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. There is no misleading coordinate singularity at the horizon. The Kruskal–Szekeres coordinates also apply to space-time around a spherical object, but in that case do not give a description of space-time inside the radius of the object. Space-time in a region where a star is collapsing into a black hole is approximated by the Kruskal–Szekeres coordinates (or by the Schwarzschild coordinates). The surface of the star remains outside the event horizon in the Schwarzschild coordinates, but crosses it in the Kruskal–Szekeres coordinates. (In any "black hole" which we observe, we see it at a time when its matter has not yet finished collapsing, so it is not really a black hole yet.) Similarly, objects falling into a black hole remain outside the event horizon in Schwarzschild coordinates, but cross it in Kruskal–Szekeres coordinates. Kruskal–Szekeres coordinates on a black hole geometry are defined, from the Schwarzschild coordinates , by replacing t and r by a new timelike coordinate T and a new spacelike coordinate : for the exterior region outside the event horizon and: for the interior region . Here is the gravitational constant multiplied by the Schwarzschild mass parameter, and this article is using units where = 1. It follows that on the union of the exterior region, the event horizon and the interior region the Schwarzschild radial coordinate (not to be confused with the Schwarzschild radius ), is determined in terms of Kruskal–Szekeres coordinates as the (unique) solution of the equation: Using the Lambert W function the solution is written as: Moreover one sees immediately that in the region external to the black hole whereas in the region internal to the black hole In these new coordinates the metric of the Schwarzschild black hole manifold is given by written using the (− + + +) metric signature convention and where the angular component of the metric (the Riemannian metric of the 2-sphere) is: Expressing the metric in this form shows clearly that radial null geodesics i.