We must express Newton's laws of motion in terms of four- vectors so that they are frame invariant and consistent with Special Relativity. The four- velocity of a particle is the tangent to its worldline of length c [ see Figure 2.1 ]:
Figure 2.1: The worldline of a particle with four- velocity 
.
This is the most natural analogue of the three- velocity. It is clearly
a four- vector since both 
and 
are invariant.
In the particle's own rest frame 
, the four- velocity is
It follows that in a general frame 
:
or
where 
is the particle's three- velocity.
For low velocities 
, 
and the spatial part is
nearly the same as .
If an observers own four- velocity is written as
in we have
We can then write the particle's three- velocity as [Assignment 2]:
This is a spacelike vector expressed in coordinate- independent
form [ although 
singles out a particular observer ].
For photons the four- velocity is not defined since 
, i.e.
there is no frame in which a photon is at rest.
The four- momentum is defined by
where 
is the rest mass of the particle i.e. the mass
in its own rest frame.
The spatial part of 
[ the three- momentum ] is
so the apparent mass exceeds the rest mass.
The time part of the four- momentum is the energy of the particle E divided by c:
so we have
For
Since the second term is the kinetic energy , we interpret the first term as the rest- mass energy of the particle .
In general 
implies that
where 
is the particle's three- momentum.
Since 
as 
, requiring infinite
energy, we infer that the particles with non- zero rest mass,
can never reach the speed of light.
For photons traveling in the x- direction
hence photons have zero rest mass.