Gauge fixing in (2+1)-gravity with vanishing cosmological constant

Abstract

We apply Dirac's gauge fixing procedure to (2+1)-gravity with vanishing cosmological constant. For general gauge fixing conditions based on two point particles, this yields explicit expressions for the Dirac bracket. We explain how gauge fixing is related to the introduction of an observer into the theory and show that the Dirac bracket is determined by a classical dynamical r-matrix. Its two dynamical variables correspond to the mass and spin of a cone that describes the residual degrees of freedom of the spacetime. We show that different gauge fixing conditions and different choices of observers are related by dynamical Poincar\'e transformations. This allows us to locally classify all Dirac brackets resulting from the gauge fixing and to relate them to a set of particularly simple solutions associated with the centre-of-mass frame of the spacetime.