[Physics] Null geodesic equation

Question

For a null geodesic curve $X^i$,
$$0=g_{ij}V^iV^j.$$
When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the geodesic equation, which is derived from space/time-like parametrization, for null geodesics?

Answer 1

Actually, you don't have to use proper time for a parametrisation of the Euler-Lagrange-Equations / geodesic equations in GR. Just take any parametrisation you want. However, if you solve the equations and use initial conditions for a time/light/space-like-path, that geodesic will stay time/light/space-like with $$ g_{ij}V^iV^j = \text{const.} $$ over the whole path $X^i$.

Proof in: Steven Weinberg - Gravitation and Cosmology (page: 76)