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I have heard that wormholes are valid solutions to the field equations, like black holes or anything else. However, for everything I've learned about relativity and what wormholes are supposed to be would seem to violate the conservation of the interval, the space-time distance between things. Here, I am ignoring all of the hows of wormholes, and just talking about the theoretical idea. Do they?

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No.

Or at least the answer is no if you're referring to the line element described by the metric. But we need to be clear exactly what we are referring to. In your question you refer to:

the space-time distance between things

but in general relativity there is no simple meaning to the space-time distance between things. The metric is used to calculate the length of a path between two spacetime points, but the length of this path obviously depends on the path you take. Even if you specify that the path is a straight line (i.e. a geodesic) there is still no unique distance because in GR it's common that two spacetime points can be connected by more than one straight line.

With a typical wormhole geometry there may be two or more straight lines connecting two points near the wormhole mouths (one line passing through the wormhole and one line going the long way round) and these two lines will in general have different path lengths. But this does not constitute non-conservation of the interval. The conservation of the interval means that for any particular path all observers will agree on the proper length of that path, and this holds for all geometries in GR.

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