Transcript Asymptotic black holes greybody factors

Asymptotic black hole
greybody factors
Jorge Escobedo
University 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:
Surface gravity   T emperature T
Area of theBH A  Bekenstein- Hawking entropy S
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:

T
2
A
S
4
Analogy between BH and thermodynamical
systems made consistent!
Black hole thermodynamics

Moreover, Hawking found that BH have a characteristic blackbody
radiation spectrum.
n


1
e

1
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.
S  ln   ln 1  0 ???
Two puzzles
Given that BH have an associated entropy, what are the
microscopic degrees of freedom that give rise to it?
S  ln 
Strominger and Vafa (1996): String theoretical derivation of
the Bekenstein-Hawking entropy.
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:
th  UPU 


Where U is an operator that acts on pure states A  U B
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!
What are greybody factors?
What are greybody factors?
Potential barrier: V
Motivation: Maldacena-Strominger
calculation
D=5 near extremal black hole:
  TH and  rH  1
F ( ) 

e   1
Motivation: Maldacena-Strominger
calculation

Results
D-brane computation (CFT) = Semiclassical computation
e   1
 ( )   L 2
(e
 1)(e  R 2  1)

Same result from a theory with gravity and one without it.

A year later (1997), Maldacena proposed the AdS/CFT
correspondence.
Asymptotic greybody factors

D=4 Schwarzschild black hole
ds2   f (r )dt2  f (r )1 dr2  r 2d22
with:
f (r )  1 

rH
2GM
 1
r
r
Tortoise coordinate:
 r
x   
 r  rH

dr  r  rH ln(r  rH )

Asymptotic greybody factors

Study propagation of a scalar field in the exterior region of the
above BH, i.e.
rH  r   or    x  

Regge-Wheeler (1957):
 d2
2


V
(
r
(
x
))



 ( x)  0
2
 dx

where:
 l (l  1) 1  j 2 
V (r )  f (r ) 2  3 
r 
 r
Asymptotic greybody factors

Solutions of the previous equation describe the scattering of
incoming or outgoing waves by the BH geometry.
Since V(x)  0 as x  ,
  e ix

Now consider:
  e
ix
  Te
Re
ix
 ix
  e

 ix
~ ix
Re
~ ix
T e
Asymptotic greybody factors
 '  T ' e
 '  e
~ i x
 '  T ' e
 ix
ix
 R' e
ix
 '  e
i x
Define the greybody factor as
~
 ()  T ()T ()
Check:
~
~
T ()T ()  R() R()  1
~  i x
 R' e
Asymptotic greybody factors

Results:
e   1
 ( )  
e 3

So, the blackbody radiation gets modified to:
1
F ( )  
e 3
Asymptotic greybody factors

D=4 Reissner-Nordstrom black hole:
 ( ) 

e 
e   1
 3e   I   2
Proposal (Neitzke, 2003):
Just as in the case of small frequencies, the results in this
regime might have dual descriptions.
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.