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I have been thinking about a concept that I was taught whilst learning GR, If I understand correctly it is that Lorentz symmetry becomes local when we consider GR. This makes sense to me as then the metric is generally speaking not Minkowski, only for a sufficiently small area with no mass/energy present.

So my question is this, as you approach a black hole is it correct to say that the local area in which lorentz symmetry holds gets smaller?

If so can it then be inferred that the symmetry is broken at event horizon?

Thanks in advance for your time :)