Mass–energy equivalence

Remarks

In the rest frame $P = (E_0/c, \vec 0)$, so $-\eta(P, P) = E_0^2/c^2$, and the definition of the invariant mass gives $E_0 = Mc^2$. So any change $\Delta E_0$ of the energy of a body at rest, such as heating it or emitting light, changes its mass by $\Delta M = \Delta E_0 / c^2$. Example: the mass of a helium-4 nucleus is smaller than the sum of the masses of two free protons and two free neutrons. The difference times $c^2$ is its binding energy. The kinetic energies of the parts count in $E_0$, so a box of gas becomes heavier when the gas is heated. Einstein derived the relation in 1905.

Going deeper

For a student

The statement follows from the energy–momentum relation in the frame where the momentum vanishes. It explains the energy released in nuclear reactions through the mass defect, and the creation of particle pairs when the available energy exceeds the sum of their rest energies. In relativistic collisions energy and momentum are conserved, while the sum of the rest masses of the particles in general is not.