Past exam of the relativity course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution

Denote the parent rest mass by $M$, the daughter rest masses by $m_i$, and their total energies in any inertial frame by $E_{\mathrm{before}}$ and $E_{\mathrm{after}}$. For an isolated disintegration with all products included, energy conservation gives $E_{\mathrm{after}}=E_{\mathrm{before}}$. The corresponding kinetic energies are $K_{\mathrm{before}}=E_{\mathrm{before}}-M{c}^2$ and $K_{\mathrm{after}}=E_{\mathrm{after}}-{\sum }_i{m}_i{c}^2$. ConsequentlyEvery mass on the right is a rest mass, hence invariant under a Lorentz transformation. This proves the kinetic energy released by relativistic decay is frame independent even though either total kinetic energy separately is not. In the parent rest frame the initial kinetic energy vanishes and the final kinetic energy equals the energy associated with this mass defect. External energy input or omission of radiation would invalidate the energy-cancellation step; if photons are products, include their energies with zero rest-mass subtraction.