Gravity in 2 + 1 dimensions coupled to point-like sources: a flat Chern-Simons gauge theory equivalent to Einstein's

Abstract

We present a classical Chern-Simons gauge theory for planar gravity coupled to point-like sources. The theory is defined in terms of flat coordinates whose relation with the space-time coordinates can be unambiguously established. Though flat, the theory is equivalent to Einstein's as can be shown in some explicit and non-trivial examples. The procedure we have developed allows us to find the space-time metric, the geodesics equations and also provides the coordinates that flatten the space-time manifold. In particular, we recover the stationary metric generated by N spinning particles and we discuss in full detail the matching conditions satisfied by the space-time coordinates that flatten the metric in the two-particle case.