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I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970)

In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity could break the Lorentz and diffeomorphism symmetries...

However, the other one didn't seem to agree, as he asked what he meant exactly by Lorentz and diffeomorphism symmetry:

Are you talking about a symmetries of the theory? The theory of general relativity is diffeomorphism (and so Lorentz) invariant. Are you talking about a symmetry of a spacetime, solution of the equations of motion of general relativity (the Einstein equations)? Defined how? Is it a change of coordinates and the corresponding change of metric components? In this case, a spacetime is trivially invariant (a change of coordinates cannot have any physical effect). Or is it an active diffeomorphism in which one keeps using the same coordinate system but moves points to new points and then compares the spacetime metric on the old set of points with the spacetime metric on the new set of points? This is the kind of symmetry the authors mention in their paper. When the authors say "spacetime without symmetries", they mean without this latter kind of symmetries. For example, Minkowski (flat) spacetime has Poincaré symmetries of this kind (e.g. it is translation invariant, but also rotation and boost invariant). The curvature at any point of the spacetime is the same. Now, imagine bending Minkowski spacetime in an arbitrary way. The result, in general, will not have translation, rotation and boost symmetry, and the curvature at any 2 arbitrary points will be, in general, different.

The thing is that I'm having trouble trying to understand how is the Poincaré/Lorentz and diffeomorphism symmetries crucial for General Relativity while there can be solutions to it corresponding to spacetimes which violate them at the same time...

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    $\begingroup$ There is no disagreement. All of your questions have been repeats of each other, and seem to stem from your misunderstanding between `physical symmetries' and gauge/diffeomorphism/local-Lorentz invariance. This type of thing will be covered in almost every GR textbook. $\endgroup$
    – Eletie
    Commented Feb 24, 2023 at 20:24
  • $\begingroup$ @Eletie the thing that I can't wrap my mind around is how can Poincaré/Diffeomorphism/Lorentz invariances are crucial for general relativity and yet there seem to be solutions to Einstein's equations corresponding to spacetimes that violate them. It's seems contradictory. How is it that the theory itself doesn't distinguish between coordinate systems, but then specific solutions of the theory (specific spacetimes) can very much distinguish between coordinate systems? $\endgroup$
    – vengaq
    Commented Feb 25, 2023 at 3:01
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    $\begingroup$ There is nothing contradictory at all. No solutions violate diffeomorphism invariance, nor local Lorentz invariance. This is the thing you need to study and understand. Any solution can be written in different coordinates, and this is one of the first things you learn when studying GR. E.g., take the FLRW solution in Cartesian vs spherical coordinates. They describe the same physics. But spacetimes have physical symmetries which are called isometries. The isometry group of flat spacetime is the group of Lorentz transformations. FLRW has less physics symmetries, so it's isometry group.. $\endgroup$
    – Eletie
    Commented Feb 25, 2023 at 9:04
  • $\begingroup$ Isn't the full Lorentz group. However, we're talking about isometries here, solutions to $\phi^* g = g$. These are real physical symmetries, not gauge symmetries (which are coordinate transformations). The gauge symmetry of GR is the group of all coordinate transformations, and this is a separate matter. This is merely the statement that GR is a theory which can be written in any choice of coordinates. Local Lorentz invariance is related to the fact that locally at each point, the tangent space is Minkowski: i.e., on small scales GR reduces to SR. $\endgroup$
    – Eletie
    Commented Feb 25, 2023 at 9:08
  • $\begingroup$ If these concepts are too unfamiliar, I'd suggest postponing your thoughts about them until you've studied it more formally. Or take some time to learn more about gauge symmetries vs physical symmetries (isometries). $\endgroup$
    – Eletie
    Commented Feb 25, 2023 at 9:10

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