A Theorem Relating Solutions of a Fourth-Order Theory of Gravity to General Relativity

Abstract

Within a fourth-order theory of gravity we give,for a static asymptotically flat spacetime, anexpression of the active mass (gravitational mass), infirst order in the coupling constant, α, of the curvature squared term in the Lagrangiandensity, a generalization of the Tolman expression forthe energy, which establishes a relationship between theactive mass and the source structure in a static spacetime. Within this approximation, we canprove that the fourth-order theory shares with Generalrelativity (GR) the property that, for sources ofcompact support, the active mass is independent of any two-dimensional surface which encloses thesupport of the matter distribution. Finally, we provethat only for conformally invariant sources thefourth-order theory and GR share the same static andasymptotically flat solutions.