We are used to ponder the information loss paradox from the point of view of external observers. [Also, black hole complementarity principle]

Transcription

1 Yen Chin Ong

2 We are used to ponder the information loss paradox from the point of view of external observers. [Also, black hole complementarity principle] Notwithstanding the firewall, horizon should not be a special place: need to understand interior spacetime --- including the resolution of the singularity. (However, with firewall, perhaps there is no interior? See e.g. Susskind arxiv: [hep-th])

3 Singularities, Firewalls, and Complementarity Susskind arxiv: [hep-th]

4 Well known can construct nontrivial geometries inside black hole, e.g. Wheeler s bag-of-gold. How about volumes inside generic black holes? Very naively, one expects, for a (classical) Schwarzschild black hole with Schwarzschild radius r h = 2M, This is NOT the case: [1] Schwarzschild interior is not static. [2] 3-volumes depend on the choice of spacelike hypersurfaces Does NOT make sense to talk about the volume. Brandon S. DiNunno, Richard A. Matzner, The Volume Inside a Black Hole, Gen. Rel. Grav. 42 (2010) 63, [arxiv: [gr-qc]]. Source: Backreaction blog Kruskal-Szekeres Maximal Extension of Schwarzschild spacetime contains another infinitely large universe on the other side.

5 A remnant looks small from the exterior point of view, but could harbor a large interior. In principle, there is enough room to store information. (Note, however, remnant scenario is independent of the information issue) Challenges: [1] What do we mean by storing information? [2] In tension with holography. ( weak form vs strong form ) [3] How generic is it for a black hole to, during the final phase of gravitational collapse, re-explode into some other region of spacetime?

6 Misner-Thorne-Wheeler: Such a process requires that the exploding end of the wormhole be built into the initial conditions of the universe, with mass and angular momentum (as measured by Keplerian orbits and frame dragging) precisely equal to those that go down the black-hole end. This seems physically implausible. So does the explosion.

7 We begin with a Schwarzschild black hole: g Sch = 1 2M r dt M r 1 dr 2 + r 2 dθ 2 + sin 2 θdφ 2. Eddington-Finkelstein: g Sch = 1 2M r dv 2 + 2dvdr + +r 2 dθ 2 + sin 2 θdφ 2 Advanced time: Marios Christodoulou, Carlo Rovelli, How Big is a Black Hole?, Phys. Rev. D 91 (2015) , [arxiv: [gr-qc]].

8 Event Horizon Asymptotically, grows linearly in time: Ingemar Bengtsson, Emma Jakobsson, Mod. Phys. Lett. A 30 (2015) , Black Holes: Their Large Interiors, [arxiv: [gr-qc]]. Marios Christodoulou, Carlo Rovelli, How Big is a Black Hole?, Phys. Rev. D 91 (2015) , [arxiv: [gr-qc]]. Bruce L. Reinhart, Maximal Foliations of Extended Schwarzschild Space, J. Math. Phys. 14 (1973) 719. Frank Estabrook, Hugo Wahlquist, Steven Christensen, Bryce DeWitt, Larry Smarr, Elaine Tsiang, Maximally Slicing a Black Hole, Phys. Rev. D 7 (1973) 2814.

9 Radius: ~ O 10 or so of Earth Moon Distance. Maximal Volume ~ can fit a million solar systems! Rotation does not change the result by much [only O(10)]. Ingemar Bengtsson, Emma Jakobsson, To appear in Modern Physics Letters A, Black Holes: Their Large Interiors, [arxiv: [gr-qc]].

10 Flat space intuition: Large area bounds large volume. This is still true for Schwarzschild geometry: However, not true in general! Yen Chin Ong, Never Judge a Black Hole by Its Area, JCAP 04 (2015) 003, [arxiv: [gr-qc]] Maximal volumes inside black holes are NOT necessarily an increasing function of the horizon areas. Which black hole is bigger?

11 A lens space is the orbit space of a free linear action of a finite cyclic group on a sphere. Construction of a 2- sphere from a disk. Area = 2π 2 /p Construction of a lens space from a solid ball.

12 Area of the horizon: Maximal Slice: Volume: 8πML 3 as p.

13 The Unspeakable But important! Classical general relativity predicts its own downfall. A Brief History of Time, S. Hawking. But what actually happens at the singularities? Usual attitude: Don t worry about it, it will go away when we have a working theory of quantum gravity. But maybe by trying to understand (and resolve?) singularities, we can get nearer to a working theory of quantum gravity? (Though I don t know how!)

14 One could argue that a singularity is not physical if one can somehow extend spacetime beyond it. However, in general, (even analytic) extensions are not unique, which physical reasons are to be used to discriminate between inequivalent extensions? Kruskal-Szekeres spacetime is the unique maximally extension of Schwarzschild solution, however if we drop some conditions like vacuum or analyticity, then other maximal extensions are possible. [Seven are explicitly given by José M. M. Senovilla, in Singularity Theorems and Their Consequences (Review), General Relativity and Gravitation 30 (1998) ]

15 For details and more references, see our review paper

16 Null areas are independent of slicing Given three nearly parallel infinitesimally separated light rays in Minkowski space moving in the direction perpendicular to their plane of separation, one can define the area they span by cutting them with a space time plane, and asking what is the area contained in the triangle they form. In Minkowski geometry the area of this triangle doesn't depend on the orientation of the plane. Same length!