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---|---|---|
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A bubble of nothing in the sense of Witten 82 is an instability of the Kaluza-Klein vacuum where a gravitational instanton mediates the decay of a -dimensional Euclidean Schwarzschild spacetime into nothing: the hole in space grows at the speed of light until it hits null infinity?, leading to the “annihilation” of spacetime.
Let us start from the spacetime , where is a ball of radius and is a circle of radius , equipped with the following Euclidean Schwarzschild metric:
where is the coordinate on the circle and is the radius of .
Let the volume of the -sphere be . We can now go back to the Minkowski signature by the Wick rotation . Thus, we have
By change of coordinates and , we see that at large radius the metric goes to the Minkowski metric
However, the coordinates parametrize all the -dimensional base manifold but the region of the bubble. The boundary of the bubble in the -dimensional base space has a radius which grows with time as
Moreover, the effective size of the circle compactification will be depend on by
so that it goes to for and it goes to for for .
Last revised on April 21, 2021 at 11:48:41. See the history of this page for a list of all contributions to it.