Kruskal-Szekeres coordinates

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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.

[edit] Definition

Kruskal-Diagram. Each blue hyperbola represents a position of constant radius.

Kruskal-Diagram - animation.

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 $(t, r,θ,φ)$ , 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 new time and radial coordinates:

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

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

for the exterior region $r > 2 G M$ , and:

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

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

for the interior region $0 < r < 2 G M$ .

In these coordinates the metric is given by

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

where r is defined implicitly by the equation

$T^2 - R^2 = \left(1-\frac{r}{2GM}\right)e^{r/2GM}.$

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

$T = \plusmn R\,$

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:

U = T - R

V = T + R,

in which the metric is given by

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

and r is defined implicitly by the equation

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

These coordinates have the useful feature that outgoing null geodesics are given by $U = c o n s t a n t$ , while ingoing null geodesics are given by $V = c o n s t a n t$ . Furthermore, the (future and past) eventhorizon(s) are given by the equation $U V = 0$ , and curvature singularity is given by the equation $U V = 1$ .

[edit] 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 (T, R) can be extended over every value possible without hitting the physical singularity. The allowed values are

$-\infty < R < \infty\,$

$T^2 - R^2 < 1.\,$

In the maximally extended solution there are actually two singularites 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.

I	exterior region	$T 2 - R 2 < 0$ and $R > 0$	$2 G M < r$
II	interior black hole	$0 < T 2 - R 2 < 1$ and $T > 0$	$0 < r < 2 G M$
III	parallel exterior region	$T 2 - R 2 < 0$ and $R < 0$	$2 G M < r$
IV	interior white hole	$0 < T 2 - R 2 < 1$ and $T < 0$	$0 < r < 2 G M$

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

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