bei48482_FM

The Lorentz Transformation 37

Appendix I to Chapter 1

The Lorentz Transformation

S

uppose we are in an inertial frame of reference Sand find the coordinates of

some event that occurs at the time tare x, y, z. An observer located in a dif-

ferent inertial frame S which is moving with respect to Sat the constant ve-

locity vwill find that the same event occurs at the time t and has the coordinates x ,

y , z. (In order to simplify our work, we shall assume that vis in the xdirection,

as in Fig. 1.22.) How are the measurements x, y, z, trelated to x , y , z , t?

Galilean Transformation

Before special relativity, transforming measurements from one inertial system to an-

other seemed obvious. If clocks in both systems are started when the origins of Sand

S coincide, measurements in the xdirection made is Swill be greater than those made

in S by the amount t, which is the distance S has moved in the xdirection. That is,

x   xt (1.26)

There is no relative motion in the yand zdirections, and so

y   y (1.27)

S

y

z

x

S′

x′

z′

y′

v

Figure 1.22Frame S moves in the xdirection with the speed relative to frame S. The Lorentz

transformation must be used to convert measurements made in one of these frames to their equivalents

in the other.

bei48482_ch01.qxd 1/15/02 1:21 AM Page 37