= Schwarzschild conformal compactification
{c}
{title2=$T=\tfrac12(\arctan V+\arctan U),\quad X=\tfrac12(\arctan V-\arctan U)$}
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 $\pm1$. 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.
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