Scalar fields in 2D black holes:

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Scalar fields in 2D black holes Exact solutions
and quasi-normal modes
Andrei Zelnikov University of Alberta
BIRS. PiTP Spin, Charge, and Topology in low
dimensions. July 30, 2006
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V. Frolov and A. Zelnikov, Phys.Rev. D63,125026
2D dilaton gravity
The 2d gravity coupled to a dilaton field with
the action

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)
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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.
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The Model
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
scalar curvature
surface gravity
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It would be convenient to use the dimensionless
form of the metric
Black hole temperature
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The field equation follows from the action
where
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Infinity
Horizon
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Schwarzschild black hole
Dilaton black hole
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Scattering modes
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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.
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A general solution of the field equation can be
obtained in terms of hypergeometric functions.
where
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The transition coefficient can be
presented as
By using relations
we get
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Radiation spectrum and energy flux
The number density of particles radiated by the
black hole to infinity is given by Hawking
expression
The corresponding energy density flux
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For massless particles (m0)
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Energy flux
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Bound State and Black Hole Instability
Solving the field equation we assumed that is
real. Besides these wave-like solutions the
system can have modes with time dependence
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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
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The Euclidean Green Function
Wicks rotation
In massless case the Green
function can be obtained in an explicit form in
terms of the Legendre function .
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Quasi-Normal Modes
QNMs 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.
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The poles of the transition coefficient give
quasi-normal frequencies.
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Vacuum polarization
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
Subtracting the UV divergent part we obtain
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Conclusion
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.