Fundamentals of Time and Relativity

Energy and mass

The mass of a body is a measure of the energy contained in it.
Albert Einstein (1905)

Equation (8.8), or equivalently (8.9), may be called the relativistic work-energy theorem. Since the left-hand side of (8.9) is the work $F^{0} = F \cdot v$ done on the particle by the force $F$ per unit of proper time, the right-hand side must be the corresponding energy change

8.11

\frac{d E}{d τ} = \frac{d p^{0}}{d τ}

This argument only determines the energy up to an additive constant. However, to obtain Newtonian mechanics in the classical limit, the additive constant must be zero. This identifies $p^{0} = γ m$ as the energy of a particle with mass $m$ :

8.12

E = \frac{m}{\sqrt{1 - v^{2}}} = m + \frac{1}{2} m v^{2} + O (v^{4})

In the non-relativistic limit, the kinetic energy term is added to the rest energy $E_{0} = m$ , which is Einstein’s famous formula of 1905. One often loosely refers to this equation as ‘ $E = m c^{2}$ ’ , but one should be aware that ‘ $E$ ’ here refers to the rest frame.

Equation (8.12) generally shows that the inertial mass of a moving particle exceeds its (rest) mass by its relativistic kinetic energy $K : = (γ - 1) m$ , naturally defined as the difference between its total and its rest energy. So kinetic energy contributes to the mass in a way that is consistent with (8.11).

The general statement of Einstein’s universal mass-energy equivalence principle is that all forms of energy have inertia, and vice versa, and that every change of energy is connected with a corresponding change of inertial mass.

The huge amount of rest energy explains why in the past mass-increases corresponding to the easily measurable kinetic energies in particle collisions never have been observed.

Newton defined mass as a quantity of matter, and asserted its conservation. Einstein redefined mass as a quantity of energy.
Einstein’s mass–energy equivalence is a fundamental principle of Nature, as it is found to be applicable to all of physics.