Kruskal-Szekeres Coordinates

The Kruskal-Szekeres coordinates are a coordinate system for the Schwarzschild metric that removes the coordinate singularity at the event horizon and reveals the full global structure of the geometry. Introduced independently by Martin Kruskal and George Szekeres in 1960, these coordinates map the entire maximally extended Schwarzschild spacetime — including the exterior, the interior, the black hole region, the white hole region, and the parallel asymptotically flat region connected by the Einstein-Rosen bridge — onto a single, non-singular coordinate patch.

In Schwarzschild coordinates, the metric components diverge at r = 2GM/c², giving the false impression that something catastrophic happens at the horizon. The Kruskal-Szekeres coordinates show that the horizon is a regular, finite point in the geometry. The divergence was a coordinate artifact, not a physical singularity. An infalling observer crossing the horizon experiences nothing locally special — the tidal forces are finite and the geometry is smooth.

The Kruskal-Szekeres diagram is one of the most important visual tools in general relativity. It displays the causal structure of the maximally extended Schwarzschild solution: the two exterior regions, the future and past singularities, the black hole interior where all paths lead to the future singularity, and the white hole interior where all paths emanate from the past singularity. The diagram makes explicit that the Schwarzschild geometry is not a single universe with a hole in it. It is a multiply connected spacetime with two asymptotic regions joined by a non-traversable wormhole.

The Kruskal-Szekeres coordinates are not merely a technical convenience. They are a demonstration that the most dramatic features of black hole geometry — the horizon, the interior, the wormhole — are not pathologies but regular structures that only appear pathological when viewed through the wrong coordinates. The singularity is real. The horizon is not.