations of motion in general relativity are given by the geodesic equation,

d2xp dx" dzx
d2 d d(5.24)
dr2 dr ddr

where r is the proper time. The coordinate accelerations can be derived from

equation (5.24) and are found to be

d2xi dx dxz dxk
dt2 = 2-r' dt rjk dt dT
+ ( 0 d + dxdx dx (5.25)
+ ^00 0i dt k dt di dt

where the indices i, j, and k range over the spatial coordinates. In the New-

tonian limit all velocities in equation (5.25) are neglected, leading to

d2xi 1 i 0 (5.26)
dt2 x 22 (5.26)

where the M and the f are the relevant mass and separation in a Newtonian

binary system. For a Newtonian system the relevant velocity v is related to

the potential energy by
S(5.27)
r







62

Thus in the Newtonian limit the accelerations given in equation (5.26) are

calculated to order (F). The post-Newtonian approximation carries the cal-

culation of the accelerations to one higher order in v2, that is to order (v).

The order to which the connection coefficients in equation (5.25) must be de-

termined can simply be read off. The metric is then expressed as a correction

to a flat background metric using an expansion series of terms in powers of 2.

The equations of motion derived in the post-Newtonian approximation

can be integrated47 for a binary system of point masses and an effective po-

tential energy constructed in terms of the separation r (in the Einstein-Infeld-

Hoffmann gauge) and total angular momentum Joal

j2,o,( ,oto,- 3j)
EPN(r, Jo lMot.tt I aM i4ta(Mtoa 3p)
2pJr2 J --- tat3, 4
2r2 r 8Mtoi3r4 (5.28)
Jt2,t(3Mota, + p) pM,2
2/r3 2r2

where the masses p and M,,,, are given by equations (5.18) and (5.17).48 For

given masses and angular momentum, the stable circular orbits are found by

solving for the zeros of the first derivative with respect to r of equation (5.28)

that also correspond to a positive second derivative with respect to r of equa-

tion (5.28). It is interesting to notice that the post-Newtonian approximation

also has the feature of an innermost stable circular orbit which is determined by

simultaneously solving for the zeros of the first and second derivatives with re-

spect to r in equation (5.28). For the case of equal masses the innermost stable

circular orbit is found at a total angular momentum of Jlotait 4.2876PLMtota

and a separation of s 19.81M1. The binding energy for this orbit is found

to be 0.85% of the total rest mass energy available when the two masses are

infinitely far away and at rest. These features of the post-Newtonian effective
















0.02 _


=-5.5
0.01

5.25

0.00



-0.01

4.2 *
-0.02



-0.03



-0.04
0 20 40 60 80
SPost-Newton/M1


100


FIGURE 7: Post-Newtonian Effective Potential for Equal Masses. Stable
circular orbits are located at the minima. Unstable circular orbits are located
at the maxima. No orbits exist for Jtor. less than z 4.2876pMo,,,r. The dotted
line locates the numerically derived stable circular orbits.







64

potential energy are illustrated in Figure 7 for several different values of the

total angular momentum Jot,, in the case of two equal masses.

Since the post-Newtonian approximation is only valid when the speed of

the particles are very small compared to the speed of light, the closer, faster,

unstable circular orbits will be less accurately determined by this approxima-

tion. Consequently, they will not be compared to numerical findings.

The numerical results of Table 5 for two black holes of equal masses reveal

several interesting features. There is an innermost stable circular orbit found

at a total angular momentum of Jo,,, 0.84MIM2. This is a considerably

smaller value of angular momentum then that of the innermost stable orbit in

the post-Newtonian approximation. However, the Newtonian approximation

has stable orbits for any non-zero angular momentum. This innermost stable

circular orbit for the numerical results is located at a separation of s 1.13M1.

The binding energy of this orbit is 17.18% of the total rest mass energy available

when the two masses are infinitely far away and at rest. This is very close to

the binding energy of 17.72% of the total rest mass energy that a Newtonian

calculation would give for this angular momentum.

The numerical results also reveal unstable circular orbits inside the sep-

aration for the innermost stable circular orbit. The position of these unstable

circular orbits have not been determined accurately since the binary system is

undoubtedly evolving rapidly through the emission of gravitational radiation.

The Hamiltonian of equation (3.16) provides an effective potential energy

for the circular orbits whereby stable circular orbits are located at the minima

and unstable circular orbits are located at the maxima. Figure 8 illustrates

these effective potential energy curves for the case of equal mass black holes

















TABLE 5: Numerical Stable Circular Orbits for Equal Masses

jo... ~ s/Mi mi/Mi M,,.,/Mi E0/M1 QM1 u0
0.84 1.13 0.750692 2.112751 1.65639436 1.24503 0.7036
0.85 1.19 0.758342 2.083652 1.66172030 1.15255 0.6857
0.90 1.37 0.778896 2.036690 1.68478020 0.93910 0.6442
1.00 1.65 0.803998 2.014120 1.72181645 0.73036 0.6022
1.30 2.40 0.849706 2.014870 1.79884909 0.44733 0.5372
1.50 2.92 0.870195 2.021581 1.83490281 0.34885 0.5088
1.60 3.19 0.878814 2.024294 1.85005175 0.31138 0.4961
2.00 4.39 0.906376 2.029113 1.89737083 0.20483 0.4493
2.20 5.10 0.917466 2.027682 1.91514164 0.16690 0.4255
2.40 5.93 0.927455 2.023912 1.92995159 0.13496 0.4000
2.50 6.40 0.932149 2.021233 1.93638889 0.12067 0.3863
2.60 6.93 0.936684 2.018113 1.94223806 0.10735 0.3719
2.70 7.52 0.941086 2.014625 1.94753545 0.09489 0.3566
3.00 9.78 0.953514 2.003297 1.96042452 0.06264 0.3063
3.25 12.38 0.962574 1.995479 1.96819869 0.04252 0.2632
3.50 15.58 0.969822 1.990978 1.97390342 0.02896 0.2256
3.70 18.46 0.974295 1.989377 1.97738415 0.02182 0.2015
3.85 20.76 0.977015 1.988904 1.97952878 0.01796 0.1864
4.00 23.17 0.979301 1.988813 1.98136239 0.01499 0.1736
4.29t 28.03 0.982772 1.989208 1.98421423 0.01097 0.1538
4.50 31.83 0.984767 1.989725 1.98589822 0.00893 0.1421
5.00 41.46 0.988223 1.991108 1.98890032 0.00584 0.1211
6.00 64.00 0.992307 1.993452 1.99255313 0.00294 0.0941
7.00 89.72 0.994488 1.995088 1.99463499 0.00174 0.0782
8.00 119.82 0.995861 1.996194 1.99594383 0.00112 0.0669
9.00 153.86 0.996771 1.996972 1.99682134 0.00076 0.0586
10.00 191.87 0.997408 1.997537 1.99744007 0.00054 0.0522
11.00 234.32 0.997875 1.997958 1.99789326 0.00040 0.0470
12.00 279.93 0.998220 1.998280 1.99823532 0.00031 0.0429
13.00 330.06 0.998490 1.998532 1.99850003 0.00024 0.0394
14.00 383.90 0.998701 1.998733 1.99870910 0.00019 0.0365
15.00 441.98 0.998871 1.998895 1.99887737 0.00015 0.0340
17.50 604.56 0.999174 1.999187 1.99917726 0.00010 0.0290
20.00 791.59 0.999369 1.999377 1.99937114 0.00006 0.0253
Note: Units with G =c = 1, M2 = M1, m2 = mi, and v0 = uO.
t Actual value of jo,,. = J.,,.,/MiM2 is 4.287597023.






66










2.4 -~.o 00
1.60

2.2


1.30
a 2.0



1.8 -
1.00


1.6 -0-84



1.4
0 2 4 6 8 10
S/M,








FIGURE 8: Numerical Effective Potential for Equal Mass Black Holes.
Stable circular orbits are located at the minima. Unstable circular orbits are lo-
cated at the maxima. No orbits exist for Jot,,ailMM2 < 0.84, where Jotal is the
total angular momentum and M1 = M2 are the irreducible masses of the black
holes. The position of the Newtonian stable circular orbits are represented by
the dotted line.


















0.000



-0.005



-0.010

+ -
-4 _
-0.015
.:


S -0.020



-0.025


0 2 4 6 8 10
Total









FIGURE 9: Numerical Binding Energy of Equal Mass Black Holes. The
binding energy per total mass (Eo M1 M2)/(M1 + M2) of two black holes
in stable circular orbits is plotted against the angular momentum jitoar
Jtotar/(MIM2) for the equal mass case M2 = M1. The dotted line represents
the Newtonian approximation. The dashed line represents the post-Newtonian
approximation.







68

for several values of total angular momentum Jto,,i. Since the extremized mass

is more accurately determined as a function of the angular momentum, it is

useful to compare the numerical binding energy derived from the variational

principle of the stable circular orbits as a function of total angular momentum

to the values obtained in the Newtonian and post-Newtonian approximations

as illustrated in Figure 9 and cataloged in Table 6.

A summary of the results for stable circular orbits found by the three

different methods, Newtonian, post-Newtonian, and numerical for equal mass

black holes are presented for comparison in Table 6. At the largest tabulated

value of the scaled angular momentum, jl,,,i = 20, the binding energies of

the three methods are within ~ 0.6% of each other. In addition, the post-

Newtonian and numerical results are within 0.03% of each other. Excluding

the Newtonian value, the separations between the two black holes are within

~ 0.04% for this large a value of jtot,i. When the compared to the New-

tonian separation, they differ by substantially larger amounts. This results

partially from the difference in the gauges. It can be shown that the post-

Newtonian's Einstein-Hoffmann-Infeld gauge relates to the Newtonian gauge

for equal masses by
2r2
r,, = (5.29)
S= (2rN + 17M)

where M is the mass of a single black hole.47 This accounts for most of the

discrepancies at large values of jtot,,. In addition all expressions for the radial

separation approach each other at very large values of jtot,,. To provide an

example of what is meant by very large values of jtot,l, the planet Mercury

orbits the Sun with roughly jot,,i 3100. This is at a separation of s w 9.6 x 106

solar mass units. Yet relativistic effect are observable for Mercury.




















TABLE 6: Comparison of Stable Circular Orbits for Equal Masses

Newtonian Post-Newtonian Numerical
jo,.., s~/Mi EB.,..,/MI s/MI EB,,ad.,g/M s/M1 Em....,/M
0.80 1.28 0.3906250
0.84 1.41 0.3543084 1.13 0.3436056
0.85 1.45 0.3460208 1.19 0.3382797
0.90 1.62 0.3086420 1.37 0.3152198
1.00 2.00 0.2500000 1.65 0.2781836
1.30 3.38 0.1479290 2.40 0.2011509
1.50 4.50 0.1111111 2.92 0.1650972
2.00 8.00 0.0625000 _4.39 0.1026292
2.50 12.50 0.0400000 6.40 0.0636111
3.00 18.00 0.0277778 9.78 0.0395755
3.50 24.50 0.0204082 15.58 0.0260966
4.00 32.00 0.0156250 23.17 0.0186376
4.29t 36.77 0.0135992 19.81 0.0170059 28.03 0.0157858
4.50 40.50 0.0123457 27.72 0.0147733 31.83 0.0141018
5.00 50.00 0.0100000 39.14 0.0113541 41.46 0.0110997
6.00 72.00 0.0069444 62.42 0.0075077 64.00 0.0074469
7.00 98.00 0.0051020 88.96 0.0053850 89.72 0.0053650
8.00 128.00 0.0039063 119.25 0.0040653 119.82 0.0040562
9.00 162.00 0.0030864 153.43 0.0031830 153.86 0.0031787
10.00 200.00 0.0025000 191.55 0.0025622 191.87 0.0025599
11.00 242.00 0.0020661 233.63 0.0021080 234.32 0.0021067
12.00 288.00 0.0017361 279.70 0.0017654 279.93 0.0017647
13.00 338.00 0.0014793 329.74 0.0015004 330.06 0.0015000
14.00 392.00 0.0012755 383.78 0.0012911 383.90 0.0012909
15.00 450.00 0.0011111 441.81 0.0011229 441.98 0.0011226
17.50 612.50 0.0008163 604.36 0.0008226 604.56 0.0008227
20.00 800.00 0.0006250 791.90 0.0006287 791.59 0.0006289
Note: Units with G = c = 1, M2 = M. Empty cells are j,,,,, without orbits.
t Actual value of j,.,., = J,.,.,/MIM2 is 4.287597023.







70

Comparisons of the smaller values of jtotl in Table 6 are restricted by the

innermost stable circular orbit of the post-Newtonian approximation at jtot,,l

4.2876. At this value of the scaled angular momentum, the numerical binding

energy differs from the post-Newtonian value by ~ 7.5%. While the same

numerical binding energy differs from the Newtonian value by ~ 14%. However,

the separations show deviations by as much as ~ 30% among individual entries

at this particular value of jtota.

By the time the scaled angular momentum has fall just below the value

of jto,,,i 4.0, at a separation of s a 23M1, the binary system satisfies the

angular momentum constraints of a Kerr black hole, Jtot, < Mta,,. The total

angular momentum is now small enough for the two black holes to coalesce

into one black hole.


5.3 Arbitrary Mass Ratios


When the irreducible masses are comparable but not equal, the only

analytic checks that can be made on the numerical results involve the same

approximations as in the equal masses case. Tables 7 through 22 list results

for irreducible mass ratios ranging from 0.005 to 0.75. All mass ratios were

found to have innermost stable orbits. These innermost stable circular orbits

will not satisfy the restrictions placed on the secular time scale except in the

test particle limit. Thus the exact location for these innermost stable circular

orbit have not been determined for these tabulate mass ratios. However, the

tables do present several other interesting features. The smallest value of the

scaled total angular moment presented rise abruptly from a mass ratio of 0.02

to the mass ratio of 0.01 used for the test particle comparisons. This reflects











TABLE 7: Numerical Masses for Mass Ratio 0.005

Total ml m2 Mtotal Eo
3.85 0.99969653 0.00469653 1.00501470 1.004780601
4.00 0.99975828 0.00475828 1.00493453 1.004805138
4.50 0.99983888 0.00483888 1.00490563 1.004856914
5.00 0.99987994 0.00487994 1.00491333 1.004888482
6.00 0.99992221 0.00492221 1.00493539 1.004925707
7.00 0.99994448 0.0049448 1.00495136 1.004946719
Note: Units with G = c = 1, and M2 = 0.005M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.


TABLE 8: Numerical Orbits for Mass Ratio 0.005

jtotal s/M1 (Eo- M)/M2 QMI u0 v_
3.85 7.736 0.9561202 0.0608471 0.00249 0.46820
4.00 9.840 0.9610275 0.0401259 0.00203 0.39281
4.50 15.014 0.9713828 0.0198039 0.00150 0.29584
5.00 20.320 0.9776964 0.0120916 0.01223 0.24447
6.00 31.633 0.9851414 0.0060107 0.00095 0.18919
7.00 44.528 0.9893438 0.0035434 0.00079 0.15700
Note: Units with G = c = 1, and M2 = 0.005M1.
M1 and M2 are irreducible masses.
Eo is the extremized mass.











TABLE 9: Numerical Masses for Mass Ratio 0.02

J0otal mi m Mtotl Eo
2.75 0.99521115 0.01521115 1.03379608 1.015516061
3.00 0.99591166 0.01591166 1.03120167 1.016760196
3.25 0.99646441 0.01646441 1.02935342 1.017763075
3.50 0.99710462 0.01710462 1.02667011 1.018555245
3.70 0.99813953 0.01813953 1.02192645 1.019007721
3.85 0.99887718 0.01887718 1.01990598 1.019143988
4.00 0.99906465 0.01906465 1.01971587 1.019235831
5.00 0.99952583 0.01952583 1.01965901 1.019560807
6.00 0.99969351 0.01969351 1.01974529 1.019707186
7.00 ~0.99978101 0.01978101 1.01980834 1.019789455
Note: Units with G = c = 1, and M2 = 0.02MI1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.


TABLE 10: Numerical Orbits for Mass Ratio 0.02

j.otal s/M1 Ebinding /M1 QM1 u v
2.75 1.581 0.0044839 0.6018193 0.03494 0.91628
3.00 1.938 0.0032398 0.4749567 0.03107 0.88941
3.25 2.320 0.0022369 0.3837010 0.02810 0.86214
3.50 2.945 0.0014448 0.2836760 0.02383 0.81166
3.70 4.866 0.0009923 0.1351638 0.01523 0.64246
3.85 8.397 0.0008560 0.0531325 0.00918 0.43696
4.00 10.182 0.007642 0.0381970 0.00786 0.38104
5.-00 20.580 0.0004392 0.0119706 0.00486 0.24149
6.00 32.118 0.0002928 0.0059197 0.00374 0.18639
7.00 45.153 0.0002105 0.0034982 0.00310 U.15485
Note: Units with G = c = 1, and M2 = 0.02Mi.
M1 and M2 are irreducible masses.
Eo is the extremized mass.











TABLE 11: Numerical Masses for Mass Ratio 0.03

Total m m2 Mtotal Eo
2.00 0.98931792 0.01931792 1.06138741 1.016584626
2.50 0.99265326 0.02265326 1.04785160 1.022291645
3.X00 0.99451994 0.02451994 1.04264424 1.025799245
3.25 0.99534476 0.02534476 1.04000908 1.027071996
3.50 0.99629503 0.02629503 1.03640449 1.028055840
3.70 0.99784213 0.02784213 1.03074025 1.028552262
3.85 0.99837031 0.02837031 1.02978813 1.028734445
4.00 0.99862660 0.02862660 1.02955457 1.028867896
5.00 0.99929551 0.02929551 1.02949354 1.029347856
6.00 0.99954520 0.02954520 1.02962145 1.029565115
7.00 0.99967752 0.02967752 1.02971454 1.029687245
Note: Units with G = c = 1, and M2 = 0.03M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.


TABLE 12: Numerical Orbits for Mass Ratio 0.03

jota s/M Ebidin /M ~ M1I u0 vo
2.00 0.895 0.0134154 1.1498122 0.06764 0.96094
2.50 1.530 0.0077084 0.6253444 0.04931 0.90771
3.00 -2.225 0.0042008 0.4026169 0.04064 0.85516
3.25 2.710 0.0029280 0.3150749 0.03613 0.81757
3.50 3.535 0.0019442 0.2201997 0.02980 0.74871
3.70 6.437 0.0014477 .0844748 0.01728 0.52652
3.85 8.690 0.0012656 0.0503511 0.01331 0.42424
4.00 10.408 0.0011321 0.0370075 0.01155 0.37361
5.00 20.777 0.0006521 0.0118638 0.00722 0.23927
6.00 32.467 0.0004349 0.0058513 0.00555 0.18443
7.00 46.000 0.0003128 0.0034044 0.00457 0.15204
Note: Units with G = c = 1, and M2 = 0.03M1.
M1 and M2 are irreducible masses.
Eo is the extremized mass.











TABLE 13: Numerical Masses for Mass Ratio 0.05

Total mI m2 Mtotal E
2.00 0.98618288 0.03618288 1.07469854 1.032107290
2.50 0.98959226 0.03959226 1.06913405 1.039332489
3.00 0.99216046 0.04216046 1.06402802 1.044097928
3.25 0.99343593 0.04343593 1.06047468 1.045815044
350 0.99510707 0.04510707 1.05491658 1.047080584
3.70 0.99681013 0.04681013 1.05033803 1.047672297
3.85 0.99741997 0.04741997 1.04950343 1.047946351
4.00 0.99779170 0.04779170 1.04921957 1.048157502
5.00 0.99885167 0.04885167 1.04916977 1.048934558
6.00 0.99925833 0.04925833 1.04938034 1.049289185
7.00 0.99947298 0.04947298 1.04953321 1.049488741_~
Note: Units with G = c = 1, and M2 = 0.05M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.


TABLE 14: Numerical Orbits for Mass Ratio 0.05

total S /M1 Ebindin/M1 QM1 u v
2.00 1.291 0.0178927 0.7622651 0.07829 0.90599
2.50 1.882 0.0106675 0.4919043 0.06696 0.85894
3.00 2.668 0.0059021 0.3210902 0.05658 0.80005
3.25 3.287 0.0041850 0.2436785 0.04970 0.75124
3.50 4.587 0.0029194 0.1491445 0.03831 0.64579
3.70 7.314 0.0023277 0.0684667 0.02537 0.47539
3.85 9.166 0.0020536 0.0464748 0.02105 0.40494
4.00 10.797 0.0018425 0.0351909 0.01856 0.36140
5.T00 21.246 0.0010654 0.0115760 0.01178 0.23417
6.00 33.183 0.0007108 0.0057129 0.00905 0.18052
7.00 46.912 0.0005113 0.0033378 0.00746 0.14912
Note: Units with G = c = 1, and M2 = 0.05M1.
M1 and M2 are irreducible masses.
Eo is the extremized mass.











TABLE 15: Numerical Masses for Mass Ratio 0.10

total mi m2 M otal Eo
1.50 0.96942701 0.06942701 1.13189937 1.055075030
t2.0 0.97766857 0.07766857 1.12568338 1.072392655
2.50 0.98301146 0.08301146 1.12164905 1.083307383
3.00 0.98746657 0.08746657 1.11471227 1.090556404
3.25 0.98984134 0.08984134 1.10917634 1.093062048
3.50 0.99270186 0.09270186 1.10230585 1.094788009
3.70 0.99447404 0.09447404 1.09944393 1.095656279
3.85 0.99530621 0.09530621 1.09867516 1.096130421
4.00 0.99590759 0.09590759 1.09834592 1.096512528
5.00 0.99781730 0.09781730 1.09840889 1.097968995
6.00 0.99858302 0.09858302 1.0988173 1.098643511
7.00 0.99899478 0.09899478 1.09910846 1.099023876
Note: Units with G = c = 1, and M2 = 0.10M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.






TABLE 16: Numerical Orbits for Mass Ratio 0.10

jtoial S/M1 E Min_,.,/Ml ~ MMI u0 v_
1.50 1.101 0.0449250 0.9359655 0.13920 0.89103
2.00 1.700 0.0276073 0.5611091 0.11946 0.83452
2.50 2.402 0.0166926 0.3693877 0.10531 0.78184
3.00 3.446 0.0094436 0.2302418 0.08783 0.70549
3.25 4.377 0.0069380 0.1626380 0.07480 0.63706
3.50 6.305 0.0052120 0.0903454 0.05583 0.51377
3.70 8.501 0.0043437 0.0543652 0.04372 0.41843
3.85 10.105 0.0038696 0.0405219 0.03825 0.37121
4.00 11.670 0.0034875 0.0317865 0.03440 0.33654
5.00 22.359 0.0020310 0.0109701 0.02241 0.22287
6.00 34.737 0.0013565 0.0054660 0.01729 0.17258
7.00 49.191 0.0009761 ~ 0.0031821 0.01424 0.14229
Note: Units with G = c = 1, and M2 = 0.10MI.
M1 and M2 are irreducible masses.
Eo is the extremized mass.











TABLE 17: Numerical Masses for Mass Ratio 0.25

total ml m2 Mtotal Eo_
1.00 0.91749572 0.16749572 1.27236282 1.119868108
1.50 0.94351167 0.19351167 1.27376855 1.172302223
2.00 0.95818493 0.20818493 1.27628112 1.201679315
2.50 0.96874165 0.21874165 1.27222378 1.220490492
3.00 0.97798722 0.22798722 1.26099682 1.232518093
3.25 0.98261884 0.23261884 1.25375920 1.236433004
3.50 0.98664235 0.23664235 1.24865925 1.239162147
3.70 0.98899046 0.23899046 1.24678896 1.240728389
3.85 0.99032198 0.24032198 1.24616044 1.241659789
4.00 0.99138940 0.24138940 1.24588521 1.242439740
5.00 0.99524254 0.24524254 1.24647167 1.245545408
6.00 0.99689374 0.24689374 1.24739243 1.247018742
7.00 0.99779160 0.24779160 1.24803711 1.247853748
Note: Units with G = c = 1, and M2 = 0.25M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.






TABLE 18: Numerical Orbits for Mass Ratio 0.25

j total s/MI1 Ebi ndig/M1 M1 U0 v
1.00- 931 0.1301319 1.2124576 0.28080 0.84839
1.50 1.616 0.0776978 0.6229878 0.23882 0.76799
2.(00 2.385 0.0483207 0.3870633 0.21371 0.70953
2.50 3.390 0.0295095 0.2453116 0.18698 0.64452
3.00 5.065 0.0174819 0.1371181 0.14972 0.54472
3.25 6.575 0.0135670 0.0903202 0.12477 0.46912
3.50 8.740 0.0108379 0.0561351 0.10096 0.38964
3.70 -10.734 0.0092716- 0.0396855 0.08680 0.33919
3.85 12.296 0.0083402 0.0315972 0.07880 0.30971
4.00 13.896 0.0075603 0.0257684 0.07240 0.28569
5.00 -25.652 0.0044546 0.0095039 0.04890 0.19489
6.00 39.618 0.0029813 0.0047840 0.03795 0.15158
7.00 55.978 0.0021463 0.0027955 0.03132 0.12517


Note: Units with G = c = 1, and
M1 and M2 are irreducible
Eo is the extremized mass.


M2 = 0.25M1.
masses.










TABLE 19: Numerical Masses for Mass Ratio 0.50

jtotai m1 m2 Mtota Eo
0.80 0.83684048 0.33684048 1.52902426 1.252472405
0.83 0.84358456 0.34358456 1.51964016 1.262647030
0.84 0.84550107 0.34550107 1.51791192 1.265832485
0.86 0.84960450 0.34960450 1.51312388 1.271950636
0.88 0.85280116 0.35280116 1.51194069 1.277757531
0.90 0.85610483 0.35610483 1.51001902 1.283287716
1.00 0.86995908 0.36995908 1.50704021 1.307633153
1.50 0.91172222 0.41172222 1.51976045 1.385011755
2.00 0.93566766 0.43566766 1.52545804 1.428501109
2.50 0.95292775 0.45292775 1.51983104 1.455911098
3.T00 0.96759739 0.46759739 1.50575813 1.472954303
3.50 0.97935488 0.47935488 1.49483587 1.482426376
4.00 0.98604687 0.48604687 1.49270298 1.487524095
5.00 0.99212891 0.49212891 1.49408400 1.492592825
6.00 0.99483686 0.49483686 1.49564738 1.495035123
7.00 0.99632427 0.49632427 1.49672629 1.496423778
Note: Units with G = c = 1, and M2 = 0.50M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.


TABLE 20: Numerical Orbits for Mass Ratio 0.50

jotal s/M Ebinding/Ml MM1 u0 v_
0.80 0.864 0.2475276 1.4966484 0.48416 0.80868
0.83 0.927 0.2373530 1.3625346 0.46896 0.79345
0.84 0.945 0.2341675 1.3270032 0.46514 0.78938
0.86 0.987 0.2280494 1.2515638 0.45611 0.77978
0.88 1.022 0.2222425 1.1978199 0.45067 0.77348
0.90 1.059 0.2167123 1.1430301 0.44449 0.76635
1.00 1.237 0.1923668 0.9363742 0.42123 0.73753
1.50 2.126 0.1149882 0.4757421 0.36085 0.65063
2.00 3.168 0.0714989 0.2860676 0.31964 0.58669
2.50 4.585 0.0440889 0.1725007 0.27509 0.51575
3.U00 6.982 0.0270457 0.0908495 0.21677 0.41751
3.50 11.370 0.0175736 0.0405442 0.15526 0.30572
4.00 17.174 0.0124759 0.0203963 0.11729 0.23300
5.00 31.016 0.0074072 0.0078222 0.08098 0.16163
6.00 47.673 0.0049649 0.0039705 0.06313 0.12616
7.00 67.266 0.0035762 0.0023254 0.05215 0.10426


Note: Units with G = c = 1, and
M1 and M2 are irreducible
Eo is the extremized mass.


M2 = 0.50M1.
masses.










TABLE 21: Numerical Masses for Mass Ratio 0.75

total mi m2 Mtotal E
0.82 0.78322430 0.53322430 1.84229277 1.446064194
0.83 0.78940386 0.53940386 1.81975573 1.450800478
0.84 0.79394315 0.54394315 1.80662714 1.455263913
0.86 0.80098080 0.55098080 1.79119016 1.463613948
0.88 0.80715847 0.55715847 1.78008201 1.471379824
0.90 0.81221052 0.56221052 1.77396050 1.478675615
1.00 0.83246008 0.58246008 1.76029885 1.510134604
1.50 0.88836870 0.63836870 1.76961585 1.607389908
2.00 0.91931986 0.66931986 1.77628956 1.661346226
2.50 0.94142692 0.69142692 1.76944950 1.695104326
3.00 0.95984375 0.70984375 1.75358352 1.715932396
3.50 0.97401575 0.7241575 1.74245215 1.727591551
4.00 0.98222554 0.73222554 1.74044445 1.734012927
5.00 0.98990091 0.73990091 1.74238088 1.740484339
6.00 0.99336337 0.74336337 1.74440238 1.743618175
7.00 0.99527678 0.74527678 1.74578981 1.745402830
Note: Units with G = c = 1, and M2 = 0.75M1.
All masses in units of the irreducible mass M1.
Eo is the extremized mass.


TABLE 22: Numerical Orbits for Mass Ratio 0.75

jtoal s/M1 EbindingM MM 0 v _
0.82 0963 0.3039358 1.4526745 0.63183 0.76752
0.83 1.011 0.2991995 1.3524250 0.61504 0.75220
0.84 1.048 0.2947361 1.2837045 0.60368 0.74153
0.86 1.109 0.2863861 1.1848341 0.58765 0.72604
0.88 1.166 0.2786202 1.1035990 0.57415 0.71267
0.90 1.216 0.2713244 1.0420676 0.56431 0.70265
1.00 1.447 0.2398654 0.8246488 0.52854 0.66476
1.50 2.540 0.1426101 0.4000700 0.44618 0.57004
2.00 -3.813 0.0886538 0.2360295 0.39338 0.50668
2.50 -5.557 0.0548957 0.1396559 0.33741 0.43859
3.00 8.484 0.0340676 0.0726496 0.26633 0.35000
3.50 13.570 0.0224084 0.0333776 0.19480 0.25813
4.00 20.232 0.0159871 0.0171888 0.14927 0.19848
5.00 36.262 0.0095157 0.0066826 0.10390 0.13842
6.00 55.633 0.0063818 0.0034033 0.08116 0.10817
7.00 78T.522 0.0045972 0.0019915 0.06703 0.08935
Note: Units with G = c = 1, and M2 = 0.75M1.
M1 and M2 are irreducible masses.
Eo is the extremized mass.







79

the trend towards tightly bound innermost stable orbits that mass ratios above

0.02 have in common with the case of equal masses. Below a mass ratio of

~ 0.01 the numerical results compare favorably to the analytic test particle

values as seen in Tables 7 and 8. In addition, it is interesting to note in Tables

19 and 20 that the mass ratio having the lowest scaled total angular momentum

is 0.50 with a scaled total angular momentum of jto,,l 0.80. The energy of the

system in these tables is presented as the binding energy in all cases except the

mass ratio of 0.005 which is represented as the energy per mass to be consistent

with the test particle description. Notice that this mass ratio has an energy

per mass of 0.9561202 for the same scaled angular momentum of jtota, = 3.85.

This is only 0.04% different from the value for the mass ratio of 0.01 given in

Table 4.


5.4 Gravitational Radiation


Introducing the global time symmetry forces the two black holes into

stationary orbits. This approximation to the true geometry of the binary

black hole system will accurately model the the actual geometry as long as the

time scale for secular effects due to gravitational radiation reaction are large

when compared to the dynamical time scale. To determine the magnitude

of the secular effects the energy and angular momentum carried away by the

gravitational radiation must be approximated.

Systems interacting through purely gravitational exchanges will not ex-

hibit dipole radiation. This is because the dipole moment couples to the linear

and angular moment of the sources and the total linear and angular moment

are conserved in a gravitational system. However, the next higher moment,






80
the quadrupole moment, will in general radiate away energy and momentum.

The reduced quadrupole moment for a distribution of mass is given by

Zjk rt m (xejxek 36jkrf)
/ (5.30)
= p xiXk bjkr2) d3x

The total power radiated, or the dimensionless luminosity, in the form of grav-

itational quadrupole radiation is given by

dEGW 1 d3Ijk d3 jk
LGW (5-31)
L dt -5 dt3 dt3(531)

The total angular momentum radiated away by the gravitational quadrupole

radiation is given by

dJw 2 Eij d2Z=j d3 ek (5.32)
dt 5 dt2 dt3

where the angle brackets ( ) represent averaging over several characteristic

cycles of the source since the energy of gravitational radiation can not be

localized to within a wavelength.

For two black holes in circular orbits about each other, the mass-energy

density p appearing in equation (5.30) is given by


p = m163(- xol) + m23( o2,) (5.33)

where ml and m2 are the masses of the two black holes located at 4o1 and

o02. Evaluating equation (5.30) for this mass energy density in Cartesian
coordinates gives the following components to the quadrupole moment for the

binary black holes:

Ixx = mi co2(Q) + m2 (cos2 (t) -C) (5.34)
\ "/ 3 /






81



yy = mi1 (si2(t) + m2z2 2(t) (5.35)



Izz = -g (mix 1 + m2x2 (5.36)




Zzy = IZy = mlx2l (cos(Qt)sin(Qt)) + m2xo2 (cos(Qt)sin(ft)) (5.37)



Zz = zx = 0 (5.38)



Iyz = Izy = (5.39)

Equations (5.31) and (5.32) for the rate of energy and angular momentum loss
require the second and third time derivatives of the quadrupole moment Ijk-

The second derivative with respect to time generates the following components:

d2 = -222 (mx2 + m2x) cos(2Qt) (5.40)
dt2 0 2


d2Y 202 (mi2 + m2x2) cos(22t) (5.41)
ddt2


d- = (5.42)



d2 = d2 = -202 (mlxo + m2x2) sin(2Qt) (5.43)
dt2 dt2 -


d2Ezz d2 zz
-d-2 -0. (5.44)
dt2 dt2






82
The third derivative with respect to time generates the following components:

d3Ix = 4 (3 (mix1 + m2x,) sin(2Qt) (5.45)



d3_ = -4_ 3 (m 2 + m2x2) sin(2 2t) (5.46)




dt3


d3y d3~ x 3 2 +z2
dt3 = = -4Q3 (mix,1 + m2x2) cos(2t) (5.48)


d22xz d2 zx
=0. (5.49)
dt2 dt2

Substitution into equation (5.31), the dimensionless luminosity is found
to be

LGW = d3 + d3Zyy d3 y+ d3 yx]
GW dt3 dt3 dt3 dt3
= 326 (mi + m2x2o)2 sin2(2Qt) + cos2(2t)) (5.50)

= 32 2 4/ 6
5
where / is the reduced mass of the system and s is the separation between the

two black holes. Next substitute into equation (5.32) to find the rate at which

the system losses angular momentum

dJw 0
w = 0 (5.51)
dt


dJw = 0 (5.52)
dt










dJzw 2 d2Iz d32xI d2Izy d3Iyy d2lyx d3Izz d2Zyy d3ily
dt 5 dt2 dt3 + dt2 dt3 dt2 dt3 dt2 dt3


32= 24Q5 .

(5.53)

The x and y components are naturally zero since the motion of the two black

holes is restricted to the x-y plane. The results for equations (5.50) and (5.53)

reveal the following relationship between the luminosity and the magnitude of

the rate of angular momentum loss for a gravitational system


Law = (dJGw (5.54)


The application of equation (5.50) to the extremized geometries having

scaled angular moment over the range of stable circular orbits allows for a

measure of the accuracy that a particular solution has to the actual physical

binary system of black holes. These dimensionless luminosities are listed in

Table 23 and plotted in Figure 10 for equal mass black holes. The luminosity

reaches a value of LGw ; 1 at a scaled angular momentum of jotai 1, and at

a separation of s 1.7M1. This is to large a luminosity to consider the results

accurate. A more reasonable limit on the luminosity might be LGW 0.01.

This occurs at jtot,, 2.40, at a separation of s ; 5.9M1.

Also useful in determining a comparison of the secular time scale to the

dynamic time scale is the approximation to the energy radiated away per orbit

given by (2 )LGW. This quantity becomes comparable to the binding energy

at a scaled angular momentum of jtotal 3.25, at a separation of s w 12.4M1.

This is where one orbit loses roughly the same amount of energy in the form

















TABLE 23: Gravitational Quadrupole Radiation for Equal Masses

jto._ s/Mi p/M1 0M1 Law (27r/))Lowt EB,..,,g
0.84 1.13 0.37535 1.25e+00 5.48e+00 2.77e+01 0.34361
0.85 1.19 0.37917 1.15e+00 4.32e+00 2.36e+01 0.33828
0.90 1.37 0.38945 9.39e-01 2.36e+00 1.58e+01 0.31522
1.00 1.65 0.40200 7.30e-01 1.16e+00 9.99e+00 0.27818
1.30 2.40 0.42485 4.47e-01 3.08e-01 4.34e+00 0.20115
1.50 2.92 0.43510 3.49e-01 1.58e-01 2.85e+00 0.16510
1.60 3.19 0.43941 3.11e-01 1.16e-01 2.34e+00 0.14995
2.00 4.39 0.45319 2.05e-01 3.60e-02 1.10e+00 0.10263
2.20 5.10 0.45873 1.67e-01 1.97e-02 7.41e-01 0.08486
2.40 5.93 0.46373 1.35e-01 1.03e-02 4.78e-01 0.07005
2.50 6.40 0.46607 1.21e-01 7.23e-03 3.76e-01 0.06361
2.60 6.93 0.46834 1.07e-01 4.95e-03 2.90e-01 0.05776
2.70 7.52 0.47054 9.49e-02 3.30e-03 2.19e-01 0.05246
3.00 9.78 0.47676 6.26e-02 8.04e-04 8.06e-02 0.03958
3.25 12.38 0.48129 4.25e-02 2.06e-04 3.04e-02 0.03180
3.50 15.58 0.48491 2.90e-02 5.23e-05 1.14e-02 0.02610
3.70 18.46 0.48715 2.18e-02 1.91e-05 5.49e-03 0.02262
3.85 20.76 0.48851 1.80e-02 9.52e-06 3.33e-03 0.02047
4.00 23.17 0.48965 1.50e-02 5.02e-06 2.10e-03 0.01864
4.29t 28.03 0.49139 1.10e-02 1.67e-06 9.54e-04 0.01579
4.50 31.83 0.49238 8.93e-03 8.07e-07 5.68e-04 0.01410
5.00 41.46 0.49411 5.84e-03 1.84e-07 1.98e-04 0.01110
6.00 64.00 0.49615 2.94e-03 1.70e-08 3.64e-05 0.00745
7.00 89.72 0.49724 1.74e-03 2.88e-09 1.04e-05 0.00537
8.00 119.82 0.49793 1.12e-03 6.34e-10 3.57e-06 0.00406
9.00 153.86 0.49839 7.62e-04 1.74e-10 1.43e-06 0.00318
10.00 191.87 0.49870 5.44e-04 5.59e-11 6.46e-07 0.00256
11.00 234.32 0.49894 4.01e-04 2.00e-11 3.13e-07 0.00211
12.00 279.93 0.49911 3.07e-04 8.12e-12 1.67e-07 0.00176
13.00 330.06 0.49924 2.39e-04 3.52e-12 9.25e-08 0.00150
14.00 383.90 0.49935 1.90e-04 1.64e-12 5.41e-08 0.00129
15.00 441.98 0.49944 1.54e-04 8.02e-13 3.28e-08 0.00112
17.50 604.56 0.49959 9.58e-05 1.65e-13 1.08e-08 0.00082
20.00 791.59 0.49968 6.39e-05 4.25e-14 4.19e-09 0.00063
25.00 1241.91 0.49980 3.24e-05 4.42e-15 8.56e-10 0.00040
50.00 4991.19 0.49995 4.01e-06 4.16e-18 6.50e-12 0.00010
Note: Units with G = c = 1.
t Actual value of j,,,., = J,,.,/MIM2 is 4.287597023.
1 In units of the irreducible mass M1.

















100 I




10-5




o 10-10


CV
10-15




10-20
0 10 20 30 40 50
J total







FIGURE 10: Quadrupole Radiation from Equal Mass Black Holes. The
quadrupole approximation to the gravitation radiation from two black holes in
circular orbits is plotted against the angular momentum jo,, = Jtotil/(M1M2)
for the equal mass case M2 = M1.






86
of gravitational quadrupole radiation as the system has lost throughout its

entire evolution.

The secular time scale can also be compared with the dynamic time

scale by comparing the radial component of the velocity of each black hole

to the tangential component of the velocity. This can be accomplished by

considering a plot of the binding energy of each stable circular orbit as a

function of the separation between the two black holes as in Figure 11, for the

equal mass black holes. As the black holes spiral in from their mutual energy

and angular momentum loses, they will slowly move inward from a quasi-stable

orbit. The change in the binding energy must account for the luminosity of

the gravitational radiation to conserve energy. Thus the following definition

emerges
ds dEbinding dEbinding 1(5.55)
dt dt ds (555)

where the (d) term on the right hand side of equation (5.55) is taken from

the plot. The (E) term is given by the dimensionless Luminosity of equation

(5.50). In the case of equal mass black holes the (4) is roughly twice the

radial component of the velocity. Thus the radial velocity is approximated by


Vradil 1 ( LGw (5.56)
2 (dEbindin

for two equal mass black holes. The numerical results for the radial and tan-

gential components of the velocity are shown in Figure 12. The radial com-

ponent is smaller than the tangential component for separations larger than

s 5.9M1. At roughly this separation the scaled angular momentum has a

value of jiotai 2.40 and the dimensionless luminosity for the quadrapole grav-

itational radiation has a value of LGw M 0.01, providing stronger justification

for the previously discussed limit for the luminosity.






87










0.00
slope AEbinding/AS
AEbinding /M1


-0.02
I I



S -0.04

I I


-0.06 -


SAS/M,

-0.08
0 5 10 15 20 25 30
S/MI








FIGURE 11: Numerical Binding Energy versus Separation. The binding
energy per mass M1 is plotted for the stable circular orbits of equal mass black
holes as a function of their separation s/M1. The slope at any point along
this curve is used in conjunction with the luminosity to determine the radial
velocity of the black holes.




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