Review of Black Hole Stability. Jason Ybarra PHZ 6607

Transcription

1 Review of Black Hole Stability Jason Ybarra PHZ 6607

2 Black Hole Stability Schwarzschild Regge & Wheeler 1957 Vishveshwara 1979 Wald 1979 Gui-Hua 2006 Kerr Whiting 1989 Finster 2006

3 Stability of Schwarzschild Black Hole Regge and Wheeler (1957) analyzed the stability of the Schwarzschild metric such that The perturbation h μν was analyzed using tensor spherical harmonics and a simplifying gauge transformation ( Regge and Wheeler 1957)

4 Stability of Scharzschild Black Hole There are 2 types of solutions for h μν : odd-parity and even-parity. Second-order wave equation for odd-parity ( Regge and Wheeler 1957)

5 Stability of Schwarzschild Black Hole There are 2 types of solutions for h μν : odd-parity and even-parity. Second-order wave equation for even-parity (Edelstein & Vishveshwara 1970) ( Regge and Wheeler 1957)

6 Stability of Schwarzschild Black Hole Regge & Wheeler found that the solutions that went to zero at r, went to zero at r = 2M Regge & 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)

7 Stability of Schwarzschild Black Hole: Kruskal Coordinates Vishveshwara (1970) expanded the analysis of the stability of the Schwarzschild Black hole by using Kruskal coordinates, thus removing the coordinate singularity at r = 2M The Schwarzschild solutions of Regge & Wheeler (1957) near r = 2M were transformed into Kruskal coordinates. (Vishveshwara 1970)

8 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. For t=0,r=2m T=0,R=0 h k 03 diverges (Vishveshwara 1970)

9 Stability of Schwarzschild Black Hole: Kruskal Coordinates Vishveshwara also analyzed real (oscillating) perturbations. Odd waves: singularity at R+T=0 Even waves: singularity at R+T = 0 and R-T=0 Divergence is removed by considering wave packets from the superposition of these waves. (Vishveshwara 1970)

10 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 (Wald 1979)

11 Stability of the Schwarzschild Black Hole: Gui-Han (2006) Gui-Han (2006) re-analyzed the stability of the Schwarzschild black hole in Kruskal coordinates. Instead of using t = 0 as the initial time, Gui-Han analyzed the behavior of the perturbations for T < 0 and T > 0 For initial T < 0, R+T = 0 at r = 2M and the perturbations h 03 and h 13 become h 03 and h 13 blow up as T 0 (Gui-Han 2006)

12 Stability of the Schwarzschild Black Hole: Gui-Han (2006) For initial T > 0, the perturbations h 03 and h 13 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 (R 2 T 2 0, T 0, R 0) Black hole connected region (R 2 T 2 0, T 0, R 0) Unstable Stable (Gui-Han 2006)

13 Stability of Kerr Black Hole a is the angular momentum where a < M. For a = 0 Schwarzschild

14 Stability of Kerr Black Hole The linear wave equation for the metric is where s is the spin The wave equation can be split into a radial equation and an angular equation

15 Stability of Kerr Black Hole For unstable modes, Whiting (1989) made the following transformations. Differential transformation: S(θ) T(θ) Integral transformation: R(r) K(r) for r + < r < Associated conserved quantity All terms under the integral and integrable modes cannot grow exponentially (Whiting 1989)

16 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 (s=0), however the decay of s=1 and s=2 has yet to be proven (Finster et al. 2006)

17 References Dotti, G., Gleiser, R. J., Francisco Ranea-Sandoval, I., & Vucetich, H. 2008, arxiv: Edelstein, L. A., & Vishveshwara, C. V. 1970, Phys. rev. D, 1, 3514 Finster, F., Kamran, N., Smoller, J., & Yau, S.-T. 2006, Commun. Math. Phys., 264, 465 Gui-Hua, T 2006, Chin. Phys. Lett., 23, 783 Regge, T, & Wheeler, J.A. 1957, Phys. Rev., 108, 1063 Vishveshwara, C. V. 1979, Phys. Rev. D, 1, 2870 Wald, R. M. 1979, J. Math. Phys, 20, 1056 Whiting, B. F. 1989, J. Math. Phys., 30, 1301