Mass in Special Relativity - The Relativistic Energy-momentum Equation

The Relativistic Energy-momentum Equation

The relativistic expressions for E and p obey the relativistic energy-momentum equation:

where the m is the rest mass, or the invariant mass for systems, and E is the total energy.

The equation is also valid for photons, which have m = 0:

and therefore

A photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always c.

For an object at rest, the momentum p is zero, therefore

The rest mass is only proportional to the total energy in the rest frame of the object.

When the object is moving, the total energy is given by

To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to, is the only four-dimensional arrow associated with the particle's motion, so that if there is a conserved four-momentum, it must be proportional to this vector. This allows expressing the ratio of energy to momentum as

,

resulting in a relation between E and v:

This results in

and

these expressions can be written as

,

,

and

When working in units where c = 1, known as the natural unit system, all relativistic equations simplify. In particular, all three quantities E, p, m have the same dimension:

.

The equation is often written this way because the difference is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. If m > 0, then there is the rest frame, where p = 0, this equation states that E = m, revealing once more that invariant mass is the same as the energy in the rest frame.