In the book Introduction to General Relativity Blackholes and Cosmology by Yvonne Choquet-Bruhat, she defines the length of a causal curve as $$\ell\equiv \int_a^b \left( -g_{\alpha \beta} \frac{d \gamma^\alpha}{d\lambda} \frac{d\gamma^\beta}{d\lambda} \right)^{\frac 12}d\lambda$$ but remarked that this definition is problematic if the the curve $\gamma$ has a null tangent vector. I have trouble interpreting this remark, in my opinion, the equation would have been problematic if the tangent vectors are spacelike, giving a square root of a negative number, but I think it's fine for null vectors, which would just give zero. But the possibility of spacelike tangent is excluded by the use of $\textbf{causal}$ curve.
I think it is alright think this integral as the action, and obtain the geodesics as the path that extremize this action. The conventional choice of Lagrangian $$\mathcal L\equiv g_{\alpha \beta}(x(\lambda))\frac{d x^\alpha}{d \lambda} \frac{d x^\beta}{d\lambda}$$ is merely a convenient choice. Is there a better way to think of this statement?
