The Geometrical Meaning of Time in the Presence of Matter

It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type g_μv = diag (1, -1, -1, -1) this is usually presented as an independent axiom of the theory, which cannot be deduced from other assumptions. The meaning of this assertion is that a specific coordinate (the temporal coordinate) is given a unique significance with respect to the other spatial coordinates. It was shown that the above assertion is a consequence of requirement that the metric of empty space should be linearly stable and need not be assumed. In this work we remove the empty space assumption and investigate the consequences of matter on the stability of Lorentzian space-time. It will be shown that in the presence of matter an upper limit scale to the size of a locally Lorentzian universe exists which incidentally is about the size of the observable universe.