Connections and geodesics in the space of metrics

Abstract

We argue that the exponential relation gµν =ḡµρ e h ρ ν 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 VilkoviskyDeWitt 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.

Gauges and functional measures in quantum gravity II: higher-derivative gravity

“…The divergences also agree, in d = 4 and in the EH limit, with earlier calculations in general gauges [29,[45][46][47][48]. For further discussions on the use of the exponential parametrization see [27,[49][50][51][52][53][54].…”

The unitary conformal field theory behind 2D Asymptotic Safety

“…Only then it is ensured that all metrics parametrized this way satisfy the signature and nondegeneracy constraint. It can be shown that there is a canonical connection on F adjusted to this constraint, and that the resulting geodesics are parametrized precisely by the exponential relation g µν =ḡ µρ (e h ) ρ ν with a symmetric tensor field h µν [59]. This explains the special status of the exponential parametrization.…”

“…Since the set of nondegenerate metrics with fixed signature forms a nonempty open subset in the space of all covariant symmetric 2-tensor fields [59], there is no a priori reason to expect that it has vanishing measure, and so this question has no obvious answer. It is known, however, that "sufficiently different" choices will lead to inequivalent theories [79].…”

“…In ref. [59] it was argued that the exponential parametrization is adapted to the nonlinear structure of the field space F of all pure metrics, i.e. objects in F are nondegenerate and have a prescribed signature.…”

“…It depends, however, on the parametrization of the metric. In the linear parametrization, g µν =ḡ µν + h µν , it is given by [1, 8, 22, 24-26, 44, 45, 48, 49] 34) while the exponential parametrization [59], g µν =ḡ µρ (e h ) ρ ν , leads to [22,26,[50][51][52][53][54][55][56][57][58] …”