inversion symmetric initial data for n charged black holes

ANNALS OF PHYSICS 165, 17-37 (1985)

Inversion Symmetric Initial Data for N Charged Black Holes

JEFFREY M . BOWEN

Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837

Received July 27, 1984

The initial value equations for gravitational and electric fields on a multiply connected manifold are addressed. The manifold considered represents two physically identical asymptotically flat universes with N throats connecting them. Matching conditions for the fields are derived for the case of a conformally flat maximal (but not time-symmetric) initial slice. A formal solution incorporating these conditions is given in terms of a simple geometric algorithm, and the method is applied to give explicit expressions for the background electric field of two oppositely charged bodies. Unlike previous methods, this procedure is general enough to allow arbitrary a priori specification of charge, angular momentum, and linear momentum for any number of separate charged moving bodies. © 1985 Academic Press, Inc.

I. INTRODUCTION

The program of the initial value problem in general relativity is based on the fact that the full four-dimensional spacetime geometry can be reconstructed from the three-dimensional geometry of spatial slices and a description of how these slices are embedded in spacetime. The three-dimensional metric gij describes the spatial geometry within a slice, while the extrinsic curvature K ~ determines the embedding. Once these data are specified on a single slice, the dynamical gravitation equations (the six Einstein equations containing time derivatives of ~(~) can be used to evolve these initial data forward in time to other slices. The task of the initial value problem then is to specify the gravitational and other fields everywhere throughout a three-dimensional space-like hypersurface such that those field equations pertain- ing to data in the surface are initially satisfied.

In developing techniques to solve the initial value problem, many authors have created multiply connected topologies on which inversion symmetric data is specified. Such methods were used by Kruskal [1] to show that a constant time slice of the Schwarzschild solution consists of two asymptotically flat universes connected by a throat. Misner's two-body wormhole solution [-2] led to a method of images [3] that could generate initial data for many stationary particles. A similar imaging process for generalized potentials was used by Lindquist [4] to solve the initial value problem for many stationary charged particles. With these results, it

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18 JEFFREY M. BOW'EN

was possible to specify most physically interesting initial data, with the restriction that they describe time-symmetric systems (systems momentarily at rest).

More recently, Bowen [5] and Bowen and York [6] have removed the time- symmetric restriction from initial data sets. The results there show that it is indeed possible to describe a moving or spinning black hole and retain the aesthetically pleasing physical equivalence of two identical universes connected by a throat. A well-defined boundary value problem is posed that, when solved numerically [7, 8], yields complete initial data for a single black hole with arbitrary linear or angular momentum. Recently, Kulkarni, Shepley, and York (KSY) [9] have generalized the method of images to the N-body problem with arbitrary spins and momenta. The remaining task in solving coupled Einstein-Maxwell systems is to develop methods for specifying charge and electric and magnetic fields that, unlike Lind- quist's approach, will be compatible with the established program that generates many-body time-asymmetric initial data. This paper goes a great deal of the way towards accomplishing that task by developing a general algorithm that produces inversion-symmetric electric fields for the case of zero magnetic field.

The major difference between this work and other previous image or inversion methods is that we work directly with the electric fields rather than with potentials. In constructing initial data that include both gravitational and electromagnetic interactions of charged black holes, it would be quite unrealistic to expect that the general solution should contain only electrostatic (curl-free) fields, even if the charges are momentarily at rest. Except for the truly static case in which electric and gravitational forces exactly balance, retardation effects will in general create non-static fields. By avoiding the use of the standard method of images (which acts on the electric potential), the formalism developed here provides electric fields that may have non-zero curl and thus could not be derived solely from a potential, inversion-symmetric or not. Similar arguments may help explain the difficulty many researchers have had in deriving inversion-symmetric extrinsic curvatures from a vector potential constructed from images.

A special property of the evolution of black hole spacetimes makes the selection of matter-free configurations particularly appropriate for inversion-symmetric initial data sets. By insisting on a matter-free situation, one need not deal with any phenomenological equations of state for the development of material sources--only the evolution equations of general relativity and electromagnetism need to be considered. The sources are thus geometric or topological models, in the spirit of Mis- ner and Wheeler [10], with electric and gravitational field lines emanating from local topological structures identified as black holes. When such initial data are evolved, only the fields outside the black holes are used, since whatever happens inside them cannot affect the external environment. In this way, all the gravitational and electromagnetic effects of material sources can be discovered without ever deal- ing directly with the sources' evolution. In practice, the numerical evolution of fields occurs only outside the apparent horizons of the spatial geometry. Since my method automatically locates by symmetry the minimal surfaces and apparent horizons of the initial geometry, the construction of inversion-symmetric initial data on a mul-

INITIAL DATA FOR CHARGED BLACK HOLES 19

tiply connected manifold is particularly suitable for spacetimes containing charged black holes.

II. CONSTRUCTING THE N-BoDY MANIFOLD

The three-dimensional manifold MN upon which physical data are to be specified consists of two asymptotically flat regions connected by N Einstein Rosen bridges. The detailed topological structure of this manifold may be found elsewhere [4, 11 ]. In outline, one takes two copies of fiat, three-dimensional space R 3, excises the interior of N non-intersecting spheres from each copy, and identifies the corresponding spherical surfaces in one copy (the upper sheet) with those in the other (the lower sheet). For N > 1, the manifold is multiply connected; great care must be exercised when describing analytic physical fields on such a structure.

To continue the discussion, we must present several definitions and introduce some notation, following Lindquist and KSY for the most part. Let Ca be the center of a sphere of radius a~. Then define the spherical inversion map J~ from R 3 - - C a tO itself by

J~(x)=(a~/r~)2(x-C~)+C~, c~=l ..... N (2.1)

where r~ - I x - C a I. We call J~(x) the inverted image of x in the sphere ~. Note that this map is its own inverse: J~(J~(x))= x.

With the mapping J~ in hand, we can map all of MN onto a subset D of R 3 (defined below), specify physical data on this subset, and map it back to MN.

Let S~ be the surface of the ~th sphere: S~ = {x ~ R31r~ = a~ }.

s 2 J

FlG. 1. The regions used to construct a two-body manifold M2. The spherical surfaces are labeled S~ and Sz, while F~ and/'2 are the shaded interiors excluding the centers C~ and C2. All image charges will appear inside the open circles or at C~ or C2 (not part of F~ and F2).

20 JEFFREY M. BOWEN

Let A be the region of R 3 outside the N spheres: zl = {x 6 R3lr~ > a~: ot = 1 .... N} . Finally, let F~ be the inversion of this region through the ~th sphere,

F ~ = { x ~ R 3 1 J ~ ( x ) ~ /I}.

This region is the interior of the ctth sphere outside the images of the other spheres and excluding the center Ca. (See Fig. 1.) The subset of R 3 that we seek is then given simply by

D = A w S~ w F~ (union over all ~ = 1 ..... N). (2.2)

Our task is then to specify appropriate initial data on D such that physical fields on M u will be equivalent (identical up to sign) on the upper and lower sheets.

III . THE INITIAL VALUE EQUATIONS

The physical data to be specified o n M N are the gravitational and electromagnetic fields that satisfy the initial value constraints of the coupled Einstein-Maxwell theory. Let us examine how these constraints arise, first for the data describing gravity, and then for electromagnetism.

The gravitational constraints can be derived from Einstein's equations by a judicious splitting of the four-dimensional spacetime quantities into separate spatial and temporal components. As mentioned previously in the Introduction, the complete description of the geometry of spacetime in the neighborhood of a space-like hypersurface is contained in the spatial metric gij and the extrinsic curvature R tj. However, these data may not be given arbitrarily. In splitting the ten Einstein equations into space- and time-components, it turns out that four of them relate the ~ij's and R~'s within a single spatial slice and thus act as constraints. These initial value equations, expressed in terms of ~'s and K's, are I-6]

~,(R'J- UR)= 8 ~ j (3.1)

+ R 2 - R ~ R ° = 161r~, (3.2)

where the covariant derivative operator compatible with g0 is denoted by g7 i, /~ is the scalar curvature of g0, and R = g o R ° is the trace of R ij. The sources, found from splitting the stress-energy tensor, are the momentum density Sj and the energy density ~.

The initial value equations for electromagnetism can be found simply in a similar fashion. We want to find those Maxwell equations that relate electric and magnetic fields in a single spatial slice. Thus the initial value equations of electromagnetism, those that involve no time derivatives, are

Vi Ei= 4rc¢3 e (3.3)

Vi Bi = 0 (3.4)

I N I T I A L D A T A F O R C H A R G E D B L A C K H O L E S 21

where E g is the electric field, /~i is the magnetic field, and Pe is the electric charge density.

Since the topic of this investigation is charged black holes with arbitrary momentum, we will consider the matter-free case, allowing topological models to replace material sources. Because we wish to apply the results of KSY to generate inversion-symmetric extrinsic curvatures, and since that method requires a homogeous momentum constraint, we must guarantee not only that material sources of momentum density vanish, but also that contributions from the electromagnetic Poynting vector vanish. This can best be accomplished by requiring the magnetic field to be zero throughtout the initial slice, and concentrating on the structure of the electric field. These conditions imply

SJ = 0, t~E = 0, B i = 0, and /~ = EiE78n.

The final reduction of the initial value equations occurs when we apply a conformal transformation to the physical metric and other quantities. O'Murchadha and York [12] have shown that this leads to considerable simplification, since a partial uncoupling of Eq. (3.1) and (3.2) results; the momentum constraint (3.1) can be solved independent of the form of the conformal factor. To make explicit closed- form solutions tractable, we follow Bowen and York, choosing a conformally fiat background space and a maximal initial slice (K=0) . Although some generality may be lost by assuming conformal flatness, this condition has been applied previously to the problem of black hole interactions, yielding significant general results (formation of minimal surfaces, generation of gravitational radiation) [13]. Writing the physical metric as

gij = I[/4 (~0" (3.5)

we find that Eq. (3.1)-(3.4) reduce to

Vi Ei= 0 (3.6)

Vi gij = 0 (3.7)

V 2 l p = __ igijgij~ I - 7 _ ~giE1 ilp - 3 ( 3 . 8 )

where ~O is the conformal factor, Vi and V 2 are the flat-space gradient and Laplacian operators, respectively, and E i and K U are the background electric field and extrinsic curvature, respectively, defined by

E i= ~16E i (or Eg = Ip2 /~ i ) (3.9)

K ~j = Ipl°R'J (or K~ = ~b2Rij). (3.10)

Equations (3.6)-(3.8) are to be solved for the background fields if, K 'j, and E;; then the physical gravitational initial data (~j and R ~j) and electric field (E ~) may be formed from them.

22 JEFFREY M. BOWEN

IV. INVERSION PROPERTIES OF THE INITIAL DATA

We now review [5, 9] the development of the symmetry properties that must be satisfied by the initial data ~k, K °, and E i in the domain D. Because M s is multiply connected and since we wish to make the upper and lower sheets physically equivalent, the gravitational and electric data will have to obey certain symmetry relations. In deriving these relations, the inversion map Ja will play an important role. For physical equivalence between the sheets, we will require that each appropriate geometric object at x ~ A be proportional to that at the inverted point J~x. Since ~ = 1, the constant of proportion must be + 1. The conformal factor will take the + sign, but the other fields, in order to have continuous normal components at the throats, will require the - sign.

The condition on 0 is derived from the fact that the inversion map J~ should be an isometry for each e = 1 ..... N. If the metric g, a symmetric bilinear form, is expressed in local coordinates as

g(x) = gi j (x ) dx i dx j,

then the isometry condition g(x) = g(J~x) becomes

g#,(x) = gij(J~x)[(Of~/Ox')(Of~/Oxm)] (4.1)

where f / = (j~x)i (a~/G)2 (X i i i = -- Ca) + C a. The derivatives are easily formed:

(O f i /Ox,) = (ajra)2 (6il 2n~ nt~) = (a~/G) 2 R~ l (4.2)

i where n, = (x i - C~)/G is the unit normal to the sphere centered at C~ and where the orthogonal matrix R" is given by

R~ = (6 i ' - 2n~ n'~).

The action of R~ on vector fields reverses the normal component , i (R~ n~ = - n~) but leaves tangential components unchanged. For our problem g is conformally flat so that (4.1) implies that the conformal factor ~9 must satisfy

O(x)=(a~/G)O(Jax), x e A ; a = l , .... N (4.3)

for each J~ to be an isometry. The condition on the components of the background extrinsic curvature is

derived similarly, but here the minus sign is chosen to agree with the known one, hole solution [6]. In analogy with (4.1), we have

j~/m(X) : ' go.(Jax)[(~fi/Ox')(OfJ/Oxm)]. (4.4)

But from (3.10), K'#dx) = ~ 2(x) Kt,,(x) and k0.(J~x) = ~ 2(j~x) Ko.(J~x) so that, combining (4.3) and (4.4), the background extrinsic curvature must satisfy

Kzm(x) = (a~/r~)6 ~il jm = (4.5) -- R~ Ki j ( J~x ) , x ~ d ; ~ 1 ..... N.


Similar considerations lead to the condition on the electric field. Following Lind- quist [4], we require that E be odd upon inversion. Thus E'(x)= - E ( J ~ x ) , which in this formulation leads to

Ei(x) = - (a~/r~) 4 R~ Et(J~x ) x e A; e = 1 ..... N. (4.6)

Note that in formulae (4.5) and (4.6), indices are raised and lowered with the flat background metric 6ij. Thus in these and succeeding formulae that deal only with background fields, indices are placed up or down as convenience dictates.

The remaining task of this section is to show that well-behaved solutions of (3.6)-(3.8) may in fact be found that also satisfy conditions (4.3), (4.5), and (4.6). For this purpose, we extend the proof by Bowen and York [6] to cover the case of the electric field. Suppose we have found an electric field that solves (3.6) everywhere in the domain D and satisfies condition (4.6). Suppose further that a symmetric extrinsic curvature has also been found. Then we may construct the appropriate conformal factor as follows. Solve the exterior boundary value problem

V2~b = _ ~KoKa~p 7 _ _ ~EiEI i~p 3 in A (4.7)

(O~b/c3r~) + (l/2a~)~b = 0 at re = a~, e = 1 ..... N (4.8)

lim ~ = 1 , ~b>0. r ~ o o

Then, to extend ~ to the rest of D, define

$(x) = (a~/r~) ~(J~x) for x e ( D - A ) . (4.9)

To see that this extension satisfies (4.7) at the inverted point, recall the inversion property of the Laplacian:

(VZf) (x) = (a~/r~) 5 ( V 2 f ) (J~x).

Fortunalety, both terms on the right of (4.7) invert with the same factor:

(gijgij~ - 7 ) ( x ) = [" - - (a~/r~) 6 R~ R~' gk,(J~x)]

x [ - (a~/r~) 6 R ~ R{" K m , ( J ~ x ) ] [ ( a ~ / r ~ ) I~(J~x)] 7

= (a~/r~)5 ( K ~ K i J $ - 7)(J~x)

since R~ is orthogonal. Similarly,

[ EiEi~9 - 3] ( x ) = [ - (a~/r~) 4 R o Ej(J~x)]

× [ - (a~/r~) 4 R ik Ek(J~x)] [(a~/r~) ~b(J~x)] -3

= (a~/r~)5 ( E i E i $ 3) (J~x).

24 JEFFREY M. BOWEN

Thus ~b satisfies (4.7) in the interior of each sphere. Furthermore, the extension (4.9) guarantees that ~ has the correct inversion symmetry and is continuous across the surface of each sphere, while the boundary condition (4.8) assures smoothness. Therefore, if we can find inversion-symmetric extrinsic curvatures and electric fields, then a symmetric conformal factor can be found. Kulkarni et aL have found such extrinsic curvatures, while the construction of inversion-symmetric electric fields is the subject of the next section.

V. INVERSION-SYMMETRIC ELECTRIC FIELDS

The problem at hand is to generate an electric field E i throughout domain D that satisfies

and

V~U(x) = O, x ~ D (5.1)

Ei(x) = - (a~/ra) 4 R ij EJ(J~x), x ~ LJ. (5.2)

We are most interested in specifying electric fields that model charged particles at the locations of the throats of MN. An observer on the upper sheet, for instance, should find net electric flux through a closed surface only if one or more of the bridges is "enclosed." The general method will be to add image charges outside the region of interest (in D's complement, R 3 - D ) . However, this differs from the method of images of electrostatics in that the inverted electric fields generated here have in general a non-zero curl! We cannot use a series of point-charge potentials, but instead build a series directly with the fields.

We begin by defining an inversion operator da that acts on vector fields:

~¢~ [ E i ( x ) ] ~ - - (aJr~) 4 R~ EJ[J~(x)]. (5.3)

The operator ~ has two important properties.

(1) The self-inverse property

J , Ja E"(x) = Ei(x). (5.4)

The proof is simple:

i ~¢a da E (x) = -- J~[(a , / ra) 4 R ~ EJ(J ,x ) ]

= + (a , / r , )4[aa/ra(Jax)]4 Rik R~J EJr_l t J xVI

= E i ( x )

since ra(Jax) = I Jax - Ca I = I(aa/r~)2( x - Ca) + Ca - Ca I = a]/r=.


(2) The divergence property

6 3 / 6 3 x i [ ~ E i ( x ) ] = - - ( a ~ / r ~ ) 6 [ 6 3 E J ( J ~ x ) / c 3 ( J ~ x ) J ] . ( 5 . 5 )

The somewhat longer proof appears in the Appendix. The divergence property shows that if E/(x) is divergence-free, then so is its inverted field J~Ei(x). This is the key property, since we may now generate solutions of the Maxwell constraint (3.6) with little difficulty.

A small digression is in order, since we often learn much from the path not taken. As a direct method of constructing the desired solution for U, one might expect that the steps used in Section IV to construct ~b would be applicable. Let us specify a well-behaved electric field E i outside the spheres. For x c A, it satisfies

V~E'(x) = 0, x c A . (5.6)

Then extend E into the interior regions F~ by the rule

E i ( x ) = J ~ E i ( x ) , X c F~. (5.7)

The divergence property (5.5) guarantees that (5.6) also holds in the interior regions and E ~ is inversion-symmetric by construction. Is this an acceptable solution? The answer is no. Whereas the extension and the boundary condition used in the construction of the scalar ~k ensured a smooth join at r~ = a , , no such guarantee is available for the electric field. The definition of the extended field (5.7), it turns out, ensures only that the normal components of the electric field will be continuous across the boundary. The correct method must at least generate an exterior field whose tangential components vanish on the spheres r~ = a~.

To avoid the problems of matching vector fields at the boundaries, we abandon methods that require us to separate the region of interest into exterior and interior. We instead seek a single expression for E i, analytic throughout the entire domain D. Misner's method of images comes to mind [3]. We will sequentially add image charges inside each sphere (but outside D) so that the tangential fields vanish on the sphere and the entire field is inversion-symmetric through that sphere. These image charges of course destroy the symmetry in the other spheres so that images of images are required to fix the field at the next sphere. And so the process continues.

All the tools are available to write down a formal solution along such lines. Define the operator J , with the ½ factor for later convenience, as

J = z 1+ ~,,~, (5.8) i} \ i = 1 /._1

where the ~ ' s take on the values 1 ..... N and the sum is over all length-n sequences of a's such that a i ¢ ~+ i. Since j 2 = 1, we have J ~ J = J for any ~. Thus for any vector field V ~, let

U = J (V ' ) . (5.9)

26 JEFFREY M. B O W E N

Then clearly E i obeys the symmetry conditions (4.6)

J ~ E i = JEot j v i = ~; V i = E i,

and the divergence property repeatedly applied guarantees that if V i is divergence- free, then so is E( Thus (5.9) is the inversion-symmetric, divergence-free field that we seek.

In order to understand exactly how the operators J~ work in practice~ let us examine their action on a particularly useful vector field. For these calculations, I will use the notation developed in Bowen, Rauber, and York (BRY) [14], as well as some intermediate results presented there. Place a point charge Q at point C2, aod look at the inverted field when the inversion sphere of radius a is centered at C1 (see Fig. 2). Choose Vi=E~ =Qn~2/r~; we will calculate JIE~, its inversion through sphere 1. Let D~:2 = C 1 - C2 be the separation vector between the centers, with D~:2 > a. Let C21 = J I ( C 2 ) be the inverted image of C2 in sphere 1. Also let D21:1 ~-- C21 - C 1 s o that D 2 1 : l = a2/Dl:2.

Before calculating ~ E2, we state special cases of two useful results. The general cases are presented in BRY.

r2(J1x)=r21(x ) Dl:2/rl(x) where r21(x)= I x - C 2 1 ] (5.10)

R] k n ~ ( J l x ) = - ( r2 ,n~- rlnk2~)/D2H where n~l = (x ~ - C~1)/r21 (5.11)

Now insert these into the definition of J~E ~ to find

J1E~ (x) = -- (a4/r 4) R~ i [Qn~ (J~ x)/r~ (J, x) ]

= - - ( a 4 / r 4 ) ( - Q/D2,;,)(r21n~-r,n~l)(r,/r2,D,:2) 2

Q(a2/Dl:2)2 (1/D2,:l) 2 2 1 = (r~r21) (r21n~--r,n~,)

= (Qn~/r~) (D2,:,/r2t) -(Qn~,/r~,)(D2,:,/r,). (5.12)

Note that the result is the difference of two point charge fields, each modulated by a simple pole. Charge + Q is located at the inversion center C1, while - Q has been placed at C21, the inverted image of the original charge. The (D/r) factors act like

FIG. 2. Orientation and labeling of image points. C2~ is the image of C 2 when inverted through the sphere centered at C~; C21 =-J~(C2).

I N I T I A L D A T A FOR C H A R G E D B L A C K H O L E S 27

point potentials whose singularities occur at the location of the opposite charge and whose strength is the distance between the poles. These interpretations will be justified in Section VI.

The final calculation of this section is to show that the inversion of a field such as (5.12) gives rise to a field of the identical form. Since we will require images of images, we introduce the following. Let ~i be an index to denote an inversion sphere, with range 1 ..... N. Following BRY [14], let S stand for any valid string (a string of ~i's such that ~ ~ c~/+ 1). Then let Csl = Jt(Cs) represent the image of Cs. when inverted through sphere I ( I= 1 ..... N). We will also need the following:

r$ -= rs (x)= I x - U s I n~ ~ n~(X)= (X i - Ci$)/r$

Ds: $, = C $ - C $ , ; D s : s, = JD$:$, ]

where $ and $' are arbitrary valid strings. Finally, let E~ ~ 2 = Qns / r s represent the field of a point charge Q located at C$. In this notation our first result (5.12) can be written as

~ E ~ = ( Ds , : / / r s , ) i E,- ( Dst:l /r i) Ei$1 . (5.13)

We wish to consider the inversion of a general expression like (5.12); that is, we will form

o¢1 [ ( D ~s :s/ G$ ) Eis - ( D ~s :s/r $ ) E~s ].

(The expression in brackets is the right side of (5.12) for a = 2, $ = 1.) The previous two intermediate results, now in general notation, together with two others derived in BRY [14], will enable us to complete the calculation.

r s [ - J l ( X ) ] --- r$i Ds:l /r I (5.14)

Ri /n~ [J1(x)] = - (n'trst i -ns:t)/Dsl:l (5.15) O$1:l = a 2 / D s j (5.16)

D~$1:$I ----- D~$:$ DsI:I/D~$:I (5.17)

Using these results, we find

~¢~ [ ( D =$:s /Gs )Eis - ( D =s :s /r s )Eis ]

= D~s:$ [ (J tEis ) /Gs ( J l x ) - (o¢lEis ) / rs (J l (x )]

= D,s:s { [ (Dsi : i / r s l )Ei l - (D$1:z/ri)Eisl ] (ri/r=siD=s:i)

- [ (D=sI :1 /Gs l )E~- (D~si: i /r l )Eis i ] ( r i / r s lDs: i ) }

= (D=s :$ D$I:I/D,$ :1) [ (rz /rs iG$z)E~ - (1/G$1)Eisi ]

- (D~s:$ D=sI:I/Ds:z) [ ( r i / G s i r s z ) E i l - (1/rst)E~s I ] i i = - [ ( D = s z : s l / G s l ) E s i - (5.18) (D~sI:$1/rsl)E~s I ].

28 JEFFREY M. BOWEN

O

FIG. 3. Physical effect of applying the inversion operator. The original charge Q, when inverted through the large sphere, induces two equal but opposite image charges, as indicated by solid arrows. The second inversion, through the smaller sphere, places another pair of image charges at the image points of the first pair, as indicated by dotted arrows. The pattern continues for further inversions.

In this notat ion, then, the act ion of the inversion opera tor ~ on a field of the form (5.13) is to replace every occurrence of $ by $ I and make an overall sign change. The physical act ion is illustrated in Fig. 3.

VI. PROPERTIES OF SYMMETRIC ELECTRIC FIELDS

The algori thm of Section V allows us to study in detail the properties of the symmetric electric field J V i whenever V i is an ordinary point-charge field or sum of such fields. In that case, after the leading point field terms, J V i is a series of terms of the form

QD~s:s[(1/r~s) i 2 i 2 (ns/r~) - (1/rs) (6.1) (n~s/r~$)],

and it is a simple matter to check, for instance, that the divergence of U vanishes term by term. We find

Vi[(1/rs ) i 2 i 2 (n=s/r=,)_(1/r=s ) = _(nis / r 2) i 2 (ns/rs)] (n=s/r~s) 71- i 2 i 2 (ns/r$) (n~s/r~s) = 0, (6.2)

except at the poles r$ = 0 or r=s = 0. Thus if we are careful to place the singularities of V i at the centers of the inversion spheres (no real restriction on the N-body problem), then all poles in the series J V ~ will be outside the domain D, and V ~ ( J V e) = 0 will be satisfied th roughout D.

To see that each successive image of an original point charge Q has in fact charge Q (or - Q ) and not some other magnitude, we look at a typical term,

Term = T ~ = (D~s:s/r~s) (Qn~/r 2) - (D~s:s/rs) (Qn~s/r2s).


Recall that D~s:s = ] C , s - C s I is just the distance between the two poles of the term shown. Thus, as we approach the singularity at r s = 0, and look at the coefficient of ( [~l,li /r2] s/ s J, we find

lira (D~s:s/r~s) = lim I t s - Cos I x~C~ x~C~ I x - f ~ s [

= 1 (6.3)

and the pole strength at C s is expected to be Q. This result is confirmed by evaluating the Gaussian surface integral of T/. For

convenience take a sphere of radius ro centered on Cs. Then we find, with details in the Appendix,

f T i d2Si i i = r T n s dO

= 4z~Q for 0 < r o < D~s:s (6.4)

= 0 for ro >D~s:s. (6.5)

Thus we find a point charge Q at C$ and a point charge - Q at Cos. The fact that every term in the series E i = J V i consists of a pair of equal but

opposite image charges has important implications. In contrast to other methods of images, the magnitude of the added charges does not diminish as more are added. However, image charges are always added in opposite pairs so that the total flux through any of the N spheres never changes as successive terms are added to the series. Thus it is a simple matter to specify in advance the total charge associated with each bridge.

Since the magnitude of the image charges remains constant, some attention should be given to questions of convergence. One observes from (5.17) that the distance between the oppositely charged pair of images diminishes quickly, while explicit calculations show that the series rapidly converges for typical separations and radii of inversion. For instance, consider two oppositely charged holes, each with radius of inversion a. Even if we place their centers at a distance of only 2.1a apart, evaluating the series at various field points shows that for most locations only one or two terms are needed before the next term becomes less than 10 4 times smaller than the preceding term. In fact, only at points less that 0.1a away from one of the spheres are as many as three such terms required; while for center separations of 4a or larger, more than two terms are never required for any field point.

30 JEFFREY M. BOWEN

VII. A COMPLETE EXAMPLE--THE ONE-HOLE CASE

In order to see the complete program operating in practise, consider the case of a single stationary charge. For the vector field V i take a point-charge field centered at the origin,

V i ~ Qni/r 2.

We wish to construct an electric field that is inversion-symmetric through a sphere of radius a centered at the origin. The series for J terminates quickly in this case: J = ½(1 + Jo), where subscript 0 refers to the origin. Now form U = JV~:

J V ~ = ½Q {n~/r 2 - (a/r) 4 R~ nJ(Jox)/rZ(Jox)} .

r(Jox ) = I (JoX)[ = I(a2/r2)xl = a2/r and ni(Jo x) = (a2xi/r2)/(aZ/r) = xi/r = n i. But Thus

J V i= ½Q {n'/r 2 - (a/r) 4 ( - n i) (r2/a4) } = Qni/r 2.

With this inversion-symmetric electric field, we now set out to solve the Hamiltonian constraint. Assuming time-symmetry (Kij = 0), we have

v2~ =-~EiE'~, -3

Substituting our solution, we find we must solve

4V20 = - (Q2/r4) ~b -3 (7.1)

for ~O. Let i f /= f l / 2 SO that (7.1) becomes

2 f V 2 f - (V f ) 2 = _ QZ/r4.

Trial and error gives the solution for f with proper asymptotic behavior,

f = 1 + Ar - l + Br -2,

where A = arbitrary constant and B = ¼(A 2 - Q2). However, A may be determined by the energy integral at infinity E = - (½~) ~ V~O. aS; we find A = E = M. Thus the solutions to the initial value equations are

= (1 + M/r + ( M 2 - Q2)/4r2)l/2 (7.2)

E i = O n i / r 2. (7.3)

The radius of inversion is related to the constants in ~ by a = ½(M 2 - Q2)1/2.


To obtain the physical metric and electric field, we use (3.5) and (3.9):

ds 2 = 1]1 4 d S 2 a t = (1 + M / r + (M 2 -- O2)/4r2)2 (dr + r 2 dg2 2)

E i = ~b - 6 E i = Orani[r 2 + M r + ( M 2 - Q2) /4] -3 (7.4)

or

E = Erer = Qra[r 2 + M r + ¼ (M 2 - Q2)] -3 er. (7.5)

We can make this look more familiar by replacing the coordinate basis vector er with the unit radial vector ee = (grr)-l/2er, and by transforming the radial coordinate. Let ? = r(1 + M / r + (M 2 - Q2)/4r2) so that d~= (1 - ( M 2 - Q2)/4r2) dr.

Then (7.4) and (7.5) become

ds 2 = (1 - 2 M / F + Q2/72)-1 d?2 + ~2 d£22

E = Qr4[r 2 + M r + ~ ( M 2 - Q2)] -2ee = Q/F2ep '

the familiar Reissner-Nordstrom solution in standard area coordinates. Note that the radius of inversion r = a = ½ ( M 2 - Q2)~/2 coincides with ? = M + ( M 2 - Q 2 ) 1/2,

the horizon for this geometry.

VIII . THE SYMMETRIC Two-BODY PROBLEM

In order to illustrate the full power of the methods of Section V, we turn to the symmetric two-body problem. Choose centers of inversion C~ and C2, separated by 2l, and let the inversion spheres each have radius a. To represent equal and opposite charges - Q and + Q at C1 and C2, respectively, we choose V i to be

Vi = V~ + V~2 = - Qn~ /r 2 + Qn~/r 2. (8.1)

The operator J becomes

J = l ( 1 +Yl + 4 + Y , 4 + 4 J , +... ), (8.2)

where there are precisely two terms with any given number of J~ facotrs. We have seen in Section VII that point-charge fields are self-inverting through their own spheres, i.e., J~ V;~ = V~,. Thus in the series J V i, every term appears twice--once, for instance, as J 2 J l ~ V~ and once as J 2 ~ V~. This factor of 2 cancels the ½ in the definition of J and all charges and images will be _ Q. Thus

E ' = J W = + V~ + V~ + 4 V~ + Y, V'~ + ...

= - Qn~/r 2 + Qn~2/rZ~ - J2 (Qn] / r~ ) + Jl(Q#2/r22) + ... (8.3)

595/165/1-3

32 JEFFREY M. BOWEN

It is possible to get a reasonably simple expression for the typical term in the series if, following Lindquist [4] , we define an operator ~ which acts on scalar functions of position by ~ [ f ( x ) ] = ( a / r z ) f ( J t ( x ) ) . Then, for instance, J2(1)=a/r2, while J l ~ ( 1 ) = J l ( a / r 2 ) = (a/IC1 - C 2 l)(a/r21)=a2/(D2:lr21) = D21:l/r21. Using these operators, we can write

Vi9 = Qniz/r2 = (Q/a) (ani2/r~)

= (Q/a) ( - a / r 2 ) , i = (Q/a) [ - J2(1)], i. (8.4)

By adding a term proportional to zero (the gradient of 1), we can give V~ a symmetric look:

Vi~ =(Q/a) {[(1)] , i [~2(1)] - [ (1)] [~2(1)], i}- (8.5)

When we act on V~ with ~ , as needed in the series (8.3), the result is

~¢~ V~2 = a D21:~ [(n~/r~) (1/r21)- (n~21/r~) (1/r~)]

= (Q/a) [( - a / r l ) , i (D21:l/rul)- (a/r1) (--Dzl:1/r21), i]

= - ( O / a ) { [ J l ( 1 ) ] , i [ J l J 2 ( 1 ) ] - [ J l ( 1 ) ] [J~J2(1)] , ~}. (8.6)

It turns out that this pattern continues--to get each succeeding term, change sign and insert a J~ or J2 as a leading operator in each bracketed factor. Therefore, the general term in the series J (V~ + V~) can be expressed in terms of Lindquist's J operators as

J m fac;ors ~2 V~ "]- Jg" m factors ~1 V~ = ( - - 1) m (Q/a) ( - [ J m" J 2 ( 1 ) ] , , [ J m'$', J l ( 1 ) ]

"+ E J "m" c~2(i)'] [ ~ m'+'l J l ( 1 ) ] ' ' + r J "m" J1(1)']' ' [ J "'" J2(1)] m

- [ J .L. ~ ( 1 ) ] [ - J m'~_' lJ2(1)] , ,} (8.7)

where the m or m + 1 below the products of J ' s indicate how many total factors of J~ or J2 are to applied to 1, and J l and J2 alternate in every product.

While the formal expressions above have a certain elegant simplicity, we would like to derive for calculational purposes coordinate expressions in terms of known functions. Establish a Cartesian coordinate system centered at the midpoint between the charges and orient the axes so that C 1 = - - ~ and C2 = +/~. Then transform to the natural coordinates for the two-hole problem, bispherical coordinates/~, 0, ~b, determined by

x = c sin 0 cos ~b(cosh/~ - cos 0) - l

y = c sin 0 sin ~b(cosh/~ - cos 0) -

z = c sinh/~(cosh/~ - cos 0) - 1,


where the pa ramete r c is given by c= (12 a2)1/2 and represents the distance f rom the origin to either accumula t ion point of image charges. Also introduce/~o, determined f rom cosh #o = l/a. The inversion spheres of radius a centered on C1 and C2 are then described by # = + / to , so that in the region outside the spheres, we

havel/~1 < #o. Recall f rom (6.1) that a typical term in any series resulting f rom a point -charge

i 2 field Vi, = Q n J G has the form

T e r m = Q D ~ s : s [ ( 1 / r ~ s ) ( n ~ / r ~ ) _ ( 1 / r s ) i 2 (n~s/r~s)] .

For the p rob lem at hand, with two equal but opposi te charges _ Q, the m t h te rm looks like

T e r m = T i ( 1 ) m Q D t s . : s . i 2 = __ __ [ ( n l s . / r l s . r $ . ) - ( n i s . / r ~ . r l s . ) ]

+ ( - - 1 ) m a o 2 $ : $ [ ( n ~ 2 s / r 2 s r s ) - (n~ / r2r2s ) ] , (8.8)

where $ is an al ternat ing string of m digits beginning with 1, and $* is its complement (obta ined f rom $ by t ransposing l ' s and 2's). The following identities, derived and presented in BRY [14] , will enable us to express (8.8) and the entire electric field series in terms of bispherical coordinates.

- 1 r l $

--1 r25

n'~ = x r ~ 1,

n~ms = z l s r~s 1,

n~$ = zzsr~s a,

Dls.:**= D2s:s = c sinh #o {s inh[ (m + 1)/~o ] s inh(m#o)} -1 (8.9)

= c - 1 sinh(npo) [cosh # - cos 0 ] 1/2 { cosh [# - ( - 1 )n 2n#o ] - cos 0 } - 1/2

(8A0)

= e - 1 sinh(n#o) [cosh p - cos 0] 1/2 { cosh [# + ( - 1 )n 2n#o ] _ COS 0 } 1/2

(8.11)

n y = Y r s I (8.12)

with Zls = c(cosh # - cos 0) ( - 1)" [ s inh(n#o)] - 1

x { c o s h ( n l a o ) c o s O - c o s h [ l ~ - ( - 1 ) " n # o ] } (8.13)

with z2$ = - e(cosh # - c o s 0) ( - 1)" [ s inh(n#o)] -1

x { c o s h ( n # o ) c o s O - c o s h [ l ~ + ( - 1 ) n n # o ] } (8.14)

In (8.10), (8.11), (8.13), and (8.14), 15 and 25 are valid strings consist ing of a total of n digits.

We calculate first the m th term in the series for the x -componen t , T~,; the o ther componen t s will then follow similarly. F r o m (8.8), (8.9), and (8.12) we have

T~, = Q D 2 s : s ( - 1 ) m x [ - r l s 3 r s - . 1 - 3 - 1 + r $ . r ls . + r ~ 3 r s 1 -3 -1 - - r s r2s ]. (8.15)

34 JEFFREY M. BOWEN

Note that 15" and 25 are strings of m + 1 digits while $* and $ have only m. Observing that cosh[/~ + ( - 1)m 2m/~ ° ] = cosh[# - ( - 1 ) '~ ( - 2m/~o) ], we can express (8.15), after much tedious algebra, as

T~ = Qc - 2 sinh #o sin 0 cos ~b (cosh/~ - cos 0) ( - 1 )"

x { - sinh2[(m + 1 ) #o ] F3- (m + 1)Fro "-]- sinh2(m#o) F3,, F - (m + 1)

+ s i n h 2 [ ( m + 1)/~o] Fmm+lF_m -sinh2(m#o)F3_mF,,,+t } (8.16)

where Fm is defined by

Fm = {cosh[/~ + ( - 1 )m 2m/~o ] -- cos 0} - 1/2. (8.17)

The first term in the curly braces of (8.16) can be obtained from the second by the rule m,--, - (m + 1) with an overall sign change; likewise for the third and fourth. Thus the full series becomes

E x = [ J (V1 + V2)] x = Q a - 2(sinh/~o)-1 sin 0 cos ~b (cosh # - c o s 0)

x ~ ( -1)msinh2(m#o) [ F 3 m F _ ~ , , + I ) - r 3 m F m + l ] . (8.18) m = - - o o

Similarly E y = E x tan ~b for this axisymmetric problem. To calculate E z requires more care since n~ is rather complicated. The result is

E z = [ J ( V 1 "[- V 2 ) ] z = Qa - 2(sinh #o) - 1 (cosh # - cos 0)

× ~ sinh(mpo )[GmF3mF (m+l )+G_mF3_mFm+l] , (8.19) m = - - o o

where Fm is as defined previously and Gm is given by

Gm = cosh[/~ + ( - 1)m m#o] _ cosh(m#o) cos 0. (8.20)

IX. DISCUSSION

This paper solves the problem of how to specify useful initial data for the coupled Einstein-Maxwell system in which there are no magnetic fields. The key approach is to specify data on a manifold M s (two asymptotically flat regions connected by N Einstein-Rosen bridges) in such a way that the top and bottom universes are physically equivalent. This restriction requires that the 3-metric, extrinsic curvature, and electric field all have the property of inversion symmetry through each of the throats. The program presented here allows one for the first time to specify indepen- dently the charge, linear momentum, angular momentum, and radius of inversion of each of N bodies.


An analysis of the known background initial data and the remaining non-linear constraint [-Eq. (4.7)] shows that the N bodies are in fact charged black holes. We have seen that net electric flux is associated with any closed surface surrounding a "hole" in the upper sheet. In addition, the conformal factor ~ that solves the boundary problem (4.7)-(4.8) describes the final physical geometry. It turns out that the boundary condition (4.8) implies that each r~ = a~ is an extremal two-surface, thus signaling the presence of a black hole.

It should be pointed out that although we have chosen B = 0 as part of the initial data, this condition cannot be maintained in the subsequent evolution. Recalling that the physical field E i is related to the background field E i by E i = ~,-6Ei, w e

have

(V X g ) p h y s = I / / - 8 [ V X E - 2(g In ~b) x E ] b a s e. (9.1)

Since the base electric fields constructed here have non-zero curl, Eq. (9.1) almost surely implies (V x J~)phys ~ 0. Then from Faraday's law, which for B = 0 at t = 0 reduces to

O 63"-7 Bphys = - - (V × E)phy s

for any choice of time direction away from the initial slice, we see that the magnetic field will not stay zero, which is reasonable since the holes will start to move.

Finally, consider the case of initially moving charged black holes. Here the requirement that B = 0 on the initial slice appears somewhat contrived. Electric fields, extrinsic curvatures, and conformal factors can still be calculated, but a more complete and realistic treatment would include the current densities generated by moving charges and their associated (initial) magnetic fields. However, the initial presence of magnetic fields as well as electric fields would bring a whole new level of complexity. Once could no longer assume a zero-valued Poynting vector, which in turn means that the gravitational momentum constraint (3.7) would become inhomogeneous throughout the region of interest and the KSY momentum solutions would no longer be valid. The implications of this extra complexity are to be the subject of a later investigation.

APPENDIX

We here complete the surface integral calculation of Section VI. We need to evaluate

[ (~ Tid2Si= ~ QD=$:$ ~ r2srsJ l

36 JEFFREY M. BOWEN

For convenience let the origin be at C$ and take the integration surface as a sphere of radius ro centered there. Then in (A.1) we have

and

r$ = r o

d2Si = &onis dg2

D~$:$ = I C~$ - C $ [ = IC~$ [ = D

2 + O 2 _ 2r ° D cos 0 r~$2 = [ x - C ~ $ [2= [ x [ 2 + [ C ~ $ 1 2 - 2 x ' C ~ $ = r o

nisn~ = (x i - Ci$) (x i - Cis)/r,$r$ = (r2o - ro D cos O)/r~$ro

where 0 is measured from the line connecting the two image charges. Inserting these into (A.1), we find

r i d 2 S i = Q D ~ [r;$1 -roni~$n~r~-~ 2 ] dO

fo = 2 ~ Q D [(&o + D 2 - 2 r o D c o s O ) - m

- (&o - roD cos O) (&o + D2 - 2roD cos 0) - 3/2] sin 0 dO.

Evaluating the 0-integral gives (2/D) for r o < D and 0 for ro > 1). Thus

f T i = 4ztQ ro < d2S~ for D

= 0 for ro > D .

For the proof of the divergence property (5.5), we will need the derivatives of R~ and the chain rule for O/Ox i. Short calculations give

(O/c3x') ( R ~ ) = c~/c~xi(6 U - 2n~n~).i j

. i ~ j _ , , j _ , j =-2(6 - - n a n , ) n ~ / r , 2 n ~ ( 6 n ~ n , ) / r ,

= - 4 n ~ / r ~

and

O/Ox'= [c3(J~x)k/c~x i] [0/c~(J~x) k] = (a~/r~) R~O/O(J~x) k

INITIAL DATA FOR CHARGED BLACK HOLES

f r o m (4.2). N o w e x p a n d the left side o f (5.5):

O/Oxi[ ~ E i (x ) ] = _ t3/Oxi[ ( a , / r~) 4 RiY EY( J~x ) ]

4 5 = 4 ( a J r ~ ) n i R i y E y - (a , /r~) 4 (t~/~3xiR iy) E y

- (a~/r~)4R~(O/Ox i) E:

4 5 j j 4 5 - 4 ( a J r ~ ) n~E + = 4(a~/r~) nY~E y

- (a~/r~) 4 (a~/r~) 2 R~ik R~iJ 0/t3(j~ x)k E j

= -- (a~/r~) 6 [agJ(J~x) / t~(J~x)J] .

37

Q . E . D .

ACKNOWLEDGMENTS

I would especially like to thank James W. York, Jr., for his continuing support and helpful comments on this work. I also thank Joel Rauber and Steve Becker for their useful discussions, Bruce Holenstein for preparing the figures, and Ann Libby for preparing the manuscript.

REFERENCES

1. M. D. KRUSKAL, Phys. Rev. 119 (1960), 1743-1745. 2. C. W. MIsr,rER, Phys. Rev. 118 (1960), 1110. 3. C. W. MISr,rER, Ann. Phys. (N.Y.) 24 (1963), 102. 4. R. W. LINDQUIST, J. Math. Phys. 4 (1963), 938. 5. J. M. BOWEN, Doctoral thesis, University of North Carolina, Chapel Hill, 1979, unpublished. 6. J. M. BOWEN AND J. W. YORK, JR., Phys. Rev. D 21 (1980), 2047. 7. T. PIRAN AND J. W. YORK, JR., in "Spacetime and Geometry, the Alfred Schild Lectures" (R.

Matzner and L. Shepley, Eds.), pp. 147-176, Univ. of Texas Press, Austin, 1982. 81 J. M. BOWEN, J. Comput. Phys. 56 (1984), 42. 9. A. KULKARNI, L. SHEPLEY, AND J. W. YORK, JR., Phys. Lett. A. 96 (1983), 228.

10. C. W. MISNER AND J~ A. WHEELER, Ann. Phys. (N.Y.) 2 (1957), 525-603. 11. A. KULKARNI, J. Math. Phys. 25 (1984), 1028-1034. 12. N. O'MURCHADKA AND J. W. YORK, JR., Phys. Rev. D 10 (1974), 437. 13. L. SMARR, A. CADEZ, B. DEWlTT, AND K. EPPLEY, Phys. Rev. D 14 (1976), 2443. 14. J. M. BOWEN, J. RAUBER, AND J. W. YORK, JR., Classical Quantum Gravity 1 (1984), 591-610.

inversion symmetric initial data for n charged black holes

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