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 :
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
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.