Gravitational Energy in Quadratic-Curvature Gravities

We define energy ($E$) and compute its values for gravitational systems involving terms quadratic in curvature. There are significant differences, both conceptually and concretely, from Einstein theory. For $D=4$, all purely quadratic models admit constant curvature vacua with arbitrary $\Lambda$, and $E$ is the “cosmological” Abbott-Deser (AD) expression; instead, $E$ always vanishes in flat, $\Lambda =0$, background. For combined Einstein-quadratic curvature systems without explicit $\Lambda$-term vacuum must be flat space, and $E$ has the usual Arnowitt-Deser-Misner form. A $\Lambda$-term forces unique de Sitter vacuum, with $E$ the sum of contributions from Einstein and quadratic parts to the AD form. We also discuss the effects on energy definition of higher curvature terms and of higher dimension.