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.

is the gravitational constant multiplied by the Schwarzschild mass parameter, and this article is using units where

), 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

Note that the metric is perfectly well defined and non-singular at the event horizon.

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".

The Kruskal–Szekeres coordinates, on the other hand, cover the entire spacetime manifold.

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).

Based on the requirements that the quantum process of Hawking radiation is unitary, 't Hooft proposed[1] 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 should be imposed, where

If we think of regions III and IV as having spherical coordinates but with a negative choice for the square root to compute

preserving the metric, this gives a well-defined Lorentzian manifold (everywhere except at the singularity).

of the interior region II corresponding to the coordinate line segment

in the Schwarzschild picture) corresponds not to a sphere but to the projective plane

The manifold is no longer simply connected, because a loop (involving superluminal portions) going from a point in space-time back to itself but at the opposite Kruskal–Szekeres coordinates cannot be reduced to a null loop.

Kruskal–Szekeres coordinates have a number of useful features which make them helpful for building intuitions about the Schwarzschild spacetime.

Chief among these is the fact that all radial light-like geodesics (the world lines of light rays moving in a radial direction) look like straight lines at a 45-degree angle when drawn in a Kruskal–Szekeres diagram (this can be derived from the metric equation given above, which guarantees that if

[2] All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the T coordinate) than 45 degrees.

The event horizons bounding the black hole and white hole interior regions are also a pair of straight lines at 45 degrees, reflecting the fact that a light ray emitted at the horizon in a radial direction (aimed outward in the case of the black hole, inward in the case of the white hole) would remain on the horizon forever.

Thus the two black hole horizons coincide with the boundaries of the future light cone of an event at the center of the diagram (at T=X=0), while the two white hole horizons coincide with the boundaries of the past light cone of this same event.

Any event inside the black hole interior region will have a future light cone that remains in this region (such that any world line within the event's future light cone will eventually hit the black hole singularity, which appears as a hyperbola bounded by the two black hole horizons), and any event inside the white hole interior region will have a past light cone that remains in this region (such that any world line within this past light cone must have originated in the white hole singularity, a hyperbola bounded by the two white hole horizons).

Note that although the horizon looks as though it is an outward expanding cone, the area of this surface, given by r is just

It may be instructive to consider what curves of constant Schwarzschild coordinate would look like when plotted on a Kruskal–Szekeres diagram.

It turns out that curves of constant r-coordinate in Schwarzschild coordinates always look like hyperbolas bounded by a pair of event horizons at 45 degrees, while lines of constant t-coordinate in Schwarzschild coordinates always look like straight lines at various angles passing through the center of the diagram.

The black hole event horizon bordering exterior region I would coincide with a Schwarzschild t-coordinate of

while the white hole event horizon bordering this region would coincide with a Schwarzschild t-coordinate of

, reflecting the fact that in Schwarzschild coordinates an infalling particle takes an infinite coordinate time to reach the horizon (i.e. the particle's distance from the horizon approaches zero as the Schwarzschild t-coordinate approaches infinity), and a particle traveling up away from the horizon must have crossed it an infinite coordinate time in the past.

This is just an artifact of how Schwarzschild coordinates are defined; a free-falling particle will only take a finite proper time (time as measured by its own clock) to pass between an outside observer and an event horizon, and if the particle's world line is drawn in the Kruskal–Szekeres diagram this will also only take a finite coordinate time in Kruskal–Szekeres coordinates.

There are actually multiple possible ways to extend the exterior Schwarzschild solution into a maximally extended spacetime, but the Kruskal–Szekeres extension is unique in that it is a maximal, analytic, simply connected vacuum solution in which all maximally extended geodesics are either complete or else the curvature scalar diverges along them in finite affine time.

[6] In the literature, the Kruskal–Szekeres coordinates sometimes also appear in their lightcone variant: in which the metric is given by and r is defined implicitly by the equation[7] These lightcone coordinates have the useful feature that radially outgoing null geodesics are given by

Kruskal–Szekeres diagram, illustrated for 2 GM =1. The quadrants are the black hole interior (II), the white hole interior (IV) and the two exterior regions (I and III). The dotted 45° lines, which separate these four regions, are the event horizons . The darker hyperbolas which bound the top and bottom of the diagram are the physical singularities. The paler hyperbolas represent contours of the Schwarzschild r coordinate, and the straight lines through the origin represent contours of the Schwarzschild t coordinate.
Kruskal–Szekeres diagram. Each frame of the animation shows a blue hyperbola as the surface where the Schwarzschild radial coordinate is constant (and with a smaller value in each successive frame, until it ends at the singularities).