The radial metric function does not identify null surfaces

Abstract

We investigate the conditions under which a hypersurface becomes null through the use of coordinate transformations. We demonstrate that, in static spacetimes, the correct criterion for a surface to be null is $g_{tt} = 0$, rather than $g^{rr} = 0$, in agreement with the results of Vollick. We further show that, if a Kruskal-like coordinate exists, the proxy condition $g^{rr} = 0$ is equivalent to $g_{tt} = 0$ if $\partial_r g_{tt} \neq 0$ and both $g^{rr}$ and $g_{tt}$ vanish at the same rate near the horizon. Our method extends naturally to axisymmetric stationary spacetimes, for which we demonstrate that the condition $\det\big(h_{ab}\big) = 0$ for the induced metric on a null hypersurface is recovered. By contrast with the induced metric approach, our method provides a physical perspective that connects the general null condition with its underlying relationship to photon geodesics.