SPEC & GEN RELATIVITY
SPEC & GEN RELATIVITY PHZ 6607
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This 17 page Class Notes was uploaded by Maureen Roob IV on Friday September 18, 2015. The Class Notes belongs to PHZ 6607 at University of Florida taught by Bernard Whiting in Fall. Since its upload, it has received 20 views. For similar materials see /class/206871/phz-6607-university-of-florida in Physics 2 at University of Florida.
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Date Created: 09/18/15
Review of Black Hole Stability Jason Ybarra PHZ 6607 Black Hole Stability Schwarzschlld Regge amp Wheeler 1957 Vishveshwara 1979 Wald 1979 GuiHua 2006 Kerr Whiting 1989 Finster2006 I Stability of Schwarzschild Black Hole Regge and Wheeler 1957 analyzed the stability ofthe Schwarzschild metric gm gm hyu such that 6RWL 0 The perturbation hW was analyzed using tensor spherical harmonics and a simplifying gauge transformation Regge and Wheeler 1957 Stability of Scharzschild Black Hole There are 2 types of solutions for hw oddparity and evenparity Secondorder wave equation for oddparity 12 mk2 ffQ0 7 7 BRA1112II 7 1 2M tl1 m Veffi T 72 7V2 Regge and Wheeler 1957 Stability of Schwarzschild Black Hole I There are 2 types of solutions for hw oddparity and evenparity Secondorder wave equation for evenparity I 3 4 2 29 2 1 3 EWl lj 2gt l W Dl1 12 2 1 I rZILL 1x TVZAMY I my Edelstein amp shveshwara 1970 Regge and Wheeler 1957 Stability of Schwarzschild Black Hole Regge amp Wheeler found that the solutions that went to zero at r gt went to zero at r 2M Regge amp Wheeler concluded that the Schwarzschild black hole is stable to perturbations using the assumption that solutions should be smoothly connected across r 2M Regge and Wheeler 1957 Stability of Schwarzschild Black Hole Kruskal Coordinates Vishveshwara 1970 expanded the analysis ofthe stability ofthe Schwarzschild Black hole by using Kruskal coordinates thus removing the coordinate singularity at r 2M The Schwarzschild solutions of Regge amp Wheeler 1957 near r 2M were transformed into Kruskal coordinates 1 1 39 7 W406 R T 2121 R T 7 RiT Vishveshwara 1970 Stability of Schwarzschild Black Hole Kruskal Coordinates Vishveshwara found that the solutions t 0 that fall of to zero at infinity also diverge at r 2M and thus physically unacceptable and were excluded T 2 03quot Q2 Ae m39 hit 3 SMQAR mMMn For tOr2M gt TORO gt hk03 diverges Vishveshwara 1970 Stability of Schwarzschild Black Hole Kruskal Coordinates Vishveshwara also analyzed real oscillating perturbations Odd waves singularity at RTO Even waves singularity at RT O and RTO Divergence is removed by considering wave packets from the superposition of these waves Vishveshwara 1970 Stability of Schwarzschild Black Hole Energy Integral of Oscillating Modes Wald 1979 found that a perturbation composed of oscillating modes is bounded for all time outside and on the horizon Energy integral lflttrl1 S lfbl dr5r1rAfbdr rfbrldrquot iftri2dr lt Ziifoii 2anii2 Wald 1979 Stability of the Schwarzschild Black Hole GuiHan 2006 GuiHan 2006 reanalyzed the stability ofthe Schwarzschild black hole in Kruskal coordinates Instead of using t 0 as the initial time GuiHan analyzed the behavior ofthe perturbations for T lt 0 and T gt 0 For initial T lt 0 RT 0 at r 2M and the perturbations hD3 and h13 become hgg 787712Alt72TJ74W I h03 and h13 blow up hi 877121472Tquotquot as T a 0 GuiHan 2006 Stability of the Schwarzschild Black Hole GuiHan 2006 For initial T gt 0 the perturbations h03 and h13 diverge for all values of T thus they are excluded and this region is stable against perturbations similar to the result of Vishveshwara 1979 White hole connected region R2 T2 2 0 T s 0 R 2 0 A U Stab39e Black hole connected region R2 T220T20 R20 4 Stab39e GuiHan 2006 Stability of Kerr Black Hole N 17 13 19112110393 zMr LUarsinz 9 zMaTIsin H E dnm ytzd mMIA V A 17 723114 12 z 13 11260329 a is the angular momentum where a lt M For a 0 a Schwarzschild Stability of Kerr Black Hole The linear wave equation for the metric is v 9 1 a a 2 a 7 K 7 2 L2 a 7 r 7 fig3 7 437 iacos a 2 a 1 2 a a 2 arose 5111 9700059 51112 9 Item 95 15096 II 7 0 where s is the spin I eildtewR5rSs6 The wave equation can be split into a radial equation and an angular equation Stability of Kerr Black Hole For unstable modes Whiting 1989 made the following transformations Differentialtransformation 86 a T6 Integral transformation Rr a Kr for r lt r lt 4 ltIgtS e MeWIx S7TS Associated conserved quantity w 3 81 3 7 y i 71 NB 7 1 i 2 2 aEdrd dubtha cos H 9 A 2 Livusb revJr 17 J liusbi39iv m 0 grow exponentially All terms under the integral and integrable a modes cannot Whiting 1989 Stability of Kerr Black Hole Stability of oscillating modes Wald energy integral method cannot be used for a Kerr black hole because the energy density is negative in the ergosphere Finster et al 2006 provided a rigorous proof for the decay of scalar waves s0 however the decay of 31 and 32 has yet to be proven Finster et al 2006 References Dotti G Gleiser R J Francisco RaneaSandoval amp Vucetich H 2008 arXiv08054306 Edestein L A amp Vishveshwara C V 1970 Phys rev D 1 3514 Finster F Kamran N Smoller J amp Yau ST 2006 Commun Math Phys 264 465 GuiHua T 2006 Chin Phys Lett 23 783 Regge T amp Wheeler JA 1957 Phys Rev 108 1063 Vishveshwara C V 1979 Phys Rev D 1 2870 Wad R M 1979 J Math Phys 20 1056 Whiting B F 1989 J Math Phys 30 1301
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