Gravitational theories with nonzero divergence of the energy-momentum tensor

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Assuming that the divergence of the energy-momentum tensor is nonzero leads to a class of theories with consistent field equations and gauge conditions as well as compatibility with the Newtonian limit of the conservation laws. We have used both the Einstein and the Brans-Dicke theories as models, but the extension to other viable theories such as vector-metric and two-metric theories is possible. One particularly interesting theory emerges that agrees with the ordinary Brans-Dicke theory except for the post-Newtonian parameter ${\ensuremath{\zeta}}_{2}$, which predicts nonconservation of total momentum. Unfortunately no accurate experimental limits for this parameter are known. It thus remains for future experiments in lunar-laser ranging to test this theory.