Non-metric gravity: I. Field equations

Abstract

We describe and study a certain class of modified gravity theories. Our starting point is the Plebański formulation of gravity in terms of a triple Bi of two-forms, a connection Ai and a ‘Lagrange multiplier’ field Ψij. The generalization we consider stems from the presence in the action of an extra term proportional to a scalar function of Ψij. As in the usual Plebański general relativity (GR) case, a certain metric can be constructed from Bi. However, unlike in GR, the connection Ai no longer coincides with the self-dual part of the metric compatible spin connection. Field equations of the theory are shown to be relations between derivatives of the metric and components of field Ψij, as well as its derivatives, the later being in contrast to the GR case. The equations are of second order in derivatives. An analog of the Bianchi identity is still present in the theory, as well as its contracted version tantamount to the energy conservation equation.