Math 2.0

Relativity and gravitational theory Definition

Kruskal–Szekeres coordinates

Builds on Schwarzschild metricLorentzian manifoldLambert W function

Definition

Fix m>0m > 0 and use units where the gravitational constant and the speed of light equal 11. Let gg be the Schwarzschild metric of mass mm in coordinates (t,r,θ,ϕ)(t, r, \theta, \phi) with r>2mr > 2m. The Kruskal–Szekeres coordinates (T,X,θ,ϕ)(T, X, \theta, \phi) on this exterior region are T=(r2m−1)1/2er/4msinh⁡t4m,X=(r2m−1)1/2er/4mcosh⁡t4m.T = \left(\frac{r}{2m} - 1\right)^{1/2} e^{r/4m} \sinh\frac{t}{4m}, \qquad X = \left(\frac{r}{2m} - 1\right)^{1/2} e^{r/4m} \cosh\frac{t}{4m}. They satisfy X2−T2=(r2m−1)er/2mX^2 - T^2 = \left(\frac{r}{2m} - 1\right) e^{r/2m}, and in them g=32m3re−r/2m(−dT2+dX2)+r2(dθ2+sin⁡2θ dϕ2),g = \frac{32 m^3}{r} e^{-r/2m}\left(-dT^2 + dX^2\right) + r^2\left(d\theta^2 + \sin^2\theta\, d\phi^2\right), where rr is now the function of (T,X)(T, X) defined by X2−T2=(r2m−1)er/2mX^2 - T^2 = \left(\frac{r}{2m} - 1\right) e^{r/2m}. The Kruskal extension is the manifold {(T,X):T2−X2<1}×S2\set{(T, X) : T^2 - X^2 < 1} \times S^2 with this Lorentzian metric.

Remarks

The exterior chart above is the region X>∣T∣X > \abs{T}. On it X2−T2>0X^2 - T^2 > 0, so r>2mr > 2m, and T/X=tanh⁡(t/4m)T / X = \tanh(t/4m) recovers tt.

Explicitly, r=2m(1+W ⁣((X2−T2)/e))r = 2m\left(1 + W\!\left((X^2 - T^2)/e\right)\right), where WW is the principal branch of the Lambert W function. This is a smooth positive function on the whole region T2−X2<1T^2 - X^2 < 1, so the metric is smooth there, also on the horizon r=2mr = 2m, which is the pair of lines T=±XT = \pm X. The hyperbola T2−X2=1T^2 - X^2 = 1 corresponds to r=0r = 0, where the curvature blows up, so it is not part of the manifold.

Non-example: the Schwarzschild coordinates are not defined at r=2mr = 2m, and the Eddington–Finkelstein coordinates cover only half of the Kruskal extension.

The four regions are the exterior X>∣T∣X > \abs{T}, a second exterior X<−∣T∣X < -\abs{T}, the black hole T>∣X∣T > \abs{X} and the white hole T<−∣X∣T < -\abs{X}. Radial null curves are the lines T=±X+constT = \pm X + \text{const}.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Lambert W function, concept, no evidence levelLambert W functionconceptLorentzian manifold, concept, no evidence levelLorentzian manifoldconceptRicci curvature, concept, no evidence levelRicci curvatureconceptSchwarzschild metric, example, no evidence levelSchwarzschild metricexampleKruskal–Szekeres coordinates, definition, no evidence levelKruskal–Szekeres coor…definition
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  1. Lambert W function concept
  2. Lorentzian manifold concept
  3. Ricci curvature concept
  4. Schwarzschild metric example
  5. Kruskal–Szekeres coordinates definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates
About this page
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Kinddefinition
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MSC83C15, 83C57
DefinesKruskal–Szekeres coordinates
UsesSchwarzschild metric, Lorentzian manifold, Lambert W function
Changed3 November 2026

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