INTRODUCTION QUANTUM BLACK HOLES: Throat Theory RST MODEL CONCLUSION AND PROSPECTS
Singularity Resolution Inside Radiating 2-D Black Holes Rikkyo University, June 2013
Gabor Kunstatter University of Winnipeg
Based on PRD85, 024025 ’12 (arXive 1111.2795) J. Gegenberg UNB, T. Taves UManitoba, GK UWinnipeg
June 20, 2013
Gegenberg, Kunstatter, Taves RST (slide 1 of 33) INTRODUCTION Motivation QUANTUM BLACK HOLES: Throat Theory Canonical Quantum Gravity RST MODEL Spherical Symmetry CONCLUSION AND PROSPECTS Classic General Relativity has ”issues”:
Singularities inevitable for large class of initial data Black holes have huge entropy: hidden degrees of freedom? Black holes evaporate thermally: what happens to initial data?
Gegenberg, Kunstatter, Taves RST (slide 2 of 33) INTRODUCTION Motivation QUANTUM BLACK HOLES: Throat Theory Canonical Quantum Gravity RST MODEL Spherical Symmetry CONCLUSION AND PROSPECTS Any viable theory of quantum gravity must:
Resolve classical singularities Account for black hole entropy Tell us the endpoint of gravitational collapse and Hawking radiation Resolve ”information loss” problem Full quantum gravity difficult → consider relevant models.
Gegenberg, Kunstatter, Taves RST (slide 3 of 33) INTRODUCTION Motivation QUANTUM BLACK HOLES: Throat Theory Canonical Quantum Gravity RST MODEL Spherical Symmetry CONCLUSION AND PROSPECTS OUTLINE OF TALK
INTRODUCTION Motivation Canonical Quantum Gravity Spherical Symmetry
QUANTUM BLACK HOLES: Throat Theory
RST MODEL Review RST Black Holes: Classical RST Black Holes: Quantum
CONCLUSION AND PROSPECTS Recap Prospects: Dirac Quantization
Gegenberg, Kunstatter, Taves RST (slide 4 of 33) INTRODUCTION Motivation QUANTUM BLACK HOLES: Throat Theory Canonical Quantum Gravity RST MODEL Spherical Symmetry CONCLUSION AND PROSPECTS Canonical Quantum Gravity
Basic Hamiltonian structure: Choose arbitrary foliation of spacetime (time slices) ij Canonical variables: spatial metric gij and conjugate K (extrinsic curvature)
Lapse N ∼ g00 and shift Ni ∼ g0i are Lagrange multipliers R n ij i Lagrangian L = d x K g˙ij − NH − Ni F Hamiltonian constraint H and diffeomorphism constraints F i generate coordinate transformations.
Gegenberg, Kunstatter, Taves RST (slide 5 of 33) INTRODUCTION Motivation QUANTUM BLACK HOLES: Throat Theory Canonical Quantum Gravity RST MODEL Spherical Symmetry CONCLUSION AND PROSPECTS
Problems with canonical quantum gravity Complicated constraint algebra Hamiltonian constraint non-polynomial physical/reduced phase space highly non-trivial Parametrized theory: no unique/natural choice of time variable Two approaches: 1) Dirac quantization: impose constraints as operator constraints on physical states; physical interpretation tricky 2) Reduced quantization: first gauge fix (choose coordinates) then quantize; coordinate dependent results
Gegenberg, Kunstatter, Taves RST (slide 6 of 33) INTRODUCTION Motivation QUANTUM BLACK HOLES: Throat Theory Canonical Quantum Gravity RST MODEL Spherical Symmetry CONCLUSION AND PROSPECTS Spherical Symmetry
Consider solvable models in which relevant gravitational modes can be rigorously quantized → spherically symmetric black holes Bohr atom of black holes: should give right physics in semi-classical limit (and may work better than expectations even for small quantum numbers) Simple dynamics: Birkhoff theorem → two dimensional physical phase space Retains beautiful geometrodynamics (Kuchar ’94) Work with invariants → Throat theory (Louko and Makela ’96). Yields most rigorous derivation of black hole mass/area spectrum.
Gegenberg, Kunstatter, Taves RST (slide 7 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory RST MODEL CONCLUSION AND PROSPECTS
Throat Quantization Louko and Makela ’96, GK, Louko, Peltola, PRD ’10, ’11; Lovelock Gravity: H. Maeda, G.K. in progress
Conformal diagram: complete spatial slice describes dynamical wormhole linking two asymptotic spaces. Throat radius shrinks to zero before anything can get through.
Gegenberg, Kunstatter, Taves RST (slide 8 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory RST MODEL CONCLUSION AND PROSPECTS
Gameplan: Start with eqn. of motion of throat radius R(τ) as function of proper time of comoving observer (inside horizon F < 0):
dτ 2 = −F (R, M)dt2 + (F (R, M))−1dr 2 + R2dΩ(n−2) dR comoving (dt = 0) → = p−F (R, M) (1) dτ Construct Hamiltonian H(R, p) that generates above eqn. of motion with H = M Quantize H(R, p) with due regard to boundary conditions at singularity (R = 0)
Gegenberg, Kunstatter, Taves RST (slide 9 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory RST MODEL CONCLUSION AND PROSPECTS
Results for GR in D-dimensions and Generic 2D gravity: 1 Equally spaced area/entropy spectrum (in semi-classical limit) 2 Singularity resolved: throat “bounces”
Gegenberg, Kunstatter, Taves RST (slide 10 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory RST MODEL CONCLUSION AND PROSPECTS
Quantum evolution of gaussian (semi-classical) state ψ(R, t):
Initial semi-classical state describes collapsing wormhole
Large fluctuations at bounce then semi-classical re-expansion.
Gegenberg, Kunstatter, Taves RST (slide 11 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory RST MODEL CONCLUSION AND PROSPECTS
How to include Hawking radiation and back reaction on metric? → 2-D Gravity
Gegenberg, Kunstatter, Taves RST (slide 12 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS A Selective History of 2-D Black Hole Models
1991 CGHS: 2-D model, with radiation back-reaction via conformal anomaly Singularity was not resolved Back-reaction term made equations more complicated 1992 RST: added local term to conformal anomaly: restored solvability, but singularity not resolved “thunderbolt” at endpoint violated energy conservation 1992– many papers on generic 2-D dilaton gravity 2008 Ashtekar et al: QFT arguments in CGHS for NO information loss. 2010 Levanony and Ori (LO) quantized near singularity homogeneous dynamics of CGHS black hole interior 2010/11: Pretorius, Ashtekar, et al: numerical study of collapse.
Gegenberg, Kunstatter, Taves RST (slide 13 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS RST Action
Local form of action (Hayward ’95)
Z ( N 1 2 √ −2φ h 2 2i −2φ X 2 I := d x −g e R(g) + 4 |∇φ| + λ e + |∇f | 2π i i=1 κ κ κ o + zR(g) − |∇z|2 − φR(g) . (2) 2 4 2 κ N where 4 = 24 for N scalar fields. z is an auxialliary field that incorporates radiation back-reaction via the one loop conformal anomaly in 2-d 1 1 eq. of motion: z ∼ − R → zz ∼ −R R (3) One loop effection action exact in large N limit. Last term is the local anomaly contribution added by RST to render theory solvable.
Gegenberg, Kunstatter, Taves RST (slide 14 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Method and Results
Method Isolate black hole sector of RST Model Quantize homogeneous interior (analogy of throat dynamics) Carefully treat boundary conditions at singularity. Results Singularity resolution verified: near singularity quantization Possible existence of bound states in the spectrum for some BC’s: → remnants? Numerically solve Schrodinger equation for physical mode → radius bounces many times with decreasing amplitude Set up formalism for exact quantization, reduced and Dirac.
Gegenberg, Kunstatter, Taves RST (slide 15 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Homogeneous Solutions
Metric: ds2 = e2ρ(t)(−dt2 + dx2). (4) Solutions:
z(t) = 2ρ(t) + z1t + z0,
ρ(t) = φ(t) + p1t + p0, 1 4 1 φ(t) = W (− e2θ(t)/κ) − θ(t), (5) 2 κ κ where W (x) is the LambertW function W (x)eW (x) = x and
θ(t) := −2e−2φ − κφ 2 2 2λ 2(p1t+p0) κ 2 z1 = 2 e + p1 − t + θ0; (6) p1 p1 4
Gegenberg, Kunstatter, Taves RST (slide 16 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Solutions (cont’d)
Parametrized theory: physical phase space 4=2x2 dim:
(θ0, p1, z0, z1)
(p0 can be eliminated by trivial shift of time coordinate) Singularity Lambert function W (x) has branch point singularity at x = −1/e where W (−1/e) = −1 This happens when κ A = 1 − e2φs = 0 − 4 κ θ = (ln(κ/4) − 1) s 2 R ∝ 1/(W + 1) so this is a physical singularity.
Gegenberg, Kunstatter, Taves RST (slide 17 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Black Hole Sector
t → −∞ corresponds to Killing horizon iff linear t term in θ z2 vanishes, i.e. κ p2 − 1 = 0 p1 1 4
2ρ (φ → φ0, R → R(φ0), e → 0 in finite proper time)
Gegenberg, Kunstatter, Taves RST (slide 18 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Connection with RST RST use Kruskal type coordinates withρ ˜ = φ and: λ2 √ M −2e−2φ − κφ = − √ x˜ x˜ + P κ ln −λ2x˜ x˜ + √ . κ + − + − λ κ . Define 1 p1x± ± x˜± = e , with x = (t ± x)/2. p1 Then:
2˜ρ → 2ρ = 2ρ − p1(x+ + x−) − 2 ln(p1) = 2φ − 2 ln(p1) 2 √ 2 −2φ λ 2p1t −λ M −2e − κφ = − 2√ e + P κ ln 2 + 2p1t + √ p1 κ p1 λ κ This is precisely the same as our black hole solution with M θ = √ 0 λ κ P = 0
Gegenberg, Kunstatter, Taves RST (slide 19 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Physical Properties of Black Holes Solutions
BH Sector corresponds to P = 0 in RST: microscopic black holes in thermal equilibrium with incoming radiation (Birnir et al hep-th/9203942) Thermal equilibrium necessary since interior homogeneous → static exterior √ Mass: M = λ κθ0 p1 Temperature: T = 2π Note: temperature/surface gravity and mass are independent.
Gegenberg, Kunstatter, Taves RST (slide 20 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Dilaton Equation in Conformal Gauge
In terms of the variable: Φ = e−2φ κ (∇Φ)2 Φ2 Φ = − + 4λ2 (7) 4 Φ(Φ − κ/4) Φ − κ/4
In conformal gauge ρ − φ = p1t is linear in time (RST simplification). The equation for Φ:
κ Φ˙ 2 Φ Φ¨ = − − 4λ2e2p1t (8) 4 Φ(Φ − κ/4) Φ − κ/4 Generated by Hamiltonian:
2 2 ΠΦ Φ 2 2p1t κ HR = 2 Φ−κ/4 + 4λ e Φ − 4 ln Φ
Gegenberg, Kunstatter, Taves RST (slide 21 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Near Singularity Dynamics
Near the singularity Φ = κ/4, the dominant term in the Hamiltonian is Π2 H = Φ (9) R (Φ − κ/4)2 This generates the leading term near the singularity:
1/2 Φ − κ/4 ∝ (t − t0) This is different from Levanony and Ori who obtained
2/3 Φ − κ/4 ∝ (t − t0) The above is consistent with series expansion of the exact solution .
Gegenberg, Kunstatter, Taves RST (slide 22 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Near Singularity Quantum Theory: Method I
Transform to y = (Φ − κ/4)2 (“Natural” Parameterization Invariant Measure). Consider functions that are L2 normalizable on (0, ∞) with measure R dy = R RdR. The quantum Hamiltonian is then d 2 Hˆ = − (10) 1 dy 2 This is just the free particle on the half line, well known: There is a one parameter family of self-adjoint extensions with bc’s: ψ(0) = λψ0(0) Scattering E > 0 spectrum continuous. For λ < 0, there exists a bound state →“stable remnant”?
Gegenberg, Kunstatter, Taves RST (slide 23 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Near Singularity Quantum Theory: Method II
Wave functions normalizable on half line with measure R dR. Standard symmetric factor ordering for the Hamiltonian: d 1 d 1 d 1 d Hˆ = − = −Φ2 2 dΦ Φ2 dΦ Φ2 dΦ Φ2 dΦ d 2 = −y 2/3 (11) y 2
where y = Φ3/3 ≥ 0. The eigenvalue equation is:
Hˆ ψ(y) = Eψ(y) → ψ00 + y −2/3Eψ = 0 (12)
Gegenberg, Kunstatter, Taves RST (slide 24 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS
The solutions are:
√ 2/3 √ 2/3 ψk (y) = C+ yJ3/4(3ky /2) + C− yJ−3/4(3ky /2) (13) Self-adjoint extension parameter λ with same BC’s free particle. Scattering states E > 0: Asymptotics ok: free particle plane waves in terms of variable R. 0 At origin, BC’s ψ(0) = λψ (0) again imply C− = λC+ Bound states:
√ −4/3 2Γ2(3/4) E = −κ2 = − √ 3π|λ| √ 3κ ψ (y) = N (λ) yK y 2/3 . (14) b b 3/4 2
Gegenberg, Kunstatter, Taves RST (slide 25 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Quantization of Full Interior
In conformal gauge dilaton decouples, dynamics described by Hamiltonian:
2 2 ΠΦ Φ 2 2p1t κ HR = 2 Φ−κ/4 + 4λ e Φ − 4 ln Φ
Canonical Transformation: κ κ κ y = Φ − ln Φ − 1 − ln (15) 4 4 4 yields (up to trivial constant shifts):
2 Πy H = + 4Λ2e2p1t+p0 y (16) 2
Gegenberg, Kunstatter, Taves RST (slide 26 of 33) INTRODUCTION Review QUANTUM BLACK HOLES: Throat Theory RST Black Holes: Classical RST MODEL RST Black Holes: Quantum CONCLUSION AND PROSPECTS Tentative Results
Figure : Evolution of initially Gauss state for RST BH interior
Gegenberg, Kunstatter, Taves RST (slide 27 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory Recap RST MODEL Prospects: Dirac Quantization CONCLUSION AND PROSPECTS Recap: Quantum Mechanics of RST Black Hole Interiors
Considered dynamics of throat area including radiation back reaction Quantization yielded ”bouncing wormhole” with decreasing radius/energy Need to do better numerics to discover end state: singular or quantum remnant?
Gegenberg, Kunstatter, Taves RST (slide 28 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory Recap RST MODEL Prospects: Dirac Quantization CONCLUSION AND PROSPECTS Prospects: Dirac Quantization
In terms of phase space variables:
2e−2φ κ y = z + − φ;Π = − (Π + 2Π ); κ y B(φ) φ z A (Πφ + Πz ) w = κ(z − 2φ); Π = κ . (17) w B(φ)
h 2 1 λ2 2β i Hamiltonian: H=2πσ κΠw − κ Πβ Πy − π2 e . Wheeler-DeWitt Equation separable, but difficult/impossible to deal with boundary conditions in this parameterization.
Gegenberg, Kunstatter, Taves RST (slide 29 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory Recap RST MODEL Prospects: Dirac Quantization CONCLUSION AND PROSPECTS Dirac Quantization (cont’d)
Simple canonical transformation √ κΩ = 2e−2φ + κφ 1 χ = √ 2e−2φ − κφ + 2κρ κ y = z − 2ρ (18)
leads to: 2 1 2 2 Πy λ2 2(χ−Ω)/κ H= 2πσ 4 (ΠΩ − Πχ) + κ − π2 e Wheeler-Dewitt Equation not separable, but boundary conditions easy to implement.
Gegenberg, Kunstatter, Taves RST (slide 30 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory Recap RST MODEL Prospects: Dirac Quantization CONCLUSION AND PROSPECTS Take Home Messages:
Quantum mechanics of spherically symmetric black holes should give robust predictions, at least in semi-classical regime. 2-D Models allow rigorous inclusion of Hawking radiation and back reaction.
Gegenberg, Kunstatter, Taves RST (slide 31 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory Recap RST MODEL Prospects: Dirac Quantization CONCLUSION AND PROSPECTS Other Prospects
More general 2-d theories “Throat” quantization (proper time instead of conformal time) 4-d black holes??
Gegenberg, Kunstatter, Taves RST (slide 32 of 33) INTRODUCTION QUANTUM BLACK HOLES: Throat Theory Recap RST MODEL Prospects: Dirac Quantization CONCLUSION AND PROSPECTS Thanks go to:
my collaborators, Jack Gegenberg, Tim Taves (soon to be Dr.), Hideki Natural Sciences and Engineering Research Council. Jorma Louko for invaluable help and collaboration on throat quantization. CECS, Valdivia for kind hospitality during completion of some of this work. Hideki and Rikkyo University for hospitality and giving me this opportunity to speak all of you for listening
Gegenberg, Kunstatter, Taves RST (slide 33 of 33)