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I appreciate that a Einstein–Rosen bridge has never been observed but since there are some valid theories in General Relativity I thought I'd ask this.

Would the start and end point of a Einstein–Rosen bridge change in an expanding Universe or would the whole thing stretch but retain it's start and end position?

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The latter is closest to the truth, although I wouldn't use the phrasing "stretch". The "mouths" of the wormhole are (more or less) fixed in comoving coordinates (i.e. the coordinate system that expands with the Universe, and in which galaxies lie approximately still). But the bridge is sort of outside our three-dimensional space, and doesn't necessarily follow the same expansion.

If the mouths retained a fixed physical distance, they would accelerate in comoving coordinates beyond bound, eventually moving through space at superluminal velocities, which is forbidden. For example, consider the Milky Way and the galaxy GN-z11 which today lies at a distance of 32 billion lightyears from us. Roughly 13.3 billion years ago, the size of the Universe was 1/10 of today's value; that is $a=0.1$, and the distance to GN-z11 was only 3.2 billion lightyears. If at that time you created a wormhole$^\dagger$ bridging MW and GN-z11, and if the distance between the mouths were fixed in physical coordinates, then today, at least one of the mouths would have moved so far outside its host galaxy that it must have traveled superluminally.


$^\dagger$Creating a wormhole 13.3 billion years ago is left as an exercise for the reader.

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