Metric symmetry by design in general relativity

TL;DR

This paper explores the implications of not assuming metric symmetry in general relativity, revealing a new antisymmetric term that could impact cosmology, gravitational energy, and quantum gravity theories.

Contribution

It demonstrates that relaxing the symmetry assumption in the Einstein-Hilbert action introduces an additional antisymmetric term, offering new perspectives on gravitational theory and quantum gravity.

Findings

Restoring Einstein's equations introduces an antisymmetric multiplier term.

The antisymmetric term may relate to cosmological angular momentum and spin currents.

Implications for energy-momentum pseudotensors and quantum gravity are discussed.

Abstract

The usual derivation of Einstein's field equations from the Einstein--Hilbert action is performed by silently assuming the metric tensor's symmetric character. If this symmetry is not assumed, the result is a new theory, such as Einstein's attempted Unified Field Theory or Moffat's Nonsymmetric Gravitational Theory. Explicitly enforcing the constraint by means of a Lagrange-multiplier term restores Einstein's field equations, but the multiplier appears as an additional, unconstrained antisymmetric term. We briefly discuss the possible significance of this term with respect to a nonvanishing cosmological angular momentum, a sourced spin current, the nonsymmetric nature of the Einstein pseudotensor characterizing the energy-momentum of the gravitational field, and possible implications on attempts to obtain a quantum theory of gravity.