Connections and geodesics in the space of metrics

Understand

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.