Formation and Evaporation of Regular Black Holes in New 2d Gravity BIRS, 2016

Transcription

1 Formation and Evaporation of Regular Black Holes in New 2d Gravity BIRS, 2016 G. Kunstatter University of Winnipeg Based on PRD90,2014 and CQG R1, 2016 Collaborators: Hideki Maeda (Hokkai-Gakuen U.), Tim Taves (C.E.C.s.), Donovan Allum and Taylor Hanson (UWinnipeg) May 16, 2016 Kunstatter BIRS 2016(slide 1 of 32)

2 Motivation Black holes exist (LIGO 16) Einstein s theory predicts singular core Black holes evaporate (Hawking 76) Loss of predictability Quantum gravity hard Can process be described by self-consistent non-singular effective theory? Kunstatter BIRS 2016(slide 1 of 32)

3 Executive Summary: new improved 2D gravity Generalizes spher. symm. Einstein-Lanczos-Lovelock gravity: higher order Lagrangian, second order equations, mass function, Birkhoff s theorem. Has as sub-class D Einstein-Lanczos-Lovelock gravity. Can be designed to produce as unique solution any spherical black hole Provides consistent model for studying effective dynamics of regular black hole formation and evaporation. Kunstatter BIRS 2016(slide 2 of 32)

4 Outline: 1. Regular black holes 2. New (improved) 2D gravity Kunstatter BIRS 2016(slide 3 of 32)

5 Static Regular black holes Regular Black Hole Formation and Evaporation Regular black holes (Sakharov, 65, Bardeen, 68, Frolov-Vilkovisky, 81) Example: derived by Poisson and Israel 88 using semi-classical arguments ( ) ( ) 1 ds 2 = 1 2MR2 (R 3 + lpl 3 ) dt MR2 (R 3 + lpl 3 ) dr 2 + R 2 dω (2) Ricci Scalar R (n) = 12lpl 3 M( R 3 + 2lpl) 3 (R 3 + lpl 3 )3 desitter core with curvature R (n) core = 12M/lpl 3 l Two horizons R + 2M and R pl l 2M pl M 0 mass gap 2M min l pl. R core (n) ; R M 0 as M Semi-classical approximation doomed from beginning! Thanks to V. Frolov for stressing this. Kunstatter BIRS 2016(slide 4 of 32)

6 Static Regular black holes Regular Black Hole Formation and Evaporation A better class of regular black holes Hayward( 06) ds 2 = ( 1 2MR2 R 3 + Ml 2 pl ) dt 2 + ( 1 2MR2 R 3 + Ml 2 pl ) 1 dr 2 + R 2 dω (2) R (n) = 24 M2 l 2 pl( R 3 + 2Ml 2 pl) (R 3 + Ml 2 pl )3 (1) desitter core: R (n) core = 12/l 2 pl Two horizons: R + 2M and R l pl Mass gap 2M min = l pl Curvature bounded above, R bounded below. Basis for consistent semi-classical model? Kunstatter BIRS 2016(slide 5 of 32)

7 Static Regular black holes Regular Black Hole Formation and Evaporation Why are we interested in regular black hole formation and evaporation? Expect qualitative changes to structure of complete semi-classical spacetime. Kunstatter BIRS 2016(slide 6 of 32)

8 Static Regular black holes Regular Black Hole Formation and Evaporation Recall: Singular Black Hole Formation Kunstatter BIRS 2016(slide 7 of 32)

9 Static Regular black holes Regular Black Hole Formation and Evaporation Singular Black Hole Formation and Evaporation Kunstatter BIRS 2016(slide 8 of 32)

10 Static Regular black holes Regular Black Hole Formation and Evaporation Regular black hole formation (Ziprick, GK 09; Maeda, Taves, GK 16) r = 0 timelike, regular. Expect mass inflation as matter piles up on inner horizon. As mass inflates, inner horizon generally shrink. Stabilizes at l pl for Hayward black holes. Kunstatter BIRS 2016(slide 9 of 32)

11 Static Regular black holes Regular Black Hole Formation and Evaporation... and evaporation Hayward 06: explicit construction by patching Vaidya spacetimes Kunstatter BIRS 2016(slide 10 of 32)

12 Static Regular black holes Regular Black Hole Formation and Evaporation...but can we find consistent dynamical equations that yield this spacetime? Outline: 1. Regular black holes 2. New (improved) 2D gravity Kunstatter BIRS 2016(slide 11 of 32)

13 Motivation and Construction Designer Black Holes Numerial Results Recall: Generic 2D Dilaton Gravity (1990 s) Einstein action in n dimensions: I E = 1 16πG d n x g (n) R(g (n) ) (2) Dimensionally Reduced Action (ds 2 = g µνdx µ dx ν + R 2 dω (n 2) ): I (2) = 1 d 2 x [ g R n 2 R l n 2 + (n 2)(n 3)R (n 4) ( R) 2 + (n 2)(n 3)R (n 4)] (3) Generalization (Generic 2D Dilaton Gravity): I G = 1 d 2 x { } g φ(r)r + h(r)(dr) 2 + V (R). (4) l n 2 Can t get bounded curvature metrics from this class of theories. Kunstatter BIRS 2016(slide 12 of 32)

14 Motivation and Construction Designer Black Holes Numerial Results Extended Spherical Einstein-Lovelock-Lanczos Gravity Recall Lovelock Action: I = 1 2κ 2 n d n x [n/2] ( 1 g α (p) 2 p δµ 1 µ pν 1 ν p p=0 ρ 1 ρ pσ 1 σ p R where κ n := 8πG n and δ µ 1 µ p ρ 1 ρ p := p!δ µ 1 [ρ 1 δ µp ρ p].. ρ 1 σ 1 µ 1 ν 1 R ) ρ pσ p µ pν p, (5) Eg: Einstein-Gauss-Bonnet Gravity (n 5): I = d n x { g R + α [ R 2 4R µνr µν + R µνρσr µνρσ]}. (6) 2 Kunstatter BIRS 2016(slide 13 of 32)

15 Motivation and Construction Designer Black Holes Numerial Results Generic spherically symmetric Lovelock terms: Shown by G.K., Maeda and Taves 13, generalizing an identity used for Einstein-Gauss-Bonnet by Taves, Leonard, GK and Mann 11: [ (n 2)! L (p) = (n 2p)! I = 1 2κ 2 n d n x [n/2] g α (p) L (p) p=0 pr 2 2p(2) R + (n 2p)(n 2p 1) + p(n 2p)R 1 2p { 1 (1 Z) p 1 } R Z Z. where: Z := R 2 { ] (1 Z) p + 2pZ }R 2p Kunstatter BIRS 2016(slide 14 of 32)

16 Motivation and Construction Designer Black Holes Numerial Results Natural extension: 1 I = l n 2 d 2 x { } g φ(r)r + H(R, Z) + χ(r, Z) R Z Z := R 2 (7) Second order equations for g µν and R Birkhoff s theorem Mass function: M such that D A M = 0 in vacuum on shell Conjecture: Most general 2D with above properties? Kunstatter BIRS 2016(slide 15 of 32)

17 Motivation and Construction Designer Black Holes Numerial Results Mass Function Integrability Condition on Lagrangian Functions: φ,rr =η,z χ,r. Guarantees existence of the mass function: Z M(R, Z) := φ,r Z + χ(r, Z)d Z. which gives D A M =(χ φ,r )D A Z 2 (φ,rr Z 12 ) η(r, Z) D A R =0 in vacuum Kunstatter BIRS 2016(slide 16 of 32)

18 Motivation and Construction Designer Black Holes Numerial Results Birkhoff Theorem Most general solution (Schwarzschild coords): ds 2 = f (R; M)dt 2 + f (R; M) 1 dr 2. where M(R, Z) = M = constant. (8) f (R; M) is determined by inverting (8) to solve for Z: Z = f (R; M) Kunstatter BIRS 2016(slide 17 of 32)

19 Motivation and Construction Designer Black Holes Numerial Results How to design your personalized black hole spacetime 1. Choose your favourite metric function f (R, M) 2. In Schwarzchild coords Z := R 2 = f (R; M): Solve for M = M(R, Z) 3. Obtain lagrangian functions: M Z =χ φ,r, M (φ R = 2,RR Z 12 ) η(r, Z). 4. Just add matter and solve collapse equations. Kunstatter BIRS 2016(slide 18 of 32)

20 Motivation and Construction Designer Black Holes Numerial Results Example: Hayward black hole in 4D metric function : f (R) = 1 2MR2 R 3 + l 2 pl M = Z. mass function: 2M = (1 Z)R3 R 2 lpl 2 (1 Z) Kunstatter BIRS 2016(slide 19 of 32)

21 Motivation and Construction Designer Black Holes Numerial Results Adding matter Massless scalar field: I matter = 1 2 d 2 xr 2 ψ 2 Straightforward to derive Hamiltonian equation. ds 2 = N 2 dt 2 + Λ 2 (dx + N r dt) 2 R 2 (x)dω (2) We work in flat slice coordinates (mostly due to inertia). R = x Λ = 1 Kunstatter BIRS 2016(slide 20 of 32)

22 Motivation and Construction Designer Black Holes Numerial Results Numerical Results: l=5, M ADM = , 150,000 iterations to T= Mass Density M,R : Kunstatter BIRS 2016(slide 21 of 32)

23 Mass Function: Regular Black Holes Motivation and Construction Designer Black Holes Numerial Results Kunstatter BIRS 2016(slide 22 of 32)

24 Motivation and Construction Designer Black Holes Numerial Results Ricci Scalar in terms of data on a slice R (n) = B (R) M,Z ψ 2 M,ZZ M 3,Z B 2 ψ 4 + M RR M,Z ( ) M,R 2M,ZR M 2,Z = + M,ZZ M,Z ( M,R M,Z ) 2 + 2(n 2) M,R 1 Z + (n 2)(n 3) M,Z R R 2 ( ) B (R)R 5 8l 2 R 5 B 2 (R) (R 3 + Ml 2 ) 2 ψ 2 ψ 4 3M2 l 2 (R ) (R 3 + Ml 2 ) 3 (R 3 + Ml 2 ) 3 ψ 2 = P2 ψ Λ 2 B 2 (R) + ψ2,x Λ 2 Kunstatter BIRS 2016(slide 23 of 32)

25 Motivation and Construction Designer Black Holes Numerial Results Ricci Scalar: Kunstatter BIRS 2016(slide 24 of 32)

26 Motivation and Construction Designer Black Holes Numerial Results Outgoing null geodesic that asymptotes to inner horizon, and plots of Mass and Ricci scalar along it. Kunstatter BIRS 2016(slide 25 of 32)

27 Motivation and Construction Designer Black Holes Numerial Results Outgoing null geodesic that asymptotes to inner horizon, and plots of Mass and Ricci scalar along it. Kunstatter BIRS 2016(slide 26 of 32)

28 Motivation and Construction Designer Black Holes Numerial Results Outline: 1. Regular black holes 2. New (improved) 2D gravity Kunstatter BIRS 2016(slide 27 of 32)

29 Radiation Terms: Polyakov Action I Poly d 4 x g (4) R (4) 1 D(4) 2 R (4) (9) Local Form: auxilliary field z gr [ ] 2 I Poly zr(g) + D A zd A z (10) So far: Eqs. obtained and code running for Bardeen-type black hole: stability (and other?) issues Working on equations and code for new 2D gravity. Kunstatter BIRS 2016(slide 28 of 32)

30 Summary Presented new class of 2D theories to model the formation and evaporation of physical regular spherically symmetric black holes. Can design lagrangian to produce any 2D black hole, including Hayward. Sub-class (eg Hayward BH) have physical interpretation as infinite dimensional Lovelock: lagrangian functions need to have Taylor expansion in 1 Z. Kunstatter BIRS 2016(slide 29 of 32)

31 Outstanding Questions: Can any of the new 2D gravity theories be obtained via dimensional reduction (Cf. Meyers, Robinson 10; Oliva, Ray 11)? Does Hayward black hole suffer (a lot) from mass inflation? (Stable inner horizon, curvature bounded.) Radiation of regular black holes should produce spacetime with compact trapping horizon. Consequences for information loss? Kunstatter BIRS 2016(slide 30 of 32)

32 Thanks for listening! Kunstatter BIRS 2016(slide 31 of 32)

33 Regular black hole formation (Ziprick, GK 09; Maeda, Taves, GK 16) Penrose Diagram Kunstatter BIRS 2016(slide 32 of 32)