A theorem relating solutions of a fourth-order theory of gravity to general relativity

Within a fourth-order theory of gravity we give, for a static asymptotically flat spacetime, an expression of the active mass (gravitational mass), in first order in the coupling constant, ct, of the curvature squared term in the Lagrangian density, a generalization of the Tolman expression for the energy, which establishes a relationship between the active mass and the source structure in a static spacetime. Within this approximation, we can prove that the fourth-order theory shares with General relativity (GR) the property that, for sources of compact support, the active mass is independent of any two-dimensional surface which encloses the support of the matter distribution. Finally, we prove that only for conformally invariant sources the fourth-order theory and GR share the same static and asymptotically flat solutions.