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.
Definition
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.e. with constant are parallel to one of the lines . In the Schwarzschild coordinates, the Schwarzschild radius is the radial coordinate of the event horizon . In the Kruskal–Szekeres coordinates the event horizon is given by . Note that the metric is perfectly well defined and non-singular at the event horizon. The curvature singularity is located at .
The maximally extended Schwarzschild solution
The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates defined for r > 2GM and can be extended, as an analytic function, at least to the first singularity which occurs at . Thus the above metric is a solution of Einstein's equations throughout this region. The allowed values are
Note that this extension assumes that the solution is analytic everywhere.
In the maximally extended solution there are actually two singularities at r = 0, one for positive T and one for negative T. The negative T singularity is the time-reversed black hole, sometimes dubbed a "white hole". Particles can escape from a white hole but they can never return.
The maximally extended Schwarzschild geometry can be divided into 4 regions each of which can be covered by a suitable set of Schwarzschild coordinates. The Kruskal–Szekeres coordinates, on the other hand, cover the entire spacetime manifold. The four regions are separated by event horizons.
The transformation given above between Schwarzschild and Kruskal–Szekeres coordinates applies only in regions I and II (if we take the square root as positive). A similar transformation can be written down in the other two regions.
The Schwarzschild time coordinate t is given by
In each region it runs from to with the infinities at the event horizons.
Based on the requirements that the quantum process of Hawking radiation is unitary, 't Hooft proposed that the regions I and III, and II and IV are just mathematical artefacts coming from choosing branches for roots rather than parallel universes and that the equivalence relation