Unimodular Einstein--Cartan gravity: Dynamics and conservation laws

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.

8 pages, 1 Appendix, matches published version

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• A clarification on prevailing misconceptions in unimodular gravity
• Is there any symmetry left in gravity theories with explicit Lorentz violation?
• Energy nonconservation and relativistic trajectories: Unimodular gravity and beyond
• On the Mathematics of Coframe Formalism and Einstein-Cartan Theory -- A Brief Review
• Spontaneous and Explicit Spacetime Symmetry Breaking in Einstein-Cartan Theory with Background Fields
• Linear and Second-order Geometry Perturbations on Spacetimes with Torsion
• A note on unimodular $N=1,d=4$ AdS supergravity
• Off-shell Noether currents and potentials for first-order general relativity
• Unimodular gravity as an initial value problem
• Revisiting cosmological diffusion models in Unimodular Gravity and the $H_0$ tension