Self-Gravitating Strings In 2+1 Dimensions

TL;DR

This paper constructs and analyzes classical self-gravitating string solutions in 2+1 dimensions, exploring their geometric, causal, and quantum properties, including limits that recover known solutions and potential for quantization.

Contribution

It introduces a family of classical self-gravitating string spacetimes in 2+1 dimensions, with a formalism applicable to various boundary conditions and limits, connecting to known solutions like Gott's.

Findings

The spacetime metric is a Minkowski space with identified worldsheets.

In the flat limit, the standard string solution is recovered.

The model includes a limit where the string reduces to Gott's solution with closed timelike curves.

Abstract

We present a family of classical spacetimes in 2+1 dimensions. Such a spacetime is produced by a Nambu-Goto self-gravitating string. Due to the special properties of three-dimensional gravity, the metric is completely described as a Minkowski space with two identified worldsheets. In the flat limit, the standard string is recovered. The formalism is developed for an open string with massive endpoints, but applies to other boundary conditions as well. We consider another limit, where the string tension vanishes in geometrical units but the end-masses produce finite deficit angles. In this limit, our open string reduces to the free-masses solution of Gott, which possesses closed timelike curves when the relative motion of the two masses is sufficiently rapid. We discuss the possible causal structures of our spacetimes in other regimes. It is shown that the induced worldsheet Liouville mode obeys ({\it classically}) a differential equation, similar to the Liouville equation and reducing to it in the flat limit. A quadratic action formulation of this system is presented. The possibility and significance of quantizing the self-gravitating string, is discussed.