Einstein gravity from the Einstein action: Counterterms and covariance

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 we 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 noncovariant connection coefficients.