Asymptotic black hole greybody factors

Jorge EscobedoUniversity of Amsterdam

Institute for Theoretical Physics

April 2008

Outline

Black hole thermodynamics

Two puzzles

What are greybody factors?

Motivation: Maldacena-Strominger

Asymptotic greybody factors

Black hole thermodynamics

Black holes (BH) are fascinating objects predicted by general relativity.

Black hole thermodynamics

Bekenstein (1973): Conjectures that BH have an associated entropy.

Bardeen, Carter and Hawking (1973):

Laws of black hole mechanics

Laws of thermodynamics

if:

SA

T

entropy Hawking-Bekenstein BH theof Area

eTemperatur gravity Surface

Black hole thermodynamics

Problem: If BH have an associated temperature, they must radiate. However, nothing can escape from a BH!

Hawking (1975): Quantum fields in a BH background. Temperature and entropy of a BH given by:

2

T4

AS

Analogy between BH and thermodynamical systems made consistent!

Black hole thermodynamics Moreover, Hawking found that BH have a characteristic blackbody

radiation spectrum.

1

1

e

n

Black hole thermodynamics Everything looks really nice, uh?

but…

Two puzzles 1. Quantum description of black holes

No-hair theorem: A BH solution is characterized only by its mass, charge and angular momentum.

Therefore, there is only one state of the BH that has the observable thermodynamical quantities mentioned above.

??? 01lnln S

Two puzzles Given that BH have an associated entropy, what are the

microscopic degrees of freedom that give rise to it?

Strominger and Vafa (1996): String theoretical derivation of

the Bekenstein-Hawking entropy.

lnS

Two puzzles

2. The information loss paradox

Pure quantum state

Thermal radiation

Two puzzles If a pure state falls into the black hole, it will be emitted as

thermal radiation (mixed state). Violation of unitarity: Pure states cannot evolve into mixed

states! In terms of density matrices:

Where U is an operator that acts on pure states

This is known as the information loss paradox: we started with quantum fields in a BH background and obtained a result that is not allowed by quantum mechanics!

UU Pth

BA U

What are greybody factors?

What are greybody factors?

Potential barrier: V

Motivation: Maldacena-Strominger calculation

1)(

e

F

D=5 near extremal black hole: 1 and HH rT

Motivation: Maldacena-Strominger calculation Results

D-brane computation (CFT) = Semiclassical computation

Same result from a theory with gravity and one without it.

A year later (1997), Maldacena proposed the AdS/CFT correspondence.

)1)(1(

1)(

22

RL ee

e

Asymptotic greybody factors

D=4 Schwarzschild black hole

with:

Tortoise coordinate:

22

22122 )()( drdrrfdtrfds

r

GM

r

rrf H 2

11)(

)ln( HHH

rrrrdrrr

rx

Asymptotic greybody factors

Study propagation of a scalar field in the exterior region of the above BH, i.e.

Regge-Wheeler (1957):

where:

xrrH or

0)())(( 22

2

xxrV

dx

d

3

2

2

1)1()()(

r

j

r

llrfrV

Asymptotic greybody factors

Solutions of the previous equation describe the scattering of incoming or outgoing waves by the BH geometry.

Now consider:

xie

xV(x)

, as 0 Since

xi

xixi

Te

eRe

xi

xixi

eT

eRe

~

~

Asymptotic greybody factors

xixi

xi

eRe

eT

'~

'

'~

'

)(~

)()( TT

xixi

xi

eRe

eT

''

''

Define the greybody factor as

Check:

1)(~

)()(~

)( RRTT

Asymptotic greybody factors

Results:

So, the blackbody radiation gets modified to:

3

1)(

e

e

3

1)(

e

F

Asymptotic greybody factors

D=4 Reissner-Nordstrom black hole:

Proposal (Neitzke, 2003): Just as in the case of small frequencies, the results in this

regime might have dual descriptions.

23

1)(

Iee

e

Conclusions

The study of greybody factors as part of perturbations around BH in classical gravity.

Moreover, the study of asymptotic greybody factors might help us in understanding the quantum nature of black holes and thus, of quantum gravity.