On a 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 gsub(ij) and the electromagnetic potential phisub(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 (J. Math. Phys.; 17: 1980 (1976)) by quite different means. (author)