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Are all wormholes gravitational instantons in the context of General Relativity?

My question concerns also the topology of spacetime in such case.

A full Wick rotation of the metric, seems to change the geometry from that of Pseudo-Riemannian to Riemannian one.

So given that topology of the Pseudo-Riemannian manifold in most general case does not match the topology with respect to the Riemannian metric, I want to know in the case of gravitational instantons, how is this situation interpreted?

There's a Euclidean hole, but there's no Lorenzian hole at the same time?

Bastam Tajik
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1 Answers1

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If one defines an instanton as a Euclidean solution, then the answer is no, because there are Lorentzian wormholes.

If one restricts to Euclidean wormholes, and stipulates that to be an instanton, it must be a critical point of the gravitational path integral, I think the answer is still no, not all wormholes are instantons.

But at this point, the discussion gets pretty technical. The answer may depend on how you define "gravitational instanton", on the specific physical theory you're using, and maybe even on your philosophy of quantum gravity. As @ACuriousMind remarks in a comment, Wick rotation in gravity is not straightforward.

It has been discussed on this site here and here.

Mitchell Porter
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