Standard treatment [of relativity]

In a standard treatment of relativity theory one way to begin is to consider setting up a square grid of points in space and time—and then to ask what kind of transformed grid corresponds to this same set of points if one is moving at some velocity v. At first one might assume that the answer would just be a grid that has been sheared by the simple transformation {t, x} -> {t, x- v t}, as in the first row of pictures below. And indeed for purposes of Newtonian mechanics this so-called Galilean transformation is exactly what is needed. But as the pictures below illustrate, it implies that light cones tip as v increases, so that the apparent speed of light changes, and for example Maxwell's equations must change their form. But the key point is that with an appropriate transformation that affects both space and time, the speed of light can be left the same. The necessary transformation is the so-called Lorentz transformation

{t, x} -> {t - v x/c^{2}, x - v t}/Sqrt[1-v^{2}/c^{2}]

And from this the time dilation factor 1/Sqrt[1-v^{2}/c^{2}] shown on page 524 follows, as well as the length contraction factor Sqrt[1-v^{2}/c^{2}]. An important feature of the Lorentz transformation is that it preserves the quantity c^{2} t^{2} - x^{2}—with the result that as v changes in the pictures below a given point in the grid traces out a hyperbola whose asymptotes lie on a light cone. Note that on a light cone c^{2} t^{2}-x^{2} always vanishes. Note also that the intersection of the past and future light cones for two events separated by a distance x in space and t in time always has a volume proportional exactly to c^{2} t^{2} - x^{2}.