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 null coordinates normalized by , the displayed field agrees with exterior Schwarzschild time translation. Its flow rescales and inversely, preserving their product, the radius and . It therefore preserves the extended metric across both horizons. Its squared norm is : it is timelike in both exterior regions, spacelike in the black-hole and white-hole interiors, and nonzero null on the horizons away from the bifurcation two-sphere. At that sphere it vanishes. The zero vector should not be mistaken for a nonzero null generator. This gives an explicit symmetry that survives where the original time coordinate ceases to be regular.
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.