Undergrad Are there non-smooth metrics for spacetime (without singularities)?

[Suekdccia]

TL;DR

Are there non-smooth metrics for spacetime (that don't involve singularities)?

Are there non-smooth metrics for spacetime (that don't involve singularities)?

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?

 

[PeterDonis]

I found this statement in a discussion

Where? Please give a reference.

 

[Vanadium 50]

I'm trying to figure out what a non-smooth spacetime is supposed to be if it is not singular at the discontinuities.

 

[PAllen]

In GR, the primary definition of singularity is geodesic incompleteness. A point of spacetime where all derivatives are undefined while continuity exists must lead to geodesic incompleteness, since geodesics require satisfaction of a differential equation. So such points necessarily lead to spacetime singularities as defined in GR.

As to the question: "Are there models or metrics of non smooth spacetimes that would be compatible with what we currently know in physics", irrespective of singularities, the answer must be no. As your quote notes, local Lorentz invariance would be violated, and all currently accepted theories require this, and all data are consistent with this.