Are there types of spacetime where no symmetries are valid?

[Suekdccia]

TL;DR Summary

Are there any types of spacetime where no symmetries, not matter how fundamental, would be valid?

We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no symmetries, no matter how fundamental, would be valid? Any special metric, geometry or shape?

 

[kimbyd]

I don't believe so. Considering only General Relativity, symmetries arise when you can shift the space-time coordinates but not impact the action. Coordinates are four numbers, while the action is one. Assuming the action can be an arbitrary function, at a minimum there must be three "conserved currents" which relate to symmetries of the system.

If you want to ask a deeper question about whether there can be a space-time where General Relativity doesn't apply because the symmetries it uses don't exist, I don't know how to even think about that.

 

[Orodruin]

Coordinates are four numbers, while the action is one. Assuming the action can be an arbitrary function, at a minimum there must be three "conserved currents" which relate to symmetries of the system.

I don’t think I understand what you are trying to imply here. The action is a functional that integrates over spacetime, which is parametrised by four coordinates that are integrated over. The extremisation is performed in the functional sense. The action is not a function of four parameters. It is a functional taking several fields on spacetime as its input.

Any spacetime will satisfy the Einstein field equations for some stress energy tensor (just set the stress energy tensor proportional to the Einstein tensor) regardless of having symmetries or not. Then one can question what kind of matter that would require and what energy conditions would be satisfied.

 

[PeterDonis]

are there any types of spacetime where no symmetries, no matter how fundamental, would be valid?

Of course. Any spacetime with no Killing vector fields has no symmetries.

 

[PeterDonis]

If you want to ask a deeper question about whether there can be a space-time where General Relativity doesn't apply because the symmetries it uses don't exist, I don't know how to even think about that.

Easy: you think about a spacetime with no Killing vector fields.

 

[Orodruin]

Easy: you think about a spacetime with no Killing vector fields.

Are you saying GR does not apply to such a spacetime or that that is a way of thinking about a spacetime without symmetries?

 

[PeterDonis]

Are you saying GR does not apply to such a spacetime or that that is a way of thinking about a spacetime without symmetries?

The latter.

 

[Orodruin]

The latter.

Then I agree. Since the quote was also containing something about GR not being applicable I just thought it would be good to clarify.

 

[Ibix]

Easy: you think about a spacetime with no Killing vector fields.

Wouldn't a spacetime with no Killing fields still have local Lorentz symmetry? My reading (and I guess @kimbyd's) of the OP was that it asked for such symmetries to be excluded.

 

[PeterDonis]

Wouldn't a spacetime with no Killing fields still have local Lorentz symmetry?

Yes, since that's part of the definition of a spacetime in GR. However, this would only be a local symmetry; no spacetime except flat Minkowski spacetime has global Lorentz symmetry.

(Note that such a spacetime could also have matter fields that obey gauge symmetries, such as EM fields; gauge symmetries are also mentioned in the OP. However, to me that's not a symmetry of the spacetime, it's a symmetry of the matter. And there is no requirement that matter fields must obey gauge symmetries.

Note also that matter can break the local Lorentz symmetry since its 4-velocity field will define a preferred timelike vector at each event within the matter. So one could imagine a spacetime with no Killing vector fields, filled with matter that satisfied no gauge symmetries; this spacetime would indeed have no symmetries even if we interpret "symmetries" in the most expansive sense we could.)

 

[Orodruin]

Note also that matter can break the local Lorentz symmetry since its 4-velocity field will define a preferred timelike vector at each event within the matter. So one could imagine a spacetime with no Killing vector fields, filled with matter that satisfied no gauge symmetries; this spacetime would indeed have no symmetries even if we interpret "symmetries" in the most expansive sense we could.)

Just to note that this will be evident already once the stress energy tensor has been computed for an arbitrary spacetime as mentioned in #3. Having a 4-velocity field still leaves local rotations around it, but a stress energy tensor with all eigenvalues different will not.

 

[kimbyd]

Easy: you think about a spacetime with no Killing vector fields.

I don't think it's that simple. See here:
https://physics.stackexchange.com/q...ry-isometry-have-an-associated-killing-vector

Furthermore, I think this depends upon what you mean by "no symmetries". GR conserves stress-energy, which stems from a symmetry of the Einstein field equations themselves independent of any metric. This is what I was, perhaps clumsily, trying to get at.

 

[PeterDonis]

I think this depends upon what you mean by "no symmetries".

Yes, agreed; it's possible to interpret "symmetries" to include things that don't correspond to Killing vector fields.

 

[vanhees71]

Here, it's gauge invariance: The (local) conservation of energy and momentum, i.e., $\vec{\nabla}_{\mu} T^{\mu \nu}=0$ follows from Einstein's field equations, $G_{\mu \nu}=\kappa T_{\mu \nu}$ from the Bianchi identity of the Einstein tensor, which is due to the gauge invariance, i.e., the invariance of GR under arbitrary changes of the coordinates (general covariance; diffeomorphism invariance).

 

[ohwilleke]

Summary: Are there any types of spacetime where no symmetries, not matter how fundamental, would be valid?

We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no symmetries, no matter how fundamental, would be valid? Any special metric, geometry or shape?

It depends to some extent of what boundaries you place on the definition of "spacetime".

You can create a mathematical structure that has or lacks any symmetries you want and call it a "spacetime".

But, if you want your mathematical structure to have any resemblance whatsoever to what we usually mean when we are talking about "spacetime" then the outer boundaries of what is necessary to be consistent with that understanding of what a "spacetime" is will set limits on what properties your mathematical description of spacetime may have and still be consistent with that definition.

For example, any reasonable definition of spacetime really ought to encompass the idea that it can contain a particles or objects that can move from one place in the spacetime to another place in the spacetime according to well defined physical laws.

Most of the other posts in this thread have implicitly assumed boundaries on what could legitimately be called a spacetime that are quite a bit more demanding and restrictive than what I believe an ordinary common sense understanding of the word spacetime could plausibly include.

Certainly, it is absolutely possible to have a theoretical construct that fairly fits a common sense understanding of what a "spacetime" is that could have symmetries which are different in some way from the symmetries that we attribute to the real world in general relativity and the Standard Model. The core theories of physics are not unique in provide potentially rigorous models of the way that any conceivably imaginable possible universe would have to be.

For example, one could easily devise a hypothetical spacetime in which general and special relativity do not exist and there is no fixed speed of light, that is otherwise fairly familiar. We call that spacetime a real four dimensional Euclidian space in which Newtonian mechanics applies.

Likewise, one would imagine a theoretical construction that has symmetries different from the commonplace gauge symmetries that we believe really govern the physical world, that is still consistent with our common sense understanding of what a "spacetime" is.

Noether's theorem is another matter, however, because that isn't a symmetry itself. Instead, Noether's theorem is a meta theory about symmetries. Noether's theorem basically states in a very general sense that has very few conditions that must be satisfied for it to apply, to oversimplify, that in a law of physics, a conserved quantity always has a corresponding symmetry, and that a symmetry always has a corresponding conserved quantity. So, what Noether's theorem means is that if you have a spacetime in which there are no symmetries, then you must also have a spacetime in which there are no conserved quantities.

It would stretch the common sense meaning of the concept of "spacetime" to include within that definition, a class of possible theories that has both no symmetries of any kind, and as a consequence of Noether's theorem, no conserved quantities either.

While I am not able to rigorous prove that it is impossible to do so (indeed, you can't even begin to rigorously come to any conclusion until you pin down a definition of what must be true for something to be a spacetime), I have a very hard time imagining a mathematical construct of that kind that could reasonably be described as a spacetime. I would be willing to entertain a suggestion, from someone more clever than I, about how that could be possible, but I just don't see it.

 

[Newton Lopes]

I don’t think I understand what you are trying to imply here. The action is a functional that integrates over spacetime, which is parametrised by four coordinates that are integrated over. The extremisation is performed in the functional sense. The action is not a function of four parameters. It is a functional taking several fields on spacetime as its input.

Any spacetime will satisfy the Einstein field equations for some stress energy tensor (just set the stress energy tensor proportional to the Einstein tensor) regardless of having symmetries or not. Then one can question what kind of matter that would require and what energy conditions would be satisfied.

Good! now I get it