Energy in first order 2+1 gravity

Abstract

We consider $Λ$=0 three dimensional gravity with asymptotically flat boundary conditions. This system was studied by Ashtekar and Varadarajan within the second order formalism -with metric variables- who showed that the Regge-Teitelboim formalism yields a consistent Hamiltonian description where, surprisingly, the energy is bounded from below and from above. The energy of the spacetime is, however, determined up to an arbitrary constant. The natural choice was to fix that freedom such that Minkowski spacetime has zero energy. More recently, Marolf and Patiño started from the Einstein-Hilbert action supplemented with the Gibbons-Hawking term and showed that, in the 2+1 decomposition of the theory, the energy is shifted from the Ashtekar-Varadarajan analysis in such a way that Minkowski spacetime possesses a negative energy. In this contribution we consider the first order formalism, where the fundamental variables are a so(2,1) connection $ω_a{^I}_J$ and a triad $e_a^I$. We consider two actions. A natural extension to 3 dimensions of the consistent action in 4D Palatini gravity is shown to be finite and differentiable. For this action, the 2+1 decomposition (that we perform using two methods) yields a Hamiltonian boundary term that corresponds to energy. It assigns zero energy to Minkowski spacetime. We then put forward a totally gauge invariant action, and show that it is also well defined and differentiable. Interestingly, it turns out to be related, on shell, to the 3D Palatini action by an additive constant in such a way that its associated energy is given by the Marolf-Patiño expression. Thus, we conclude that, from the perspective of the first order formalism, Minkowski spacetime can consistently have either, zero, or a negative energy equal to -1/4G, depending on the choice of consistent action employed as starting point.