Throat quantization of the Schwarzschild-Tangherlini(-AdS) black hole ICGAC13-IK15 @ 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, 115009 (2014) [2] "Exact time-dependent states for throat quantized toroidal AdS black holes“, HM & G. Kunstatter, arXiv:1706.01906 [gr-qc]

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

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

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

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

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

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)

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

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

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

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)

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

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

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