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I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics:

Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity gets involved. While in SR, quantities maintain Lorentz (or Poincare) symmetry via Lorentz (or Poincare) transforms, in GR they obey general covariance which is symmetry under arbitrary differentiable and invertible transformations (aka diffeomorphism). If a spacetime was not smooth, and didn't allow local Lorentz symmetry, it would break the principle of equivalence which is the bedrock assumption in GR.

I would like to know if there are possible spacetimes where they would not be smooth. The only problem is that this usually involves singularities. Are there models or metrics of non smooth spacetimes that would be compatible with what we currently know in physics but that don't necessarily involve singularities?

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  • $\begingroup$ This isn’t really a good answer because it’s rather speculative, so I’ll leave it in the comments, but there has often been discussion of spacetime being jagged and discontinuous at subatomic scales. There is no widely accepted theory of quantum gravity however, so what this means or if it’s even true is still an active and hard to probe area of study. $\endgroup$
    – Justin T
    Mar 25, 2023 at 22:35

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It's a basic assumption of general relativity that a tangent space must exist at every point on the manifold. Therefore, the metric needs to be $C^1$. But "smooth" normally means at least $C^2$.

The stress-energy tensor is a function of second derivatives of the metric, which suggests that the metric needs to be $C^2$, but there are actually important exact "solutions" to GR that fail to be $C^2$, such as

  • solutions describing a uniform-density spherically symmetric gravitating body in vacuum (such as the Schwarzschild interior and Schwarzschild exterior solutions grafted together at some $r$)

  • solutions describing a ball of dust collapsing into a black hole (FLRW interior and Schwarzschild exterior)

  • swiss-cheese cosmologies (FLRW exterior and Schwarzschild, or non-vacuum spherically symmetric, interior)

These solutions are $C^\infty$ except on the 3D hypersurfaces where the regions are grafted together, but only $C^1$ on those boundaries.

The same thing happens in Newtonian gravity, where Poisson's equation ($\nabla^2\phi = -4πρ$) suggests that $\phi$ must be twice differentiable, but, e.g., the potential of a uniform-density spherically symmetric body ($\sim r^2$ on the inside, $\sim -1/r$ on the outside) is only once differentiable at the boundary, and that doesn't really cause problems in practice.

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  • $\begingroup$ but then, in these solutions, would we have a non-smooth region of space where symmetries like the Lorentz symmetry are broken? @benrg $\endgroup$
    – vengaq
    Mar 27, 2023 at 12:23
  • $\begingroup$ @vengaq "Smooth" normally means at least $C^2$, but the local Lorentz symmetry only needs $C^1$, so yes to the first half of your question but no to the second. I added a clarification. $\endgroup$
    – benrg
    Mar 27, 2023 at 16:29

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