Gravitation & Cosmology, Vol. 0 (00) (1999), No. 1, pp. 001–003 (cid:13)c 1998 RussianGravitational Society Dilatonic Black Holes Time Stability O.S. Khovanskaya1 Sternberg Astronomical Institute, Universitetskii Prospekt, 13, Moscow 119992, RUSSIA Received 1 February 2002 2 0 0 The stability under small time perturbations of the dilatonic black hole solution near the determinant curvature 2 singularity is proved. This fact gives the additional arguments that the investigated topological configuration can n realise in nature. In the frames of this model primordial black hole remnants are examined as time stable objects, a which can form an significant part of a dark matter in theUniverse. J 3 Oneratherimportantresultofthe stringgravityis particular point is the determinant c urvature singu- 2 arestrictionuponthe minimalblackholemass[1],[2], larity, which has the infinite derivatives of the metric 1 [3] in frames of the model with the 4D effective string functions [2]. v action, containing graviton, dilaton and higher order Primordialblackholestabilityontheeventhorizon 5 cur vature corrections. Let me consider the action in was investigated in [8], [9]. 7 the following form [1]: It was obtained (P. Kanti [8] at al.) that the dila- 0 1 tonic blackholesarestable nearthe blackholeregular 1 20 S = 16π Z d4x√−g(cid:20)m2Pl(cid:18)−R+2∂µφ∂µφ(cid:19) whohriiczhonderphenudndoenrolinnleyarontiemrea-ddieaplepnadreanmtepteerrt.uTrbhaetmionets-, 0 / + λ e−2φ(R Rijkl 4R Rij ric parameterisationwas c (cid:18) ijkl − ij q ds2 =eΓ(r,t)dt2 eΛ(r,t)dr2 r2(dθ2+sin2 θdψ2) − − - + R2) +... , (1) r and the asymptotic forms of metric components and g (cid:19) (cid:21) dilaton field near the regular horizon r r were : ≈ h v where m is Plank mass, λ is the string coupling i constant,Pφl isthedilatonicscalarfield. (Thesystemof e−Λ(r) = λ1(r rh)+λ2(r rh)2+..., X − − r units where h¯ =c=G=1 and mPl =1 is used. The eΓ(r) = γ1(r−rh)+γ2(r−rh)2+..., a community of the consideredproblem is not restricted φ(r) = φ +φ′ (r r )+φ′′(r r )2+..., by choosing λ=1). h h − h h − h where λ =2/(λeφhφ′/g2+2r ). Therestrictionupontheminimalblackholemassis 1 h h P. Kanti at al. considered perturbing equations by absentinthe SchwarzschildsolutionofEinstein’s clas- time-dependent linear perturbations of the form: sical gravity. This ”mathematical” result can be put into practice in modern cosmology to study the rem- Γ(r,t) = Γ(r)+δΓ(r,t)=Γ(r)+δΓ(r)eiωt, nants of primordialblack holes [4], [5]. Such remnants Λ(r,t) = Λ(r)+δΛ(r,t)=Λ(r)+δΛ(r)eiωt, canrepresentthefinalstageofHawkingevaporationof primordial black holes, formed in the early Universe, φ(r,t) = φ(r)+δφ(r,t)=φ(r)+δφ(r)eiωt, and are considered as dark matter candidates [6], [7]. where the variations δΓ(r,t),δΛ(r,t) and δφ(r,t) The search of the exact solutions or at least the were assumed to be small. The stability problem was numericalonesintheofferedmodel(1)withthemetric reduced to one-dimensional Schrodinger problem [8]. depended on two parameters, the radial co-ordinate The regular horizon stability was investigated also andthetime,isknowntobeverydifficult[1],[2],[3]. N inthe paperbyT.ToriiandK.-i.Maeda[9]. Theyused everthelessonecanreceivethegeneralpropertiesofthe the catastrophe theory and compared it with linear time-evolution of such solutions by study its stability perturbation analysis. Generally the catastrophe the- about time-parameter in all particular points. ory is a mathematical tool to investigate a variety of The blackhole solutioninthe framesofconsidered some physical states, T.Torii and K.-i.Maeda showed model has only two particular points. The first one is this method is also applicable to the stability analysis the usual coordinate singularity r , which represents h of various types of non-Abelian black holes [10]. the event regular horizon of a black hole. The second Itisnecessarytomakemorecarefulstabilityanaly- 1e-mail: [email protected] sisunderthehorizonwiththehelpofsuitablechoiceof 2 η asymptotic forms that approximate the metric func- σ =4 , 0 1 η2 tions. In this work I investigate the dilatonic black − hole stability near the determinant curvat ure singu- η2 δ = 16 , larity r = rs [2]. After the definition of a rest point, 0 − (1 η2)2 this problem can be reduced to a one-dimensional − 16 σ (η 1) Schrodinger problem under the variation of the field δ = 2 − equation. One can prove that the small time pertur- 3 − 3 rs(1+η)2 bations do not increase. S o in that case the solution and of the dilatonic black hole is stable near r . s 1 √2(1 η2)2 To investigate the stability under time perturba- θ = − . tions of the dilatonic black hole near the singularity 32 η2 r rs, I use the model (1) with the following non- It is possible to note that η ( 1,0). We have sta≈tic,asymptoticallyflatsphericallysymmetricmetric three free parameters η, σ and r∈ 2−. 2 s [1]: The exact field equations for (1)-(2) which depend on r and t are in Appendix. Transforming (14)-(17), σ2 ds2 = ∆dt2 dr2 r2(dθ2+sin2 θdψ2), (2) one can obtain the autonomous (over t) equation sys- − ∆ − tem of the first order in the following form: where ∆=∆(r,t), σ =σ(r,t), φ=φ(r,t). ∆˙ = α AccordingtoKanti’smethod[8]Iproducethemet-  σ˙ = β ric functions in the form: ∆(r,t)=∆(r)+δ∆(r,t)=∆(r)+δ∆(r)eiωt, (3)  αφ˙˙ == γG (9) − β˙ = 2G/Λ σ(r,t)=σ(r)+δσ(r,t)=σ(r)+δσ(r)eiωt, (4)  γ˙ = F ′ ′′ ′ ′ ′′ where G,F arefunctionsof r,∆,∆,∆ ,σ,σ ,φ,φ,φ φ(r,t)=φ(r)+δφ(r,t)=φ(r)+δφ(r)eiωt, (5) and α,β,γ are additional variables. The function Λ = 2∆/σ. Using the asimptotic wherethevariations δ∆(r,t),δσ(r,t) and δφ(r,t) are form (6)-(7) one can o−btained that Λ = 8η/(1 η2) assumed to be small 1. for r =r . Λ =0,Λ= near the singularity r− r s s Thusforsuchdefinitionofvariables r and t in(3)- becausethe dil6atonfu6nc∞tion φ(r) (8)is limited in≈this (5) the asymptotic forms of metric functions near the region [2]. singularity r rs depend on only radial coordinate r Let me examine the equilibrium states of the au- ≈ and do not depend on time: tonomous system (9). ∗ ∗ ∗ ∗ ∗ ∗ Let the point (∆ , σ , φ , α , β , γ ) is some rest φ (r r ) ∆(r)=δ + 2 − s +δ (r r )3/2+..., (6) point of the system (9), that is for 0 3 s θ − f α, β, γ, G, 2G/Λ, F , i ∈{ − } σ(r)=σ +σ √r r +..., (7) 0 2 s − i=1,2,...,6 φ(r)=φ +φ (r r )+φ (r r )3/2+..., (8) the condition 0 2 s 3 s − − ∗ ∗ ∗ ∗ ∗ ∗ where f (∆ , σ , φ α , β , γ )=0 i 3 1 φ = ln(2) ln( η)+ln(1 η) ln(r ), is executed. The trivial ”equilibrium-like” solution 0 s 4 − 2 − − − which corresponds to the given rest point is asymp- totically stable if the first order system is stable. It 1 (η 1)√2 φ = − , occurs when the all roots s of the characteristicequa- 2 2 ηr s tion df φ3 = −96r1η4 σ2(η−1)2√2(−8η+8η3 deth(cid:16)dyik(cid:17)|restpoint−s·δkii = 0 (10) s 20η2+4η4 4 23/4η2+3 23/4η4+23/4), where yk ∆∗, σ∗, φ∗, α∗, β∗, γ∗ (k = 1,2,...,6) − − · · ∈ { } have the negative real parts. 1In this work I only study the small time perturbations of thediagonalmetriccomponentsbecauseingeneralIinvestigate 2Inthiscaseitisconvenientlytochoose η, σ2 and rs asfree. the case of spherically symmetric metric. I’m not interesting There are direct three free parameters which can be reduced in some rotating effects, which can appear because of non-zero to the usual free parameters: the black hole mass, the dilaton non-diagonal metriccomponents. chargeandthedilatonvalueatinfinity[2]. 3 Appendix Itispossibletofindtherestpointofthesystem(9) usingtheasymptoticforms(6)-(8)nearthesingularity The exact field equations for (1)-(2) which depend on r r from the condition: r and t are thefollowing unwieldy form: s ≈ F = 0 0 = 2e2φr2φ′2σ3∆2−8φ′′σ3∆2 (cid:26) G|rreessttppooiinntt = 0 (11) − 2e2φrσ′σ2∆2+16φ′2σ3∆2 | + 8φ′σ′σ2∆2+2σ5e2φr2φ˙2 ∗ In the rest point α = β = γ 0 and η = η = C , σ =σ (r )=C /r1/2 <0,w≡here C and C are + 8φ˙σ˙ σ4−8φ˙σ˙σ2∆ 1 2 2 s 2 s 1 2 − 16φ˙2σ3∆+16σ5φ˙2 the number coefficients. By solving(10)one canreceivethe pure imaginary − 8σ5φ¨+8φ¨σ3∆ roots s. Thus some additional investigations for de- − 16∆3φ′2σ−24∆3φ′σ′+8∆3φ′′σ, (14) termine the stability are required. Usingthemethodoflinearstabilityonecanrewrite 0 = −2e2φσ4∆2+2e2φr∆′σ2∆2 the field equations (14)-(17) with the variations: − 2∆3e2φr2φ′2σ2+24∆3φ′∆′+8φ˙∆˙ σ2∆ 0 = A δφ¨+A δφ˙+A δφ+A δφ′ − 8φ′∆′σ2∆2+2∆3e2φσ2 i1 i2 i3 i4 + A δφ′′+B δ∆¨ +B δ∆˙ +B δ∆ − 16φ¨σ∆2+32φ˙2σ2∆2 i5 i1 i2 i3 ′ ′′ − 32φ˙2σ4∆+16φ¨σ4∆−8φ˙∆˙ σ4 + B δ∆ +B δ∆ +C δσ¨+C δσ˙ i4 i5 i1 i2 ′ ′′ − 2e2φr2φ˙2σ4∆, (15) + C δσ+C δσ +C δσ , (12) i3 i4 i5 where i=1..4. Nonzero coefficients are in Appendix. 0 = −8∆˙2σ3∆−8∆′′σ3∆3−16σ¨σ2∆3 In the vicinity of finding from (11) rest point − 8∆˙ σ˙σ2∆2+4σ5e2φr2φ¨∆2+8∆4∆′′σ O+(0,(r rs)), (r rs) 10−3 it is possible to eval- − 8∆4e2φrφ′σ3−24∆4∆′σ′+8∆′σ′σ2∆3 − − ∼ uate the coefficients in front of the variations in (12) + 32σ˙2σ∆3+8∆¨ σ3∆2+8∆′2σ∆3 using the asymptotic forms (6)-(8). The simplified − 8σ5∆¨ ∆+16σ5∆˙2−24∆˙ σ˙ σ4∆ equation from (12) becomes the following: + 16σ¨σ4∆2−4∆4e2φr2φ′′σ3 − 4e2φ∆′r2φ′σ3∆3+4∆4e2φr2φ′σ′σ2 A(δφ)′′+B(δφ)′+C(δφ) = ω2(δφ), (13) + 4e2φσ˙r2φ˙σ4∆2−4σ5e2φr2φ˙∆˙ ∆, (16) where A,B and C near the singularity r ≈rs are: 0 = 2e2φr∆′′∆3σ3+128φ˙σ˙ φ′∆3σ2+16φ′∆¨ ∆2σ3 a a (r r )1/2 − 32∆4φ′2∆′σ+32φ˙2∆′∆2σ3+32φ˙′∆˙ ∆2σ3 1 2 s A = + − +O(r r ), rs rs3/2 − s + 16φ˙∆′σ˙ ∆2σ2−64φ˙∆˙φ′∆2σ3+16∆4φ′∆′′σ b b (r r )−1/2 − 16φ¨∆′∆2σ3−16φ′∆˙2∆σ3−2e2φr∆′σ′∆3σ2 B = rcs12 + c2(r−rs5r/2s)−1/2 +O((r−rs)1/2), −+ 342σφ5¨eσ2′φ∆r3∆σ˙22+−146e2∆φ4rφσ¨′′∆∆2′σσ4−−1648φ˙∆∆˙4σφ′′∆∆2′σσ2′ C = rs13 + 2 −rs5/2s +O((r−rs)1/2), −− 464σφ5˙2e2σφ′∆rφ3˙2σ∆2−2+646φ4˙′φσ˙′∆σ˙23∆σ23σ++164∆e2′2φφ∆′∆′∆33σσ3 where a1,a2,b1,b2,c1,c2 depend on σ2 and η. For − 32φ′σ¨∆3σ2−16φ′∆˙ σ˙ ∆2σ2 r =rs these coefficients are number constants. + 6e2φσ˙r∆′∆σ4+2σ5e2φr∆¨ ∆−4∆4e2φσ′σ2 The eigenvalue ω2 in(13)has only positivevalues. + 4∆4e2φrφ′2σ3, (17) Thus, the small time perturbations do not increase (theycanoscillate). Sointhatcasethesolutionissta- Nonzero coefficients for the field equations (14)-(17) bleinthevicinityoffindingrestpoint (O+(0,(r rs)). with thevariations (12): − Taking into account both results: the stability of A = −2∆′σ3∆2+4σ′σ2∆3, the solution under time perturbations at the regular 41 event horizon r [8] and at the determinant curva- A43 = ∆4rφ′2σ3e2φ+∆′σ3e2φ∆3−∆4σ′σ2e2φ h ture singularity rs, it is possible to conclude that the + 1r∆′′σ3e2φ∆3− 1r∆′σ′σ2e2φ∆3, 2 2 solution of dilatonic black ho le is stable in all partic- ular points. In application to cosmology this fact can A44 = ∆4rφ′σ3e2φ−6∆4∆′σ′+2∆4∆′′σ confirm the existence of remnants of primordial black − 8∆4φ′∆′σ+2∆′2σ∆3, holes, which are stable during time evolution. A = 2∆4∆′σ, 45 I would like to gratefully acknowledge to Prof. B = 2φ′σ3∆2+ 1σ5re2φ∆, M.V. Sazhin and Dr. S.O. Alexeyev for useful com- 41 4 ments. B = −24∆3φ′∆′σ′+8∆3φ′′∆′σ 43 4 + 6∆′2φ′σ∆2−16∆3φ′2∆′σ B = −3∆2r2φ′2σ2e2φ− 1σ4e2φ∆ + 2∆3rφ′2σ3e2φ−2∆3σ′σ2e2φ 23 4 2 − 3r∆′σ′σ2e2φ∆2+ 3r∆′′σ3e2φ∆2 + 12r∆′σ2e2φ∆−2φ′∆′σ2∆+ 43∆2σ2e2φ 4 4 + 9∆2φ′∆′, + 23∆′σ3e2φ∆2+8∆3φ′∆′′σ, B24 = 3∆3φ′−φ′σ2∆2+ 14rσ2e2φ∆2, B44 = 2∆4φ′′σ−6∆4φ′σ′− 14rσ′σ2e2φ∆3 C = 1∆3σe2φ− 1∆3r2φ′2σe2φ 23 2 2 − 4∆4φ′2σ+ 12σ3e2φ∆3+4∆′φ′σ∆3, − σ3e2φ∆2+ 1r∆′σ′e2φ∆2−2φ′∆′σ∆2, 2 B45 = 2∆4φ′σ+ 14rσ3e2φ∆3, A11 = −σ5+σ3∆, C41 = −4φ′σ2∆3− 21rσ4e2φ∆2, A13 = 12r2φ′2σ3e2φ∆2− 21rσ′σ2e2φ∆2, C43 = −21r∆′σ′σe2φ∆3+ 32∆4rφ′2σ2e2φ A14 = −3∆3σ′+σ′σ2∆2+ 12r2φ′σ3e2φ∆2 + 3∆′σ2e2φ∆3−∆4σ′σe2φ − 4∆3φ′σ+4φ′σ3∆2, 2 A = −σ3∆2+∆3σ, + 2∆4φ′∆′′−4∆4φ′2∆′+2∆′2φ′∆3 B15 = −1rσ′σ2e2φ∆−2φ′′σ3∆ + 2∆4φ′′∆′+ 3r∆′′σ2e2φ∆3, 13 2 4 + 1r2φ′2σ3e2φ∆+3∆2φ′′σ C = −1r∆′σ2e2φ∆3− 1∆4σ2e2φ−6∆4φ′∆′, 2 44 4 2 − 9∆2φ′σ′+2φ′σ′σ2∆ 1 A31 = 2σ5r2e2φ∆2, − 6∆2φ′2σ+4φ′2σ3∆, A = −∆4r2φ′′σ3e2φ−2∆4rφ′σ3e2φ C13 = ∆3φ′′−2∆3φ′2−3φ′′σ2∆2 33 + ∆4r2φ′σ′σ2e2φ−∆′r2φ′σ3e2φ∆3, + 2φ′σ′σ∆2− 1rσ′σe2φ∆2 2 A34 = −∆4rσ3e2φ+ 12∆4r2σ′σ2e2φ + 3∆2r2φ′2σ2e2φ+6φ′2σ2∆2, 4 − 21∆′r2σ3e2φ∆3, C14 = φ′σ2∆2− 41rσ2e2φ∆2−3∆3φ′. 1 A = − ∆4r2σ3e2φ, 35 2 B31 = σ3∆2−σ5∆, References B = 3∆′2σ∆2−4∆3rφ′σ3e2φ−2∆3r2φ′′σ3e2φ 33 + 4∆3∆′′σ+3∆′σ′σ2∆2−12∆3∆′σ′ [1] S.Alexeev, M.Pomazanov, Phys.Rev. D55, 2110 − 3∆′′σ3∆2− 3∆′r2φ′σ3e2φ∆2 (1997). 2 [2] S.Alexeyevand M.Sazhin, Gen. Relativ.and Grav. 8, + 2∆3r2φ′σ′σ2e2φ, 1187 (1998). B = −3∆4σ′− 1r2φ′σ3e2φ∆3+σ′σ2∆3 [3] S.Alexeev, O.Khovanskaya, Gravitation and Cosmol- 34 2 ogy, Vol.6No 1, 14 (2000). + 2∆′σ∆3, [4] M.Markov, JETP 51 878 (1966). B = ∆4σ−σ3∆3, 35 [5] M.Markov and V.Frolov, Pisma v Astron. Zhurnal C31 = −2σ2∆3+2σ4∆2, (Astronomy Letters) 29 372 (1979). C33 = ∆′2∆3−3∆4rφ′σ2e2φ− 23∆4r2φ′′σ2e2φ [6] S.Rubin, M.Khlopov, A.Sakharov, Phys. Rev. D62 08350 (2000). + 2∆′σ′σ∆3−3∆′′σ2∆3 − 3∆′r2φ′σ2e2φ(r)∆3+∆4∆′′ [7] SM.A.Slaezxheiyne,v”,BAla.cBkarHraoule,RGel.iBctosudinouSlt,rinOg.KGhroavvaitnys:kalaysat, 2 + ∆4r2φ′σ′σe2φ, Stages of HawkingEvaporation” (in preparation). C34 = ∆′σ2∆3−3∆4∆′+ 21∆4r2φ′σ2e2φ, [8] PE..KWainntsit,anNle.yM,aPvhryosm.aRtoevs,. DJ.5R7i,zo6s2,55K(.1T9a9m8)v.akis and A21 = 2σ4∆−2σ2∆2, [9] T.Torii, K.-i.Maeda, Phys. Rev. D58, 084004-1 A = 1r∆′σ2e2φ∆2− 1σ4e2φ∆2 (1998). 23 2 2 [10] K.-i.Maeda, T.Tachizawa, T.Torii and T.Maki, Phys. + 1∆3σ2e2φ− 1∆3r2φ′2σ2e2φ, Rev.Lett 72, 450 (1994). 2 2 A = −1∆3r2φ′σ2e2φ+3∆3∆′−∆′σ′2∆′2, 24 2