(adsbygoogle = window.adsbygoogle || []).push({}); The "hole argument" and diffeomorphism invariance

First, let me give a summary of my understanding of the "hole argument": Consider a space-time completely filled with matter with exception of a finite space-time volume that contains no matter (a hole). The hole is located between two spatial hypersurfaces. On the first spatial hypersuface the initial conditions for the hole are defined. One may be able to perform a active smooth transformation that leaves unchanged the metric and matter outside the hole, but that changes only the metric within the hole. These two different configurations differ only within the hole that has, however, same initial conditions. This means that both configurations must be the same physical solution in general relativity. Otherwise general relativity would not be deterministic, because matter and initial conditions should determine one unique solution to the equations of motion.

Now to my question. Consider the same situation, but instead of performing a smooth transformation, take a more general transformation. For example, a transformation that leaves unchanged the outside, but exchanges of two points (permutation) within the hole. The result and conclusions of the hole argument should be the same. However, general relativity is based "only" on the principle that two solutions are equivalent if they are related via smooth space-time transformations...?

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# The hole argument and diffeomorphism invariance

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