Kruskal spacetime

The Kruskal spacetime is the maximal analytic extension of the positive-mass vacuum Schwarzschild spacetime. It contains two exterior regions, a black hole, and a white hole. Kruskal–Szekeres coordinates remove both horizon coordinate singularities, while $r=0$ remains a curvature singularity.

In Kruskal–Szekeres coordinates, every future-directed radial causal curve has nondecreasing $U$ and $V$. The right exterior has $U<0,V>0$, while the left has $U>0,V<0$. No future-directed causal curve can travel between the exteriors because it would have to decrease one null coordinate. The spatial bridge on a time-symmetric slice is therefore not a traversable passage.

This radial spacetime diagram uses Kruskal–Szekeres coordinates to display both exteriors, the black hole and the white hole of maximally extended Schwarzschild spacetime. Each point represents a symmetry two-sphere. Radial null geodesics have slopes $\pm 1$ in the displayed coordinates; the horizons are $T_K=\pm X_K$ and the singularities satisfy $T_K^2-X_K^2=1$ in the usual normalization. A further compactification gives the corresponding Penrose diagram.

Every maximal radial timelike geodesic in positive-mass Kruskal spacetime is incomplete in at least one direction. Its equations are ${\dot{r}}^2=E^2-1+2M/r$ and $\ddot{r}=-M/r^2$. A finite turning point is a maximum, and each trajectory reaches $r=0$ at one or both ends. The remaining proper time is finite because $d\tau \sim \sqrt{r/(2M)}dr$. Complete circular timelike geodesics exist, but they have nonzero angular momentum.