Information Loss in the CGHS Model

Transcription

1 Information Loss in the CGHS Model Fethi Mübin Ramazanoğlu PRINCETON UNIVERSITY DEPARTMENT of PHYSICS ILQGS, 03/09/2010 collaborators Abhay Ashtekar, PennState Frans Pretorius, Princeton

2 2 / 30 Outline 1 A Quick Look at Informaton Loss 2 CGHS Model 3 Numerical Solution 4 Results: Macroscopic BH Finiteness of y Bondi Mass and Hawking Radiation Diminishing of the Bondi Mass 5 Results: Planck Scale BH 6 Recent 7 Conclusions and Future

3 A Quick Look at Informaton Loss A Quick Overview of Information Loss I + L y I + R 3 / 30 z I L I R Fixed Background

4 A Quick Look at Informaton Loss A Quick Overview of Information Loss I + L y I + R 4 / 30 z I L I R Fixed Background Quantum Gravity

5 A Quick Look at Informaton Loss A Quick Overview of Information Loss I + L y I + R 5 / 30 z I L I R Fixed Background MFA Quantum Gravity

6 6 / 30 Information Loss in the CGHS Model CGHS Model Why CGHS Why 1+1 D? Conformal flatness Easy calculation at 1-loop: Trace anomaly Local action More manageable numerics Qualitatively similar to reduced 3+1 D, yet analytical solutions.

7 CGHS Model Action and Classical Equations of Motion S(g, φ, f ) = 1 d 2 Ve 2φ ( R + 4g ab a φ b φ + 4κ 2) G 1 N d 2 Vg ab a f i b f i 2 i=1 S (4) (g, φ, f ) = 1 ( d 2 Ve 2φ R + 2g ab a φ b φ + 2e 2φ κ 2) G 1 N d 2 Ve φ g ab a f i b f i 2 i=1 Callan, Giddings,Harvey, Strominger, Phys. Rev. D (1992) 7 / 30

8 CGHS Model Equations of Motion Φ = e 2φ g ab = Θ 1 Φη ab ATV2008 (g) f = 0 (η) f = 0 + Φ + κ 2 Θ = GT + = 0 Φ + ln Θ = GT + = Φ + + Φ + ln Θ = GT ++ 2 Φ + Φ ln Θ = GT 8 / 30

9 CGHS Model Eternal Black Hole = c = 1 G = 1 κ = 1 f = 0 Φ = M κ κ2 x + x Θ = 1 9 / 30

10 CGHS Model Classical Collapsing Shell 1 2 +f + f = Mδ(z + ) Φ = e κz+ e κz M ( ) e κz / 30

11 CGHS Model Classical Collapsing Shell: Affine coordinate T µ µ = N 24 R y + = z + e κy = e κy M y (z s = ) =! dy dz 1 z s z 11 / 30

12 CGHS Model Hawking Radiation: T y y (y+ ) N [ ] 1 1 (1+Me κy ) 2 T y y (external field) / 30 z c

13 CGHS Model MFA Equations of Motion T µ µ = N 24 R (g) f = 0 (η) f + Φ + κ 2 Θ = GT + = N 24 + ln ( ΦΘ 1) Φ + ln Θ = GT + = N 24 + ln ( ΦΘ 1) 13 / 30

14 CGHS Model Asymptotic behavior Φ = A(z )e κz+ + B(z ) + O(e κz+ ) Θ = A(z )e κz+ + B(z ) + O(e κz+ ) κy = ln A κ <? Ashtekar, Taveras, Varadarajan Phys. Rev. Lett. (2007) 14 / 30

15 CGHS Model Bondi Mass and Black Hole Evaporation Bondi mass and Hawking radiation is connected to the asyptotic behavior of Φ near I + R {}}{ d [ db NG (d 2 y dy + κb + ) ] (dy dy 24 dz 2 dz ) 2 = NG [d 2 y ] (dy 24 dz 2 dz ) 2 2 }{{} MFA Flux on I + R M B Area = Φ N / 30

16 CGHS Model Some Remarks Scalable: Only M N matters! (Φ, Θ, N, f ) (αφ, αθ, αn, f ) Dimensionless [G ], chosen to be / 30

17 Numerical Solution Numerical Solution Regularize the fields Φ(z +, z ) = e κz+ κz (1 + φ(z +, z )) + Φ 0 (z + ) (i,j) Θ(z +, z ) = e κz+ κz (1 + θ(z +, z )) Scaling and compactification Unigrid mesh z = z z /2 z 10 9/2 + z s,e z = e tan(πzc π/2) (z c 1) vacuum boundary I z L i (i,j 1) (i 1,j) I R j z + I R + I R (i 1,j 1) z + = C tan πz + c z + c [0, 1 2 ], z+ c [0, 1] Discretization, recasting into a polynomial vacuum boundary condition from matter fields 17 / 30

18 i Information Loss in the CGHS Model Numerical Solution Numerical Solution (i,j) (i,j 1) (i Very high resolution near the last ray Very small truncation errors vacuum boundary I R I R + (i 1,j 1) I z L j z + I R vacuum boundary condition from matter fields 18 / 30

19 Results Finiteness of y Macroscopic BH Asymptotic Killing Coordinate y y z c 19 / 30

20 Results Finiteness of y Macroscopic BH: Power Law for dy dz d ln(dy /dz ) /d ln(z sing z ) ln (z sing z) 20 / 30

21 Results Bondi Mass and Hawking Radiation Macroscopic BH: Bondi Mass M Bondi (natural units) z c 21 / 30

22 Results Bondi Mass and Hawking Radiation Macroscopic BH: Hawking Radiation 1.4 MFA energy flux (natural units) z c 22 / 30

23 Results Diminishing of the Bondi Mass Macroscopic BH: Comparison of Area and M B Area/M B z c 23 / 30

24 Results Planck Scale BH: Asymptotic Killing Coordinate y 2 x y z c 24 / 30

25 Results Planck Scale BH: Bondi Mass M Bondi (natural units) z c 25 / 30

26 Results Planck Scale BH: Hawking Radiation 0.06 MFA energy flux (natural units) z c 26 / 30

27 Recent Hayward Mass M Hayward z c 27 / 30

28 Recent Universal Curve on A-M plane A M B A =\Phi \frac{n}{12} 28 / 30

29 Recent Mass at the last ray m * (mass at last ray) data fitted curve M * (mass at z = ) 29 / 30

30 Conclusions and Future Conclusions y is finite, strong evidence for unitarity. For all macroscopic BHs, Bondi mass at the last ray order of unity (in natural units). If the initial mass itself is of the order unity or smaller, evaporation process is very different from the standard (external field) picture. High resolution numerics near singularity is necessary to describe the overall behavior. Future directions (with Amos Ori) Behavior of the flux near last ray Comparison to more detailed analytical results Beyond the singularity 30 / 30