Kruskal–Szekeres coordinates

In general relativity Kruskal–Szekeres coordinates, named for 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.

Definition

Kruskal-Diagram. Each blue hyperbola represents a set of events of constant radius in Schwarzschild coordinates.

Kruskal-Diagram - animation (for different constants r)

Conventions: In this article we will take the metric signature to be (− + + +), and we will work in units where c = 1. The gravitational constant G will be kept explicit. We will denote the characteristic mass of the Schwarzschild geometry by M.

Recall that in Schwarzschild coordinates $(t,r,\theta,\phi)$, the Schwarzschild metric is given by

$ds^{2} = -\left(1-\frac{2GM}{r} \right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1}dr^2+ r^2 d\Omega^2$,

where

$d\Omega^2\ \stackrel{\mathrm{def}}{=}\ d\theta^2+\sin^2\theta\,d\phi^2$

is the line element of the 2-sphere $S^2.$

Kruskal–Szekeres coordinates are defined by replacing t and r by a new time coordinate V and a new radial coordinate U:

$V = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)$

$U = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)$

for the exterior region $r>2GM$, and:

$V = \left(1 - \frac{r}{2GM}\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)$

$U = \left(1 - \frac{r}{2GM}\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)$

for the interior region $0<r<2GM$.

In these coordinates the metric is given by

$ds^{2} = \frac{32G^3M^3}{r}e^{-r/2GM}(-dV^2 + dU^2) + r^2 d\Omega^2,$

where r is defined implicitly by the equation

$V^2 - U^2 = \left(1-\frac{r}{2GM}\right)e^{r/2GM}$

or equivalently by

$\frac{r}{2GM} = 1 + W \left( \frac{U^2 - V^2}{e} \right)$

where W is the Lambert W function.

The location of the event horizon (r = 2GM) in these coordinates is given by

$V = \plusmn U\,$

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

In the literature the Kruskal-Szekers coordinates sometimes also appear in their lightcone variant:

$\tilde{U} = V - U$

$\tilde{V} = V + U,$

in which the metric is given by

$ds^{2} = -\frac{32G^3M^3}{r}e^{-r/2GM}(d\tilde{U} d\tilde{V}) + r^2 d\Omega^2,$

and r is defined implicitly by the equation

$\tilde{U} \tilde{V} = \left(1-\frac{r}{2GM}\right)e^{r/2GM}.$

(some sources use an alternate notation where the regular Kruskal-Szekeres coordinates are labeled T and R instead of V and U, and the Kruskal-Szekeres lightcone coordinates are labeled u and v rather than $\tilde{U}$ and $\tilde{V}$)

These lightcone coordinates have the useful feature that outgoing null geodesics are given by $\tilde{U} = constant$, while ingoing null geodesics are given by $\tilde{V} = constant$. Furthermore, the (future and past) event horizon(s) are given by the equation $\tilde{U} \tilde{V} = 0$, and curvature singularity is given by the equation $\tilde{U} \tilde{V} = 1$.

The maximally extended Schwarzschild solution

The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates is defined for r > 0, r ≠ 2GM, and −∞ < t < ∞, which is the range for which the Schwarzschild coordinates make sense. However, the coordinates (V, U) can be extended over every value possible without hitting the physical singularity. The allowed values are

$-\infty < U < \infty\,$

$V^2 - U^2 < 1.\,$

In the maximally extended solution there are actually two singularites at r = 0, one for positive V and one for negative V. The negative V 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.

I	exterior region	$V^2 - U^2 < 0$ and $U > 0$	$2GM < r$
II	interior black hole	$0 < V^2 - U^2 < 1$ and $V > 0$	$0 < r < 2GM$
III	parallel exterior region	$V^2 - U^2 < 0$ and $U < 0$	$2GM < r$
IV	interior white hole	$0 < V^2 - U^2 < 1$ and $V < 0$	$0 < r < 2GM$

The transformation given above between Schwarzschild and Kruskal–Szekeres coordinates applies only in regions I and II. A similar transformation can be written down in the other two regions.

The Schwarzschild time coordinate t is given by

$\tanh\left(\frac{t}{4GM}\right) = \begin{cases}V/U & \mbox{(in I and III)} \\ U/V & \mbox{(in II and IV)}\end{cases}$

In each region it runs from −∞ to +∞ with the infinities at the event horizons.

Qualitative features of the Kruskal-Szekeres diagram

Kruskal-Szekeres diagram showing the four regions bounded by event horizons (solid straight lines at 45 degrees passing through the center of the diagram), with dotted lines representing curves of constant Schwarzschild r-coordinate (the dotted hyperbolas) and constant Schwarzschild t-coordinate (the dotted straight lines at different angles passing through the center) drawn in. I and III are the two exterior regions, while II is the black hole interior region (with the two solid 45-degree lines bordering it being the two black hole horizons) and IV is the white hole interior region (with the two solid 45-degree lines bordering it being the two white hole horizons). The solid hyperbola at the top (i.e. the one drawn with a solid line rather than a dashed line) represents the black hole singularity, and the solid hyperbola at the bottom represents the white hole singularity. Note that the creator of this diagram has used a different convention for labeling the Kruskal-Szekeres radial and time coordinates.

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 $dU = \plusmn dV\,$ then $ds = 0$). All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the V coordinate) than 45 degrees. So, a light cone drawn in a Kruskal-Szekeres diagram will look just the same as a light cone in a Minkowski diagram in special relativity.

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 U=0 and V=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).

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 Schwarzchild 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.

The Schwarzschild coordinate system can only cover a single exterior region and a single interior region, such as regions I and II in the Kruskal-Szekeres diagram. The Kruskal-Szekeres coordinate system, on the other hand, can cover a "maximally extended" spacetime which includes the region covered by Schwarzschild coordinates. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": any geodesic path can be extended arbitrarily far in either direction unless it runs into a gravitational singularity. Technically, this means that a maximally extended spacetime is either "geodesically complete" (meaning any geodesic can be extended to arbitrarily large positive or negative values of its 'affine parameter', which in the case of a timelike geodesic could just be the proper time), or if any geodesics are incomplete, it can only be because they end at a singularity. In order to satisfy this requirement, it was found that in addition to the black hole interior region (region II) which particles enter when they fall through the event horizon from the exterior (region I), there has to be a separate white hole interior region (region IV) which allows us to extend the trajectories of particles which an outside observer sees rising up away from the event horizon, along with a separate exterior region (region III) which allows us to extend some possible particle trajectories in the two interior regions. 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.

See also

• Schwarzschild coordinates
• Eddington-Finkelstein coordinates
• Isotropic coordinates
• Gullstrand-Painlevé coordinates

References

• Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.