Connections and geodesics in the space of metrics

TL;DR

This paper advocates for the exponential parametrization of metrics in the space of metrics, deriving the associated connection, and discusses its implications for quantum gravity, highlighting differences between Euclidean and Lorentzian signatures.

Contribution

It introduces the exponential metric parametrization as the natural choice for geodesics in the space of metrics and explores its geometric and quantum gravity implications.

Findings

Exponential parametrization describes geodesics naturally in the space of metrics.

The derived connection relates to Levi-Civita and Vilkovisky-DeWitt connections.

Euclidean metrics can all be connected by geodesics, unlike Lorentzian metrics.

Abstract

We argue that the exponential relation $g_{\mu\nu} = \bar{g}_{\mu\rho}\big(\mathrm{e}^h\big)^\rho{}_\nu$ is the most natural metric parametrization since it describes geodesics that follow from the basic structure of the space of metrics. The corresponding connection is derived, and its relation to the Levi-Civita connection and the Vilkovisky-DeWitt connection is discussed. We address the impact of this geometric formalism on quantum gravity applications. In particular, the exponential parametrization is appropriate for constructing covariant quantities like a reparametrization invariant effective action in a straightforward way. Furthermore, we reveal an important difference between Euclidean and Lorentzian signatures: Based on the derived connection, any two Euclidean metrics can be connected by a geodesic, while this does not hold for the Lorentzian case.