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.

We apply the Dirac gauge fixing procedure to Chern-Simons theory with gauge group ISO(2, 1) on manifolds R × S, where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural requirements, this yields an explicit description of the Poisson structure on the moduli space of flat ISO(2, 1)-connections on S in terms of classical dynamical r-matrices for iso(2, 1). We show that the Poisson structures and classical dynamical r-matrices arising from different gauge fixing conditions are related by dynamical ISO(2, 1)-valued transformations that generalise the usual gauge transformations of classical dynamical r-matrices. By means of these transformations, it is possible to classify all Poisson structures and classical dynamical r-matrices obtained from such gauge fixings. Generically these Poisson structures combine classical dynamical r-matrices for non-conjugate Cartan subalgebras of iso(2, 1). 1

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 Poincaré 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.