Gravitational Energy in Quadratic Curvature Gravities

TL;DR

This paper develops a framework for defining and computing gravitational energy in quadratic curvature gravity theories, revealing how energies relate to known solutions and how they differ from Einstein gravity, especially in various dimensions and with cosmological constants.

Contribution

It introduces a method to define gravitational energy in quadratic curvature gravities and analyzes their properties across different dimensions and background curvatures.

Findings

Pure quadratic models in 4D have vacua of arbitrary constant curvature.

Energies in these models vanish in asymptotically flat spaces.

Energy expressions are proportional to the Abbott-Deser energy in constant curvature backgrounds.

Abstract

We define the notion of energy, and compute its values, for gravitational systems involving terms quadratic in curvature. While our construction parallels that of ordinary Einstein gravity, there are significant differences both conceptually and concretely. In particular, for D=4, all purely quadratic models admit vacua of arbitrary constant curvature. Their energies, including that of conformal (Weyl) gravity, necessarily vanish in asymptotically flat spaces. Instead, they are proportional to that of the Abbott-Deser (AD) energy expression in constant curvature backgrounds and therefore also proportional to the mass parameter in the corresponding Schwarzschild-(Anti) de Sitter geometries. Combined Einstein-quadratic curvature systems reflect the above results: Absent a cosmological constant term, the only vacuum is flat space, with the usual (ADM) energy and no explicit contributions from the quadratic parts. If there is a Lambda term, then the vacuum is also unique with that Lambda value, and the energy is just the sum of the separate contributions from Einstein and quadratic parts to the AD expression . Finally, we discuss the effects on energy definition of both higher curvature terms and higher dimension.