On theories of gravitation with higher-order field equations

Abstract

Weyl and Eddington suggested three alternative general relativistic theories of gravitation with fourth-order field equations which in empty space admit the Schwarzschild metric as a solution. These theories, Like Einstein's, follow from a variational principle and thus imply differential identities. If, as in Einstein's theory, the sources are taken to be proportional to the energy-momentum tensorT μν, these identities imply the vanishing of the covariant divergence ofT μv. It is shown here that in the presence of extended sources, Weyl's and Eddington's theories (as well as all other higher-order metric theories derivable from an action principle) contradict Newton's law of gravitation in the nonrelativistic limit. To entail this law would require a modification of the source term of the field equations which in general is not compatible withT μv ;v alternatively, one could require only asymptotic agreement with Newton's law, which is compatible with supplementary higher-order terms in Einstein's equations, but which requires the introduction of universal constants of the dimensions of length. None of the generalizations of Einstein's equations considered here admits Birkhoff's theorem.