Special Theory of Relativity
17
In terms of the velocity u of the particle,
0
1/ 2
2
2
0
0
1/ 2
2
2
1
1
=


−




=


−




m
u
c
m c
p
u
c
u
p
(1.52)
Since ∆τ is an invariant scalar, it follows that p µ = (p, p 0 ) is a 4-vector and
its transformation relations are
0
1/ 2
2
2
( )
1
−
′ =


−




x
x
v
p
p
c
p
a
v
c
;
( )
y
y
z
z
p
p p
p b
′
′
=
=
(1.53)
0
0
1/ 2
2
2
( )
1
−
′ =


−




x
v
p
p
c
p
c
v
c
The vector p is the relativistic generalization of the Newtonian momentum
vector m 0 u. For the interpretation of p 0 , it is noted that for u << c,
4
2
2
0
0
0
0 2
1
3
...
2
8
u
cp m c
m u
m c
=
+
+
+
(1.54)
where the second term is the Newtonian kinetic energy. Therefore cp 0 may be
defined as the energy of the particle, m 0 c
2
being the rest energy and the remaining
terms being the relativistic generalization of the Newtonian kinetic energy. If T
denotes the kinetic energy, then
T = cp 0 – m 0 c
2
(1.55)
The rest energy does not play a significant role if there is no change of mass
in a process, but becomes important if there is a change of mass. Finally, it is
noted that the scalar product p.p is
p 0
2
– p.p = m 0
2
c
2
(1.56)
an invariant scalar as expected.