Download Scalar fields in 2D black holes: Exact solutions and quasi

V. Frolov and A. Zelnikov, Phys.Rev. D63,125026
The 2d gravity coupled to a dilaton field with the action
1
Sg 
2
d
2
x ge
2 


R  4( )  4
2
2


has black hole solutions with the properties similar to those of the (r,t) sector
of the Schwarzschild black hole.
This action ( CGHS ) arises in a low-energy asymptotic of string theory
models and in certain models with a scalar matter.
Mandal, Sengupta, and Wadia (1991);
Witten (1991),
Callan, Giddings, Harvey, and Strominger (1992)
Most of the papers on the quantization of matter fields on the 2D
dilaton gravity background consider conformal matter on the 2d
black hole spacetimes. It is important to know greybody factors to
study, e.g., the Hawking radiation and vacuum polarization effects.
For the minimally coupled scalar fields in 2d there is no potential
barrier and greybody factors are trivial.
In this talk we address to the problem of quantization of nonconformal fields. This problem is more complicated but much
more interesting since nonconformal fields interact with the
curvature and feel the potential barrier, which plays an important
role in black hole physics. Qualitative features of the potential
barrier of the Schwarzschild black hole are very close to that of the
string inspired 2d gravity model we consider here.
Our purpose is to study a quantum scalar massive field in a spacetime
of the 2-dimensional black hole. The black hole solution in the CGHS
dilaton gravity reads
2
dr
dS 2   f dT 2 

f
f  1 e
surface gravity
1 df
1

r r0 

2 dr
2r0
 rr
0

r
 
r0
scalar curvature
d2 f
1  r r0
R 2  2 e 
dr
r0
It would be convenient to use the dimensionless form of the metric
2
dx
r02 dS 2  ds 2   fdt 2 

f
1
x
x



R

e

f  1 e ,
2

1
r0 TBH 

.
Black hole temperature
2 4
By introducing a new variable
z  1  e x
we get
2
dz
ds 2   z dt 2 

2
z (1  z )
t  T r0
x  r r0
The field equation follows from the action W [ ]  
(   m 2   R)   0 
1
 dX 2 g1 2  2  (m2   R)  2  
2
 2 2

 2  2  U    0 
 t x

where
U  f ( x) (  2   R)
2 




2
1  exp( x ) (1  exp( x ))
The ”tortoise” coordinate
The dimentionless mass
dx
x  
 ln (e x  1) 
1 f ( x)
  mr0 ,
x
z  1  exp( x)  1 (1  exp( x ).
U ( z )  z ((  2   )   z ) 
0
1
z  (01)
2 1
1
m 
16
2
m 0
2
10
0
20
x*
10
0
Dilaton black hole
m2  1
20
x*
10
0
20
x*
Schwarzschild black hole
x 

x
1
dx
 ln (e x  1) 
f ( x)
Rin ( x   )
Rup ( x   )

 1 2 T ei x 
1 


4  1 2 
i x
 i x 


R
e

e







 1 2  ei x  r e i x  



1 

4 
 1 2 t e i x 


x  
x  
x  
x  
 R 2   T 2  1
 r 2   t 2  1
T  t 
  2  2 ,
  m r0 .
Rin ( x   )

 1 2 T ei x 
1 


4  1 2 
i x
 i x 


R
e

e







x  
e
T e  i x
 R 2   T 2  1
x  
 i x
R ei x
2
x*  
x*  
In the absence of bound
states the modes and and
their complex conjugated
form a complete set (basis).
Rout ( x   )  ( Rin ( x   )) 
Rdown ( x   )  ( Rup ( x   )) 
Only two of four solutions are
linearly independent.
Rin ( x   )  T Rdown ( x   )  R Rout ( x   ) 
Rup ( x   )  r Rdown ( x   )  t Rout ( x   ) 
A general solution of the field equation can be obtained in terms of
hypergeometric functions.
R( x   )  z
 i
(1  z)
 i
u( z   ) 
d 2u
du
z (1  z ) 2  [c  (a  b  1) z ]
 ab u  0 
dz
dz
where
a      b      c  1  2i  c  1  2i 
1
1
   i(   )   
 
2
4
T
Rin ( x   ) 
z i (1  z ) i F (a b c z ) 
4
t
Rup ( x   ) 
z i (1  z ) i F (a b c1  z ) 
4
T i
Rout ( x   ) 
z (1  z )i F (a  b  c  z ) 
4
t
Rdown ( x   ) 
z i (1  z )i F (a  b  c 1  z ) 
4
1  b
for real  

a  

1  a for imaginary  

for real  
1  a

b  
1  b for imaginary  

1
  
4
The transition coefficient
T
can be presented as
 (a) (b)
T 

 (c) (c  1)
By using relations
( ) (1   ) 
we get


 ( ) ( )  

sin( )
 sin( )
2sinh(2  ) sinh(2  )
 T  

cosh[2 (   )]  cos(2  )
2
  2  2 ,

1

4
The number density of particles radiated by the black hole to infinity is given
by Hawking expression
 T 2
dn( )
exp(2 )sinh(2 )



d
exp(4 )  1 cosh[2 (   )]  cos(2 )
The corresponding energy density flux
dE
1

dt
2 r02


d   exp( 2 ) sinh(2 )

cosh[2 (   )]  cos(2 )
For massless particles (m=0)
dE
dt

 0
1
32 r
3 2
0
e
4 1
dilog(1  e
4 1
 4 1
e
)  dilog(1  e
1
dilog( z )   
z
1
4 1
)

ln t
dt 
t 1
Flux  192 r02  dE
dt
Solving the field equation we assumed that is real. Besides these wave-like
solutions the system can have modes with time dependence
exp( t )
If the radial function has decreasing asymptotics both at the horizon and at
infinity, then the corresponding states are called bound states. For ()  0
this condition can be satisfied only if both   1 4   and  are real. The
1
number of bound states is defined by the integer part of the quantity [     ] .
2
1
A new bound state appears when     2
reaches a new integer number value. The
transition from a pure continuous spectrum to
a spectrum with a single bound state occurs
when     (   1) 


1
1
2 
n  (   n  ) 


2
2  n 1

2
 1
Let us calculate now the stress-energy tensor for the first bound state mode.
 (v z )  e v (1  z ) B ,
1
1
1
B 
  .
2
4
2
0
Here
v  t  x*  t  ln( z  (1  z ))
In the presence of a bound state B 
event horizon
Tvv H
is the advanced time coordinate.

and the energy density flux through the
B2   2
2
2
2
2
2
 2

exp((
B


)
(
t

x
)

B
)
B
(4
B

6
B

3)


(2
B

1)


*
2


4B
is positive and grows exponentially with the advanced time parameter. This
behavior reflects instability of the quantum system in the presence of bound states.
Wick’s rotation
g 
t  i
2


  R  G ( , z |  ', z ')   (   )  ( z  z '),
E
  0 the Green function can be obtained in an explicit form
In massless case
in terms of the Legendre function P ( ) .
1
G( , z |  ', z ') 
P 1   ( ) 
4cos( ) 2
    



  (1  2 z )(1  2 z )  4 zz (1  z )(1  z ) cos 

 2 

1

4
QNM’s are the vibrational modes
of perturbations in the spacetime
exterior to the event horizon. They
are defined as solutions to the
wave equation for perturbations
with boundary conditions that are
ingoing at the horizon and
outgoing at spatial infinity.
The frequency spectrum of QNM
is discrete and complex. The
Imaginary part of QNM describes
damping.
For Schwarzschild black holes in
the high damping limit.
1
2
 n  kTBH [ 2 i (n  )  ln 3 ]
n
The poles of the transition coefficient give quasi-normal frequencies.
2sinh(2  ) sinh(2  )

cosh[2 (   )]  cos(2  )
  2  2



n
1
2 
 2 i  n    


1
TBH
2

n  

2

1
  
4
 T 2 
If
1
all

4
n
are imaginary.
If

1
quasi-normal modes
4
n
acquire a real part.




2
(n )
1 


 2   1 
2

TBH
4
 1   1 
  n       
4 
  2 

2
on the horizon ( X  X 0 )
Because the Euclidean horizon is a fixed point of the Killing vector field
the Green function does not depend on time and only zero-frequency
 2 ren . The Green function then reads
mode contributes to
G( X  X 0 )  
1
4

1
1




x
ln(1

e
)

2
















 

2
2





Subtracting the UV divergent part we obtain
G div ( X  X )  
1
4
 1 2



ln


(
X

X
)

2


 2


 


 2 ( x  0)
ren








1 
1
1
1
1




2 ln      
       
    

4 
2
4
2
4








We studied quantum nonminimal scalar field in a two-dimensional
black hole spacetime. For a string motivated black hole we found
exact analytical solutions in terms of hypergeometric functions.
A explicit expression for greybody factors and Hawking radiation are
calculated.
For negative values of nonminimal coupling constant  the field besides
usual scattering modes can have bound states. These bound states lead
to instability. This effect is accompanied by exponentially growing
positive energy fluxes through the black hole horizon and to infinity.
This kind of instability occurs for any theory with negative  for solutions
describing evaporating black holes, since r0  0 and there is a moment
of time when the parameter   mr0 meets the condition of formation of
a bound state.
1


Exact formula for Quasi-Normal Modes is obtained. If
then all
4
QNM are imaginary. If   1 Quasi-Normal frequencies have a real
4
part.