Information Loss in the CGHS Model

Information Loss in theCGHS Model

Fethi Mubin Ramazanoglu

PRINCETON UNIVERSITYDEPARTMENT 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 BHFiniteness of y−

Bondi Mass and Hawking RadiationDiminishing of the Bondi Mass

5 Results: Planck Scale BH

6 Recent

7 Conclusions and Future

2 / 30


A Quick Look at Informaton Loss

A Quick Overview of Information Loss

I+RI+L

I−RI−L

z−

y−

Fixed Background

Quantum GravityMFA

3 / 30




I+RI+L

I−RI−L

z−

y−

Fixed Background Quantum Gravity

MFA

4 / 30




I+RI+L

I−RI−L

z−

y−

Fixed Background Quantum GravityMFA

5 / 30


CGHS Model

Why CGHS

Why 1+1 D?Conformal flatnessEasy calculation at 1-loop: Trace anomaly⇒ Local actionMore manageable numerics

Qualitatively similar to reduced 3+1 D, yet analytical solutions.

6 / 30


CGHS Model

Action and Classical Equations of MotionS(g, φ, f ) =1G

∫d2Ve−2φ (R + 4gab∇aφ∇bφ+ 4κ2)

−12

N∑i=1

∫d2Vgab∇afi∇bfi

S(4)(g, φ, f ) =1G

∫d2Ve−2φ

(R + 2gab∇aφ∇bφ+ 2e−2φκ2

)−1

2

N∑i=1

∫d2Ve−φgab∇afi∇bfi

Callan, Giddings,Harvey, Strominger, Phys. Rev. D (1992)7 / 30


CGHS Model

Equations of Motion

Φ = e−2φ ATV2008

gab = Θ−1Φηab

�(g) f = 0⇔ �(η)f = 0

∂+ ∂−Φ + κ2Θ = GT+− = 0

Φ∂+ ∂− ln Θ = −GT+− = 0

−∂2+ Φ + ∂+ Φ∂+ ln Θ = GT++

−∂2−Φ + ∂−Φ∂− ln Θ = GT−−

8 / 30


CGHS Model

Eternal Black Hole

~ = c = 1

G = 1

κ = 1

f = 0

Φ =Mκ− κ2x+x−

Θ = 1

9 / 30


CGHS Model

Classical Collapsing Shell

12∂+f∂+f = Mδ(z+)

Φ = eκz+e−κz− −M(

eκz+ − 1)

10 / 30


CGHS Model

Classical Collapsing Shell: Affine coordinate→∞

〈Tµµ 〉 =N24

R

⇒y+ = z+

e−κy− = e−κy− −M

y−(z−s =∞) =∞!

dy−

dz−∼ 1

z−s − z

11 / 30


CGHS Model

Hawking Radiation: 〈Ty−y−(y+ →∞)〉 → N48

[1− 1

(1+Meκy−)2

]

−1.5 −1 −0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z−c

Ty− y− (

exte

rnal

fiel

d)

12 / 30


CGHS Model

MFA Equations of Motion

〈Tµµ 〉 =N24

R

�(g) f = 0⇔ �(η)f

∂+ ∂−Φ + κ2Θ = GT+− =N24∂+ ∂− ln

(ΦΘ−1)

Φ∂+ ∂− ln Θ = −GT+− = − N24∂+ ∂− ln

(ΦΘ−1)

13 / 30


CGHS Model

Asymptotic behavior

Φ = A(z−)eκz+ + B(z−) + O(e−κz+)

Θ = A(z−)eκz+ + B(z−) + O(e−κz+)

κy− = − lnAκ

<∞?

Ashtekar, Taveras, Varadarajan Phys. Rev. Lett. (2007)

14 / 30


CGHS Model

Bondi Mass and Black Hole Evaporation

Bondi mass and Hawking radiationis connected to the asyptoticbehavior of Φ near I+R

ddy−

MB︷︸︸︷[ dBdy−

+ κB +NG~24

(d2y−

dz−2 (dy−

dz−)−2 ) ]

= − NG~24[d2y−

dz−2 (dy−

dz−)−2 ]2︸︷︷︸

MFA Flux on I+R

”Area” = Φ− N12

15 / 30


CGHS Model

Some Remarks

Scalable: Only MN matters !

(Φ,Θ,N, f )→ (αΦ, αΘ, αN, f )

Dimensionless [G~], chosen to be 1.

16 / 30


Numerical Solution

Numerical Solution

Regularize the fields

Φ(z+, z−) = eκz+−κz−(1 + φ(z+, z−)) + Φ0(z+)

Θ(z+, z−) = eκz+−κz−(1 + θ(z+, z−))

Scaling and compactification⇒ Unigrid mesh

z− = z−z− − 1

109/2

z− − 109/2 + z−s,e

z− = −e− tan(πzc−π/2) + 4.096× 10−9(zc − 1)

z+ = C tanπz+c z+

c ∈ [0,12

] , z+c ∈ [0, 1]

Discretization, recasting into a polynomial

I R−

boun

dary

cond

ition

from

mat

ter f

ields

vacu

um b

ound

ary

I R−

I L−

(i−1,

j)

(i−1,

j−1)

(i,j)

(i,j−1

)

vacu

um

I R+

z−

ij

z+

17 / 30


Numerical Solution

Numerical Solution

Very high resolution near the last ray

Very small truncation errorsI R−

boun

dary

cond

ition

from

mat

ter f

ields

vacu

um b

ound

ary

I R−

I L−

(i−1,

j)

(i−1,

j−1)

(i,j)

(i,j−1

)

vacu

um

I R+

z−

ij

z+

18 / 30


Results

Finiteness of y−

Macroscopic BH Asymptotic Killing Coordinate y−

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5

0

5

10

15

20

25

30

z−c

y−

19 / 30


Results

Finiteness of y−

Macroscopic BH: Power Law for dy−dz−

−36 −34 −32 −30 −28 −26 −24 −22 −20−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

d ln

(dy− /d

z− ) /d

ln(z

− sing

−z− )

ln (zsing

−z) 20 / 30


Results

Bondi Mass and Hawking Radiation

Macroscopic BH: Bondi Mass

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

z−c

MB

ondi

(na

tura

l uni

ts)

21 / 30


Results

Bondi Mass and Hawking Radiation

Macroscopic BH: Hawking Radiation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

z−c

MF

A e

nerg

y flu

x (n

atur

al u

nits

)

22 / 30


Results

Diminishing of the Bondi Mass

Macroscopic BH: Comparison of “Area” and MB

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.5

0.6

0.7

0.8

0.9

1

1.1

z−c

Are

a/M

B

23 / 30


Results

Planck Scale BH: Asymptotic Killing Coordinate y−

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−16

−14

−12

−10

−8

−6

−4

−2

0

2x 10−3

z−c

y−

24 / 30


Results

Planck Scale BH: Bondi Mass

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

z−c

MB

ondi

(na

tura

l uni

ts)

25 / 30


Results

Planck Scale BH: Hawking Radiation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

z−c

MF

A e

nerg

y flu

x (n

atur

al u

nits

)

26 / 30


Recent

Hayward Mass

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4

6

8

10

z−c

MH

ayw

ard

27 / 30


Recent

”Universal Curve” on A-M plane

0 2 4 6 8 10 12 14 16 18−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

A =\Phi −\frac{N}{12}

A−

MB

18

14

12

10

8

28 / 30


Recent

Mass at the last ray

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M*(mass at z− = −∞)

m* (m

ass

at la

st r

ay)

data

fitted curve

29 / 30


Conclusions and Future

Conclusions

y− is finite, strong evidence for unitarity.

For all macroscopic BHs, Bondi mass at the last ray ≈ order ofunity (in natural units).

If the initial mass itself is of the order unity or smaller,evaporation process is very different from the standard (externalfield) picture.

High resolution numerics near singularity is necessary todescribe 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

Information Loss in the CGHS Model

Documents

Transcript of Information Loss in the CGHS Model