The Komar charge in presence of the Holst term and the gravitational Witten effect

TL;DR

This paper investigates how the addition of a Holst term in the first-order formalism of gravity modifies the Komar charge, revealing a gravitational Witten effect where NUT charge influences mass through the Barbero parameter.

Contribution

It demonstrates that the Holst term alters the Komar charge, leading to a parity-breaking effect that couples NUT charge to mass, extending the understanding of topological modifications in gravity.

Findings

The Komar charge is modified by a topological term involving the Barbero parameter.

A non-zero NUT charge induces a non-zero mass proportional to the Barbero parameter.

The modification resembles the Witten effect in gauge theories.

Abstract

In the first-order formalism, the Einstein--Hilbert action can be modified by the addition of a Holst term multiplied by the Barbero parameter $\alpha$. This modification breaks parity although it does not affect the equations of motion. We show that the standard Komar charge is also modified by the addition of a topological term multiplied by the Barbero parameter $\alpha$. For the Killing vector that generates time translations, the value of the Komar integral at infinity is modified by the addition of a term proportional to the NUT charge $N$ and the parity-breaking Barbero parameter. Thus, as in the standard Witten effect, a non-vanishing NUT charge $N$ induces a non-vanishing mass $\alpha N$.