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.