Elements of Modern Physics
8
These are the celebrated Lorentz equations. In obtaining them, the
positive root has been chosen, so that v = 0 implies x′ = x, t′ = t. It is also noted
that for small
v
c
and
,
<<
x ct the Galilean transformations x′ = x – vt and
t′ = t are recovered. The transformations in Eq. (1.20) can be inverted to express
x, y, z and t in terms of x′, y′, z′ and t′
1/ 2
2
2
;
;
1
x vt
x
y y z z
v
c
′
′
+
′
′
=
=
=


−




(1.21)
2
1/ 2
2
2
1
v
t
x
c
t
v
c
′
′
+
=


−




From these equations, it is seen that frame F moves with velocity –v with
respect to F′ so that the relative velocities of the frames are reciprocal.
It should be noted that the deviations of the Lorentz transformations from
the Galilean transformations are second order in or
v
x
c
ct
and hence the
experiments which can test Lorentz transformations must be accurate enough
to detect these second order terms. The Michelson-Morley experiment did have
such an accuracy and could prove the inadequacy of Galilean transformations.
Lorentz transformations, though they differ only slightly from Galilean
transformations in most physical situations, bring in a profoundly new concept in
the kinematics of the universe. They remove the universal character of time
and treat it on the same footing as space coordinates. They require that physical
space be treated as a 4-dimensional space of space and time coordinates. As
might be expected, this mixing of space and time coordinates leads to some
unfamiliar consequences. A few of them are discussed here.
1.6 SIMULTANEITY AND TIME DILATION
It follows from Lorentz relations (1.20) that events which are simultaneous in
frame F but take place at different positions are not simultaneous in frame F′.
For example, if two events take place in frame F at
t 1 = t 2 = 0
(1.22)