38 Appendix to Chapter 1
z z (1.28)In the absence of any indication to the contrary in our everyday experience, we fur-
ther assume thatt t (1.29)The set of Eqs. (1.26) to (1.29) is known as the Galilean transformation.
To convert velocity components measured in the Sframe to their equivalents in the
S frame according to the Galilean transformation, we simply differentiate x , y , and
z with respect to time: (^) xx (1.30)
(^) yy (1.31)
(^) zz (1.32)
Although the Galilean transformation and the corresponding velocity transfor-
mation seem straightforward enough, they violate both of the postulates of special
relativity. The first postulate calls for the same equations of physics in both the S
and S inertial frames, but the equations of electricity and magnetism become very
different when the Galilean transformation is used to convert quantities measured
in one frame into their equivalents in the other. The second postulate calls for the
same value of the speed of light cwhether determined in Sor S. If we measure the
speed of light in the xdirection in the Ssystem to be c, however, in the S system
it will be
c c
according to Eq. (1.30). Clearly a different transformation is required if the postulates
of special relativity are to be satisfied. We would expect both time dilation and length
contraction to follow naturally from this new transformation.
Lorentz Transformation
A reasonable guess about the nature of the correct relationship between xand x is
x k(xt) (1.33)
Here kis a factor that does not depend upon either xor tbut may be a function of .
The choice of Eq. (1.33) follows from several considerations:
1 It is linear in xand x , so that a single event in frame Scorresponds to a single event
in frame S , as it must.
2 It is simple, and a simple solution to a problem should always be explored first.
3 It has the possibility of reducing to Eq. (1.26), which we know to be correct in
ordinary mechanics.
dz
dt
dy
dt
dx
dt
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