PC1672 Advanced dynamics

1.5 Relativistic momentum and energy

The 4-momentum of a particle is defined as its mass times its 4-velocity

$\begin{displaymath}{\sf p}=m{\sf v}=m\gamma(v)(v_1,v_2,v_3,{\rm i}c)\end{displaymath}$

In terms of the momentum ${\bf p}$ and energy

of the particle, the 4-momentum is

$\begin{displaymath}{\sf p}=\left(p_1,p_2,p_3,{\rm i}{E\over c}\right)\end{displaymath}$

Note that the mass used here is the only unambiguous mass of a relativistic particle. It is what is often called the ``rest mass''.

The relativistic dynamical law which we will focus on here is:

The total 4-momentum of a system of isolated particles is conserved.

It contains conservation of both momentum and energy. These were two separate concepts in Newtonian dynamics but we see here that they are aspects of a single 4-vector (like space and time).

The square of any 4-momentum is an invariant. For a single particle it is just

$\begin{displaymath}{\sf p}\cdot{\sf p}=-m^2c^2\end{displaymath}$

which gives us the relativistic relation between energy and momentum:

$\begin{displaymath}p^2-{E^2\over c^2}=-m^2c^2\end{displaymath}$

The total energy of a freely moving relativistic particle can be written in terms of its momentum as

$\begin{displaymath}E=\sqrt{m^2c^4+p^2c^2}\end{displaymath}$

or in terms of its speed as

$\begin{displaymath}E=m\gamma(v)c^2\end{displaymath}$

The kinetic energy of the particle is the difference between its total energy and its rest energy (Einstein's famous

)

$\begin{displaymath}T=E-mc^2\end{displaymath}$

For a slowly moving particle, this becomes

$\begin{displaymath}T\simeq{mv^2\over 2}\end{displaymath}$

as in nonrelativistic dynamics.

Mass: In relativity there is no unique definition of the ``mass'' of a moving particle; for example, the inertial mass is different for accelerations parallel and perpendicular to the direction of motion. Hence only the rest mass has a clear, unambiguous definition, as Einstein himself was careful to point out.

Conservation laws: Conservation of momentum is in fact a consequence of translational invariance. This link between a conservation law and a symmetry principle follows from Noether's theorem. In a similar way conservation of energy can be derived from the fact that our laws of physics are independent of time.

Textbook references

K&K 14.4 (see also 13.1-2)
M 10.4
M&T 14.7-9

Mike Birse
16th January 2001