The <proper time> elapsed along the smooth <timelike curve> is
$$
\boxed{T[\gamma]=\int_0^1
\sqrt{-g_{\mu\nu}(x(s))\frac{dx^\mu}{ds}\frac{dx^\nu}{ds}}\,ds}.
$$
The expression is invariant under every orientation-preserving <reparametrization> of the curve.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 1 / a / i / Solution

{wiki=Parametric_equation#Reparametrization}

A reparametrization composes a parametrized curve with a smooth change of parameter. An orientation-preserving reparametrization has positive derivative.


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