Special Theory of Relativity
13
and
2
1/ 2,
2
2
1
v
dt
dx
c
dt
v
c
−
′ =


−




(1.36)
Dividing the position intervals by the time interval,
2
1
x
x
x
u v
u
vu
c
−
=
′
−
1/ 2
2
2
2
1
1
y
y
x
v
u
c
u
vu
c


−




=
′
−
(1.37)
1/ 2
2
2
2
1
1
z
z
x
v
u
c
u
vu
c


−




=
′
−
where the subscript describes the components of the velocities. These
expressions relate the velocities of a particle measured in different inertial frames.
As an application of the above formulae, consider the relation between u and
′
u . It follows from Eq. (1.37) that
2
2
2
2
2
2
2
2
2
(
)(
)
1
−
−
′ − =


−




c v u c
u
c
vux
c
c
(1.38)
For v
2
< c
2
, which is required for the Lorentz transformations to be physically
meaningful, the following important results are obtained:
if
u c
u c
′ <
<
(1.39)
if
u c
u c
′ =
=
(1.40)
if
u c
u c
′ >
>
(1.41)
The first result implies that the relativistic addition of velocities with the
speed of each being less than c will again give a velocity with speed less than c.