Schwarzschild conformal compactification ($T=\frac{1}{2}(\arctan V+\arctan U),X=\frac{1}{2}(\arctan V-\arctan U)$)

For the Kruskal–Szekeres coordinates of a positive-mass Schwarzschild spacetime, $UV=(1-r/(2M))e^{r/(2M)}$. The displayed conformal compactification makes radial null geodesics lines of slope $\pm 1$. The Killing horizons are $T=\pm X$, and the spacelike curvature singularities $r=0$ lie at $T=\pm \pi /4$, with $|X|<\pi /4$. Two asymptotically flat exterior regions each have past null infinity, null infinity to the future, timelike infinity and spacelike infinity. The two exterior regions cannot communicate by any causal curve; future-directed curves that cross a future horizon instead end at the black-hole singularity.