Vector-tensor field theories and the Einstein-Maxwell field equations

Abstract

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of the metric tensor g ij and its first two derivatives together with the first derivative of a vector field ψ i are investigated. In general, the Euler-Lagrange equations obtained by variation of g ij are of fourth order in g ij and third order in ψ i . It is shown that in a four dimensional space the only Euler-Lagrange equations which are of second order in g ij and first order in ψ i are the Einstein field equations (with an energy-momentum term). Various conditions are obtained under which the Einstein-Maxwell field equations are then an inevitable consequence.