Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 1 / a

Use coordinates $X^{\mu }=(ct,\mathbf{x})$ and the mostly-plus Minkowski metric ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$. Along a future-directed timelike world line, proper time satisfiesThe four-velocity and four-acceleration areThe Minkowski norm is fixed: $U^{\mu }{U}_{\mu }=-c^2$. Differentiating with respect to proper time, using the constant Minkowski metric, givesIn the instantaneous rest reference frame, $U^{\mu }=(c,0,0,0)$, so the orthogonality says $A^0=0$ there. This construction uses a timelike world line; a massless particle has no rest-frame proper time parameter.