## Vtv Vtmtmv64

which gives a property of the rotation or transformation matrix MTM = 1. The transpose of a matrix interchanges the rows and columns by Mj ^ Mji. It is then possible to use this information to find the entries of the matrix M.

The unchanged length of a vector under this transformation, called its invariance, means that the components of V and V' are ct, x, ,y, z and ct', x', y', z' respectively with

-(ct')2 + (x')2 + (y')2 + (z')2 = -(ct)2 + x2 + y2 + z2. (6.5)

This problem is simplified by considering the transformation as a velocity in the x direction. This means that y' = y and z' = z. As a trial solution consider x' = y(x — vt) and x = y'(x' + vt'). The factors 7, y' are to be determined. By reversing the roles of x and x' it is evident that y = y'. For a particle or photon moving with x = ct then x' = y(c — v)t. Further, since the speed of light is regarded as an invariant x = ct implies x' = ct'. This leaves the two equations, ct' = Yt(c — v), ct = Yt'(c + v).

At this point it is easy to see that by multiplying these two together and dividing out the tt' term, we obtain

which is the famous Lorentz-FitzGerald contraction factor. This was derived by Lorentz and FitzGerald to contract the length of a body moving in the aether to cancel out the influence of the aether. This leaves the following Lorentz transformations t' = y(t — vx/c), x' = y(x — vt), y' The transformation matrix then has the form