Past exam of the physics 1 paper of the University of Oxford / 2013 resit / 9 / a

For a frame moving at velocity $V\hat{\mathbf{x}}$, put $\Gamma =(1-V^2/c^2{)}^{-1/2}$. The Lorentz transformation of the energy-momentum four-vector isExpansion givesThe transverse components are unchanged, hence $\boxed{E^{\prime 2}-c^2|{\mathbf{p}}^{\prime }{|}^2=E^2-c^2|\mathbf{p}{|}^2}$. For a particle of rest mass $m$ this invariant is $m^2{c}^4$; the same algebra also applies to the total four-momentum of a system.