Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry

TL;DR

This paper analyzes (2+1)-dimensional gravity as a constrained system, applying Dirac gauge fixing to derive brackets and interpret the spacetime geometry, especially in observer-related conical spacetimes with particles.

Contribution

It provides explicit Dirac brackets for (2+1)-gravity with physical gauge conditions and links them to the geometry of conical spacetimes with particles.

Findings

Explicit Dirac brackets derived for (2+1)-gravity.

Physical gauge fixing conditions interpreted as observer choices.

Conical spacetime parameters linked to particle velocities and distances.

Abstract

We consider (2+1)-gravity with vanishing cosmological constant as a constrained dynamical system. By applying Dirac's gauge fixing procedure, we implement the constraints and determine the Dirac bracket on the gauge-invariant phase space. The chosen gauge fixing conditions have a natural physical interpretation and specify an observer in the spacetime. We derive explicit expressions for the resulting Dirac brackets and discuss their geometrical interpretation. In particular, we show that specifying an observer with respect to two point particles gives rise to conical spacetimes, whose deficit angle and time shift are determined, respectively, by the relative velocity and minimal distance of the two particles.