A simple thought experiment to discuss the mass–energy equivalence in the special theory of relativity

Abstract

Abstract Einstein’s relation between mass and energy is perhaps the most famous equation of Physics. Despite its simplicity, the meaning of E 0 = m c 2 is not easy to grasp. Furthermore, its traditional derivations rely either on the integral of momentum, on properties of electromagnetic radiation, or even on the expression for transformation of energy. In the present work, we provide a simple thought experiment with an inelastic collision between two particles observed from two inertial reference frames. We show that for the conservation of relativistic momentum to hold, the mass of the system must increase after the collision. We also show that the increase of mass relates to the loss of kinetic energy according to the equation Δ K = − Δ m c 2 , which enables us to define the equation for relativistic energy ( E = m c 2 / 1 − v 2 / c 2 ), rest energy ( E 0 = m c 2 ) and relativistic kinetic energy ( K = E − E 0 ) . There are two main advantages in this presentation: first, it relies only on simple algebra, not depending on differential calculus and on any property of radiation; second, it leads directly to a comprehensible physical meaning of the relation of equivalence, which can sometimes be too obscure in more formal derivations.