The radial metric function does not identify null surfaces

TL;DR

The paper tackles the problem of reliably identifying null hypersurfaces across coordinate systems by using EF-like and Kruskal-like transformations. It shows that for static spacetimes the physically correct criterion is $g_{tt}=0$ (not $g^{rr}=0$) and derives the conditions under which Kruskal-like coordinates recover the Vollick proxy, while acknowledging cases where such coordinates are unavailable. The approach generalises to axisymmetric stationary spacetimes, where the null condition is captured by det$(h_{ab})=0$ on the induced metric, and is demonstrated through applications to Ellis drainhole and traversable wormholes (including rotating Teo-type geometries). Overall, the method provides a physically intuitive link between null surfaces and photon geodesics and offers a coordinate-robust framework for horizons in diverse spacetimes.

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.

Figures (1)

• Figure 1: Different metric functions (left and right) are used to color cylindrical embeddings of an Eling--Jacobson wormhole with $\Kb{}=1$ (outer surfaces) for comparison with a Schwarzschild black hole of the same asymptotic mass (inner surfaces). The upper halves show the asymptotically flat wormhole near side and black hole exterior, which would share the same geometry at infinity (here truncated). The wormhole throat lies at a dilated radial coordinate compared to the black hole horizon, and on this diagram the throat and horizon are aligned vertically. The condition $\tensor{g}{^{rr}}=0$ is met both at the wormhole throat and black hole horizon. The condition $\tensor{g}{_{tt}}=0$ is met only at the black hole horizon, and so the throat of the wormhole is not a null surface. The lower halves show the 'flare-out' of the wormhole far side and the 'pinch-off' of the black hole interior at the singularity. A curvature singularity would also occur at infinite radius on the far side of the wormhole (here truncated). This singularity would be null, as evidenced by the steadily decreasing value of $\tensor{g}{_{tt}}$ in the visible region.