Gravity asymptotics with topological parameters

Abstract

In four-dimensional gravity theory, the Barbero-Immirzi parameter has a topological origin, and can be identified as the coefficient multiplying the Nieh-Yan topological density in the gravity Lagrangian, as proposed by Date et al. [ Phys. Rev. D 79 044008 (2009)]. Based on this fact, a first order action formulation for spacetimes with boundaries is introduced. The bulk Lagrangian, containing the Nieh-Yan density, needs to be supplemented with suitable boundary terms so that it leads to a well-defined variational principle. Within this general framework, we analyze spacetimes with and without a cosmological constant. For locally anti–de Sitter (or de Sitter) asymptotia, the action principle has nontrivial implications. It admits an extremum for all such solutions provided the SO(3,1) Pontryagin and Euler topological densities are added to it with fixed coefficients. The resulting Lagrangian, while containing all three topological densities, has only one independent topological coupling constant, namely, the Barbero-Immirzi parameter. In the final analysis, it emerges as a coefficient of the SO(3,2) [or SO(4,1)] Pontryagin density, and is present in the action only for manifolds for which the corresponding topological index is nonzero.