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
neverending 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 SpaceTime ..................................... .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 PostNewtonian 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 spacetime geometry
VV Covariant derivative associated with spacetime metric
Lt Lie derivative in the direction of the fourvector t
(M, gi,) Spacetime manifold with metric gpv
E, Spacelike hypersurface embedded in spacetime 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 xaxis in corotating frame
Xo2 Position of second black hole along xaxis 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 spacetime
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 timelike Killing vector field is assumed to exist over
a finite region of the spacetime 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 FermiDirac 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 Xray 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 EinsteinMaxwell 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 spacetime 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, spacelike,
timelike, and null infinities, and the orientation of light cones at all events for
a black hole's spacetime 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 spacetime 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 spacetime 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 spacetime geometry as
approximately time independent. Making the time independence into a global
feature of the spacetime 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 spacetime
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 spacetime
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 SpaceTime
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 spacetime manifold. The elegant statement of Ein
stein's equations, that given the mass energy distribution then solve for the
geometry of spacetime, 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
spacelike 3geometry 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 4geometry at that one time, and then evolve the mass energy fields and
4geometry 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
spacetime 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 spacelike hypersurface viewed as a Cauchy problem.33 That
the spacetime 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 timelike 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 spacetime,
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
spacetime manifold (M, gap) is foliated by Cauchy surfaces, spacelike hyper
surfaces Er, parametrized by a universal time function r. Each Cauchy surface
will have a timelike 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 oneform
w = wa,' = wadxa (2.8)
where "' = dxa in a coordinate basis, and w satisfies the relation
(n,w) = 1 (2.9)
The oneform w describes the foliation of spacetime and is related to the
universal time function r by the gradient
wO = aVer (2.10)
where V is the covariant derivative of the spacetime 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 fourvelocity 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 spacetime 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 spacetime 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 spacetime metric gap of M induces the spacelike metric 7ya on the
hypersurface Er given by
Ta# = gag + UwaWz (2.14)
with the inverse spacelike 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 spacetime 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 spacetime 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 = 7y7 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 3geometry 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 = yv'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
= 27 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
spacetime 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 spacetime is characterized by (Er, ya, Kag), and provides the
initial data for the spacetime manifold (M, ga,).
As previously mentioned, the 3geometry is found first, followed by the
determination of the 4geometry for that one value of the universal time r.
15
The geometry of the hypersurface is related to the spacetime geometry by the
GaussCodazzi equations
Sa^ AI 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
spacetime 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 spacetime 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 GaussCodazzi 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 spacetime 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 spacetime geometry. The GaussCodazzi equations relate the Riemannian
tensor of the spacetime geometry to the initial data of the hypersurface. In
order to impose Einstein's equations, the GaussCodazzi 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 spacetime
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 spacetime 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 spacetime Riemannian curvature tensor.
The 6 components of the spacetime curvature tensor involving second order
derivatives in a timelike 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 spacetime Riemannian tensor
is replaced by expressions involving the spacetime 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 + 47rypTAA
D(2.35)
+ Rap + KKapK\ 2KA, 1DDpa 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 spacetime
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 spacetime 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 spacetime geometry must be zero. The initial
value formalism applied to vacuum spacetimes 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 spacetime 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 spacetime
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 spacetime 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 spacetime 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 spacetime for the inertial frame is
at
N = an u = .. (2.52)
The Killing vector field for the asymptotically flat region of the spacetime
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 spacetime 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\Kap2aKKXp 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, p2Kh,(Dp),aRapaK 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 spacetime 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
EinsteinRosen 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)
roo
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)
roo 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 spacelike 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 spacetime manifold which is asymptotically flat the mass and
angular momentum of the geometry is found using two dimensional spherical
surface integrals near spacelike 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 antisymmetric 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 spacetime 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 spacelike 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 spacetime 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 roo, 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 supermetric
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 spacetime geom
etry is given by
ds2 = gp dxdx" (4.15)
Once the four dimensional spacetime 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 quasiCartesian
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 xdirection and 3velocity of the
black hole of mass mi, and the x02 and v are the position along the xdirection
and 3velocity 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 spacetime 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 corotate each point on the hy
persurface Er with the orbiting black holes. In this quasiCartesian 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 2yy 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)
+ 9119) + (Yi )
+ + (
(27yy7zz7zz) (2xyzz Yxxyy)
(28) ( Y )
(27yxxYyyTzz) (27'y7zyyYzz)
(7z ) ( (yY)
+ (+yy+zzyzz)
(2yyuyy~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 quasiCartesian
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 al1
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 xaxis 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, postNewtonian 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 restmass 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
600 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
restmass 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,
postNewtonian 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 postNewtonian 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 = 2r' 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 postNewtonian 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 postNewtonian 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 EinsteinInfeld
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 postNewtonian 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 postNewtonian 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
SPostNewton/M1
100
FIGURE 7: PostNewtonian 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 postNewtonian 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 postNewtonian approximation. However, the Newtonian approximation
has stable orbits for any nonzero 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
1.6 084
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 postNewtonian
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 postNewtonian 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, postNewtonian, 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 EinsteinHoffmannInfeld 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 PostNewtonian 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 postNewtonian approximation at jtot,,l
4.2876. At this value of the scaled angular momentum, the numerical binding
energy differs from the postNewtonian 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 (531)
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 massenergy
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
d2 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 xy 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 9.39e01 2.36e+00 1.58e+01 0.31522
1.00 1.65 0.40200 7.30e01 1.16e+00 9.99e+00 0.27818
1.30 2.40 0.42485 4.47e01 3.08e01 4.34e+00 0.20115
1.50 2.92 0.43510 3.49e01 1.58e01 2.85e+00 0.16510
1.60 3.19 0.43941 3.11e01 1.16e01 2.34e+00 0.14995
2.00 4.39 0.45319 2.05e01 3.60e02 1.10e+00 0.10263
2.20 5.10 0.45873 1.67e01 1.97e02 7.41e01 0.08486
2.40 5.93 0.46373 1.35e01 1.03e02 4.78e01 0.07005
2.50 6.40 0.46607 1.21e01 7.23e03 3.76e01 0.06361
2.60 6.93 0.46834 1.07e01 4.95e03 2.90e01 0.05776
2.70 7.52 0.47054 9.49e02 3.30e03 2.19e01 0.05246
3.00 9.78 0.47676 6.26e02 8.04e04 8.06e02 0.03958
3.25 12.38 0.48129 4.25e02 2.06e04 3.04e02 0.03180
3.50 15.58 0.48491 2.90e02 5.23e05 1.14e02 0.02610
3.70 18.46 0.48715 2.18e02 1.91e05 5.49e03 0.02262
3.85 20.76 0.48851 1.80e02 9.52e06 3.33e03 0.02047
4.00 23.17 0.48965 1.50e02 5.02e06 2.10e03 0.01864
4.29t 28.03 0.49139 1.10e02 1.67e06 9.54e04 0.01579
4.50 31.83 0.49238 8.93e03 8.07e07 5.68e04 0.01410
5.00 41.46 0.49411 5.84e03 1.84e07 1.98e04 0.01110
6.00 64.00 0.49615 2.94e03 1.70e08 3.64e05 0.00745
7.00 89.72 0.49724 1.74e03 2.88e09 1.04e05 0.00537
8.00 119.82 0.49793 1.12e03 6.34e10 3.57e06 0.00406
9.00 153.86 0.49839 7.62e04 1.74e10 1.43e06 0.00318
10.00 191.87 0.49870 5.44e04 5.59e11 6.46e07 0.00256
11.00 234.32 0.49894 4.01e04 2.00e11 3.13e07 0.00211
12.00 279.93 0.49911 3.07e04 8.12e12 1.67e07 0.00176
13.00 330.06 0.49924 2.39e04 3.52e12 9.25e08 0.00150
14.00 383.90 0.49935 1.90e04 1.64e12 5.41e08 0.00129
15.00 441.98 0.49944 1.54e04 8.02e13 3.28e08 0.00112
17.50 604.56 0.49959 9.58e05 1.65e13 1.08e08 0.00082
20.00 791.59 0.49968 6.39e05 4.25e14 4.19e09 0.00063
25.00 1241.91 0.49980 3.24e05 4.42e15 8.56e10 0.00040
50.00 4991.19 0.49995 4.01e06 4.16e18 6.50e12 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
105
o 1010
CV
1015
1020
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 quasistable
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.
