general relativity - Why is light described by a null geodesic?

I'm trying to wrap my head around how geodesics describe trajectories at the moment.

I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must move along a timelike geodesic. And a timelike geodesic can be defined as a geodesic that lies within the light cone.

I want to know why exactly null geodesics define the light cone. Or, why null geodesics define the path of light.

Also, if there's a better explanation why matter follows timelike geodesics, that would also be welcome.

Answer

Even in curved spacetime, you can perform a coordinate transformation at any location ("move to a freely falling frame") such that your metric is locally flat , and takes the form \begin{equation} ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2\end{equation}

If you consider a null trajectory where $ds^2 = 0$, then the above equation takes the form

\begin{equation} cdt = \sqrt{dx^2 + dy^2 + dz^2}. \end{equation}

This is the statement that "the speed of light times the differential time interval, as measured by an observer in a freely falling frame at the location in consideration, is equal to the differential physical distance traveled along the trajectory, measured by that same observer." From Einstein's equivalence principle, this is precisely the way that light must behave.