2 2 2 214( ) 4

2gS d x ge R

The 2d gravity coupled to a dilaton field with the action

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)

has black hole solutions with the properties similar to those of the (r,t) sector of the Schwarzschild black hole.

V. Frolov and A. Zelnikov, Phys.Rev. D63,125026

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 non-conformal 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

22 2 dr

dS f dTf

0

0

1rr r

f er

0

0

2

2 20 0

1 1 1

2 2r r

r r

df d fR e

dr r dr r

surface gravity scalar curvature

It would be convenient to use the dimensionless form of the metric

22 22 2

0

dxds

fr S fdtd 0t T r

0x r r

1 ,xf e 1

2xR e

By introducing a new variable we get 1 xz e

22 2

2(1 )

dzds z dt

z z

Black hole temperature 0

1.

2 4BHr T

The field equation follows from the action 22 1 2 2 21[ ] ( )

2W dX g m R

2( ) 0m R

2 2

2 20U

t x

2( ) ( )U f x R 2

21 exp( ) (1 exp( ))x x

where

The ”tortoise” coordinate

0 ,mr 1

ln ( 1)( )

x xdxx e

f x The dimentionless mass

1 exp( ) 1 (1 exp( ).z x x

2( ) (( ) )U z z z

2 1 (0 1)z 0 1

2 0m

0*x

10 20

2 1

16m

10 0 20*x

10 0 20*x

2 1m

Schwarzschild black holeDilaton black hole

1ln ( 1)

( )

x xdxx e

f x

1 2

in

1 2

1( )

4

i x

i x i x

T e x

R x

R e e x

1 2

up1 2

1( )

4

i x i x

i x

e r e x

R x

t e x

2 2 ,

0.mr

2 2 1R T 2 2 1r t

T t

i xR e

i xT e

i xe

2 2 1R T

*x *x

1 2

in

1 2

1( )

4

i x

i x i x

T e x

R x

R e e x

2

out in

down up

( ) ( ( ))

( ) ( ( ))

R x R x

R x R x

In the absence of bound states the modes and and their complex conjugated form a complete set (basis).

Only two of four solutions are linearly independent.

in down out( ) ( ) ( )R x T R x R R x

up down out( ) ( ) ( )R x r R x t R x

A general solution of the field equation can be obtained in terms of hypergeometric functions.

( ) ( ) ( )1i iR x z u zz 2

2(1 ) [ ( 1) ] 0

d u duz z c a b z abu

dz dz

where

1 2 1 2a b c i c i

1 1( )

2 4i

in ( ) (1 ) ( )4

i iTR x z z F a b c z

up ( ) (1 ) ( 1 )4

i itR x z z F a b c z

out ( ) (1 ) ( )4

i iTR x z z F a b c z

down ( ) (1 ) ( 1 )4

i itR x z z F a b zc

1 for real 1 for real

1 for imaginary 1 for imaginary

b a

a b

a b

1

4

( ) ( )

( ) ( 1)

a bT

c c

The transition coefficient can be presented as T

By using relations ( ) (1 ) ( ) ( )sin( ) sin( )

2 2sinh(2 ) sinh(2 )

cosh[2 ( )] cos(2 )T

we get

2 2 ,

1

4

The number density of particles radiated by the black hole to infinity is given by Hawking expression

2( ) exp( 2 )sinh(2 )

exp(4 ) 1 cosh[2 ( )] cos(2 )

Tdn

d

The corresponding energy density flux

20

1 exp( 2 )sinh(2 )

2 cosh[2 ( )] cos(2 )

dE d

dt r

4 1 4 1 4 1

3 2 4 10 0

1 dilog(1 ) dilog(1 )

32 1

dE e e e

dt r e

For massless particles (m=0)

1

lndilog( )

1

z tz dt

t

20Flux 192 dE

dtr

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 this condition can be satisfied only if both and are real. The number of bound states is defined by the integer part of the quantity

( ) 0 1 4

1[ ] .

2

21 1( )

12 22

n nn

A new bound state appears when reaches a new integer number value. The transition from a pure continuous spectrum to a spectrum with a single bound state occurs when

1

2

( 1)

1

Let us calculate now the stress-energy tensor for the first bound state mode.

0( ) (1 ) ,v Bv z e z

Here is the advanced time coordinate. * ln( (1 ))v t x t z z

1 1 1.

2 4 2B

2 22 2 22 2

*2

2exp(( ) ( (4 6 3) (2 1)) )

4vv HB t x

BT B B B B

BB

In the presence of a bound state and the energy density flux through the event horizon

is positive and grows exponentially with the advanced time parameter. This behavior reflects instability of the quantum system in the presence of bound states.

B

2 ( , | ', ') ( ) ( '),Eg R G z z z z

t iWick’s rotation

In massless case the Green function can be obtained in an explicit form in terms of the Legendre function .

0

12

1( , | ', ') ( )

4cos( )G z z P

(1 2 )(1 2 ) 4 (1 )(1 ) cos2

z z zz z z

( )P

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.

1[ 2 ( ) ln 3 ]

2n BHn

kT i n

For Schwarzschild black holes in the high damping limit.

2 2sinh(2 ) sinh(2 )

cosh[2 ( )] cos(2 )T

The poles of the transition coefficient give quasi-normal frequencies.

1

4

2 2

212

122

n

BH

i nT n

If all are imaginary.1

4 n

2

2

( ) 12 1

4 1 12 4

n

BHT n

If quasi-normal modes acquire a real part.1

4 n

2 on the horizon

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 mode contributes to . The Green function then reads

2

ren

0( )X X

0

1 1 1( ) ln(1 ) 2

4 2 2xG X X e

div 21 1( ) ln ( ) 2

4 2G X X X X

Subtracting the UV divergent part we obtain

2

ren

1 1 1 1 1( 0) 2ln

4 2 4 2 4x

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.

Exact formula for Quasi-Normal Modes is obtained. If then all QNM are imaginary. If Quasi-Normal frequencies have a real part.

1

4

1

4

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 and there is a moment of time when the parameter meets the condition of formation of a bound state.

0 0r

0mr