Gravitational theory with a dynamical space time

A gravitational theory involving a vector field Xμ which has the properties of a dynam-ical space-time, is studied. The variation of the action with respect to Xμ gives the covariant conservation of an energy-momentum tensor T ^μV(X). Studying the theory in a background that has Killing vectors and Killing tensors, we find appropriate shift symmetries of the field Xμ which lead to conservation laws. The energy-momentum that is the source of gravity T ^μV (G) is dierent but related to T ^μV (X) and the covariant conservation of T ^μV (G) determines in general the vector field Xμ. When T^μV (X) is chosen to be proportional to the metric, the theory coincides with the Two Measures Theory, which has been studied before in relation to the Cosmological Constant Problem. When the matter model consists of point particles, or strings, the form of T ^μV (G), solutions for Xμ are found. In a locally inertial frame, the vector field corresponds to the locally at coordinates. For the case of a string gas cosmology, we find that the Milne universe can be a solution, where the gas of strings does not curve the space-time since although T ^μV (X) 6= 0, T ^μV (G) = 0, as a model for the early universe, this solution is also free of the horizon problem. There may also be an application to the "time problem" of quantum cosmology.