What do physicists mean by "local degrees of freedom"?

[willidietomorrow]

When physicists talk about a theory having local degrees of freedom, what is exactly meant by that statement? What are examples of theories with local degrees of freedom and what are examples of theories with no local degrees of freedom?

 

[Vanadium 50]

Why is this in Relativity? Can you clarify your question, particularly in the context of relativity?

 

[willidietomorrow]

Why is this in Relativity? Can you clarify your question, particularly in the context of relativity?

Hi sorry, you're right. I usually hear discussions about local degrees of freedom when talking about gravity and string theory, so I naively thought I might post it here. I'm not exactly sure what category in the forum might my question best.

 

[PeterDonis]

I usually hear discussions about local degrees of freedom when talking about gravity and string theory

If by "gravity" you mean "General Relativity", that would be on topic in this forum. But a discussion of string theory would be more suitable in the Beyond the Standard Model forum.

Can you give a specific reference?

 

[willidietomorrow]

Can you give a specific reference?

For example, this wikipedia page CGHS model - Wikipedia mentions " In 2+1D, general relativity becomes a topological field theory with no local degrees of freedom, and all 1+1D models are locally flat."

 

[PeterDonis]

For example, this wikipedia page CGHS model - Wikipedia mentions " In 2+1D, general relativity becomes a topological field theory with no local degrees of freedom, and all 1+1D models are locally flat."

The Wikipedia article claim doesn't really make sense to me as it is stated. All spacetimes in GR are "locally flat" in the sense of the equivalence principle--a small enough patch of any spacetime looks like a small patch of flat Minkowski spacetime. So I don't know what the article means by saying that all 1+1D models are "locally flat", as if models in higher dimensions were not. (The linked article on topological field theory has the same kind of issue: I don't know what it means by saying that in 2+1D vacuum spacetimes can be locally de Sitter or locally anti-de Sitter instead of locally flat; de Sitter and anti-de Sitter spacetimes are still locally flat in the sense I gave above.)

The topic of the article you linked to in general is really an "A" level topic, not an "I" level one, so you might not have the background knowledge to dig into it very deeply. What brought you to this topic? Was there something else you were researching when you came across it?

 

[atyy]

There are no local propagating degrees of freedom in 3D gravity, analogous to there being no gravitational waves in 3D.
https://ncatlab.org/nlab/show/3d+quantum+gravity

 

[PeterDonis]

local propagating degrees of freedom

Do you know of a mathematical definition of what "propagating" means in this context? I have seen the claim quoted above stated in words in multiple sources, but none of them have defined "propagating" mathematically.

 

[atyy]

Do you know of a mathematical definition of what "propagating" means in this context? I have seen the claim quoted above stated in words in multiple sources, but none of them have defined "propagating" mathematically.

Classically, it means that there are no gravitational waves because the spacetime has constant curvature. In the quantum case it means that the theory is a TQFT.
https://math.ucr.edu/home/baez/planck/node3.html
https://arxiv.org/abs/gr-qc/0409039

The nLab link in post #7 has some more references.

 

[PeterDonis]

Classically, it means that there are no gravitational waves because the spacetime has constant curvature.

A spacetime doesn't have to have constant curvature to not have gravitational waves. There are no gravitational waves in Schwarzschild spacetime or matter-dominated FRW spacetime, for example, but neither of those spacetimes have constant curvature.

The description in the Baez article you link to is a little different; Baez says:

If spacetime has 4 or more dimensions, Einstein's equations imply that the metric has local degrees of freedom. In other words, the curvature of spacetime at a given point is not completely determined by the flow of energy and momentum through that point: it is an independent variable in its own right. For example, even in the vacuum, where the energy-momentum tensor vanishes, localized ripples of curvature can propagate in the form of gravitational radiation. In 3-dimensional spacetime, however, Einstein's equations suffice to completely determine the curvature at a given point of spacetime in terms of the flow of energy and momentum through that point. We thus say that the metric has no local degrees of freedom. In particular, in the vacuum the metric is flat, so every small patch of empty spacetime looks exactly like every other.

In somewhat more technical language, I would rephrase this as: in 4 or more spacetime dimensions, the Riemann tensor contains more information than the Einstein tensor; the extra information, over and above the Einstein tensor, is in the Weyl tensor. But the Einstein tensor is the only piece of the Riemann tensor that is directly related to the local stress-energy content, via the Einstein Field Equation. So in 4 or more spacetime dimensions, knowing the degrees of freedom in the local stress-energy is not sufficient to know the entire spacetime geometry--that only tells you the Einstein tensor, not the Weyl tensor. The extra degrees of freedom in the Weyl tensor would be the local degrees of freedom of the spacetime geometry (as opposed to the local degrees of freedom of the matter, contained in the stress-energy tensor and which determine the Einstein tensor).

In 3 or fewer spacetime dimensions, however, the Einstein tensor contains all of the information that is in the Riemann tensor. There is no additional information contained in the Weyl tensor. So there are no local degrees of freedom of the spacetime geometry, over and above the local degrees of freedom in the stress-energy tensor. But this in itself would not require that the curvature be constant--the Einstein tensor/stress-energy tensor could still vary from point to point. There would just be no additional variation that wasn't captured in the Einstein tensor/stress-energy tensor.

 

[atyy]

Yes, that's right. The constant curvature happens for the vacuum 3D equations.

 

[willidietomorrow]

The topic of the article you linked to in general is really an "A" level topic, not an "I" level one, so you might not have the background knowledge to dig into it very deeply. What brought you to this topic? Was there something else you were researching when you came across it?

I'm primarily interested in getting the jist of what is meant by that. The way I came across was in a philosophy of physics seminar where the topic was briefly mentioned.