Minkowski space and relativistic units
First, we consider relativistic units in which c is a dimensionless constant and takes the value 1, ie assume a photon at the speed of light, for him. It is holds

c = dv/dt = (dx2 + dy2 + dz2) /dt2 = 1 => -dt2 + dx2 + dy2 + dz2 = 0

This gives us the idea consider the space-time as a semi-riemannian variety fouth-dimension. In this variety is defined as the inner product

v1 = (t, x, y, z) v2 = ( t', x', y', z)

= (-tt', xx', yy', zz')

vNote that the inner product is not positive definite (hence the prefix "semi"). Thus we have the space-time is a variety semi-riemannian with metric

ds2 = -dt2 + dx2 + dy2 + dz2

Note also that the norm generated by this space in the form

|| v || = (<v, v>)1/2

It is not a true normbut we look the other way and talk here scalar product norm.

In Minkowski space we define some things for special relativity theory, like

x, Vector temporary or timelike => <x, x> < 0

x, Vector spatial or spacelike => <x, x> > 0

x, Genus ligh or zero vector => <x, x> = 0

A referral system acceptable S = {e0, e1, e2, e3} in Minkowski space if it holds that
1) e0 es un vector temporal y e1, e2, e3 are spatial
2) All have norm1 (Minkowski!).
3)Are orthogonal to each other (with the inner product of Minkowski!).

x,Future vector is a vector such as <x, e0> < 0 and is said to have past if<x, e0> > 0