It is obvious that there is a difference between time and space. In space we can go either left or right, up and down. But as much as one wants to he cannot visit yesterday. General relativity teaches us that space-time is a four dimensional manifold. If so, why there is such a profound difference between one particular dimension of this manifold and another? Mathematically speaking one of the main differences between time and space is encapsulated in the flat metric of our space-time which is  locally of the Lorentzian type ημν =  diag  (1,-1,-1,-1). But this is an empirical fact or a mathematical postulate thus unexplained. One can imagine also other flat metrics such as the Euclidian metric:  diag (+1,+1,+1,+1). Naturally an Euclidian metric does not discriminate between dimensions hence one can visit yesterday or tomorrow as easy as he can go left or right. There is also no restriction on the speed of any moving body as the speed of light restricts the speed of propagation only in the presence of a Lorentzian metric. Why is our space-time Lorentzian and not Euclidean? The answer is that only the Lorentzian metric is stable [1] for an (almost) empty space. If space-time is not empty some generic solutions of Einstein's equations have different metric signatures in different regions of space-time. The main points of this talk are as follows: the possible constant metrics which are not equivalent to each other by trivial manipulations are listed. It is shown that among possible metrics only Lorentz metrics are linearly stable. We show that generic solutions of the Einstein equations in the presence of matter change signature. We discuss some possible implications of our results, in particular the conditions under which a particle can travel in speeds exceeding the speed of light. Finally some possible physical implications are suggested pertaining to the horizon problem and the problems of dark energy and dark matter.

[1] Asher Yahalom "The Geometrical Meaning of Time" [“The Linear Stability of Lorentzian Space-Time” Los-Alamos Archives - gr-qc/0602034, gr-qc/0611124] Foundations of Physics http://dx.doi.org/10.1007/s10701-008-9215-3 Volume 38, Number 6, Pages 489-497 (June 2008).

[2] Asher Yahalom “On the Difference between Time and Space” Cosmology 2014, Vol. 18. 466-483. Cosmology.com.