1. Spacetime solutions and their understandings
Dec. 21, 2015 at the Int’l Conference on General Relativity: Centennial Overviews and Future Perspectives, Ewha Womans U., Korea
Spacetime solutions and their understandings
Gungwon Kang (KISTI)
w/ Sungwon Kim (Ewha Womans U)

2. Outline Stationary black hole solutions Formation of black holes
Black hole uniqueness theorem
Stability
Other solutions

3. 0. Introduction Spacetime (ST) is a “manifold” of events.
It is characterized by many features, including topology, dimensionality, signature, being continuous or discrete, being compact of infinite, singularity, being classical or quantum, etc..
It is important to understand the STs allowed by the nature.
Assuming Einstein’s general relativity (GR),
we try to see some properties of them.
For simplicity, we consider some simple solutions only.

4. I. Stationary black hole solutions
Static vacuum sol. w/ spherical symmetry: Karl Schwarzschild (1915)
Note:
In isotropic coordinates,
For an ingoing light, as the light approaches to r=2M.

5. Actually, the t-coordinate
becomes BAD in the vicinity of
r=2M, namely, being degenerate.
It took more than 45 years to get a correct understanding of its geometrical property: Kruskal (1960) & Szekeres

6. Rotating charged black holes: Newman et al. (1965)
Three parameter family of solutions: (M, J, Q)
Reissner-Nordstrom BH ( ): a  0
Kerr BH (1963): e  0

8. II. Formation of black holes

9. Black hole as an end product of a star collapse:
Credit: YS Myung

11. III. Black hole uniqueness theorem
Topology theorem: “The two-dimensional surface formed by the intersection of the horizon of a stationary BH with a Cauchy surface must have the topology of sphere.”
 Stationary black ring does not exist in 4D.

12. “The Kerr BHs are the only possible stationary vacuum black holes.”
Hawking: A stationary vacuum BH must be static or axisymmetric.
Israel (‘67): The only BHs that are static, vacuum, topologically spherical are the Schwarzschild solutions.
Carter (‘71) & Robinson (‘75): All stationary axisymmetric BHs are uniquely characterized by two parameters which appear in the boundary conditions.
Note that the Kerr solutions exhaust all possible values of these parameters.

13. All these results have been generalized to the electrovac case easily.
“The charged Kerr solutions are the only stationary, axisymmetric electrovac solutions.”
Bekenstein (‘72): Include other types of classical fields around a BH such as scalar, EM, etc.
 “Black hole no hair theorem”

14. IV. Stability
In order for a solution to remain for a long time, it should be stable, at least, under small perturbations.

15. “Odd Waves”
“Even Waves”
 No unstable mode solutions!

16. Such study had been extended to charged Kerr solutions, resulting in stable behaviors.
Some of higher dimensional BH solutions are NOT stable; Black strings/branes, Black rings, Some of AdS BHs, etc.

17. C.H. Lee; “Coupled gravitational and electromagnetic perturbation equations with source terms,” Nuovo Cimento B41 (1977).

18. V. Other solutions More realistic STs  Talks by Hee-Il and Chang-Hwan
Cosmological STs:  Talks by Jinn-Ouk and Yong-Seon
“Quantum” STs:  Talks by Dong-han, Ohkyung and Hyun Seok.
STs in other gravity theories:  Talk by Hyeong-Chan
STs in higher dimensions: ...include black strings/branes
Wormhole STs, ST warping, etc.
Resolution of ST singularities
Practically, realistic ST solutions for gravitational wave sources