- Kruskal-Szekeres coordinates
In

general relativity **Kruskal-Szekeres coordinates**, named forMartin Kruskal andGeorge Szekeres , are acoordinate system for theSchwarzschild geometry for ablack hole . These coordinates have the advantage that they cover the entire spacetimemanifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity.**Definition**"Conventions": In this article we will take the

metric signature to be (− + + +) and we will work in units where "c" = 1. Thegravitational constant "G" will be kept explicit. We will denote the characteristic mass of the Schwarzschild geometry by "M".Recall that in

Schwarzschild coordinates $(t,r,\; heta,phi)$, the Schwarzschild metric is given by:$ds^\{2\}\; =\; -left(1-frac\{2GM\}\{r\}\; ight)\; dt^2\; +\; left(1-frac\{2GM\}\{r\}\; ight)^\{-1\}dr^2+\; r^2\; dOmega^2$,where:$dOmega^2\; stackrel\{mathrm\{def\{=\}\; d\; heta^2+sin^2\; heta,dphi^2$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::$T\; =\; left(frac\{r\}\{2GM\}\; -\; 1\; ight)^\{1/2\}e^\{r/4GM\}sinhleft(frac\{t\}\{4GM\}\; ight)$:$R\; =\; left(frac\{r\}\{2GM\}\; -\; 1\; ight)^\{1/2\}e^\{r/4GM\}coshleft(frac\{t\}\{4GM\}\; ight)$for the exterior region $r>2GM$, and::$T\; =\; left(1\; -\; frac\{r\}\{2GM\}\; ight)^\{1/2\}e^\{r/4GM\}coshleft(frac\{t\}\{4GM\}\; ight)$:$R\; =\; left(1\; -\; frac\{r\}\{2GM\}\; ight)^\{1/2\}e^\{r/4GM\}sinhleft(frac\{t\}\{4GM\}\; ight)$for the interior region$02gm\; math>.$

In these coordinates the metric is given by:$ds^\{2\}\; =\; frac\{32G^3M^3\}\{r\}e^\{-r/2GM\}(-dT^2\; +\; dR^2)\; +\; r^2\; dOmega^2,$where "r" is defined implicitly by the equation:$T^2\; -\; R^2\; =\; left(1-frac\{r\}\{2GM\}\; ight)e^\{r/2GM\}$or equivalently by:$frac\{r\}\{2GM\}\; =\; 1\; +\; W\; left(\; frac\{R^2\; -\; T^2\}\{e\}\; ight)$where "W" is the

Lambert W function .The location of the

event horizon ("r" = 2"GM") 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:$ds^\{2\}\; =\; -frac\{32G^3M^3\}\{r\}e^\{-r/2GM\}(dU\; dV)\; +\; r^2\; dOmega^2,$and "r" is defined implicitly by the equation:$U\; V\; =\; left(1-frac\{r\}\{2GM\}\; ight)e^\{r/2GM\}.$These coordinates have the useful feature that outgoing null

geodesics are given by $U\; =\; constant$, while ingoing null geodesics are given by $V\; =\; constant$. Furthermore, the (future and past) eventhorizon(s) are given by the equation $U\; V\; =\; 0$, and curvature singularity is given by the equation $U\; V\; =\; 1$.**The maximally extended Schwarzschild solution**The transformation between Schwarzschild coordinates and Kruskal-Szekeres coordinates is defined for "r" > 0, "r" ≠ 2"GM", 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. TheKruskal-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. A similar transformation can be written down in the other two regions.

The Schwarzschild time coordinate "t" is given by:$anhleft(frac\{t\}\{4GM\}\; ight)\; =egin\{cases\}T/R\; mbox\{(in\; I\; and\; III)\}\; \backslash R/T\; mbox\{(in\; II\; and\; IV)\}end\{cases\}$In each region it runs from −∞ to +∞ with the infinities at the event horizons.

**ee also***

Schwarzschild coordinates

*Eddington-Finkelstein coordinates

*Isotropic coordinates

*Gullstrand-Painlevé coordinates

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