Artikel,
Effective Stringy Description of Schwarzschild Black Holes
(16.03.2004)
Zusammenfassung
We start by pointing out that certain Riemann surfaces appear rather
naturally in the context of wave equations in the black hole background. For a
given black hole there are two closely related surfaces. One is the Riemann
surface of complexified ``tortoise'' coordinate. The other Riemann surface
appears when the radial wave equation is interpreted as the Fuchsian
differential equation. We study these surfaces in detail for the BTZ and
Schwarzschild black holes in four and higher dimensions. Topologically, in all
cases both surfaces are a sphere with a set of marked points; for BTZ and 4D
Schwarzschild black holes there is 3 marked points. In certain limits the
surfaces can be characterized very explicitly. We then show how properties of
the wave equation (quasi-normal modes) in such limits are encoded in the
geometry of the corresponding surfaces. In particular, for the Schwarzschild
black hole in the high damping limit we describe the Riemann surface in
question and use this to derive the quasi-normal mode frequencies with the
log(3) as the real part. We then argue that the surfaces one finds this way
signal an appearance of an effective string. We propose that a description of
this effective string propagating in the black hole background can be given in
terms of the Liouville theory living on the corresponding Riemann surface. We
give such a stringy description for the Schwarzschild black hole in the limit
of high damping and show that the quasi-normal modes emerge naturally as the
poles in 3-point correlation function in the effective conformal theory.
