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

• I
• Suekdccia
In summary, there is currently no evidence or accepted models for non-smooth spacetimes that do not involve singularities. The principle of equivalence in general relativity and the requirement for local Lorentz invariance make it incompatible with non-smooth spacetimes. While there may be discussions and theories about non-smooth metrics, there is no widely accepted concept of a non-smooth spacetime that is compatible with known physics.
Suekdccia
TL;DR Summary
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?

Suekdccia said:
I found this statement in a discussion

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

Last edited:
Hornbein
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.

timmdeeg

## 1. What is a non-smooth metric for spacetime?

A non-smooth metric for spacetime refers to a mathematical model that describes the curvature of spacetime without any singularities. It is a metric that is continuous but not differentiable at certain points, meaning that it does not have a well-defined slope or tangent at those points.

## 2. Why is the existence of non-smooth metrics for spacetime important?

The existence of non-smooth metrics for spacetime is important because it challenges our understanding of the structure of spacetime and the laws of physics. It also has implications for the behavior of matter and energy in extreme conditions, such as near black holes.

## 3. How are non-smooth metrics for spacetime related to singularities?

Non-smooth metrics for spacetime are related to singularities in that they provide an alternative mathematical description of the curvature of spacetime without relying on the concept of singularities. This allows for a more complete and consistent understanding of the structure of spacetime.

## 4. Can non-smooth metrics for spacetime be observed or measured?

Currently, there is no experimental evidence for the existence of non-smooth metrics for spacetime. However, some theories, such as loop quantum gravity, predict the existence of non-smooth structures in spacetime that may be observable in the future with advanced technology.

## 5. What are the implications of non-smooth metrics for spacetime on our understanding of the universe?

The existence of non-smooth metrics for spacetime challenges our current understanding of the universe and may lead to new insights and discoveries in the field of theoretical physics. It also has potential implications for the behavior of matter and energy in extreme conditions, which could have practical applications in fields such as astrophysics and cosmology.

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