Gravitational Energy in Quadratic Curvature Gravities

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Abstract
We define and compute the energy of gravitational systems involving terms quadratic in curvature. While our construction parallels that of Einstein gravity, there are significant differences both conceptually and concretely. In particular, for D=4, all purely quadratic models admit (zero energy) vacua of arbitrary constant curvature. The energy of all quadratic models, including conformal Weyl gravity, necessarily vanishes in asymptotically flat spaces. Instead, in cosmological backgrounds, the energy expressions are proportional to the usual (AD) form of cosmological Einstein gravity, and therefore to the mass parameter in the corresponding asymptotic Schwarzschild-(Anti) deSitter geometry. Combined Einstein-quadratic curvature systems reflect the above results: Absent a cosmological constant term, the only vacuum is flat space, with (ADM) Einstein energy. With an explicit Lambda term, the energy is just the sum of the separate AD contributions. We also discuss higher curvature invariants and D > 4 spaces, where obstacles to a useful energy arise.

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