Gravity Asymptotics with Topological Parameters

TL;DR

This paper explores the topological origins of the Barbero-Immirzi parameter in four-dimensional gravity, analyzing its role in the action principle for spacetimes with boundaries and asymptotic conditions, and linking it to topological densities.

Contribution

It introduces a first order action formulation incorporating topological densities and boundary terms, clarifying the role of the Barbero-Immirzi parameter in asymptotic gravity.

Findings

The Barbero-Immirzi parameter appears as a coefficient of the SO(3,2) or SO(4,1) Pontryagin density.

Adding topological densities ensures a well-defined variational principle for spacetimes with boundaries.

The action admits an extremum for asymptotically Anti de Sitter or de Sitter spacetimes with fixed topological coefficients.

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.[1]. 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 analyse spacetimes with and without a cosmological constant. For locally Anti de Sitter (or de Sitter) asymptotia, the action principle has non-trivial 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 non-zero.