Modified Brans-Dicke gravitational theory with nonzero divergence of the energy-momentum tensor

The Brans-Dicke theory of gravitation is modified by assuming that the divergence of the energy-momentum tensor is proportional to the covariant derivative of the scalar curvature. No ad hoc additions to the usual Brans-Dicke field equations are required as in Rastall's modification of the Einstein theory or as in the steady-state theories, of which this is a natural possibility. Three parameters emerge from the theory: the unnormalized gravitational constant $\mathcal{G}$, the usual Brans-Dicke parameter $\omega$, and the proportionality constant $\sigma$. In the post-Newtonian approximation, these parameters can be fixed by experiment. However, there exists a certain choice of the parameters for which the theory reduces to an Einstein theory with constant scalar curvature.