KruskalSzekeres

Kruskal–Szekeres coordinates are a coordinate system for the Schwarzschild solution in general relativity, named after Alison Kruskal and George Szekeres who introduced the maximal analytic extension in 1960. They reveal the global structure of the Schwarzschild spacetime and remove the coordinate singularity that occurs at the Schwarzschild radius r = 2M (in geometric units G = c = 1).

In these coordinates, one defines the tortoise coordinate r* by r* = r + 2M ln(|r/2M − 1|). Then, for the exterior region r > 2M, the Kruskal–Szekeres null coordinates are

U = − exp[−(t − r*)/(4M)]

V = exp[(t + r*)/(4M)].

The Schwarzschild metric becomes regular across the horizon and takes the form

ds^2 = − (32 M^3 / r) e^{−r/(2M)} dU dV + r^2 dΩ^2,

where r is implicitly defined by UV = (r/2M − 1) e^{r/(2M)} and dΩ^2 is the metric on the unit 2-sphere.

The construction covers four regions in the Kruskal diagram: region I and region III are external, asymptotically flat regions; region II is the black-hole interior; region IV is a white-hole interior. The event horizon corresponds to U = 0 or V = 0, and the singularity at r = 0 lies elsewhere in the diagram.

Kruskal–Szekeres coordinates are widely used to study the causal structure of black holes, analyze geodesics crossing horizons, and illustrate the maximal analytic extension of the Schwarzschild solution beyond the limitations of standard Schwarzschild coordinates.