Supplement 1.5: Mass, Energy and Momentum of Particles and Photons (1/2)

Mass

We must distinguish between the rest mass (or: the intrinsic mass) from the relativistic mass of particles. The rest mass corresponds to the mass of a non-moving particle, we denote it with the symbol m_o. If the particle velocity is is much smaller than the velocity of light (which holds for all objects in our daily life), the relativistic mass is reduced to the rest mass.

However, if the speed v approaches the speed of light c, the mass increases indefinitely and would become infinitely large at the speed of light. According to the special theory of relativity, the following applies to this moving mass:

m = \frac{m_{o}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

With $v \to c$ , the denominator approaches 0 and therefore $m \to \infty$ . The increasing mass of the particle with increasing speed is referred to as relativistic mass increase.

Photons cannot be at rest but their velocity is always the speed of light. Therefore their rest mass is zero, and there is no increasing relativistic mass. Apparently photons do not have a mass at all, which makes them very different from other particles!

The particle momentum

The momentum of a particle depends on its mass an velocity:

p = m v

This equation holds also with increasing particle velocity, even if the velocity increases toward the speed of light. However, besides the velocity the relativistic increase of mass must be considered as well:

p = m v = \frac{m_{o} v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

The quantity m_o is again the rest mass of the non-moving particle.

The particle energy

The kinetic energy of a particle having small velocity is:

E = \frac{1}{2} m v^{2}

If the velocity increases and gets close to the speed of light, then this equation does not hold anymore. Energy and mass must then be considered as two sides of the same coin; this equivalence of mass and energy is given by the following equation from the theory of special relativity:

E = m c^{2}

Here again m is the relativistic mass, and it follows:

E = m c^{2} = \frac{m_{o} c^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

Hence, the relativistic energy of a particle also includes also a rest energy, since with v=0 it follows E=m_oc². This part of the energy follows from the rest mass of the particle.

At a velocity v=c, the energy and also the momentum of a particle is infinity. However, particles cannot have an infinitely high energy and momentum. Therefore, the velocity of particles having a non-zero rest mass cannot reach the speed of light, which makes them very different from photons!

The square root term can be expanded as a power series: