Zero-Energy Theorem for Scale-Invariant Gravity

The general scale-invariant Lagrangian, $-\frac{1}{4}\int d^{4}x{\mathrm{}(-g)\mathrm{}}^{\frac{1}{2}}[\alpha c^{\mu \nu \lambda \sigma }{c}_{\mu \nu \lambda \sigma }+\beta R^2]$, with $\alpha \beta >~0$, is considered. Although this theory admits linearized solutions with negative energy, it is shown that all exact solutions representing isolated systems have precisely zero energy. This result also holds in the presence of arbitrary matter. It can be understood as resulting from a confinement of energy. The implications of this result for quantum gravity are discussed.