Mathematics of Physics and Engineering

The Lorentz Transformation 99

2.4.2 The Lorentz Transformation and Special Relativity

Consider two frames, the fixed frame O and the moving frame 0. We
choose cartesian coordinate systems in both frames and assume that, at
each time moment, the frame 0\ is a parallel translation of the frame O
so that the vectors i, i\ are always on the same line; see Figure 2.4.1. As
usual, we represent the points in the frames O and 0\ with vectors of the
form xi + yj + zk and x\ %\ + y\ j 1 + z\ k\, respectively.

3\

O l / Ox ^l

R, ki

Fig. 2.4.1 Fixed and Moving Frames

Thus, 0\ is moving to the right relative to O with constant velocity v.
A linear transformation of the space-time coordinates is

x = axi+0ti, y = 3/1, z = zi (2.4.1)

and

t = 7H + ati (2.4.2)

for some real numbers a, (3, ~f,a. Equation (2.4.2) means that time is no
more absolute than space: each frame has its own time coordinate as well
as space coordinates. Equations (2.4.1) and (2.4.2) also demonstrate that
space and time are bound together in one space-time continuum. We will
see that Postulate (i) cannot hold without relation (2.4.2) with appropriate
7 and a.
Consider point 0\, the origin of the moving frame. It has the f i coordi-
nate x\ = 0 in the 0\ frame at all times t. In the fixed O frame, by (2.4.1),
the i coordinate of 0\ is x = @t\; by (2.4.2), we also have t = at. Note
that /3 > 0. If v = ||u|| is the speed of 0\ relative to O, then v = x/t = f3/a.
A light impulse emitted at the instant when O = 0\ will reach, at time t,
a point P, where the coordinates (x, y, z) of P in O satisfy

x^2 +y^2 + z^2 = c^2 t^2. (2.4.3)

By Postulate (i), if (xi,yi,zi) are the coordinates of the same point P in