Why does $ds^2=0$ for a light signal's worldline in general relativity? I know that in special relativity, the invariant interval $ds^2$ for a light signal's worldline is $$ds^2=\eta_{\mu\nu}dx^\mu dx^\nu=0$$ where the flat metric  $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$.
How does this result extend to
$$ds^2=g_{\mu\nu}dx^\mu dx^\nu=0$$
for the worldline of a light signal in general relativty, where the metric $g_{\mu\nu}$ is arbitrary?
 A: There is nothing special about $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$ in special relativity - if we were working in oblique co-ordinates or in spherical co-ordinates then the metric tensor would have different elements. The key points are:

*

*The metric tensor is a tensor so we know how its elements transform if we change our system of co-ordinates.

*In special relativity the metric tensor is constant and has zero curvature i.e spacetime is globally flat, and we can find a system of co-ordinates in which the metric tensor has the form $g_{\mu\nu} = \eta_{\mu\nu}$ everywhere.

Light rays follow null geodesics - paths along which $ds^2=g_{\mu\nu}dx^\mu dx^\nu=0$ at each point. This follows from the postulate that the speed of light is invariant. And in special relativity, since spacetime is flat, we can simply extend this local definition of null geodesics to a non-local one i.e. $\Delta s^2=0$.
In general relativity the restriction to globally flat spacetime is relaxed. The metric tensor can now depend on where we are in spacetime. However, spacetime is always locally flat - there is always a system of co-ordinates in which the metric tensor has the form $\eta_{\mu\nu}$ locally. And the postulate that the speed of light is invariant still holds, so light rays still follow null geodesics along which $ds^2=g_{\mu\nu}dx^\mu dx^\nu=0$ at each point.
A: At a single point $x$, you can always choose coordinates so the metric has the form $g_{\mu\nu}(x)=\eta_{\mu\nu}$. In these coordinates, by the equivalence principle, it is clear that a null geodesic passing through $x$ will satisfy $\eta_{\mu\nu} p^\mu p^\nu=0$, where $p^\mu$ is a tangent vector to the null geodesic at $x$. However, this is a tensor equation, and therefore is valid in any coordinate system. Thus we conclude $g_{\mu\nu}p^\mu p^\nu=0$ at $x$ in any coordinate system. Since our choice of the point $x$ was arbitrary, this relation holds at every point of the null geodesic.
A: $ds^2$ is a scalar, so it will have the same value in any coordinate system, and you know that it is locally zero for light, by assumption,  in Minkowski coordinates, and thus, in every coordinate.
