1. Singularity Resolution in Scolymerized (BTZ) Black Holes Gabor Kunstatter ESI, 3-D Gravity Workshop April, 2009 Based on collaborative work: A. Peltola and G.K. -- arXiv:0902.1746arXiv:0902.1746 -- arXiv:0811.3240arXiv:0811.3240 J. Ziprick and G.K. -- arXiv:0902.3224arXiv:0902.3224 -- arXiv:0812.0993arXiv:0812.0993 J. Gegenberg and G.K. -- gr-qc/0606002v2 -- in preparation Inspired by work of (with apologies for distortions): Bohmer and Vandersloot ‘07 Campiglia, Gambini and Pullin ’07 Gambini and Pullin ‘08

2. Introduction Quantum gravity should address: –Classical Singularities –Endpoint of gravitational collapse and Hawking radiation –Information loss Basic Idea: –Choose simple, but generic models that retain some essential features of full theory; –Explore semi-classical limit; –In general hard to justify from first principles, but sometimes gives interesting results.

3. Polymerization—underlying motivation General Relativity, quantum mechanics, electromagnetism assume space is smooth to arbitrarily small scales Quantum gravity will almost certainly mess this up on tiny (i.e. Planck) scales –String theory: loops, matrices, non-commutive geometry… –Loop quantum gravity: spin networks, discrete area spectrum Polymer/Bohr quantization: unitarily inequivalent to Schrodinger; starts with discrete topology on real line  generic consequences of this (independent of specific microscopic theory)?

4. OUTLINE Introduction Polymer/Bohr Quantization Semi-Classical polymerization (“Scolymerization”)‏ Scolymerized Schwarzschild (Peltola and GK)‏ BTZ (Gegenberg and GK)‏ Dynamical Singularity Resolution (Ziprick and GK)‏ --time permitting: neat movies Conclusions Singularity Resolution in Scolymerized BTZ Black Holes

5. Polymer Quantization A quantization that naturally incorporates fundamental discreteness of spatial geometry at the microscopic level. Motivated by loop quantum gravity, but distinct. Unitarily inequivalent to Schrodinger quantization Fundamental difference: gives real line a discrete topology  p=-I d/dx doesn’t exist as self-adjoint operator Must build observables out of Ashtekar et al ’93, Halverson ‘91

6. Polymer Quantization Ashtekar et al ’93, Halverson ‘91 Has been applied to simple quantum systems: Harmonic Oscillator Ashtekar, Fairhurst and Willis ‘’02 Coulomb Potential Husain and Louko ’06 1/X^2 Potential Louko, Ziprick and GK ’08 Black hole exterior, Gegenberg, Small and GK ‘06 Spectrum generally differs near ground state, agrees semi-classically Has been shown to lead to singularity resolution in Cosmology Ashtekar, Pawlowski and Singh ‘06 Black hole interiors Ashtekar and Bojowald ’06

7. Scolymerization In the limit that Planck’s constant goes to zero but the discretization scale stays small but finite we can “remove hats”: (see Husain and Winkler ‘07 for derivation in terms of coherent states)‏ Quantum turning point at prevents momentum from getting too large Has been applied to black hole interiors: Modesto ’06 Boehmer and Vandersloot ’07 Pullin et al ‘07 “Singularity avoidance”

8. Spherically Symmetric Black Hole Interiors Start with spherically symmetric gravity in D=n+2 dimensions in 2-D dilaton form (with apologies to workshop organizers)‏ Simplifies equations, but just a canonical transformation after all is said and done (with apologies to W. Kummer). l is an arbitrary length scale. Convenient to take l=l planck  2G=1

9. 3-D Gravity: Rotating BTZ Same general form of reduced action, but without need for conformal factor in front of physical metric: The vector potential has been solved for and absorbed into the “dilaton potential”

10. Hamiltonian: homogeneous interior Since we are inside the black hole x  t This is a parametrized theory describing a single pair of dynamical phase space degree of freedom Instead of fixing time coordinate, we first use Hamilton-Jacobi theory to find general solution in terms of physical constants of motion Yields standard Schwarzschild interior, which can be extended across horizon to reproduce complete black hole spacetime ADM Parametrization Action: Hamiltonian:

11. Scolymerized Interior Consider 2 variations: Version 1, in which both variables scolied gives results similar to those of Campigni et al.: singularity is resolved as expected, at cost of adding more Horizons to the spacetime. Version 2, gives different results, so this is the one we will concentrate on

12. A Hamilton-Jacobi Primer

13. A Hamilton-Jacobi Primer (cont’d)‏

14. Quantum Corrected 4-d Schwarzschild: In 4-d the integral can easily be done. Can write the metric using r as the time coordinate, we get: Interior

15. Quantum Corrected 4-d Schwarzschild: In 4-d the integral can easily be done. Can write tthe metric using r as the time coordinate, we get: Exterior:

16. Complete Quantum Corrected 4-d Schwarzschild Spacetime ?: Exterior: asymptotically flat, with O(k 2 /r 2 ) corrections that violate energy conditions Interior: bounce at r=k, expands to macroscopic Kantowski-Sachs cosmology Reminiscent of “Universe creation inside a black hole” Frolov, Markov and Mukhanov, (1990); Easson, Brandenberger, (2001).

17. Higher Dimensions: eg 5-d Schwarzschild: fall-off in asymptotic region too slow for Poincare generators to be well defined  not asymptotically flat in strictest sense This seems generic in all higher dimensions. Not so nice! NOTE:

18. Back to the BTZ Black Hole Black holes requires cosmological constant and rotation: Generic quantum bounce now gives a minimum and maximum radius: Ruins asymptotic behaviour !

19. Dynamical Singularity Resolution Einstein gravity is always attractive  gravitational potential goes to minus infinity at r=0. LQG suggests that at Planck scale gravitational potential should be repulsive, and finite at the origin: Ziprick, GK ’08 Quantum corrected Classical Husain ‘08 used quantum corrected potential to look at critical collapse in double null coordinates; found mass gap of order of quantum scale We used P-G coordinates; verified mass gap and found singularity resolution

20. Subcritical Collapse: No black hole

21. Classical Black Hole Formation

22. Distance from center P-G Time Infalling matter Outgoing Light rays Event horizon Trapped region --all light rays move inward singularity

23. Near Critical: Choptuik Scaling

24. Quantum Corrected Black Hole Formation J. Ziprick and GK ‘08

25. Quantum Corrected Black Hole Formation J. Ziprick and GK ‘08

26. Quantum Corrected Black Hole Formation J. Ziprick and GK ‘08 Classical spacetime: red Line marks boundary of trapped region. Ends at the singularity and at infinity. Quatum corrected spacetime: red Compact trapping region. Horizon eventually shrinks due to radiation No singularity!

27. Summary and Conclusions Scolymerization in 4-d yields a interesting semi-classical solution: Singularity free Single horizon (no mass inflation)‏ Realizes Universe Generation inside Black Hole Seems to work only in 4-d: good for LQG? Approach also yields intriguing results for dynamical black holes Results: Weakness and Future Prospects (Scolymerization): Need formulation that extends beyond horizon (Pullin et al.)‏ Connect mini-superspace model to full QG Extend to non-asymptotically flat space times and higher D? Eg. assuming scolymerized solution only valid in interior; match smoothly to Schwarzschild-(A)dS interior and extend across horizon (Maeda, GK in progress)‏ OR: time to move to a different lamp post?