 Kruskal extension of the five-dimensional Schwarzschild-Tangherlini metric

ID: kruskal-extension-of-the-five-dimensional-schwarzschild-tangherlini-metric

r_{*} = r + (r_{h} /2) lo g ∣ (r - r_{h}) / (r + r_{h}) ∣

u = t - r_{*}

and

v = t + r_{*}

, then

U = - e^{- u / r_{h}}

and

V = e^{v / r_{h}}

in the right exterior. The extended radial metric is

- r_{h}^{2} (r + r_{h})^{2} e^{- 2 r / r_{h}} d U d V / r^{2}

, regular at

r = r_{h}

. The horizons are

U V = 0

, while

r = 0

gives

U V = 1

and is a spacelike curvature singularity. Compactification by

arctan U, arctan V

displays two exteriors, a future black hole and a past white hole.

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