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

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.