Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals

TL;DR

This paper explores the conditions under which the Barbero-Immirzi parameter influences physical observables in Euclidean Quantum Gravity, linking topological invariants and manifold properties to quantum effects in gravitational path integrals.

Contribution

It establishes topological and geometric conditions for the Barbero-Immirzi parameter to have physical effects and evaluates its impact on specific gravitational solutions.

Findings

The Holst action's surface term is non-zero only if the metric is non-diagonalizable and the Pontryagin number is non-zero.

The Barbero-Immirzi parameter affects the energy and entropy in Taub-NUT-ADS solutions.

Topological invariants like the Euler characteristic influence black-hole merger dynamics.

Abstract

The Barbero-Immirzi parameter $\gamma$ appears in the \emph{real} connection formulation of gravity in terms of the Ashtekar variables, and gives rise to a one-parameter quantization ambiguity in Loop Quantum Gravity. In this paper we investigate the conditions under which $\gamma$ will have physical effects in Euclidean Quantum Gravity. This is done by constructing a well-defined Euclidean path integral for the Holst action with non-zero cosmological constant on a manifold with boundary. We find that two general conditions must be satisfied by the spacetime manifold in order for the Holst action and its surface integral to be non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryagin number of the manifold has to be non-zero. The latter is a strong topological condition, and rules out many of the known solutions to the Einstein field equations. This result leads us to evaluate the on-shell first-order Holst action and corresponding Euclidean partition function on the Taub-NUT-ADS solution. We find that $\gamma$ shows up as a finite rotation of the on-shell partition function which corresponds to shifts in the energy and entropy of the NUT charge. In an appendix we also evaluate the Holst action on the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that case as well $\gamma$ shows up in the energy and entropy of the NUT and bolt charges. We also present an example whereby the Euler characteristic of the manifold has a non-trivial effect on black-hole mergers.