  We propose an extension of General Relativity with two different metrics. To
each metric we define a Levi-Cevita connection and a curvature tensor. We then
consider two types of fields, each of which moves according to one of the
metrics and its connection. To obtain the field equations for the second metric
we impose an exchange symmetry on the action. As a consequence of this ansatz,
additional source terms for Einstein's field equations are generated. We
discuss the properties of these additional fields, and consider the examples of
the Schwarzschild solution, and the Friedmann-Robertson-Walker metric.
