Gravitational Memory?

Transcription

1 Gavitational Memoy? a Petubative Appoach T. Haada 1, Depatment of Physics, Waseda Univesity, Shinjuku, Tokyo , Japan B.J. Ca 2,andC.A.Goyme 3 Astonomy Unit, Queen May and Westfield College, Univesity of London, London E1 4NS, England Abstact It has been pointed out that the value of the gavitational constant in the ealy univese may be diffeent fom that at pesent. In that case, it was conjectued that pimodial black holes may emembe the value of the gavitational constant in the ealy univese. The pesent analysis shows that this is not the case, at least in cetain contexts. 1 Intoduction In the ealy univese, black holes may have been fomed due to inhomogeneous initial conditions, phase tansitions, o othe mechanisms. They ae called pimodial black holes (PBHs). The mass of the PBHs is of ode the mass contained within the Hubble hoizon at the fomation epoch. Hawking [1] discoveed that black holes emit adiation due to the quantum effects of the cuved spacetime. PBHs theefoe lose mass and those lighte than 1 15 g may contibute to the cosmic gamma-ay backgound and spoil the success of the big bang nucleosysthesis scenaio. PBHs heavie than that may dominate the density of the univese at pesent. Theefoe the faction β of the univese which may have gone into PBHs can be constained fom vaious cosmological obsevations (Ca [2]). It has been pointed out that gavity in the ealy univese may have been dilatonic, since this aises natually as a low-enegy limit of sting theoy. In such theoies, the gavitational constant is given by a function of a scala field which couples non-minimally with gavity. In paticula, the gavitational constant in the ealy univese may have been diffeent fom that at the pesent epoch. The Bans-Dicke theoy is the simplest such theoy of gavity. Baow [3] consideed PBHs in Bans-Dicke theoy and poposed two scenaios. In scenaio A, the gavitational constant at the black hole event hoizon is always the same as that at the cosmological paticle hoizon, i.e., G EH (t) =G PH (t). In scenaio B, the gavitational constant at the black hole event hoizon is constant with time and theefoe always the same as that at the fomation epoch, i.e., G EH (t) =G EH (t f )=G PH (t f ). Scenaio B coesponds to what is called the gavitational memoy, while scenaio (A) entails no gavitational memoy. Baow and Ca [4] obtained obsevational constaints on β in both scenaios. They assumed that the enegy emission due to the Hawking adiation is detemined by the tempeatue T H =(8πG EH (t)m) 1, so that the mass loss ate is Ṁ (G EH (t)m) 2. The esults fo the two scenaios can be significantly diffeent. Subsequently, Ca and Goyme [5] agued that the tue situation may be intemediate between Scenaio A and Scenaio B and this allows othe possibilities. Theefoe, in ode to deduce the histoy of the ealy univese fom the pesent obsevations, it is vey impotant to examine which scenaio is ealized. Recently, Jacobson [6] has agued that thee is no gavitational memoy fo a Schwazschild black hole with a time-vaying bounday condition which mimics the cosmological evolution of a scala field. 1 haada@gavity.phys.waseda.ac.jp 2 B.J.Ca@qmw.ac.uk 3 C.A.Goyme@qmw.ac.uk 1

2 Howeve, this agument applies only if the black hole is much smalle than the paticle hoizon, so we must conside moe geneal situations. The pupose of this aticle is to solve the poblem self-consistently fo a Fiedmann-Robetson-Walke (FRW) backgound. 2 Basic Equations The field equations in Bans-Dicke theoy ae given by R µν 1 2 g µνr = 8π φ T µν + ω φ 2 ( φ,µ φ,ν 1 2 g µνφ,α φ,α ) + 1 φ ( µ ν φ g µν φ), (1) µ T µν =, (2) φ = 8π 3+2ω T µ µ, (3) whee the constant ω is the Bans-Dicke paamete and φ is the Bans-Dicke scala field, which is elated to the gavitational constant by G φ 1. In the limit ω, the theoy goes to geneal elativity with a minimally coupled scala field. The constaint ω > 33 is equied by ecent obsevations of light deflection [7]. Howeve, if we conside moe geneal scala-tenso theoies, ω also becomes a function of φ. In this case, ω may have been small in the ealy univese, even though it is lage today. Hee we calculate the time evolution of the gavitational constant by using the following post-genealelativistic expansion. Fist we assume that the scala field is constant, i.e., φ = φ. Then we find the geneal elativistic solution (g GR,T GR ) which satisfies Eqs. (1) and (2) with G = φ 1. Next we put (g, T) =(g GR,T GR ) in Eq. (3) and assume that δφ = φ φ is of ode ω 1 fo ω 1. We can then obtain φ to this ode, denoting it as φ (1), by solving the following equation: GR φ (1) = 8π 3+2ω T GR µ µ, (4) whee GR denotes the d Alembetian in the geomety given by g GR. Substituting the solution φ (1) of Eq. (4) into the ight-hand side of Eq. (1), we can then detemine g (1) and T (1) by solving Eqs. (1) and (2). Then, φ (2), which is the solution up to O(ω 2 ), is detemined by solving Eq. (3) fo the backgound (g (1),T (1) ). Repeating this pocess, we can constuct the Bans-Dicke solution fom the geneal elativity solution ode by ode. Hee we tuncate the expansion fo the scala field at O(ω 1 ). This implies that we only have to solve Eq. (4) and we can neglect the effect of the scala field on the backgound cuvatue. This appoximation was shown to be vey good fo the geneation and popagation of gavitational waves fom the collapse to a black hole in asymptotically flat spacetime fo Bans-Dicke theoy with ω > 4[8]. 3 Lemaite-Tolman-Bondi solution We adopt the Lemaite-Tolman-Bondi model as the backgound solution (g GR,T GR ) in which the scala field evolves. This model is an exact solution of geneal elativity which descibes a spheically symmetic inhomogeneous univese with dust. The line element in the synchonous comoving coodinates is given by ds 2 = dt 2 + A 2 (t, )d 2 + R 2 (t, )dω 2, (5) whee dω 2 = dθ 2 +sin 2 θdφ 2. (6) The function R(t, ) isgivenby ( ) 1/3 9F () R = (t t s ()) 2/3 fo f =, (7) 4 2

3 R = F () (cosh η 1) 2f() t t s () = R = F () fo f>, (8) 2f 3/2 () (sinh η 1) F () (1 cos η) 2( f)() t t s () = F () fo f<, (9) 2( f) 3/2 () (η sin η) whee the pime and dot denote the deivatives with espect to and t, espectively. The function A(t, ) is given by A 2 (t, ) = (R ) 2 1+f(). (1) The functions t s (), f() > 1 andf () ae abitay and one of them coesponds to the gauge feedom. The fist two elate to the big bang time and the total enegy pe unit mass fo the shell at adius. The thid elates to the mass within adius, the density of the dust being given by 4 Method ρ = F 8πR 2 R. (11) We ae inteested in the behavio of the scala field long afte the black hole has fomed. In this case, the chaacteistic method is suitable. This method was fist applied to the Lemaite-Tolman-Bondi backgound by Iguchi, Nakao and Haada [9]. If we intoduce the etaded time u, the line element becomes ds 2 = α(u, )du 2 2α(u, ᾱ)a(u, )dud + R 2 (u, )dω 2, (12) whee α must satisfy α = Ȧα. (13) In tems of the deivatives, d/du and /, the d Alembetian is given by [ φ = 2 dϕ αar du A A 3 R ϕ + 1 ( ) ] R (AṘ) φ, (14) AR A ϕ = (Rφ). (15) We can then integate Eq. (4) along the chaacteistic cuves, i.e., null geodesics. With this choice of coodinate system, we can calculate the behavio of the scala field long afte black hole fomation without having to impose bounday conditions on the event hoizon. 5 Models As we have seen, the Lemaite-Tolman-Bondi solution has thee abitay functions. We want to obtain models which descibe a PBH in a flat FRW univese by fixing these functions appopiately. Hee we adopt the following assumptions: the big bang occus at the same time eveywhee; the model is asymptotically flat FRW, which equies the ovedense egion to be compensated by a suounding undedense egion; the model is fee fom naked singulaities. In ode to satisfy these assumptions, we fist choose t s () =, (16) 3

4 and f() = ( ( c c ) 2 fo < w ) [ 2 ( ) ] 4 w exp w fo w, (17) whee c and w ae the cuvatue scale and the size of the ovedense egion. Using the abitay function F (), we then set the adial coodinate to be = R(t,) (18) whee t = t is the initial time. Befoe integating Eq. (4), we have to fix the initial data fo φ, so we choose the cosmological homogeneous solution given by ( φ = φ ω 3 ln t ). (19) t We set the initial null hypesuface as the null cone whose vetex is at (t, ) =(t, ) and egad the cosmological solution as the initial data on this hypesuface. Although the value of the scala field at the cosmological paticle hoizon must be given by this solution, the value in the petubed egion may be diffeent fom this. We have theefoe examined the effects of othe initial data which ae diffeent fom the cosmological solution in the petubed egion. Howeve, the esults ae not changed much. 6 Results We adopt units in which the asymptotic value of the Hubble paamete at t = t is unity. We set the paametes which specify the initial data as c =2and w = 1, while the Bans-Dicke paamete is chosen as ω = 5. It is noted that c must satisfy c > w, else the ovedense egion is isolated fom the est of the univese, as Ca and Hawking [1] pointed out. The esults ae shown in Figs In Figs. 1 (a) and (b), a set of initial data fo the backgound geomety is shown. The initial density petubation δ is defined as ρ(t, ) ρ(t, ) δ(t, ). (2) ρ(t, ) In Fig. 2, the tajectoies of outgoing light ays in this backgound geomety ae shown. It is shown that the event hoizon is fomed at R 1. The esults of integating Eq. (4) ae shown in Figs. 3(a) and (b), whee φ is defined as φ φ(t, ) φ. (21) φ It is found that the value of the scala field aound the black hole is slightly smalle than the asymptotic value because of the undedensity of the suounding egion. Nevetheless, the scala field as a whole is almost spatially homogeneous at each moment, in spite of the existence of the black hole. The evolution of the scala field nea the event hoizon is well descibed by the cosmological homogeneous solution. 7 Summay We have calculated the evolution of the Bans-Dicke scala field in the pesence of a pimodial black hole fomed in a flat FRW univese. We have found that the value of the scala field at the event hoizon almost maintains the cosmological value at each moment. This suggests that pimodial black holes foget the value of the gavitational constant at thei fomation epoch. Howeve, it should be stessed that this esult has only been demonstated fo a dust univese in which the scala field does not appeciably affect the backgound cuvatue. It emains to be seen whethe the same conclusion applies when these assumptions ae dopped. We ae gateful to T. Nakamua fo helpful discussions and useful comments. 4

5 f (a) δ (b) Figue 1: (a) Specific enegy f() and (b) initial density petubation δ at t = t. 5

6 t Refeences R Figue 2: Tajectoies of outgoing light ays in the backgound geomety. [1] S.W. Hawking, Commun. Math. Phys. 43, 199 (1975). [2] B.J. Ca, Astophys. J. 21, 1 (1975). [3] J.D. Baow, Phys. Rev. D 46, 3227 (1992). [4] J.D. Baow and B.J. Ca, Phys. Rev. D 54, 392 (1996). [5] B.J. Ca and C.A. Goyme, Pog. Theo. Phys. 136, 321 (1999); C.A. Goyme and B.J.Ca, Phys. Rev. D., in pess. [6] T. Jacobson, Phys. Rev. Lett. 83, 2699 (1999). [7] T.M. Eubanks et al., Bull. Am. Phys. Soc., Abstact #K 11.5 (1997); C.M. Will, g-qc/ [8] T. Haada, T. Chiba, K. Nakao and T. Nakamua, Phys. Rev. D 55, 224 (1997). [9] H. Iguchi, K. Nakao and T. Haada, Phys. Rev. D 57, 7262 (1998); H. Iguchi, T. Haada and K. Nakao, Pog. Theo. Phys. 11, 1235 (1999); Pog. Theo. Phys. 13, 53 (2). [1] B.J. Ca and S.W. Hawking, MNRAS 168, 399 (1974). 6

7 R= φ (a) t t= φ (b) R Figue 3: φ (φ φ )/φ (a) along the wold lines R = const and on the hypesufaces t =const. 7