1. Throat quantization of the Schwarzschild-Tangherlini(-AdS) black hole
Seoul, 3rd July 2017
Throat quantization of the Schwarzschild-Tangherlini(-AdS) black hole
Hideki Maeda
(Hokkai-Gakuen U, Sapporo)
References
[1] “Throat quantization of the Schwarzschild-Tangherlini(-AdS) black hole”,
G. Kunstatter & HM, CQG 31, (2014)
[2] "Exact time-dependent states for throat quantized toroidal AdS black holes“,
HM & G. Kunstatter, arXiv: [gr-qc]

2. Goal and method Goal: Derive mass spectrum for vacuum BH with symmetry
Method: Midisuperspace approach
Quantize only spacetime with symmetry
Vacuum spacetime with spherical, planar, hyperbolic symmetry
Birkhoff’s theorem in GR: Classical solution is locally characterized only by a single parameter M (mass)
Method: Reduced phase-space quantization
GR = Constrained dynamical system
Quantization on the constraint surface

3. Maximally extended BH spacetime
Asymptotically flat
Asymptotically AdS
Red: Orbit of the wormhole throat
Classical solution:

4. Throat quantization (Louko & Makela ‘96)
Important: Following 2 actions are formally identical
Action for the dynamics of the wormhole throat (with proper time t) in the maximally extended Schwarzschild BH spacetime
A reduced action starting from the ADM action in vacuum GR with spherical symmetry by canonical transformations (where t is the proper time at one spacelike infinity)
Note: t = t holds in this correspondence
Throat quantization: Quantize this dynamics

5. Symmetric spacetime in vacuum
n(>2)-dim action:
Symmetric spacetime
General metric:
Kn-2: Maximally symmetric space
We assume it is compact with area
Misner-Sharp mass (k=1,0,-1: Gauss curvature of Kn-2)
where

6. Dimensionally reduced action
ADM coordinates:
2-dim effective action:
Constraint eqs: H=0 & Hr=0 (Hopeless to solve)
Canonical transformation:
M: Misner-Sharp mass
S: Areal radius

7. Kuchar reduced action New action together with the boundary term:
Solutions of the constraint eqs.
Spatial integration on the constraint surface
where
Prescribed functions (not varied)
where
Kuchar reduced action (1-dim) (Kuchar ’94)

8. Throat variables Set such that Take another canonical transformation
t is the proper time in one asymptotic region (x=+infty)
Take another canonical transformation
New action & Hamiltonian:
Equivalent to the action for throat dynamics with areal radius a (with t as a proper time on the throat)
Dynamics of a： 0 (singularity) => ah(horizon) => 0 (singularity)
Hamiltonian = Misner-Sharp mass

9. Canonical quantization
Schrodinger equation:
E is identified as BH mass
Hamiltonian operator:
Inner product:
Effective mass & potential：
Laplace-Beltrami operator-ordering (Christodoulakis & Zanelli 1984)

10. Schrodinger eq. on the half-line
Our Shrodinger equation：
Domain of x is [0,∞)
Hamiltonian operator must be self-adjoint
Well-defined quantum theory
Time-evolution is unique and unitary
Our H admits infinite number of self-adjoint extentions
Boundary condition at x=0: (at classical singularity)
One real parameter L (L=0: Dirichlet, L=∞: Neumann)
Different L => Different quantum theory (different spectrum)
Unitarity

11. Exact spectrum for k=0 AdS BH
Toroidal (k=0) AdS BH = Quantum harmonic oscillator
Mass spectrum with Dirichlet (L=0) boundary condition
(Classical) Mass-entropy relation:
Mas is equally spaced
Entropy is not equally spaced
Same holds for large mass AdS BH with k=1,-1 (by WKB)

12. Asymptotically flat BH (k=1)
WKB approximation for large mass BH
With Dirichlet (L=0) boundary condition:
Mass-entropy relation:
Mass is not equally spaced
Entropy is equally spaced

13. Summary: BH mass spectrum
Generalization of the work by Louko & Makela ‘96
Quantity equally spaced
Asymptotically flat BH: Entropy
Asymptotically AdS BH: Mass
Our boundary condition at the origin is Dirichlet
Neumann: Similar result
Robin: Unknown (interesting but difficult)
Comparison to the result in Loop Quantum Gravity
Eigenvalues of the Area operator: Equally spaced
Is it true even in the AdS case? (There are 2 length scales)
What happens if the AdS and Planck length are comparable?
FIN

14. Spectrum for spherical (k=1) AdS BH
Classical relation
Same as
k=0 AdS BH
Same as
Asymp. flat BH
Unknown
Horizon radius