Unimodular Einstein--Cartan gravity: Dynamics and conservation laws

TL;DR

This paper develops a first-order formulation of unimodular gravity incorporating torsion and spin density, deriving field equations and conservation laws, and compares results with Einstein-Cartan theory, highlighting how the cosmological constant emerges as an integration constant.

Contribution

It introduces a novel unimodular gravity framework with torsion and spin, deriving new field equations and conservation laws, and analyzing the role of matter spin in this context.

Findings

Torsion is algebraically related to spin density, similar to Einstein-Cartan theory.

The vierbein field equation remains traceless in unimodular gravity.

Massless Dirac spinors are used as a specific example to illustrate the theory.

Abstract

Unimodular gravity is an interesting approach to address the cosmological constant problem, since the vacuum energy density of quantum fields does not gravitate in this framework, and the cosmological constant appears as an integration constant. These features arise as a consequence of considering a constrained volume element 4-form that breaks the diffeomorphisms invariance down to volume preserving diffeomorphisms. In this work, the first-order formulation of unimodular gravity is presented by considering the spin density of matter fields as a source of spacetime torsion. Even though the most general matter Lagrangian allowed by the symmetries is considered, dynamical restrictions arise on their functional dependence. The field equations are obtained and the conservation laws associated with the symmetries are derived. It is found that, analogous to torsion-free unimodular gravity, the field equation for the vierbein is traceless; nevertheless, torsion is algebraically related to the spin density as in standard Einstein-Cartan theory. The particular example of massless Dirac spinors is studied, and comparisons with standard Einstein-Cartan theory are shown.