Past exam of the short options course of the University of Oxford / 2023 / 8 / a

Use the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ and coordinates $x^0=ct$. At a point, a smooth path is timelike if its tangent $\dot{x}$ is a timelike vector, here meaning ${\eta }_{ab}{\dot{x}}^a{\dot{x}}^b>0$. It is lightlike if its tangent is a null vector: the same expression is zero and the tangent is nonzero. In these coordinates, a future-directed timelike tangent has spatial speed less than $c$; a lightlike tangent has speed $c$.

For increasing parameter $s$ along a future-directed timelike curve, the elapsed proper time isor $\int \sqrt{1-|\mathbf{v}{|}^2/c^2}dt$ when coordinate time is the parameter. A Lorentz transformation satisfies $L^T\eta L=\eta$, so replacing $\dot{x}$ by $L\dot{\tilde{x}}$ leaves the expression under the square root unchanged. Hence the integral is invariant. A monotone change of path parameter also leaves it unchanged by the ordinary substitution rule.