Information Loss in the CGHS Model Fethi Mubin¨ Ramazanoglu˘
PRINCETON UNIVERSITY DEPARTMENT of PHYSICS
ILQGS, 03/09/2010
collaborators Abhay Ashtekar, PennState Frans Pretorius, Princeton Information Loss in the CGHS Model
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
2 / 30 MFA Quantum Gravity
Information Loss in the CGHS Model A Quick Look at Informaton Loss
A Quick Overview of Information Loss
I+ I+ L − R y 3 / 30
z− − − IL IR
Fixed Background MFA
Information Loss in the CGHS Model A Quick Look at Informaton Loss
A Quick Overview of Information Loss
I+ I+ L − R y 4 / 30
z− − − IL IR
Fixed Background Quantum Gravity Information Loss in the CGHS Model A Quick Look at Informaton Loss
A Quick Overview of Information Loss
I+ I+ L − R y 5 / 30
z− − − IL IR
Fixed Background MFA Quantum Gravity 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.
6 / 30 Information Loss in the CGHS Model CGHS Model
Action and Classical Equations of Motion S(g, φ, f ) = 1 Z d2Ve−2φ R + 4gab∇ φ∇ φ + 4κ2 G a b N 1 X Z − d2Vgab∇ f ∇ f 2 a i b i i=1
S(4)(g, φ, f )= 1 Z d2Ve−2φ R + 2gab∇ φ∇ φ + 2e−2φκ2 G a b N 1 X Z − d2Ve−φgab∇ f ∇ f 2 a i b i i=1
Callan, Giddings,Harvey, Strominger, Phys. Rev. D (1992) 7 / 30 Information Loss in the CGHS Model CGHS Model
Equations of Motion
Φ = e−2φ ATV2008 gab = Θ−1Φηab
(g) f = 0 ⇔ (η)f = 0 2 ∂+ ∂− Φ + κ Θ = GT+− = 0
Φ∂+ ∂− ln Θ = −GT+− = 0
2 −∂+ Φ + ∂+ Φ∂+ ln Θ = GT++ 2 −∂− Φ + ∂− Φ∂− ln Θ = GT−−
8 / 30 Information Loss in the CGHS Model CGHS Model
Eternal Black Hole
~ = c = 1 G = 1 κ = 1
f = 0 M Φ = − κ2x+x− κ Θ = 1
9 / 30 Information Loss in the CGHS Model CGHS Model
Classical Collapsing Shell
1 ∂ f ∂ f = Mδ(z+) 2 + + + − + Φ = eκz e−κz − M eκz − 1
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Classical Collapsing Shell: Affine coordinate → ∞
N hTµi = R µ 24 ⇒ y+ = z+ − − e−κy = e−κy − M − − y (zs = ∞)= ∞! dy− 1 − ∼ − dz zs − z
11 / 30 Information Loss in the CGHS Model CGHS Model
+ N 1 Hawking Radiation: hTy−y− (y → ∞)i → 1 − 2 48 (1+Meκy− )
1
0.9
0.8
0.7
0.6
0.5
0.4 (external field) − y −
y 0.3 T
0.2
0.1
0 −1.5 −1 −0.5 0 0.5 1 1.5 12 / 30 z− c Information Loss in the CGHS Model CGHS Model
MFA Equations of Motion
N hTµi = R µ 24
(g) f = 0 ⇔ (η)f N ∂ ∂ Φ + κ2Θ = GT = ∂ ∂ ln ΦΘ−1 + − +− 24 + − N Φ∂ ∂ ln Θ = −GT = − ∂ ∂ ln ΦΘ−1 + − +− 24 + −
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Asymptotic behavior
+ + Φ¯ = A(z−)eκz + B(z−) + O(e−κz ) + + Θ¯ = A(z−)eκz + B(z−) + O(e−κz ) A κy− = − ln < ∞? κ
Ashtekar, Taveras, Varadarajan Phys. Rev. Lett. (2007) 14 / 30 Information Loss in the CGHS Model CGHS Model
Bondi Mass and Black Hole Evaporation
Bondi mass and Hawking radiation is connected to the asyptotic + behavior of Φ near IR
MB z }| { d dB NG d2y− dy− + κB + ~ ( )−2 dy− dy− 24 dz−2 dz− NG d2y− dy− = − ~ ( )−2 2 24 dz−2 dz− | {z } + MFA Flux on IR
N ”Area” = Φ − 12
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Some Remarks
M Scalable: Only N matters ! (Φ, Θ, N, f ) → (αΦ, αΘ, αN, f )
Dimensionless [G~], chosen to be 1.
16 / 30 Information Loss in the CGHS Model Numerical Solution
Numerical Solution
Regularize the fields
+ − + − κz −κz ˜ + − + (i,j) Φ(z , z ) = e (1 + φ(z , z )) + Φ0(z ) + − κz+−κz− ˜ + − Θ(z , z ) = e (1 + θ(z , z )) (i,j−1) (i−1,j)
+ Scaling and compactification ⇒ Unigrid mesh I R (i−1,j−1)
vacuum boundary − − 1 I ¯z − R z− = ¯z− 109/2 + z− ¯z− − 109/2 s,e − I L + − − tan(πzc−π/2) −9 − z − ¯z = −e + 4.096 × 10 (z − 1) j z I c R i + + + 1 + z = C tan πz z ∈ [0, ] , z ∈ [0, 1] vacuum c c 2 c boundary condition from matter fields Discretization, recasting into a polynomial 17 / 30 Information Loss in the CGHS Model Numerical Solution
Numerical Solution
(i,j)
(i,j−1) (i−1,j) Very high resolution near the last ray + I R (i−1,j−1)
vacuum boundary Very small truncation errors − I R
− I L + − z j z − I R i
vacuum
boundary condition from matter fields
18 / 30 Information Loss in the CGHS Model Results Finiteness of y− Macroscopic BH Asymptotic Killing Coordinate y− 30
25
20
15 − y 10
5
0
−5 0 0.1 0.2 0.3 0.4 0.5− 0.6 0.7 0.8 0.9 1 z 19 / 30 c Information Loss in the CGHS Model Results Finiteness of y− dy− Macroscopic BH: Power Law for dz− 0
−0.1 ) − −0.2 −z −0.3 − sing
−0.4
−0.5 ) /d ln(z − −0.6 /dz − −0.7
−0.8 d ln(dy −0.9
−1 −36 −34 −32 −30 −28 −26 −24 −22 −20 ln (z −z) 20 / 30 sing Information Loss in the CGHS Model Results Bondi Mass and Hawking Radiation Macroscopic BH: Bondi Mass
18
16
14
12
10
8 (natural units) 6 Bondi
M 4
2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z− c 21 / 30 Information Loss in the CGHS Model Results Bondi Mass and Hawking Radiation Macroscopic BH: Hawking Radiation
1.4
1.2
1
0.8
0.6
0.4
MFA energy flux (natural units) 0.2
0 0 0.1 0.2 0.3 0.4 0.5− 0.6 0.7 0.8 0.9 1 z 22 / 30 c Information Loss in the CGHS Model Results Diminishing of the Bondi Mass
Macroscopic BH: Comparison of “Area” and MB 1.1
1
0.9
B 0.8
0.7 Area/M
0.6
0.5
0.4 0 0.1 0.2 0.3 0.4 0.5− 0.6 0.7 0.8 0.9 1 z 23 / 30 c Information Loss in the CGHS Model Results
Planck Scale BH: Asymptotic Killing Coordinate y− x 10−3 2
0
−2
−4
−6 − y −8
−10
−12
−14
−16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z− 24 / 30 c Information Loss in the CGHS Model Results
Planck Scale BH: Bondi Mass 0.05
0.045
0.04
0.035
0.03
0.025
(natural units) 0.02
Bondi 0.015 M 0.01
0.005
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z− 25 / 30 c Information Loss in the CGHS Model Results
Planck Scale BH: Hawking Radiation 0.06
0.05
0.04
0.03
0.02
0.01 MFA energy flux (natural units)
0 0 0.1 0.2 0.3 0.4 0.5− 0.6 0.7 0.8 0.9 1 z 26 / 30 c Information Loss in the CGHS Model Recent
Hayward Mass 10
8
6
4
Hayward 2 M
0
−2
−4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z− 27 / 30 c Information Loss in the CGHS Model Recent
”Universal Curve” on A-M plane 0.4
0.2
0
−0.2 B
A−M −0.4 18 −0.6 14 12 10 −0.8 8
−1 0 2 4 6 8 10 12 14 16 18 28 / 30 A =\Phi −\frac{N}{12} Information Loss in the CGHS Model Recent
Mass at the last ray 1
0.9
0.8
0.7
0.6
0.5
0.4 (mass at last ray) * 0.3 m
0.2 data fitted curve 0.1
0 0 2 4 6 8 10 12 14 16 18 29 / 30 M*(mass at z− = −∞) Information Loss in the CGHS Model 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