On theories of gravitation with higherorder field equations.
Abstract
Weyl and Eddington suggested three alternative general relativistic theories of gravitation with fourthorder 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 energymomentum 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 higherorder 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 higherorder 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.
 Publication:

General Relativity and Gravitation
 Pub Date:
 August 1977
 DOI:
 10.1007/BF00756315
 Bibcode:
 1977GReGr...8..631H
 Keywords:

 Cosmology;
 Field Theory (Physics);
 Gravitation Theory;
 Relativity;
 Conservation Laws;
 Gravitational Constant;
 Green'S Functions;
 Laplace Equation;
 Newton Theory;
 Astrophysics