Singularity Resolution Inside Radiating 2-D Black Holes Rikkyo University, June 2013

Transcription

1 Singularity Resolution Inside Radiating 2-D Black Holes Rikkyo University, June 2013 Gabor Kunstatter University of Winnipeg Based on PRD85, (arxive ) J. Gegenberg UNB, T. Taves UManitoba, GK UWinnipeg June 20, 2013 Gegenberg, Kunstatter, Taves RST (slide 1 of 33)

2 Motivation Canonical Quantum Gravity Spherical Symmetry 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)

3 Motivation Canonical Quantum Gravity Spherical Symmetry 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)

4 Motivation Canonical Quantum Gravity Spherical Symmetry OUTLINE OF TALK INTRODUCTION Motivation Canonical Quantum Gravity Spherical Symmetry Review RST Black Holes: Classical RST Black Holes: Quantum Recap Prospects: Dirac Quantization Gegenberg, Kunstatter, Taves RST (slide 4 of 33)

5 Motivation Canonical Quantum Gravity Spherical Symmetry Canonical Quantum Gravity Basic Hamiltonian structure: Choose arbitrary foliation of spacetime (time slices) Canonical variables: spatial metric g ij and conjugate K ij (extrinsic curvature) Lapse N g 00 and shift N i g 0i are Lagrange multipliers Lagrangian L = d n x ( K ij ġ ij NH N i F i) Hamiltonian constraint H and diffeomorphism constraints F i generate coordinate transformations. Gegenberg, Kunstatter, Taves RST (slide 5 of 33)

6 Motivation Canonical Quantum Gravity Spherical Symmetry 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)

7 Motivation Canonical Quantum Gravity Spherical Symmetry 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)

8 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)

9 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)dt 2 + (F (R, M)) 1 dr 2 + R 2 dω (n 2) comoving (dt = 0) dr dτ = F (R, M) (1) 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)

10 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)

11 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)

12 How to include Hawking radiation and back reaction on metric? 2-D Gravity Gegenberg, Kunstatter, Taves RST (slide 12 of 33)

13 Review RST Black Holes: Classical RST Black Holes: Quantum 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 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)

14 Review RST Black Holes: Classical RST Black Holes: Quantum RST Action Local form of action (Hayward 95) I := 1 d 2 x { [ g e 2φ R(g) + 4 φ 2 + λ 2] N e 2φ + f i 2 2π i=1 + κ 2 zr(g) κ 4 z 2 κ } 2 φr(g). (2) where κ = N for N scalar fields z is an auxialliary field that incorporates radiation back-reaction via the one loop conformal anomaly in 2-d eq. of motion: z 1 R z z R 1 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)

15 Review RST Black Holes: Classical RST Black Holes: Quantum 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)

16 Review RST Black Holes: Classical RST Black Holes: Quantum Homogeneous Solutions Metric: Solutions: ds 2 = e 2ρ(t) ( dt 2 + dx 2 ). (4) z(t) = 2ρ(t) + z 1t + z 0, ρ(t) = φ(t) + p 1t + p 0, φ(t) = 1 2 W ( 4 κ e2θ(t)/κ ) 1 θ(t), (5) κ where W (x) is the LambertW function W (x)e W (x) = x and θ(t) := 2e 2φ κφ = 2λ2 e 2(p 1t+p 0 ) + κ ( p p z 1 2 p 1 4 ) t + θ 0; (6) Gegenberg, Kunstatter, Taves RST (slide 16 of 33)

17 Review RST Black Holes: Classical RST Black Holes: Quantum Solutions (cont d) Parametrized theory: physical phase space 4=2x2 dim: (θ 0, p 1, z 0, z 1) (p 0 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 A = 1 κ 4 e2φs = 0 θ s = κ (ln(κ/4) 1) 2 R 1/(W + 1) so this is a physical singularity. Gegenberg, Kunstatter, Taves RST (slide 17 of 33)

18 Review RST Black Holes: Classical RST Black Holes: Quantum Black Hole Sector t corresponds ( to Killing ) horizon iff linear t term in θ κ vanishes, i.e. p 1 p1 2 z2 1 4 = 0 (φ φ 0, R R(φ 0 ), e 2ρ 0 in finite proper time) Gegenberg, Kunstatter, Taves RST (slide 18 of 33)

19 Review RST Black Holes: Classical RST Black Holes: Quantum Connection with RST RST use Kruskal type coordinates with ρ = φ and: 2e 2φ κφ = λ2 x + x + P κ ln ( λ 2 x ) + x + M κ λ κ.. Define x ± = 1 e p 1x ±, with x ± = (t ± x)/2. p 1 Then: 2 ρ 2ρ = 2ρ p 1(x + + x ) 2 ln(p 1) = 2φ 2 ln(p 1) 2e 2φ κφ = λ2 e 2p1t + P } κ {ln λ2 + 2p 1t + M p1 2 κ p1 2 λ κ This is precisely the same as our black hole solution with θ 0 = M λ κ P = 0 Gegenberg, Kunstatter, Taves RST (slide 19 of 33)

20 Review RST Black Holes: Classical RST Black Holes: Quantum 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/ ) Thermal equilibrium necessary since interior homogeneous static exterior Mass: M = λ κθ 0 Temperature: T = p 1 2π Note: temperature/surface gravity and mass are independent. Gegenberg, Kunstatter, Taves RST (slide 20 of 33)

21 Review RST Black Holes: Classical RST Black Holes: Quantum Dilaton Equation in Conformal Gauge In terms of the variable: Φ = e 2φ Φ = κ 4 ( Φ) 2 Φ(Φ κ/4) + 4λ2 Φ 2 Φ κ/4 In conformal gauge ρ φ = p 1t is linear in time (RST simplification). The equation for Φ: Φ = κ Φ 2 4 Φ(Φ κ/4) 4λ2 e 2p 1t Φ Φ κ/4 Generated by Hamiltonian: ( ) 2 H R = Π2 Φ Φ 2 Φ κ/4 + 4λ 2 e 2p 1t ( Φ κ 4 ln Φ) (7) (8) Gegenberg, Kunstatter, Taves RST (slide 21 of 33)

22 Review RST Black Holes: Classical RST Black Holes: Quantum Near Singularity Dynamics Near the singularity Φ = κ/4, the dominant term in the Hamiltonian is Π 2 Φ H R = (9) (Φ κ/4) 2 This generates the leading term near the singularity: Φ κ/4 (t t 0) 1/2 This is different from Levanony and Ori who obtained Φ κ/4 (t t 0) 2/3 The above is consistent with series expansion of the exact solution. Gegenberg, Kunstatter, Taves RST (slide 22 of 33)

23 Review RST Black Holes: Classical RST Black Holes: Quantum Near Singularity Quantum Theory: Method I Transform to y = (Φ κ/4) 2 ( Natural Parameterization Invariant Measure). Consider functions that are L 2 normalizable on (0, ) with measure dy = RdR. The quantum Hamiltonian is then Ĥ 1 = d 2 (10) 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) = λψ (0) Scattering E > 0 spectrum continuous. For λ < 0, there exists a bound state stable remnant? Gegenberg, Kunstatter, Taves RST (slide 23 of 33)

24 Review RST Black Holes: Classical RST Black Holes: Quantum Near Singularity Quantum Theory: Method II Wave functions normalizable on half line with measure dr. Standard symmetric factor ordering for the Hamiltonian: Ĥ 2 = d 1 d dφ Φ 2 dφ = 1 d 1 d Φ2 Φ 2 dφ Φ 2 dφ = y 2/3 d 2 (11) y 2 where y = Φ 3 /3 0. The eigenvalue equation is: Ĥψ(y) = E ψ(y) ψ + y 2/3 Eψ = 0 (12) Gegenberg, Kunstatter, Taves RST (slide 24 of 33)

25 Review RST Black Holes: Classical RST Black Holes: Quantum The solutions are: ψ k (y) = C + yj3/4 (3ky 2/3 /2) + C yj 3/4 (3ky 2/3 /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. At origin, BC s ψ(0) = λψ (0) again imply C = λc + Bound states: ( ) 2Γ E = κ 2 2 4/3 (3/4) = 3π λ ψ b (y) = N b (λ) yk 3/4 ( 3κ 2 y 2/3 ). (14) Gegenberg, Kunstatter, Taves RST (slide 25 of 33)

26 Review RST Black Holes: Classical RST Black Holes: Quantum Quantization of Full Interior In conformal gauge dilaton decouples, dynamics described by Hamiltonian: ( ) 2 H R = Π2 Φ Φ 2 Φ κ/4 + 4λ 2 e 2p 1t ( Φ κ 4 ln Φ) Canonical Transformation: y = Φ κ 4 ln Φ κ 4 ( 1 ln κ ) 4 (15) yields (up to trivial constant shifts): H = Π2 y 2 + 4Λ2 e 2p 1t+p 0 y (16) Gegenberg, Kunstatter, Taves RST (slide 26 of 33)

27 Review RST Black Holes: Classical RST Black Holes: Quantum Tentative Results Figure : Evolution of initially Gauss state for RST BH interior Gegenberg, Kunstatter, Taves RST (slide 27 of 33)

28 Recap Prospects: Dirac Quantization 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)

29 Recap Prospects: Dirac Quantization Prospects: Dirac Quantization In terms of phase space variables: Hamiltonian: y = z + 2e 2φ κ φ; Π y = κ B(φ) (Π φ + 2Π z); w = κ(z 2φ); Π w = (Π φ + A Πz) κ. (17) B(φ) [ ] H=2πσ κπ 2 w 1 Π κ βπ y λ2 e 2β. π 2 Wheeler-DeWitt Equation separable, but difficult/impossible to deal with boundary conditions in this parameterization. Gegenberg, Kunstatter, Taves RST (slide 29 of 33)

30 Recap Prospects: Dirac Quantization Dirac Quantization (cont d) Simple canonical transformation κω = 2e 2φ + κφ χ = 1 ) (2e 2φ κφ + 2κρ κ leads to: y = z 2ρ (18) ( H= 2πσ 1 4 (Π2 Ω Π2 χ) + Π2 y κ λ2 π 2 e 2(χ Ω)/κ ) Wheeler-Dewitt Equation not separable, but boundary conditions easy to implement. Gegenberg, Kunstatter, Taves RST (slide 30 of 33)

31 Recap Prospects: Dirac Quantization 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)

32 Recap Prospects: Dirac Quantization 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)

33 Recap Prospects: Dirac Quantization 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)