Causal nature and dynamics of trapping horizon in black hole collapse

Transcription

1 Causal nature and dynamics of trapping horizon in black hole collapse Ilia Musco (CNRS, Observatoire de Paris/Meudon - LUTH) KSM FIAS (Frankfurt) July 2017 Classical and Quantum Gravity Vol. 34, No. 13, (2017) Collaborators: John Miller (Oxford) Alexis Helou (Paris/Munich)

2 I must say I find it brave of the authors to invest so much time and effort in recreating numerical models that were thoroughly investigated 50 years ago, with little prospect of discovering anything new, but their approach to the problem is fresh and interesting. The Anonymous Referee

3 Outline Introduction to the Misner-Sharp formalism Trapping horizons in spherical symmetry (R=2M): black hole horizon / cosmological horizon Causal Nature/velocity of the horizons: Oppenheimer-Snyder / polytropic star collapse Simulations of classical gravitational collapse LTB collapses Horizon phase space diagram (BH - hyperbola) Conclusions & Future perspectives

4 Introduction Spherically symmetric metric in comoving coordinates with t cosmic time : ds 2 = a 2 (r, t)dt 2 + b 2 (r, t)dr 2 + R 2 (r, t)d 2 Proper time and proper distance operators: D t ) U D t R D r ) D rr Perfect Fluid: T µ =(e + p)u µ u + pg µ Constraint equation (integrating G00) : Mister-Sharp Mass : M = Z 4 er 2 dr 2 =1+U 2 2M R D t M = 4 pr 2 U

5 Trapping Horizons Expansion of ingoing/outgoing null-rays : k a /l a = 1 a, ±1 b, 0, 0 =) ± = h cd r c k d = 2 R (U ± ) h ab = g ab (k al b + l a k b ) k a l a = 2 Black Hole / Cosmological horizon : ± =0 ) 1 a dr dt ± ) 2 = U 2 R =2M The horizon condition is independent of the slicing and holds also within a non-vacuum moving medium The so-called apparent horizon of a black hole (which is a future trapping horizon) is the outermost trapped surface for outgoing radial null rays while the trapping horizon for an expanding universe (which is a past trapping horizon) is foliated by the innermost anti-trapped surfaces for ingoing radial null rays.

6 Causal Nature L v v { L nv v α > 0 : space-like α = 0 / 1 : null α < 0 : time-like Lie Derivatives: { L + v = L k v = k a v = L v = L l v = l a v = 1 a @r v v L ± v =(D t ± D r ) v = 4 R2 (e + p) 1 4 R 2 (e p) H

7 Horizon Velocity 3-velocity of the horizon with respect the matter: v H b a dr dt H v =0 ) D t v + b a dr dt D r v =0 v H v H = v H = D t v ) v H = D t D r v D r ( 2 L + v + L v L + v L v H U H { 1+8 R 2 p 1 8 R 2 e H 2 U 2 U 2 ) H ) v H = ± 1+ 1 v H > 1: space-like v H =1: null v H < 1: time-like

8 Schwarzschild Black Hole space-time R =2M

9 G µ =8 T µ T µ =(e + p)u µ u pg µ COSMIC TIME ds 2 = a 2 dt 2 + b 2 dr 2 + R 2 d 2 D t 1 a t D r 1 b r Proper time / space derivative U D t R D r R D t U = apple (e + p) D rp + M +4 Rp R2 D t = R 2 D r(r 2 U) D t e = e + p D t M = D r a = D r M =4 R 2 D t 4 R 2 pu a e + p D rp e 2 =1+U 2 2M R 4-velocity & Lorentz factor Euler equation Continuity equation Mass conservation dm = audt + b dr Lapse equation / pressure gradients Constraint equation

10 Equation of State energy density: e = (1 + ) pressure: p =( 1) rest mass density adiabatic index - particle degree of freedom specific internal energy (velocity dispersion) p = we w 2 [0, 1] Barotropic fluid (no rest mass density): with - radiation dominated era: w =1/3 RADIATION ( =4/3) - matter dominated era: w =0 DUST ( = 1) Polytropic fluid: p = K(s) ( =5/3, 4/3, 2) - If the fluid is adiabatic (no entropy change): (constant) K(s) =K

11 Oppenheimer-Snyder (1939) homogenous collapse May & White (1966): non homogenous collapse 2M R = 8 3 R2 e p =0 p = K ( =5/3) in = 3 v H = 1/2 out =0 v H =1

12 General Scheme for in/out-going horizon evolution t3 t2 t1

13 p = K ( =5/3, HOM I.C.) = 4 R2 (e + p) H 1 4 RH 2 (e p) v H = 1+8 R2 H p 1 8 RH 2 e

14 p = K ( =4/3, HOM I.C.) = 4 R2 (e + p) H 1 4 RH 2 (e p) v H = 1+8 R2 H p 1 8 RH 2 e

15 p = K ( =5/3, TOV I.C.) = 4 R2 (e + p) H 1 4 RH 2 (e p) v H = 1+8 R2 H p 1 8 RH 2 e

16 Simulation Summary = 4 R2 H (e + p) 1 4 RH 2 (e p) v H = 1+8 R2 H p 1 8 R 2 H e ( =1)) e = 1 2A H

17 Black Hole Horizons - Phase Diagram v H = +1 1 A. Helou, I.M., J. Miller - CQG (2017) V. Faraoni, G. Ellis, J. Firouzjaee, A. Helou, I.M. - PRD (2017)

18 LTB collapse ( inhomogeneous profile - p=0 ) SINGLE HORIZON: If the singularity forms before reaching the R=2M condition, the horizon come out from the center, expanding through the matter. In this case there is only one ingoing horizon, with the second one degenerate..

19 LTB collapse THREE HORIZONS: If the ingoing horizon is not reaching the center when the singularity is forming, a second outgoing horizon is originated from the singularity which is going to annihilate with the ingoing horizon at =1and v H = 1. =2.0

20 Black Hole Horizons - Phase Diagram v H = +1 1 Negative Presssure A. Helou, I.M., J. Miller - CQG (2017) V. Faraoni, G. Ellis, J. Firouzjaee, A. Helou, I.M. - PRD (2017)

21 Conclusions & Future perpectives With the Misner-Sharp equation (cosmic time slicing) we have studied the causal nature of trapping horizons appearing in gravitational collapse forming black holes. Within the classical regime of GR we have observed space-like outgoing horizons and space-like/time-like ingoing horizons (equation of state and initial conditions for density). Pressure plays a key role - Cosmic Censorship. The conditions of horizon formation and annihilation are independent of the initial conditions. The formalism developed to show the possibility of incorporating quantum effects within the classical formulation of the GR-hydro equations modifying the equation of state accordingly to quantum gravity. Is it possible to obtain non singular BH? C. Bambi, D. Malafarina & L. Modesto (2013) C. Rovelli & F. Vidotto (2014); A. Helou, D. Malafarina & I.M. (2017) - in progress The formalism can be also to the cosmological horizon, studying causal nature evolution for a non homogenous Universe. I.M, A. Helou, G. Ellis (2017) - in progress