The Two-Dimensional Stringy Black-Hole: A New Approach and a Pathology

TL;DR

This paper investigates string propagation in a two-dimensional black hole, solving classical and quantum dynamics, revealing a new quantum phenomenon affecting the speed of light, and analyzing scattering and partition functions.

Contribution

It introduces a new approach to analyze string dynamics in 2D black holes, including exact solutions and the discovery of a quantum renormalization of the speed of light.

Findings

Scattering matrix reproduces coset CFT results

Quantum renormalization of the speed of light observed

No critical Hagedorn temperature found

Abstract

The string propagation in the two-dimensional stringy black-hole is investigated from a new approach. We completely solve the classical and quantum string dynamics in the lorentzian and euclidean regimes. In the lorentzian case all the physics reduces to a massless scalar particle described by a Klein-Gordon type equation with a singular effective potential. The scattering matrix is found and it reproduces the results obtained by coset CFT techniques. It factorizes into two pieces : an elastic coulombian amplitude and an absorption part. In both parts, an infinite sequence of imaginary poles in the energy appear. The generic features of string propagation in curved D-dimensional backgrounds (string stretching, fall into spacetime singularities) are analyzed in the present case. A new physical phenomenon specific to the present black-hole is found : the quantum renormalization of the speed of light. We find $c_{quantum} = \sqrt{{k\o{k-2}}}~c_{classical}$, where $k$ is the integer in front of the WZW action. This feature is, however, a pathology. Only for $ k \to \infty$ the pathology disappears (although the conformal anomaly is present). We analyze all the classical euclidean string solutions and exactly compute the quantum partition function. No critical Hagedorn temperature appears here.