Einstein gravity from the Einstein action: Counterterms and covariance

TL;DR

This paper demonstrates how to derive Einstein gravity from the Einstein action with a new counterterm that ensures covariance and removes divergences, offering insights into gravity's fundamental description.

Contribution

It introduces a novel counterterm for the Einstein action that maintains covariance and simplifies the derivation of Einstein's equations from a quadratic connection-based action.

Findings

A new counterterm removes divergences in the Einstein action.

Covariance is achieved through simultaneous transformation of spacetime and reference tetrads.

Different perspectives on gravity emerge from the use of reference tetrads.

Abstract

The field equations of general relativity can be derived from the Einstein action, which is quadratic in connection coefficients, rather than the standard action involving the Gibbons-Hawking-York term and counterterm. We show that it is possible to construct a new counterterm directly for the Einstein action, which removes divergences and naturally introduces a flat reference spacetime. The total action is then covariant under simultaneous transformation of both the spacetime and reference tetrads, and argue that this is analogous to the Gibbons-Hawking action. We then explore different perspectives arising naturally from different uses of the reference tetrad, and explore implications of viewing gravity as fundamentally described in terms of non-covariant connection coefficients.