A gravitational theory involving a vector field $\chi^{\mu}$, whose zero
component has the properties of a dynamical time, is studied. The variation of
the action with respect to $\chi^{\mu}$ gives the covariant conservation of an
energy momentum tensor $T^{\mu \nu}_{(\chi)}$. Studying the theory in a
background which has killing vectors and killing tensors we find appropriate
shift symmetries of the field $\chi^{\mu}$ which lead to conservation laws. The
energy momentum that is the source of gravity $T^{\mu \nu}_{(G)}$ is different
but related to $T^{\mu \nu}_{(\chi)}$ and the covariant conservation of $T^{\mu \nu}_{(G)}$ determines in general the vector field $\chi^{\mu}$. When $T^{\mu \nu}_{(\chi)}$ 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 $T^{\mu \nu}_{(G)}$, solutions for
$\chi^{\mu}$ are found. 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 spacetime since although $T^{\mu \nu}_{(\chi)} \neq 0$, $T^{\mu
\nu}_{(G)}= 0$, as a model for the early universe, this solution is also free
of the horizon problem. There may be also an application to the "time problem"
of quantum cosmology.Comment: 21 pages, discussions extended, some more explicit proofs included,
more references include