LAGRANGIAN FOR NON-MINIMALLY COUPLED GRAVITATIONAL AND ELECTROMAGNETIC-FIELDS

An arbitrarily chosen Lagrangian L for non-minimally coupled gravitational and electromagnetic fields will usually lead to higher-order field equations, in the sense that the functional derivatives of L with respect to the gravitational potential gij and the electromagnetic potential phi i will involve at least the third, instead of merely the second, derivatives of these quantities. By temporarily contemplating a five-dimensional formalism this paper uncovers an exceptional case in which one is led to second-order equations. The result obtained is in agreement with the conclusions reached by Horndeski (1976) by quite different means.