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The question is almost self-explanatory. What I am trying to say is that if there are no space-time outside our universe, how does collisions even happen. I am having a hard time explaining this question, I hope it is understandable. When talking about multiverse(the bubble one) do we assume that there is space-time outside our universe?

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I assume you're thinking about bubble universes in the context of inflation. In this case the universe consists of an (infinite?) inflating manifold with the bubble universes embedded in it. So the bubble universes are just a region of the whole manifold.

The terminology is a bit confusing - the problem is that there isn't a clear definition of the term universe. My own view is that the term means the whole manifold i.e. both the inflating region and the bubbles, though I guess this is normally referred to as the multiverse with the word universe being used to describe just the bubble. If you insist on using the terms this way then yes there is spacetime outside the universe because the spacetime consists of the whole manifold along with the metric that describes its geometry.

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