The Cauchy problem for the <i>R</i>+<i>R</i>2 theories of gravity without torsion

Abstract

The exterior Cauchy problem is discussed for the fourth-order theories of gravity derived from the Lagrangian densities ℒ=(−g)1/2 (R+ (1/2)aR2+bRμν Rμν) −κℒm. When b≠0, the Cauchy problem can be solved by the standard method already used in general relativity. When b=0, the problem cannot be formulated as in the case where b≠0, since the corresponding fourth-order theory is shown to be equivalent to a second-order scalar–tensor theory. This scalar–tensor theory is proved to coincide with one of the models of gravity proposed by O’Hanlon in order to present a covariant version of the massive dilaton theory suggested by Fujii. This result is generalized: The models of O’Hanlon are shown to be indistinguishable from the fourth-order theories derived from the Lagrangian densities ℒ=(−g)1/2 F(R)−κℒm, where F is any real function such that F″(R) does not identically vanish.