Reparametrisation of geodesic

Question

Suppose that we have a lorentzian manifold M, with a geodesic $\gamma_1:I_1\to M$ that is not constant. Here $I_1$ is an open interval in the real line. Now suppose that $\gamma_2:I_2\to M$ (where $I_2$ is an open interval) is a reparametrisation of $\gamma_1$, i.e. we have $\gamma_2=\gamma_1\circ f$ for some smooth $f:I_2\to I_1$. Then I want to prove that $\gamma_2$ is also a geodesic, if and only if $f$ is an affine function, i.e. that $f(x)=mx+b$ for some $m,b\in\mathbb{R}$. Not sure where to even start on this one.

Answer 1

By the chain rule, we have$$\dot{\gamma}_2(t)=f'(t)\dot{\gamma}_1(f(t)).$$ Hence, $$\frac{D^2}{dt^2}\gamma_2(t)=\frac{D}{dt}\dot{\gamma}_2(t)=f''(t)\dot{\gamma}_1(f(t))+f'(t)\frac{D}{dt}\dot{\gamma}_1(f(t)).$$ The second summand vanishes as $\gamma_1$ is a geodesic (depending on your definition for geodesic, this statement may need some reasoning).