Singularity Resolution in Scolymerized (BTZ) Black Holes

Gabor KunstatterESI, 3-D Gravity Workshop April, 2009

Based on collaborative work:

A. Peltola and G.K. -- arXiv:0902.1746 -- arXiv:0811.3240

J. Ziprick and G.K. -- arXiv:0902.3224 -- arXiv:0812.0993

J. Gegenberg and G.K. -- gr-qc/0606002v2-- in preparation

Inspired by work of (with apologies for distortions):

Bohmer and Vandersloot ‘07Campiglia, Gambini and Pullin ’07Gambini and Pullin ‘08

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.

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 linegeneric consequences of this (independent of specific microscopic theory)?

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

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 topologyp=-I d/dx doesn’t exist as self-adjoint operator

•Must build observables out of

Ashtekar et al ’93, Halverson ‘91

x∣x ⟩=x∣x ⟩U μ∣x ⟩=∣xμ ⟩

⇒ p μ=U μ− U−μ

2iμ≈eiμp−e−iμp

2iμ=

sin μpμ

Polymer QuantizationAshtekar 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

Scolymerization

H scol=sin2 μp

2Mμ 2 V x ⇒Mdxdt

=sin μp cos μp μ

•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

p= π2μ

“Singularity avoidance”

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).

S=12G ∫ d2 x φR g V φ

l2 ; V φ ∝φ−1n≃l

r φ

ds2phys=1

j φ gμν dx

μdxνφ x ,t d n

j φ =∫d φV φ ; l is an arbitrary length scale. Convenient to take l=lplanck2G=1

3-D Gravity: Rotating BTZ

• Same general form of reduced action, but without need for conformal factor in front of physical metric:

S=1

2G3 ∫ d 2 x rR g V r ;

V r =rΛ−J2

r3;

dsphys

2 =gμνdxμ dxνr2 x ,t dϑ Aμ dx

μ 2

The vector potential has been solved for and absorbed into the “dilaton potential”

Hamiltonian: homogeneous interior

ds2=e2ρ t

jφ −σ2 t dt2dx2 φ t d n

I=∫ dt π ρ ρ¿

πφφ¿

−σH H=Gπ ρπφ

e2ρV φ 2l2G

≈0

•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:

Scolymerized Interior

• Consider 2 variations:

H1=1

μ2 sin μGπ ρ sin μGπ φ e2ρV φ 2l2 ≈0

H2=1μGπ ρ sin μGπφ

e2ρV φ 2l2

≈0

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 moreHorizons to the spacetime.

Version 2, gives different results, so this is the one we will concentrate on

A Hamilton-Jacobi Primer

¿ Look for Hamilton Jacobi Function: S ρ ,φ such that: S ,A=Gπ A

Ham. Constraint ⇒ S,ρsin μS,φ μ

e2ρV φ

2l2=0

¿ The solution is:

S ρ , φ =−α4G

e2ρlμ∫ dφarcsin μVα lG C

where α is a constant of integration .

The general solution is obtained from: ∂S∂α

=−β

where β is the constant of motion conjugate to α¿ Recall: spher . symm . gravity has 2 physical phase space degrees of freedom: M ,PM

A Hamilton-Jacobi Primer (cont’d)

This allows us to express all four phase space variables in terms of one free function choice of time :14G

e2ρ=β−lμI n φ

sin μπφ =μα lG

V φ ;

π ρ=−μα lG

e2ρ ∣V∣=lr≤α lGμ

⇒ r≥μαG

≡k Bounce!

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:

¿quantum bounce at r=k . Change coordinate to:r=k cosh y ⇒metric regular at y=0

¿after bounce metric asympotes to anisotropic Kantowski-Sachs cosmology:

ds2−1−2GMr

O 1/ r2 dT 2T 2d12GMr

O 1 /r2 dx2

Interior

ds2=−1

2G4 Mr

−ε 1−k2

r2 1−k2

r2 dr22G4 M

r−ε 1−k

2

r2 dx2r2d2

where ε=1 before the quantum bounce at r=k and -1 after the bounce .

ds2=−1

2G4 Mr

−ε 1−k2

r2 1−k2

r2 dr22G4 M

r−ε 1−k

2

r2 dx2r2d2

where ε=1 before the quantum bounce at r=k and -1 after the bounce .

Quantum Corrected 4-d Schwarzschild:

¿continue across single horizon¿exterior described by Schwarzschild metric with corrections of order k2/ r2

¿ violations of energy conditions to this order as well

ds2 −1−2GMr

O k2/r2 dt 212GMr

O k2/r2 dr2r2d

¿ the quantum corrected horizon is at:

rH= 2GM 2k2

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

Exterior:

Complete Quantum Corrected 4-d Schwarzschild Spacetime ?:

•Exterior: asymptotically flat, with O(k2/r2) 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).

Higher Dimensions: eg 5-d Schwarzschild:

ds2=−1

8G5 M

3πr2−ε 1−k

2

r2−ε

k 2

r2ln rk r2

k 2−1

dr28G 5 M

3πr2 −ε 1−k2

r2 −ε k2

r2 ln rk r2

k 2 −1 dx2r2d2

•fall-off in asymptotic region too slow for Poincare generators to be well definednot asymptotically flat in strictest sense•This seems generic in all higher dimensions. Not so nice!

grr r∞

1− 2GMk 2 ln r r2 . . .NOTE:

Back to the BTZ Black Hole

∣V∣=rl− j2l3

r3≤α lGμ

⇒ j2 l3μα lG 1 /3

≤r≤αl2Gμ

•Black holes requires cosmological constant and rotation:

V= rl− j2 l3

r3r∞

∞

•Generic quantum bounce now gives a minimum and maximum radius:

•Ruins asymptotic behaviour !

Dynamical Singularity Resolution

• Einstein gravity is always attractivegravitational 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 correctedClassical

•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

Subcritical Collapse: No black hole

Classical Black Hole Formation

Distance from center

P-GTime

Infalling matter

OutgoingLight rays

Event horizon

Trapped region--all light rays move inward

singularity

Near Critical: Choptuik Scaling

Quantum Corrected Black Hole FormationJ. Ziprick and GK ‘08

Quantum Corrected Black Hole FormationJ. Ziprick and GK ‘08

Quantum Corrected Black Hole FormationJ. Ziprick and GK ‘08

Classical spacetime: redLine marks boundary of trapped region. Ends at the singularity and at infinity.

Quatum corrected spacetime: redCompact trapping region.Horizon eventually shrinks due to radiationNo singularity!

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?