Surface terms, asymptotics and thermodynamics of the Holst action

Abstract

We consider a first-order formalism for general relativity derived from the Holst action. This action is obtained from the standard Palatini–Hilbert form by adding a topological-like term and can be taken as the starting point for loop quantum gravity and spin foam models. The equations of motion derived from the Holst action are, nevertheless, the same as in the Palatini formulation. Here we study the form of the surface terms of the action for general boundaries as well as the symplectic current in the covariant formulation of the theory. Furthermore, we analyze the behavior of the surface terms in asymptotically flat spacetimes. We show that the contribution to the symplectic structure from the Holst term vanishes and one obtains the same asymptotic expressions as in the Palatini action. It then follows that the asymptotic Poincaré symmetries and conserved quantities such as energy, linear momentum and relativistic angular momentum found here are equivalent to those obtained from the standard Arnowitt, Deser and Misner formalism. Finally, we consider the Euclidean approach to black hole thermodynamics and show that the on-shell Holst action, when evaluated on some static solutions containing horizons, yields the standard thermodynamical relations.