The geometrical meaning of time

It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type η _μν = diag(1,-1,-1,-1) this is usually presented as an independent axiom of the theory, which can not 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. In this work it is 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.