Creating a wormhole connecting two universes with different spacetime metrics?
A civilization wishes to create a wormhole connecting its universe to a universe with a different spacetime metric. For instance if we describe a timelike dimension with $+$ symbol and a spacelike dimension with $-$ symbol then this wormhole could connect a universe with the spacetime metric $(---+)$ to a universe with the spacetime metric $(++++)$. Alternatively if the civilization was in a universe with the spacetime metric $(++++)$ the wormhole might connect to a universe with a spacetime metric $(+++++)$.
Would such a wormhole be possible assuming that wormholes are possible? If such a wormhole is possible how would the geometry right at the border between the two universes with different spacetime metrics work? Would it be possible to send macroscopic objects through such a wormhole or only elementary particles?
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Within general relativity at least, I think this is impossible. The metric tensor of a pseudo-Riemannian manifold is usually taken to be everywhere nondegenerate (see e.g. this source. A metric is nondegenerate when its determinant is nonzero, or equivalently when it has no $0$ eigenvalues), but for it to change signature from one spacetime point to another it would have to pass through the subspace of singular metrics somewhere along the path connecting them.
In other words, to go from $+$ to $-$ smoothly one has to pass through $0$, and a signature with $0$s is not allowed in ordinary GR.
This does not mean that signature-changing spacetimes haven't been at least considered in the literature (see for example this paper, this paper or this bachelor thesis). However, to treat these spacetimes we must necessarily extend ordinary general relativity, or relax one or more of its assumptions. Since you are probably interested in the worldbuilding aspects, here are some possible options you might like:
Allow for degenerate metrics: this case entails important limitations on what you can do. For example, you can't even define Christoffel symbols since they involve the inverse of the metric, and $1/0$ is not defined.
Allow for discontinuities in the metric: you can discontinuously jump from one signature to another without passing through a degenerate one. However, a spacetime like this would look nothing like a wormhole, it would be more like two separate regions "pasted" together with some junction conditions. Again there are important limitations; for example, anything involving derivatives of the metric is not defined in the junction hypersurface.
Allow for complex numbers: here you can go from $+$ to $-$ and bypass $0$ by going around it in the complex plane. In fact, all nondegenerate signatures become equivalent in this setting. I know this is used sometimes in quantum field theory as a calculational "trick" called Wick rotation, but I am not aware of any accepted physical meaning of complex coordinates, aside from Hawking's famous cosmological proposal involving imaginary time.
This post was sourced from https://worldbuilding.stackexchange.com/a/145720. It is licensed under CC BY-SA 4.0.
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