Kruskal–Szekeres coordinates extend the Schwarzschild solution smoothly through its future and past horizons. Constant Schwarzschild radius curves are hyperbolae, while the horizons are null lines through the bifurcation sphere.
For , suppose and are analytic at , , and . The tortoise coordinate obeys the displayed expansion with analytic . For and in the right exterior, after fixing an additive constant. The inverse function theorem makes analytic in through zero, while has a finite nonzero radial coefficient there. This supplies the local analytic extension and its bifurcate Killing horizon.
A radial geodesic orthogonal to the time-symmetric Kruskal slice starts at with zero Schwarzschild energy and reaches after proper time .
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Kruskal–Szekeres coordinates are a specific set of coordinates used in the context of general relativity, particularly to describe the Schwarzschild solution, which describes the spacetime surrounding a spherically symmetric, non-rotating mass such as a black hole. These coordinates are particularly useful because they allow for a smooth and complete description of the Schwarzschild black hole, including regions that might be singular or undefined in standard Schwarzschild coordinates.