Kruskal–Szekeres coordinates

In general relativity Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity.

Kruskal–Szekeres coordinates are defined, from the Schwarzschild coordinates ${\displaystyle (t,r,\theta ,\phi )}$, by replacing t and r by a new time coordinate T and a new spatial coordinate X:

for the exterior region ${\displaystyle r>2GM}$, and:

for the interior region ${\displaystyle 0<r<2GM}$. Note GM is the gravitational constant multiplied by the Schwarzschild mass parameter, and this article is using units where c = 1.

...
Wikipedia