Mark A. Scheel. November is dynamical even in spherical symmetry, where it can contain gravitational radiation. Our code is

Transcription

1 Collapse to Black Holes n Brans-Dcke Theory: I. Horzon Boundary Condtons for Dynamcal Spacetmes Mark A. Scheel Center for Radophyscs and Space Research and Department of Physcs, Cornell Unversty, Ithaca, New York Stuart L. Shapro and Saul A. Teukolsky Center for Radophyscs and Space Research and Departments of Astronomy and Physcs, Cornell Unversty, Ithaca, New York November 1994 ABSTRACT: We present a new numercal code that evolves a sphercally symmetrc conguraton of collsonless matter n the Brans-Dcke theory of gravtaton. In ths theory the spacetme s dynamcal even n sphercal symmetry, where t can contan gravtatonal radaton. Our code s capable of accurately trackng collapse to a black hole n a dynamcal spacetme arbtrarly far nto the future, wthout encounterng ether coordnate pathologes or spacetme sngulartes. Ths s accomplshed by truncatng the spacetme at a sphercal surface nsde the apparent horzon, and subsequently solvng the evoluton and constrant equatons only n the exteror regon. We use our code to address a number of long-standng theoretcal questons about collapse to black holes n Brans-Dcke theory.

2 I. INTRODUCTION In recent years, there has been renewed nterest n scalar-tensor theores of gravtaton. One reason s that these theores are mportant for cosmologcal naton models[1], n whch the scalar eld allows the natonary epoch to end va bubble nucleaton wthout the need for ne-tunng cosmologcal parameters (the \graceful ext" problem). In addton, scalar-tensor gravtaton (\dlaton gravty") arses naturally from the low-energy lmt of superstrng theores[; 3]. Fnally, wth the constructon of LIGO, t may be possble to test scalar-tensor theores to hgh precson[4] by lookng for monopole and dpole gravtatonal radaton from astrophyscal sources. Qute apart from ther potental physcal sgncance, scalar-tensor theores play another very useful role: they provde an deal laboratory for testng new algorthms for numercal relatvty. In general relatvty, numercal methods for treatng spacetmes contanng gravtatonal radaton requre at least two spatal dmensons, snce a tme-varyng quadrupole moment s needed to produce gravtatonal waves. In scalartensor theores, one can study many of the same strong-eld phenomena that occur n general relatvty, ncludng gravtatonal radaton and dynamcal black holes, whle stll workng n sphercal symmetry. We have developed a numercal code that solves the coupled matter and gravtatonal eld equatons for the evoluton of a sphercally symmetrc conguraton of nonnteractng partcles n Brans-Dcke[5] gravtaton, the smplest of the scalar-tensor theores. We use ths code to study gravtatonal collapse to a black hole n Brans-Dcke theory. Ths process has been dscussed extensvely n the lterature[6], but these studes have been lmted to addressng the nal state of the black hole after collapse, or have used lnearzed approxmatons of the eld equatons. Other than an early smulaton by Matsuda and Nara[7], t s only very recently[4] that ths process has been calculated n any detal. In constructng numercal models of gravtatonal collapse n Brans-Dcke theory, we have been forced to address the same dculty that has plagued the eld of numercal relatvty for the last 30 years: how does one handle the spacetme sngularty at the orgn that nevtably develops durng the formaton of a black hole? The tradtonal approach has been to utlze the \many-ngered tme" gauge freedom of general relatvty to avod the sngularty altogether. Speccally, one chooses coordnates such that the passage of proper tme grnds to a halt near the orgn before the sngularty appears, whle weak-eld regons of spacetme farther from the orgn evolve farther nto the future. Ths sngularty-avodng (SA) method works well for short tmes, but eventually pathologes develop n the transton regon between the \frozen" nteror and the \evolvng" exteror. These typcally take the form of steep gradents or spkes n the metrc functons, and wll eventually cause the numercal code to crash[8]. Countermeasures such as ncreasng the grd resoluton produce lttle mprovement because the pathologes ncrease exponentally wth tme. Our soluton to ths problem s to use an apparent horzon boundary condton (AHBC) method after the formaton of a black hole. The basc dea of ths approach s to truncate a black hole spacetme at a surface nsde the apparent horzon (AH) that cannot causally nuence the exteror. One then dscards the sngular nteror entrely, and only evolves the physcally relevant exteror. Sedel and Suen[9] have mplemented ths dea n general relatvty for the case of a black hole wth a Klen-Gordon eld n sphercal symmetry. Ther method nvolves a coordnate system that s locked to the AH, so that the coordnate speed of radally outgong lght rays nsde the AH s negatve. Ths enables them to use a causal derence scheme, smlar to the Causal Reconnecton Scheme of Alcuberre and Schutz[10], to solve evoluton equatons n such a way that nformaton does not escape from the black hole, and no explct boundary condton s needed on the AH. Our AHBC method s derent from that of Sedel and Suen. Although we also use a coordnate system that s locked to the AH, we solve the wave equaton for the Brans-Dcke scalar eld usng an mplct derencng scheme motvated by the work of Alcuberre[11]. In addton, we solve for the metrc varables usng the ellptc constrant equatons rather than the evoluton equatons. We obtan the requred boundary condtons for ths approach from asymptotc atness, propertes of the apparent horzon, and by solvng a sngle evoluton equaton only on the AH, as explaned n Secton IV.

3 Wth our code we are able to follow accurately Brans-Dcke collapse to black holes and the assocated generaton of monopole gravtatonal waves. We are able to ntegrate the equatons to arbtrarly late tmes nto the future, when all of the radaton has propagated out to large dstances and the central black hole has settled down to nal equlbrum. Usng our code we are able to resolve a number of long-standng, theoretcal questons about collapse n Brans-Dcke theory. We are also able to rene a promsng technque for evolvng black hole spacetmes wth radaton by ntegratng only n the observable regons of spacetme. Ths paper s prmarly concerned wth numercal methods. The reader not nterested n the numercal detals should read secton II and then skp to Paper II[1], n whch we dscuss Brans-Dcke gravtaton n more detal, and we show how black holes formed n Brans-Dcke theory behave derently than those n general relatvty. II. BASIC EQUATIONS A. Brans-Dcke Theory The acton for Brans-Dcke gravtaton s[5] Z I = L BD (0g) 1= d 4 x; (:1) where the Lagrangan densty s L BD = g ab R ab + 16 c 4 L 0! b : (:) The couplng constant! s dmensonless, and the scalar eld has dmensons of G 01, where G s Newton's gravtatonal constant. The Lagrangan densty L for matter and nongravtatonal elds depends on the metrc g ab but not on. The Rcc tensor R ab s related to the metrc n the usual way. In general relatvty, the thrd term n Eq. (.) s absent, and one sets = G 01. In other scalar-tensor theores[13], the Lagrangan s more complcated. In partcular, the couplng parameter! can be a functon of. In general relatvty we are free to choose our unts of mass and tme such that G = c = 1. In Brans- Dcke theory, the nverse of the scalar eld 01 plays the role of G, so multplyng by a global scalng factor changes the unt of mass. We choose unts such that where 1 s the value of the scalar eld far from any sources. c = 1 = 1; (:3) Varaton of Eq. (.1) wth respect to g ab and yelds the Brans-Dcke eld equatons, whch can be wrtten n the form = 8T 3 +! ; (:4) G ab = 8T 0 ab ; (:5) where 8T 0 ab 1 8T ab +! 0 ra r b 0 1 g abr a r a 1 + r a r b 0 g ab : (:6) 3

4 Here r denotes covarant derentaton wth respect to the metrc g ab, s the covarant Laplacan r a r a, and G ab s the usual Ensten tensor. The symmetrc tensor T ab s the energy-momentum tensor for matter and nongravtatonal elds, and T s ts trace: T ab g ab L 0 L ; (:7) gab T T a a = T ab g ab ; (:8) r b T ab = 0: (:9) Although we have wrtten the eld equatons (.5) n a form that resembles Ensten's equatons, we emphasze that Brans-Dcke gravtaton s not the same as general relatvty wth a Klen-Gordon scalar eld. The derence s the factor of n the rst term n the Lagrangan densty (.), whch leads to second dervatves of n the eld equatons (.5). Physcally, ths manfests tself as a volaton of the weak equvalence prncple for massve bodes[14]. Ths s dscussed n more detal n Paper II. Notce that the matter stress-energy s conserved (Eq..9), even though T ab s not equal to 8G ab (the quantty r b T 0ab also vanshes because of the Banch dentty r b G ab = 0). As a result, the equatons of moton of matter do not nvolve the scalar eld; test partcles move on geodescs of the metrc. Notce also that t s the trace of the matter stress-energy tensor T ab, not the eectve tensor T 0 ab, that appears n the wave equaton (.4). In vacuum, = constant s a soluton of Eq. (.4) for any!. In ths case, T 0 ab reduces to T ab and the eld equatons (.5) become Ensten's equatons. Therefore, any vacuum soluton of Ensten's equatons s also a vacuum soluton of the Brans-Dcke equatons. In addton, many (but not all) Brans-Dcke solutons wth j!j! 1 have! constant and obey Ensten's equatons. It s therefore sad (but not rgorously correct See Paper II) that Brans-Dcke theory reduces to general relatvty n the lmt j!j! 1. B. Equatons (.5) n (3+1) form Adoptng the usual ADM[15] (3 + 1) decomposton, we ntroduce a set of spacelke hypersurfaces, or tme slces, and a tmelke vector eld n a normal to these hypersurfaces. We adopt the conventon that the 4-metrc g ab has sgnature (0 + ++), so that n a n a = 01: (:10) Spatal dstances on a partcular tme slce are measured by the 3-metrc ab, dened by ab g ab + n a n b : (:11) The extrnsc curvature K ab descrbes the rate of change of the three-metrc along n a (the \tme" drecton): K ab 0 1 $ n ab : (:1) Here $ denotes a Le dervatve. The eld equatons are splt nto spatal constrants that relate K ab and ab on each tme slce, and rst-order (n tme) evoluton equatons that take K ab and ab from one slce to the next. We work n sphercal symmetry, and we choose the maxmal tme slcng and sotropc radal coordnate condtons. Ths gauge s dened by the sotropc lne element ds = 0( 0 A ) dt + A dr dt + A (dr + r d ) (:13) and the maxmal slcng condton K = K ;t = 0: (:14) 4

5 Here K s the trace of the extrnsc curvature K ab. The 3-dmensonal metrc j on each t = constant tme slce s gven by (3) ds = A (dr + r d ): (:15) Eqs. (.10) and (.11) mply n a = (0; 0; 0; 0): (:16) The lapse functon measures the rato of proper tme to coordnate tme for a normal observer: d = dt: (:17) The shft s the velocty of the spatal coordnates wth respect to normal observers. It s a vector quantty, but n sphercal symmetry only the radal component s nonzero ( a = 0 for a 6= r), so we wrte r. The shft s crucal for our AHBC method: n order for a coordnate grd pont to have no causal eect on the regon exteror to that pont, the coordnate speed of radally outgong photons must not be postve, and ths requres a (postve) shft. The maxmal slcng condton (.14) s mportant for our conventonal SA method because t causes the lapse functon to become small n the strong-eld regon of a spatal slce that s about to ht a sngularty. Ths slows down the passage of proper tme n ths regon whle other regons on the slce propagate farther nto the future. In a typcal black hole spacetme, maxmal slces wll never encounter the sngularty at r = 0. For the AHBC method, maxmal slcng s not at all necessary, but t s convenent because t elmnates a component of K ab from the equatons. We choose the sotropc spatal gauge prmarly for convenence. The most general sphercally symmetrc metrc can be wrtten n the form ds = 0( 0 A ) dt + A dr dt + A dr + B r d : (:18) The sotropc gauge condton A = B elmnates B from the eld equatons. Ths choce s not necessary for ether the SA or AHBC methods. However, there are gauge choces (for nstance, 0) that would spol the AHBC method. Note that Sedel and Suen[9] do not use sotropc coordnates, but nstead choose the metrc (.18) and the gauge = 0. Ths choce s useful because the proper dstance between two coordnate rad remans xed n tme, but t does not seem to be necessary for the success of the AHBC method, at least n sphercal symmetry. In fact, t tends to complcate the analyss after matchng onto the SA method, especally n the lnearzed regme where t produces a frozen coordnate wave that adds gauge terms to the metrc varables. In maxmal sotropc gauge, the eld equatons (.5) break up nto two evoluton equatons, K T;t 0 K T;r = and four spatal constrant equatons A ;t A 0 A ;r A = r 0 1 K T; (:19) 8S r r K T 0 A ;r A 3 A ;r A + r 1 r (A 1= ) r ;r = 0 A5= ;r 4 0 ;r A (Ar) ;r Ar ; (:0) K T ; (:1) K T;r = 8S 0 (Ar) r 0 ;r 3K T Ar ; (:) 1 0 Ar 1 A 3 r ;r ;r K T T 0 ; (:3) r = 0 3 r ;r K T: (:4) 5

6 Here and elsewhere n ths paper, commas denote partal dervatves and K T K = K : (:5) Eqs. (.1) and (.) are the Hamltonan and momentum constrants. Eqs. (.3) and (.4) result from the maxmal slcng condton K T = 0K r r and the sotropc coordnate condton A = B, respectvely. It s mportant to note that Eqs. (.19){(.4) are not all ndependent: one may use the Banch denttes to elmnate two of the sx equatons. The eectve source terms appearng n Eqs. (.19){(.4) are dened by T 0 g ab T 0 ab; 0 n a n b T 0 ab; S 0 a b a nc T 0 bc ; S 0 a b an c T 0 bc; S 0 ab c a d b T 0 cd; (:6) where T 0 ab s gven by Eq. (.6). Smlarly, matter source terms, T, S a, and S ab (wthout the prme) are dened as n Eqs. (.6), wth the unprmed T ab appearng on the rght hand sde of each equaton. If one replaces the prmed source terms n Eqs. (.19){(.4) wth ther unprmed counterparts, one recovers the Ensten equatons. C. Scalar Feld To solve the scalar wave equaton (.4) numercally, t s useful to dene the varables In our coordnate system, Eq. (.7) reduces to and the scalar wave equaton (.4) becomes 5 0$ n ; (:7) 8 ;r : (:8) ;t = 8 0 5; (:9) 8 ;t = 8 ;r + ;r 8 0 (5) ;r ; (:30) 5 ;t = 5 ;r + 8T 3 +! Ar 8 1 A 3 r : ;r (:31) Usng 5 rather than ;t as a dynamcal varable elmnates explct tme dervatves of and n Eq. (.31). Furthermore, lke the extrnsc curvature K ab, the quantty 5 s dened geometrcally (by Eq..7) on each tme slce, ndependent of how one gets from one slce to the next: n other words, ts value on a gven slce s ndependent of and. Ths s not true for ;t. Ths s mportant when makng the transton from the SA method to the AHBC method: the lapse and shft are dscontnuous across the tme slce that denes ths transton. The quantty 8 s a useful dynamcal varable for the AHBC method because t elmnates explct second dervatves from the wave equaton. However, n the SA method t s sucent to use as a dynamcal varable and to explctly compute ;r and ;rr by nte derencng when necessary. 6

7 Evaluatng the eectve source terms (.6) usng Eq. (.6) and the metrc (.13) yelds 8T 0 = 16T ( + 3=!) +! A ; (:3a) 8 0 = 8 +! A + 1 A 3 r 0 8Ar 1 ;r ; 8S r r 0 = 8S r r 0 8T (3 +!) +! A + 5K T + 1 A 8 A ;r ; (:3b) (:3c) 8S 0 r = 8S r 0 85! 0 5 ;r 0 8K T : (:3d) D. Lnearzed Equatons n Vacuum In the weak-eld regme, we can use lnearzed theory to descrbe the gravtatonal eld. By matchng our numercal varables to the lnearzed soluton, we can determne the gravtatonal radaton seen by an observer at nnty, and we can set boundary condtons at the outer edge of our nte-derence grd. These boundary condtons, ncludng the ones mposed on ellptc equatons, are vald even whle the wave s passng through the boundary. Such a matchng technque was ntroduced n numercal relatvty by Abrahams and Evans[16]. Dene the new varables a A 0 1; (:33) 0 1; (:34) 0 1: (:35) If we set all matter terms to zero n Eqs. (.19){(.4) and Eqs. (.9){(.3), and f we keep only terms lnear n a, K T,,,, 8, and 5, we obtan ;tt = 1 r 0 r ;r 1;r ; (:36) 5 = 0 ;t ; (:37) 8 = ;r ; (:38) a ;t = r 0 1 K T; (:39) K T;t = 0 ;r r 0 a ;r + 8 ;r ; (:40) r a ;rr = 0 a ;r r + 8 ;r 0 8 r ; (:41) K T;r = 0 3K T 0 5 ;r ; (:4) r 0 r ;r 1;r = 0 r 1 ;r ; (:43) ;r r = 0 3 r ;r K T: (:44) If only outgong waves are present, the soluton of Eq. (.36) s r = f(t 0 r); (:45) 7

8 where f s an arbtrary functon of (t 0 r). We now nsert Eq. (.45) nto Eqs. (.39){(.44), and mpose the boundary condtons = a = 0 at r = 1. We nd the soluton: A = 1 + a = 1 + M T r = 1 + = 1 + C(t) r = rk T K T = f0 (t 0 r) r f(t 0 r) 0 ; (:46) + r f(t 0 r) ; (:47) r 0 f0 (t 0 r) ; (:48) Z t MT + 3f(~ t 0 r) + C( t) ~ d t: ~ (:49) 3f(t 0 r) r + 1 r 3 Here C(t) s an arbtrary functon of tme, M T s a constant, and a prme denotes a dervatve wth respect to the argument. The gauge functon C(t) results from the presence of a nonzero shft. 0 E. Masses In order to dene the mass of a sphercal body n Brans-Dcke theory, t s convenent to transform the soluton (.45){(.49) nto a smpler gauge. Let where 0 = 1 r Z t h LT ab = h MI ab + a;b + b;a ; (:50) 0 M T + 3 f( ~ t 0 r) + C( ~ t) d ~ t; (:51) = 0: (:5) Here h MI ab s the metrc perturbaton n maxmal sotropc gauge, and hlt ab Thorne (LT) gauge[16]. In LT gauge, we have s the perturbaton n Lorentz- h LT 00 = M T f(t 0 r) + ; r r (:53) h LT 0 = 0; (:54) M h LT j = T j 0 r = 1 + Note that there s no shft n LT gauge. f(t 0 r) r ; (:55) f(t 0 r) : (gauge nvarant) (:56) r For a statc stuaton, f s a constant, so t wll appear as an addtonal \mass" n the metrc. We wll therefore wrte f M S = constant; tme-ndependent case: (:57) Hence, 00 = M T + M S ; r (:58) = 0; (:59) h LT h LT 0 h LT j = j M T 0 M S r ; (:60) = 1 + M S r : (:61) 8

9 We see from Eq. (.58) that a test partcle n a Kepleran orbt measures a total mass M equal to M T + M S. The \scalar mass" M S s the porton of the actve gravtatonal mass produced by the scalar eld. As dscussed further n Paper II, the \tensor mass" M T s the mass measured by a test black hole n a Kepleran orbt. In general relatvty, M S = 0 and M T = M. Lee[17] has derved expressons and conservaton laws for the tensor and scalar masses n terms of superpotentals. He denes the gauge-nvarant quanttes M T 1 Z (0g) 0 g 00 g j 0 g 0 g 0j13 16 ;j d 6 ; (:6a) M T 0 M S 1 16 Z (0g) 0 g 00 g j 0 g 0 g 0j13 ;j d 6 ; (:6b) whch n the tme-ndependent case reduce to M T and M T 0 M S, respectvely. Here we assume Cartesan coordnates, and 6 s the two-dmensonal area element n the asymptotc rest frame of the source. Evaluatng Eqs. (.6) usng the metrc (.13), we obtan M T = 1 16 M T 0 M S = 1 16 Z 0 A 41 ;r r d = 0 r 4 Z 0A 4 1 ;r r d = 0 r 4 0 A 41 ;r ; (:63a) 0 A 4 1 ;r : (:63b) In the lnearzed regme, we can expand these expressons to rst order n ( 0 1) and (A 0 1). The result s M T = 0r A ;r 0 1 r 8; M S = 0 1 r 8: (:64a) (:64b) Further smplcaton usng the lnearzed equatons (.45){(.49) yelds M T = M T ; (:65a) M S = f + rf0 : (:65b) Although these reduce to M T and M S n the tme-ndependent case, the scalar mass s not well-dened durng dynamcal epochs: M S approaches rf 0 (t 0 r)= as r! 1. The scalar mass has other unusual propertes, whch are dscussed n detal by Lee[17]. For example, the scalar mass s not postve dente, and scalar mass carred by a gravtatonal wave does not curve up the background spacetme. Because the tensor mass M T does not suer from such dcultes, t s the quantty that most deserves to be called \mass". The quantty M T s postve dente, can only decrease by the emsson of gravtatonal radaton, and has other energy-lke propertes, unlke the scalar mass or the actve gravtatonal mass M. Gauge-nvarant formulas for the uxes of M T and M S can be expressed n terms of the Landau-Lfshtz pseudotensor and other pseudotensors nvolvng the scalar eld[17]. In the case of sphercal symmetry, t s easer to obtan these expressons by derentatng Eqs. (.63) wth respect to tme and usng the eld equatons to elmnate second dervatves of the metrc. The two methods must gve the same answer. The result, to second order n the ampltudes, s 0M T;t = r r (6A ;r + 38) 0 5 8K T + 58(! 0 1) 0 (5 + K T )(4A ;r + ;r ); (:66a) 0(M T 0 M S ) ;t = 6A ;r 0 8 r ;r K T +!58 0 K T (4A ;r + ;r ) r r + 5 ;r ( a); (:66b) 0M S;t = 58 r r + 8 ;r 0 7 8K T (4A ;r + ;r ) 0 5 ;r ( a); (:66c) 9

10 where the varables, a and are dened by Eqs. (.33){(.35), and we assume that the uxes are evaluated n vacuum. Notce that M T;t = 0 to rst order. Ths s because M T s strctly constant n lnearzed theory. However, there s a nonzero rst-order contrbuton to M S;t (the 5 ;r term). If we convert the surface ntegrals n Eqs. (.63) to volume ntegrals by Gauss' theorem, and we elmnate second dervatves of A usng the Hamltonan constrant, we obtan M T = 1 Z 1 8~A A6 (K T ) +! A 4 +! 5 A A 3 A ;r 0 7 A (A ;r ) r dr; (:67a) M T 0 M S = 1 Z 1 8~A A6 (K T ) +!A4 0 5 A A3 A ;r 0 7A (A ;r ) + A4 8 ;r + A4 8 r dr: r (:67b) These expressons wll gve us an addtonal check on our numercal code. Because the regon of ntegraton contans the orgn, ths check s only useful n the SA method. F. Lght Rays and Apparent Horzons From the metrc (.13), one can determne the coordnate velocty of radal lght rays and the ngong and outgong radal null vectors dr dt = 6 0 ; (:68) 6 A N a 6 = 1; 6 A 0 : (:69) A margnally trapped surface s dened by where A s the area of a (t; r = const) surface: In maxmal sotropc gauge, Eq. (.70) takes the form where we have used Eq. (.19) to elmnate the tme dervatve of A. r N +A = 0; (:70) A = 4r A : (:71) 1 r + A ;r A = 1 AK T; (:7) The apparent horzon s the outermost margnally trapped surface. In our standard SA method, we use Eq. (.7) to locate trapped surfaces. In the AHBC method, we place a grd pont on the apparent horzon and use Eq. (.7) as a boundary condton at that grd pont. For the AHBC method, we also need a relaton that forces the apparent horzon to reman at a constant grd pont as the metrc varables and scalar eld evolve n tme. Such an equaton can be obtaned by derentatng Eq. (.7) wth respect to t (holdng r constant), and then usng Eqs. (.19){(.) to elmnate 10

11 tme dervatves, second radal dervatves of A, and rst radal dervatves of K T. After some algebra, we arrve at the result A = where the prmed source terms are gven by Eq. (.3) A r ( 0 + Sr=A) A r (S r r 0 + S 0 (:73) r =A); III. SINGULARITY-AVOIDING (SA) METHOD Here we present our sngularty-avodng numercal method for solvng the Brans-Dcke equatons for a sphercally symmetrc dstrbuton of dust. Except for the addton of the Brans-Dcke source terms n the eld equatons, ths method s very smlar to that of Ref. [18]. We use a mean eld partcle smulaton scheme n whch the collsonless matter dstrbuton s sampled by a nte number of nonnteractng partcles, and the gravtatonal eld varables are dened on a nte number of grd zones. At each tme step, the partcles are moved accordng to the geodesc equaton. By bnnng the partcles, we calculate the source terms, S r, and T n each grd zone. The source term T s used to determne the scalar eld va the wave equaton, whch s solved by nte derencng on the grd. All three source terms appear n the constrant equatons, from whch we determne the metrc varables. The process s then repeated for the next tme step. A. Matter Source Terms We sample the matter by a nte number of nonnteractng partcles. If each partcle has rest mass m, 4-velocty u a, and comovng number densty, the stress-energy tensor s gven by T ab = m u a u b ; (3:1) where the sum s over all of the partcles. The source terms dened by Eq. (.6) are = m (u 0 ) ; (3:) T = 0 m ; (3:3) S r = 0 m (u 0 )u r ; (3:4) S r r = m u r =A : (3:5) The comovng number densty of each partcle s proportonal to a delta functon for pont partcles, but can be treated as a contnuous quantty f the number of partcles s large and the partcle dstrbuton s smooth. In evaluatng the source terms, we average over all partcles n each grd zone, so that we can thnk of each partcle as occupyng the entre volume of the zone. Therefore, we wrte 1 = 4(u 0 )A 3 r 1r; (3:6) where 1r s the wdth of a grd zone. Notce that the source terms n Eqs. (3.){(3.5) depend on the metrc functon A. Snce we use these source terms to solve for the metrc functons, t s helpful to dene quanttes that can be calculated from 11

12 the partcle varables alone, wthout referrng to the metrc. Ths s especally mportant when solvng the nonlnear Hamltonan constrant (.1) for A. We dene ~ A 3 = ~T T A 3 = 0 ~S r S r A 3 = 0 ~S r r S r ra 5 = m(u 0 ) 4r 1r ; (3:7) m 4(u 0 )r 1r ; (3:8) mu r 4r 1r ; (3:9) mu r 4(u 0 )r 1r : (3:10) Although the metrc functon stll appears n these expressons, t s always multpled by u 0. The quantty u 0 depends only weakly on the metrc (Eq. 3.1). Smply evaluatng the sums n Eqs. (3.7){(3.10) n each grd zone gves reasonable values for the source terms n that zone. Even better s to use the algorthm of Hockney and Eastwood[19] to dstrbute each partcle's rest mass among ts own grd zone and each of the two neghborng zones. Ths procedure reduces the stochastc uctuatons assocated wth havng only a nte number of partcles, gvng us smoother results[0]. B. Geodesc Equatons Snce the matter s made up of partcles, the equaton of moton r a T ab = 0 reduces to the geodesc equaton for each partcle. Usng the metrc (.13), we wrte the geodesc equaton as dr dt = u r A (u 0 ) 0 ; du r dt The normalzaton of four-velocty gves us (3:11a) = 0(u 0 ) ;r + u r ;r + u r A ;r (u 0 ) A + u 1 3 (u 0 )A r r + A ;r A : (3:11b) u 0 = 1 + u r A + u A r 1= : (3:1) These equatons are solved for the varables r, u r, and u 0 for each partcle at each tme step by an embedded fourth-fth order Runge-Kutta scheme wth adaptve step sze[1]. The quantty u s a constant of the moton. Dervatves of the metrc that appear n Eqs. (3.11){(3.1) are obtaned on the grd by nte derencng. The values of these dervatves and the metrc functons at the partcle poston are then determned by nterpolatng from the grd. C. Scalar Wave Equaton After movng the partcles and determnng the source terms, we proceed to the wave equaton. In order to mnmze roundo error n the nearly GR regme ( close to unty), we use the varable 0 1: (3:13) 1

13 The wave equaton can then be wrtten ;t 0 r ;r = 05; 5 ;t 0 r5 ;r = (3:14a) 8 T ~ A 3 (3 +!) r 3 1 A A 3 ;r ;r 3 : (3:14b) These are Eqs. (.9) and (.31) wrtten wthout explct use of the varable 8. It s not necessary to use 8 as a dynamcal varable untl we ntroduce the AHBC method n Secton IV. The regularty condton at the orgn s Snce and 5 behave lke ;r = 5 ;r = 0: (3:15) C 1 + C r ; C 1 ; C constant (3:16) for small r, we take radal dervatves wth respect to r rather than r n Eqs. (3.14). Ths ensures that our nte derence equatons yeld the correct result near the orgn[]. The outgong-wave condton at nnty s (ry ) ;r + (ry ) ;t = 0; (3:17) where Y s ether 5 or. For sphercal symmetry, Eq. (3.17) s exact n lnearzed theory, as one can verfy by substtutng Eq. (.45). We solve Eqs. (3.14) by an explct staggered leapfrog scheme that determnes and 5 at tmestep n + 1 gven ther values at tmesteps n and n01. No nformaton at tmestep n+1 s needed to solve the equatons. After determnng and 5, we calculate from Eq. (3.13). Fnte derence approxmatons are presented n Appendx A.1. D. Constrants After determnng the quanttes, 5, and, we then solve for A and K T. By ntroducng the new varables Z A 3 r 3 K T ; (3:18) A 1= ; (3:19) we can rewrte the momentum constrant (.) and the Hamltonan constrant (.1) n the form 5Z ;r 5 = 8 Sr ~ r 6 r 3 1= 1= ;r = 0 3 ;r 3 16r 6 7 Z 0 ~ ;r 0 6!5 ;r ; (3:0) 0! ! r ;r r 3 ;r 1 ;r 3 : (3:1) In general relatvty 5 = ;r = 0, so one can determne Z from Eq. (3.0) and then compute from Eq. (3.1). Ths s the reason for usng the varable Z nstead of K T. In Brans-Dcke theory, Eqs. (3.0) and (3.1) are coupled because of the last two terms n Eq. (3.0), so we must solve these equatons smultaneously. 13

14 Eq. (3.0) requres a sngle boundary condton. For regularty, we set Z = 0 at the orgn. Eq. (3.1) requres two boundary condtons. The regularty condton at the orgn s ;r = 0: (3:) At nnty, we match to the lnearzed soluton (.46). Derentatng ths equaton yelds whch does not requre knowledge of the tensor mass M T. (r ) ;r = 1 0 r5 4 ; (3:3) We determne the varable Z usng the momentum constrant (3.0), and we solve the nonlnear Hamltonan constrant (3.1) for usng the teratve scheme descrbed n Appendx A.. Because appears n Eq. (3.0), we must recompute Z at each step n the teraton. We obtan an ntal guess for from the evoluton equaton (.19), whch we wrte as ;t 0 ;r = r K T: (3:4) After determnng Z and, the varables A and K T are found from Eqs. (3.18) and (3.19). Fnte derence forms of Eqs. (3.0){(3.4) are presented n Appendx A.. E. Lapse and Shft Havng determned the scalar eld varables and the spatal metrc varables, we can now calculate and. The lapse equaton (.3) can be wrtten 6 0 r 3 1 A A 3 ;r = 6 0 ;r r A A 3 ;r + 8 ;r 3 ~ + ~T + 3=! +!5 + 3 K T: (3:5) The regularty condton at the orgn s ;r = 0: (3:6) We match to the lnearzed soluton to obtan the boundary condton at nnty: by derentatng Eq. (.47), we obtan (r) ;r = 1 + r5: (3:7) Ths condton s ndependent of the gauge functon C(t) that appears n Eq. (.47). After solvng for, we calculate from the shft equaton (.4). boundary condton. We mpose Eq. (.48) at the outermost grd pont: Ths equaton requres a sngle = rk T + r5 : (3:8) Fnte derence forms of Eqs. (3.5){ (3.8) are gven n Appendx A.3. F. Grd Our numercal grd extends from r 1 = 0 to r max = r max. All varables are centered at half-grd ponts r +1=. In order to obtan a nearly constant number of partcles n each grd zone, we dvde the grd nto nner and outer regons. The nner regon, whch contans all the partcles, extends from r 1 to r part where 14

15 r part s the grd pont just outsde the outermost partcle. The grd pont r s chosen so that t contans a fracton 1= part of the rest mass. Spacng between other grd ponts n the nner regon s geometrc n r 3 : r r 3 r 3 0 r01 3 = r3 + 0 r 3 r r 3 : (3:9) Ths ensures that the grd spacng s close to unform n r 3, so that for a unform partcle dstrbuton each zone has approxmately the same number of partcles. The outer grd, whch extends from r part to r max, s matched smoothly onto the nner grd. The spacng between outer grd ponts s also geometrc n r 3. The grd s allowed to move at every tme step so that t follows the partcle dstrbuton. Each tme the grd s moved, all gravtatonal eld varables are nterpolated from the old grd onto the new one. Movng the grd allows us to place many grd ponts where they are requred to mantan accuracy. If the grd remaned statonary durng gravtatonal collapse, the partcles would soon end up only n the nnermost grd zones, nvaldatng our nte derence approxmatons. G. Identfyng Apparent Horzons To determne the locaton of an apparent horzon, we evaluate the quantty = 1 r + ;r 0 1 K T (3:30) at each grd pont. By Eq. (.7), all grd ponts wth < 0 are contaned wthn a trapped surface. Therefore, f r AH s the outermost grd pont wth < 0, then the apparent horzon les between r AH and r AH+1. We obtan the approxmate value of r AH by r r AH = r AH 0 AH+1 0 r AH AH : (3:31) AH+1 0 AH H. Intal Data Our ntal tme slce occurs at a moment of tme symmetry, so that K T = 5 = = 0. We rst calculate the ntal values of the metrc and scalar eld usng a method smlar to that of Matsuda[3]. Ths method s dscussed n Appendx B. It nvolves only ODEs, whch can be solved to arbtrary accuracy by standard numercal methods. After obtanng the ODE soluton, we re-solve for the ntal data usng the mean-eld partcle scheme. We rst randomly place partcles n the nteror accordng to the rest mass functon M rest (r) obtaned from the ODE soluton. We then compute the matter source terms by bnnng the partcles, and we solve for usng Eq. (3.14b ) wth 5 = = 0. Next we solve for and usng the Hamltonan constrant (3.1) and the lapse equaton (3.5). By comparng,, and wth the result of the ODE soluton, we obtan an mportant check on our method. I. Dagnostcs There are two evoluton equatons, Eqs. (.19) and (.0), that we dd not use n solvng the Brans-Dcke equatons (Although we used Eq. (.19) as an ntal guess for, we rened our guess usng the Hamltonan 15

16 constrant). We can test the accuracy of our code by comparng the left and rght-hand sdes of these equatons at each tme step. Another check s obtaned by mass conservaton. In the lnearzed regon far from the orgn, we calculate the quanttes M S and M T by Eqs. (.65), and compare wth the result gven by Eqs. (.66) and wth the volume ntegrals (.67). Ths s more useful than evaluatng the evoluton equatons at each tme step because t s senstve to errors that accumulate over tme. IV. APPARENT HORIZON BOUNDARY CONDITION (AHBC) METHOD Here we present our AHBC method of solvng the Brans-Dcke equatons for a spacetme that contans a black hole. Our method depends on two prmary ngredents. The rst s the exstence of what we call a coordnate causal horzon (CCH), whch s a coordnate surface nsde of whch the coordnate velocty of radally outgong lght rays s negatve, so that nstantaneously any pont nsde the CCH cannot causally nuence a pont n the exteror. Note that a CCH s coordnate dependent, and can exst even n Mnkowsk space by choosng a coordnate system that moves faster than lght. The second ngredent s a horzon-locked coordnate system, smlar to that of Sedel and Suen[9]. We place the AH at a xed radal coordnate r AH, and requre t to reman there at all tmes. We truncate our spacetme at some radal coordnate r 1 such that 0 < r 1 r AH, and we dscard the nteror regon. As long as a CCH exsts for some r r 1, dscardng the nteror cannot n prncple aect the evoluton of the exteror. In general relatvty, a horzon-locked coordnate system guarantees that the AH s on or nsde a CCH, snce a lght ray at the AH cannot propagate outwards. In ths case, one can set r 1 = r AH. In Brans-Dcke theory, we nd that ths s not the case, as dscussed further n Paper II. We therefore leave a small buer zone between r 1 and r AH, and check that a CCH s always present n the porton of spacetme that we retan. Even n general relatvty, ths buer zone s also useful for computng radal dervatves at the AH. We solve the wave equaton n a manner that takes advantage of the causal structure of the spacetme. We dene a \causal boundary" at some radal grd pont r CB whch concdes wth ether the AH or the CCH, whchever s smaller. Gven any r 0 r CB, our derence scheme ensures that the scalar eld at r 0 depends only on quanttes at r > r 0. Furthermore, no explct boundary condton s needed at the nner grd pont r 1. To obtan the metrc varables, A,, and K T, we solve spatal constrant equatons that requre a total of sx boundary condtons. Three of these are provded by matchng to the lnearzed soluton at the outer grd pont. The other three, n the SA method descrbed n the last secton, were obtaned by regularty at the orgn. In the AHBC method, however, we exclude the orgn from our spacetme, so these three boundary condtons must be mposed at the AH. We obtan two condtons by lockng the AH to a xed radal grd pont. One s the margnally trapped surface relaton (.7), and the other, Eq. (.73), s the requrement that the AH reman at a constant coordnate radus for all tme. A thrd condton could be obtaned by settng the tensor mass, M T, of the black hole to the value obtaned by mass conservaton, Eq. (.66a ). However, ths would prevent us from usng mass conservaton as a check on our numercal ntegratons. Instead, we use a derent approach: we use the evoluton equaton (.19) to update A on the AH. Another possblty would be to evolve the value of K T on the AH usng Eq. (.0), but ths would be more complcated. A. Matter As n the SA method, the partcles can be moved accordng to the geodesc equatons and the matter source terms can be computed by bnnng partcles nto zones. However, n the case of gravtatonal collapse 16

17 startng wth a unform densty prole, we wll nd that an AH does not form untl after all matter has fallen nto the black hole. For ths reason, we only need to solve the AHBC equatons n vacuum t s not necessary to nclude matter partcles n the code. Ths s a great advantage n terms of ecency, snce n the SA scheme most of the computer tme s spent movng the partcles, and t also gves us exblty n the grd choce, snce we no longer need to choose grd zones that are approxmately constant n volume. We emphasze that there s no fundamental restrcton that prevents us from ncludng matter n the AHBC method, and for some physcal stuatons not nvestgated n ths paper, e. g., the collapse of a dstrbuton wth a core-halo structure, an AH may form whle there s stll matter n the exteror regon. In such a case, ncludng partcles n the code would not be very dcult. For ths reason, n the followng sectons we wll contnue to nclude the matter source terms n our equatons. B. Grd For accuracy, t s usually desrable to choose the nte derence grd to be unformly spaced. However, a grd unformly spaced n r does not yeld enough grd coverage near the AH. Ths s because the ntal data for the AHBC method s determned by matchng onto the SA method after a horzon s formed, and the SA method typcally produces an apparent horzon radus r AH much less than the outer grd radus and metrc quanttes that vary exponentally wth r near the AH. It s therefore convenent to choose a logarthmcally spaced grd, and to take all radal dervatves wth respect to ln r rather than r. Ths choce places many grd ponts near the AH where they are needed, and stll allows us to take nte derences on a unformly spaced (n ln r) grd. For convenence, we dene the new varable ln r: (4:1) Our radal grd then conssts of the max ponts ( 1 ; ;...; AH;...; max). The AH s located at the grd pont = AH. The pont r max s chosen to be far from the black hole, n a regon where lnearzed theory s vald. All varables except 8 are dened on the grd ponts; 8 s dened on the half-grd ponts ( 3= ;...; max+1=). The staggerng of 8 s essental for our method of solvng the wave equaton. Unfortunately, spacng the grd unformly n ln r tends to produce too sparse a grd at large rad. A consequence of ths s that outgong waves get partally reected at the outer edge of the grd. To mnmze ths eect, we pck a maxmum threshold 1r 0 for the spacng between grd ponts. We choose our grd unform n unless ths choce would yeld a grd spacng (n r) greater than 1r 0 at the outer boundary. In the latter case, we set r max 0 r max01 equal to 1r 0, and choose the other grd ponts so that = : (4:) Ths gves us a grd that s as close as possble to beng unform n whle stll obeyng r +1 0 r all. 1r 0 for C. Wave equaton The wave equaton can be wrtten n the form ;t = 8 0 5; (4:3) 8 ;t = 1 r [8 ; + ; 8 0 (5) ; ] ; (4:4) 5 ;t = r 5 ; + 8T 3 +! ; ; r + ; : (4:5) 17

18 We solve Eqs. (4.4) and (4.5) for 5 and 8 by the causal method descrbed n Appendx C. Because ths method takes nto account the causal structure of the spacetme, t s not necessary to mpose explct boundary condtons at the nner edge of the grd. Boundary condtons at the outer edge of the grd are obtaned by matchng both 5 and 8 to the outgong-wave lnearzed soluton. Eqs. (.37), (.38), and (.45) yeld 5 ;t + 1 r 8 ; + 8 r 8 ;t + 1 r 8 ; + 8 r 0 5 r = 0; (4:6) = 0: (4:7) After determnng 8 and 5, we solve for usng Eq. (4.3). Ths equaton s an ODE n tme and requres no boundary condtons. The scalar eld s then calculated from Eq. (3.13). D. Constrants After solvng the wave equaton, the Hamltonan and momentum constrants are solved smultaneously for the varables and Z, from whch we can compute A and K T. These equatons are coupled because scalar wave terms contanng appear n the momentum constrant (.). Eqs. (.1) and (.) can be wrtten as ; + ; Z ; =8r 4 ~ Sr 0 6 r 3 5 ; + 58!r 1 + 8r = Z 7 r 0 ~r 4 0! 5 5 r 8 0! 8 r 8 ; (4:8) 0 r 4 (8 ; + 8) : (4:9) Eq. (4.8) requres a sngle boundary condton. At the apparent horzon, we mpose the margnally trapped surface condton (.7), whch we rewrte usng the varables, Z, and : ; + = 1 4 Z 3 r : (4:10) Eq. (4.9) requres two boundary condtons. At the outermost grd pont, we match to lnearzed theory by mposng Eq. (3.3): (r ) ; = r 0 r 5 4 : (4:11) Because there s no extra equaton for the nner boundary condton, we use the evoluton equaton (.19) to compute the value of at the apparent horzon. Ths equaton takes the form ;t = ; r + r K T: (4:1) Eqs. (4.8) (4.11) comprse a coupled system of nonlnear equatons. We solve them smultaneously by the teraton scheme descrbed n Appendx D. E. Lapse and Shft After obtanng and Z, we solve the lapse and shft equatons smultaneously for and. In the SA method, these equatons are solved ndependently; here they are coupled because of a boundary condton 18

19 that we now mpose at the apparent horzon. We wrte these equatons as ; + ; 1 + ; = 3 0 KT r 1 + 8r ~ + ~ T + 3=! +! 4 5 r + r 8 ; ; ; (4:13) r ; = 0 3 K T: (4:14) At the outer grd pont, we mpose Eqs. (3.7) and (.48), whch take the form (r) ; = r + r 5; (4:15) = rk T + r5 : (4:16) For the nner boundary condton on we use Eq. (.73), whch can be wrtten as where = 1=r 0 F 1 0 F 3 1=r + F + F 3 + F 4 ; (4:17) F 1 8 ~ + ~ S r ; (4:18a) F 8 ~S r r 0 ~T 3 +! + ~ S r F 3! ; r + 8 ; r 0 8 ; r ; F 4 K TA ; (4:18b) (4:18c) (A5 0 8) : (4:18d) Ths condton forces a grd pont to reman at the apparent horzon. It also couples the lapse and shft equatons (4.13) and (4.14), so that we must solve these equatons smultaneously. Fnte-derence forms of Eqs. (4.14){(4.18) are presented n Appendx D. Notce that n vacuum wth constant scalar eld, Eq. (4.17) reduces to = A. Combnng ths wth Eqs. (4.10) and (4.1), we see that n ths case ;t = 0 on the apparent horzon. As a result, all quanttes are manfestly tme ndependent: Gven the value of on the AH, whch s constant n tme, the quanttes A,,, and K T are completely determned by coupled spatal constrant equatons wth no source terms. Ths tme ndependence should come as no great surprse because the vacuum soluton wth constant scalar eld s smply the exteror Schwarzschld soluton. However, ths result s remarkable from a practcal pont of vew: When one uses standard sngularty-avodng schemes (ncludng our SA scheme) to compute sphercal collapse n general relatvty, one does not obtan a tme-ndependent system at the end of the smulaton. Instead, one nds that the metrc functons and ther dervatves are changng exponentally wth tme nsde the AH. However, usng the AHBC method, any system that results n a Schwarzschld black hole wll become manfestly tme ndependent. Ths s the great benet of the AHBC method: One can ntegrate black-hole spacetmes arbtrarly far nto the future wthout encounterng sngulartes and wthout causng metrc functons or ther dervatves to blow up. F. Intal Data The ntal tme slce for the AHBC method can be any maxmal sotropc tme slce that contans an apparent horzon. We use a slce provded by the SA method. Because s a scalar and, K T, and 5 are 19

20 three-dmensonal geometrc objects on the slce, and because we use the same 3-metrc for both the SA and AHBC methods, the values of these varables on the ntal slce can be smply read o from the SA slce. After layng down a new spatal grd as descrbed n secton A, we spatally nterpolate these varables from the old SA grd to the new one. We then determne 8 from Eq. (.8). For self-consstency, we then freeze the value of at the AH and recalculate and K T from Eqs. (4.8){(4.11). Fnally, we calculate and by solvng Eqs. (4.14){(4.18). Although the ntal AHBC tme slce concdes wth an SA slce, subsequent AHBC slces do not concde wth those that would have been produced by contnung the SA method, because the two methods have a derent lapse and shft. These derences are due soley to the derent nner boundary condtons used n the two methods. G. Dagnostcs In determnng the metrc and the scalar eld, we never use the K T evoluton equaton (.0) or the denton of 8, Eq. (.8). Furthermore, whle we need the A evoluton equaton (.19) to determne a boundary condton at the AH, ths equaton s not used at any other grd pont except as an ntal guess. We therefore have three equatons that can serve as dagnostcs at each tme step. In addton, we can compare the results of the masses calculated by Eqs. (.65) and (.66). Unlke the step-by-step comparson of equatons (.0), (.8), and (.19), ths method s senstve to errors that accumulate over tme. Because we exclude the orgn from our spacetme, we cannot evaluate the volume ntegrals (.67) n the AHBC method. V. NUMERICAL RESULTS In ths secton we treat a few select collapse scenaros to calbrate our code. We use these cases to compare the standard SA scheme wth the AHBC method. For a more complete summary of physcal results and an evaluaton of collapse to black holes n Brans-Dcke theory, see Paper II. A. Oppenhemer-Snyder Collapse n General Relatvty A useful test of our code s Oppenhemer-Snyder[4] collapse to a black hole n general relatvty. We start wth a momentarly statc, unform sphere of dust wth an ntal areal radus R s = 10M, where M s the total mass of the conguraton as measured by Kepleran orbts far from the orgn. We follow the collapse of ths object to a black hole usng our SA scheme wth! = Such a large value of! puts us well nto the general relatvstc lmt. We use 41 nteror grd ponts, 87 exteror grd ponts, and 100 partcles. Our outer grd boundary s at r = 100M. Fgures 1,, and 3 show snapshots of metrc parameters at selected values of coordnate tme. Also shown are the exact results obtaned[5] by transformng the analytc soluton[4] nto the maxmal sotropc gauge. The SA method agrees well wth the exact soluton untl very late tmes. The \collapse of the lapse" can be seen n Fgure 1: nsde r s = 1:5M, the lapse functon decreases exponentally wth tme, freezng the clocks of normal observers n ths regon. The values of A and nsde r s = 1:5M also change rapdly wth tme, as can be seen n Fgures and 3. Because of the large values of A, coordnate rad become very small compared to sotropc rad. For example, the areal radus of the matter surface at t = 87M s about r s = 1:5M, whle ts coordnate radus s only r M. In addton, the gradents of, A, and become very steep near r s = 1:5M. Trackng partcles at r M wth a grd that extends to r = 100M and resolvng the metrc functon A whch drops from > 10 7 to between r = M and r = M requres an enormous dynamc range. 0

21 gure 1 The lapse functon versus areal radus r s on selected maxmal tme slces for general relatvstc Oppenhemer-Snyder collapse from R s = 10M. Calculatons by the SA method (dotted lne) are compared wth the exact soluton (dashed lne). Tme and dstance are measured n unts of M. gure The metrc component A 1= versus areal radus r s on selected maxmal tme slces. Labelng s the same as n Fgure 1. Because our nner grd follows the partcles at each tme step, grd ponts tend to accumulate at the lmt (matter) surface, r s 1:5M. Although ths allows us to resolve the large metrc gradents better than wth a statonary grd, our code eventually loses accuracy as the gradents grow. Ths can be seen n Fgures 1{3: 1

22 gure 3 The radal shft versus areal radus r s on selected maxmal tme slces. Labelng s the same as n Fgure 1. gure 4 Oppenhemer-Snyder collapse n maxmal slcng, calculated by the SA method. The ve sold lnes represent world lnes of matter elements contanng 0, 40, 60, 80, and 100 of the nteror rest mass. Tme and areal radus are measured n unts of M. The dotted lne s the apparent horzon, whch forms at about t = 44M, at whch pont t already concdes wth the event horzon. the SA method and the exact soulton begn to dsagree at about t = 60M, and ths dscrepancy ncreases exponentally wth tme. Even f we were able to mantan accuracy, numercal overow would eventually

23 cause our SA code to termnate. A spacetme dagram of Oppenhemer-Snyder collapse s shown n Fgure 4. Because of the \collapse of the lapse", the matter partcles st at a constant areal radus after t = 50M. An apparent horzon, whch forms at t = 44M outsde of the matter, has an ntal areal radus of r s = M, n agreement wth the Schwarzschld soluton. It ncreases slghtly n sze after about t = 70M because of numercal errors. gure 5 Metrc quanttes for t > 45M for Oppenhemer-Snyder collapse n general relatvty, calculated by the AHBC method. We remove the regon of spacetme nsde the apparent horzon at r s = M and only retan the vacuum exteror. All quanttes are constant n tme. The above numercal dcultes do not occur n the AHBC method because we no longer use such a pathologcal coordnate system. In Fgure 5 we show the metrc coecents obtaned from the AHBC method after matchng onto an SA tme slce at t = 45. We use 18 grd ponts, and place the outer boundary at r = 100M. The apparent horzon s located at the grd pont = 4, whch s at coordnate radus r = 0:78M and areal radus r s = M. The nnermost grd pont s at r = 0:7M 0. Because all the matter has fallen past the AH by t = 45M, we no longer need to move the partcles we have dscarded the regon of spacetme that contans them. In ths case the metrc s manfestly statc. B. Scalar Waves on a Schwarzschld Background Consder a small scalar perturbaton about a Schwarzschld black hole. If the perturbaton s so small that ts contrbuton to the metrc s neglgble, then t s not necessary to solve the coupled Brans-Dcke equatons (.4) and (.5) to determne the future evoluton of. Instead, one only needs to solve the wave equaton (.4) n vacuum on a Schwarzschld background. Ths equaton can be wrtten n the form[6; 7] where u ;tst s = u ;zz + u; (5:1) 3

24 u r s ; (5:) r 3 s 1 0 r s ; (5:3) z r s + ln(r s = 0 1); (5:4) r s s the areal radus, and t s s the Schwarzschld tme coordnate. The varable z s the famlar \tortose" coordnate, whch runs from z = 01 (at r s = ) to z = 1 (at r s = 1). The mass of the black hole has been set equal to unty. Eq. (5.1) provdes an mportant check for our AHBC code. Ths equaton s not dcult to solve numercally t s smply a one-dmensonal at space wave equaton n a statc potental. We set up the followng test case: At t s = 0, the metrc s Schwarzschld and the scalar eld s gven by = 1 + C r s exp (r s 0 r 0 ) ; (5:5) where r 0 = 80, C = 10 06, and = 5. We set ;ts = 0 ntally. As tme progresses, the ntal pulse dvdes nto two peces: one moves outwards to nnty, whle the other moves toward the black hole and s partally reected by the Schwarzschld potental. We calculate as a functon of r s and t s for ths case by two ndependent methods: we solve the full Brans-Dcke equatons usng the AHBC method, and we solve the perturbaton equaton (5.1) usng a staggered leapfrog nte derence scheme. The AHBC method requres metrc coecents on an ntal tme slce; these are provded by matchng onto Oppenhemer-Snyder collapse (calculated from the exact soluton followng Petrch et al.[5]) after the matter has fallen nto the black hole. Because the AHBC method and the perturbaton method use derent coordnate systems, we must be careful n settng up the ntal data. Eq. (5.5) s dened on the Schwarzschld tme slce t s = 0. Ths slce does not correspond to the maxmal slce t = 0, or any other t = constant slce n maxmal sotropc gauge. However, the ntal Gaussan pulse (5.5) s far from the black hole, where t = constant and t s = constant slces nearly concde. Therefore, f we choose the slce t = 0 to concde wth t s = 0 at r s = 80, we can use Eq. (5.5) to determne at t = 0 for the AHBC method wthout ntroducng much error. Although t = constant and t s = constant slces nearly concde at r s 80 where the ntal wave s nonzero, these slces der consderably n the strong eld regon near the black hole. If we wsh to compare the results of the AHBC and perturbaton methods n ths regon, we must use coordnate nvarant quanttes. We therefore ntroduce a set of statonary observers at speced values of the areal radus r s, and we record the values of and ; measured by each of these observers as a functon of, the proper tme measured by the clock of each observer. To synchronze the clocks n a manner ndependent of the choce of tme slcng, we ntroduce an ngong lght sgnal that passes r s = 80 at t s = t = 0. Each observer starts hs clock runnng from = 0 when he sees the sgnal, e. g., at the ntal tme slce, an observer at r s = 80 reads = 0, an observer at r s = 90 reads 10 (plus (1=r s ) correctons), and an observer at r s = 70 reads 010. Fgure 6 shows and ; as measured by two derent observers. The observer at r s = 100M sees two peaks of scalar radaton. The rst travels outward from r s = 80M, startng at the observer's proper tme 0M, and passes the observer at 40M. The second travels nward from r s = 80M, s partally reected by the black hole, and then moves outward, passng the observer at 0M. The observer at r s = 5M does not see the outgong radaton pulse, but nstead sees a combnaton of the ngong pulse and ts reecton. The agreement between the two numercal methods s excellent, demonstratng that the AHBC scheme s able to handle gravtatonal radaton wthout producng large numercal reectons at the apparent horzon or at the outer grd boundary. In addton, the above case extends from t = 0 to t = 300M, much longer than a tradtonal SA scheme would allow. Fgure 7 shows the coordnate velocty of outgong lght rays, dr dt = 0 : (5:6) A 4