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Just as the Lorentz symmetry holds globally in Minkowski spacetime, could the opposite also occur? That is, are there any spacetimes where the Lorentz symmetry would be broken (locally, not just globally)?

Even more, are there any spacetimes where the Poincaré, Lorentz, CPT, time-translational, spatial-translation, diffeomorphism....etc symmetries (and even internal symmetries like gauge symmetries) would be broken or would not hold (even locally)?

Could there be a vacuum state that would have such spacetimes with no symmetries?

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    $\begingroup$ Symmetry breaking is not magical or unusual -- our universe already breaks Lorentz symmetry, because the CMB defines a preferred rest frame. $\endgroup$
    – knzhou
    Feb 18 at 22:51
  • $\begingroup$ @knzhou what s the difference between CMB breaking Lorentz and, say, a gas nebula or any other kind of matter? Only QFT vacuum is Lorentz invariant as far as I know. $\endgroup$
    – Quillo
    Feb 19 at 11:05
  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/748785/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Apr 6 at 12:29

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Your question seems a bit confused, as you're asking two different things, so I'll try and clarify them here. Firstly, you seem to be asking about the symmetries of spacetimes within the framework of GR. Having a spacetime with no physical symmetries is absolutely fine and causes no issues. The key point is that these are physical symmetries, but the general covariance as well as the local Lorentz invariance of the theory still remains intact. This is always the case in GR.

On the other hand, breaking local Lorentz invariance (or equivalently diffeomorphism invariance) necessarily means some theory beyond GR. One can easily construct such a theory by hand, but it's pretty ad hoc and there's no current evidence for such a theory.

To drill this point home, that these are two different questions, note that physical spacetime configurations cannot break the local Lorentz invariance which is built into General Relativity. I explained this on one of your previous questions here.

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  • $\begingroup$ so is it impossible to find a type of spacetime in the entire theory of relativity that breaks the Lorentz invariance (not only globally)? @Eletie $\endgroup$
    – vengaq
    Feb 20 at 13:18
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    $\begingroup$ Local Lorentz invariance is built into GR: it is always there. Global Lorentz invariance is essentially implying flat space (SR), which is a special case of GR. $\endgroup$
    – Eletie
    Feb 20 at 14:45

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