Witten–Nester energy in topologically massive gravity

Abstract

We formulate topologically massive supergravity with a cosmological constant in the first-order formalism and construct the Noether supercurrent and superpotential associated with its local supersymmetry. Using these results, we construct in ordinary topologically massive gravity the Witten–Nester integral for conserved charges containing spinors which satisfy a generalized version of the Witten equation on the initial value surface. We show that the Witten–Nester charge, represented as an integral over the boundary of the initial value surface, produces the Abbott–Deser–Tekin energy for asymptotically anti-de Sitter spacetimes. We consider all values of the Chern–Simons coupling constant, including the critical value known as the chiral point, and study the cases of standard Brown–Henneaux boundary conditions, as well as their weaker version that allows a slower fall-off. Studying the Witten–Nester energy as a bulk integral over the initial value surface instead, we find a bound on the energy, and through it the sufficient condition for the positivity of the energy. In particular, we find that spacetimes of Petrov type N that admit globally well-defined solutions of the generalized Witten equation have positive energy.