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.