Relativistic action for a massive point particle is defined to be
$$S=-mc\int d\sigma \sqrt{-g_{\mu \nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}.$$
In David Tong's lecture notes on GR
If all we want to do is derive the geodesic equation for some metric, then we can ignore all the shenanigans and simply work with the action $$S_{useful}=\int d\tau g_{\mu \nu}(x)\frac{d{x}^{\mu}}{d\tau}\frac{d{x}^{\nu}}{d\tau}.\tag{1.32}$$
Massless particles will follow null geodesics so geodesic equations
$$g_{\mu \nu}\dot{x}^{\mu}\dot{x}^{\nu}=0.$$
hence the above action will be zero. If one considers the action as the area under the Lagrangian, what does it mean for action to be zero for massless particles?
