Semiclassical Unimodular Gravity

TL;DR

This paper investigates various formulations of unimodular gravity beyond the classical level, analyzing their quantum and semiclassical properties, and comparing them to General Relativity, especially regarding the cosmological constant and horizon entropy.

Contribution

It explores non-covariant and covariant formulations of unimodular gravity at quantum and semiclassical levels, revealing conditions under which they agree or differ from GR.

Findings

Quantum theory of fixed-determinant metric matches GR in infinite volume backgrounds.

Semiclassical instanton calculations agree with GR for infinite volume but differ for finite volume.

Allowing variable 4-volume restores full Einstein equations, including the cosmological constant.

Abstract

Classically, unimodular gravity is known to be equivalent to General Relativity (GR), except for the fact that the effective cosmological constant $\Lambda$ has the status of an integration constant. Here, we explore various formulations of unimodular gravity beyond the classical limit. We first consider the non-generally covariant action formulation in which the determinant of the metric is held fixed to unity. We argue that the corresponding quantum theory is also equivalent to General Relativity for localized perturbative processes which take place in generic backgrounds of infinite volume (such as asymptotically flat spacetimes). Next, using the same action, we calculate semiclassical non-perturbative quantities, which we expect will be dominated by Euclidean instanton solutions. We derive the entropy/area ratio for cosmological and black hole horizons, finding agreement with GR for solutions in backgrounds of infinite volume, but disagreement for backgrounds with finite volume. In deriving the above results, the path integral is taken over histories with fixed 4-volume. We point out that the results are different if we allow the 4-volume of the different histories to vary over a continuum range. In this "generalized" version of unimodular gravity, one recovers the full set of Einstein's equations in the classical limit, including the trace, so $\Lambda$ is no longer an integration constant. Finally, we consider the generally covariant theory due to Henneaux and Teitelboim, which is classically equivalent to unimodular gravity. In this case, the standard semiclassical GR results are recovered provided that the boundary term in the Euclidean action is chosen appropriately.