Signature, boundary conditions and topology

[lucas_]

It is said that the metric tensor in GR is generally covariant and obey diffeomorphism invariance.. but the signature, boundary conditions and topology are not. What would be GR like if these 3 obey GC and DI too? Is it possible?

 

[PeterDonis]

but the signature, boundary conditions and topology are not.

Can you give a reference for this? It doesn't look right to me. Certainly the metric signature is invariant; and once you've chosen a given spacetime, the topology is invariant too AFAICT. As for boundary conditions, calling them "invariant" or "not invariant" doesn't even seem right; they're inputs to the problem, not results you get from solving it.

 

[lucas_]

Can you give a reference for this? It doesn't look right to me. Certainly the metric signature is invariant; and once you've chosen a given spacetime, the topology is invariant too AFAICT. As for boundary conditions, calling them "invariant" or "not invariant" doesn't even seem right; they're inputs to the problem, not results you get from solving it.

reference:
https://www.physicsforums.com/threads/emergent-gravity-spacetime.393318/
message #10 (author: tom.stoer)

"So it's not that you want to get rid of background independence but that you want to enlarge it.

You are right, there are aspects like the signature, boundary conditions and topology that are fixed. If you eliminate spacetime from the setup of your theory (like you do it in LQG) you automatically get rid of some of these fixed aspects (they may still be hidden in the theory)."

 

[PeterDonis]

reference:
https://www.physicsforums.com/threads/emergent-gravity-spacetime.393318/
message #10 (author: tom.stoer)

This isn't talking about GR, it's talking about quantum gravity theories. Your OP referred to GR. In GR, as I said (and as the post you linked to also says, you even quote it), the signature, boundary conditions, and topology are fixed. So the answer to the question in your OP is that, if those things were fixed, GR would be exactly like it is now, since those things are fixed in GR as it is now.

 

[lucas_]

This isn't talking about GR, it's talking about quantum gravity theories. Your OP referred to GR. In GR, as I said (and as the post you linked to also says, you even quote it), the signature, boundary conditions, and topology are fixed. So the answer to the question in your OP is that, if those things were fixed, GR would be exactly like it is now, since those things are fixed in GR as it is now.

Huh. You stated two message above while we were talking of pure GR that "Certainly the metric signature is invariant; and once you've chosen a given spacetime, the topology is invariant too AFAICT." And now you are stated they were really fixed. So are they not fixed (earlier you said) or fixed (above)? Please decide. We are talking of pure GR. Thanks.

 

[PeterDonis]

You stated two message above while we were talking of pure GR that "Certainly the metric signature is invariant; and once you've chosen a given spacetime, the topology is invariant too AFAICT." And now you are stated they were really fixed. So are they not fixed (earlier you said) or fixed (above)?

"Invariant" and "fixed" are the same thing in this context; they both mean "unchanged under coordinate transformations on the same spacetime". In other words, basically the same thing that you meant by "generally covariant and obey diffeomorphism invariance".

The only reason I put in the qualifier "once you've chosen a given spacetime" regarding topology is that different spacetimes (i.e., different solutions of the Einstein Field Equation) that are used in GR can have different topologies, whereas they all have the same metric signature. But mathematically, you can have different solutions of the EFE with different metric signatures as well. The ones that don't have a Lorentzian signature simply aren't used for practical purposes in GR.

 

[lucas_]

"Invariant" and "fixed" are the same thing in this context; they both mean "unchanged under coordinate transformations on the same spacetime". In other words, basically the same thing that you meant by "generally covariant and obey diffeomorphism invariance".

The only reason I put in the qualifier "once you've chosen a given spacetime" regarding topology is that different spacetimes (i.e., different solutions of the Einstein Field Equation) that are used in GR can have different topologies, whereas they all have the same metric signature. But mathematically, you can have different solutions of the EFE with different metric signatures as well. The ones that don't have a Lorentzian signature simply aren't used for practical purposes in GR.

No. "Invariance" in their discussions means background independent while fixed means background depending. According to a certain Atty:

Well, GR is background independent in the sense that the metric is dynamic. Any theory of quantum gravity must produce have some classical limit whose solutions are those of general relativity. However, GR has elements that are not background independent in the larger sense of the term, for example, the signature.

So the signature is not background independent. What would it take to make the signature background independent and what would happen to GR if this were done?

 

[PeterDonis]

"Invariance" in their discussions means background independent while fixed means background depending. According to a certain Atty:

I don't see how you get this out of what you quoted. He doesn't use either of the words in question ("invariance" or "fixed") at all, let alone define them the way you are defining them. If what you're actually concerned about is what is "background independent", that wasn't at all clear from your previous posts. We can't tell what you want to know if you don't use words that describe it.

For the record, "invariant" does not mean "background independent". "Invariant" means what I said it meant in my previous post, and invariance has nothing to do with the issue of background independence, which is a separate question. Background independence is about what things emerge dynamically in a solution to the underlying equations, versus what things are specified before the solution is derived. *(See below.) Invariance refers to what things are independent of the choice of coordinates within a particular solution--i.e., after it's already been derived. At that point, background independence is no longer an issue; you already know what solution you're working with, and how you arrived at it is immaterial. Also, at that point, the metric signature is an invariant, just like anything else that is independent of the choice of coordinates; the fact that the signature was specified in advance instead of being derived as part of the solution is immaterial. (This illustrates, btw, why "invariant" and "background independent" are not the same thing; the signature is background-dependent, but it is also invariant.)

(I see on re-reading the tom.stoer quote you gave in post #3 that he does use the word "fixed" to basically mean "background dependent". However, I was using it in post #4 to mean the same as "invariant", because I thought you were asking about general covariance/diffeomorphism invariance, since that's what you referred to in your OP. As noted above, "invariant" does not mean "background independent".)

the signature is not background independent.

Meaning, in GR, it does not emerge dynamically from a solution to the underlying equations, the way the spacetime geometry does. Yes, that's correct; in GR, we only use solutions with a particular signature, the one corresponding to a geometry that is locally Minkowski. That's because, physically, locally Minkowski geometry is what we actually observe. Mathematically, the Einstein Field Equation has solutions with other signatures; GR just doesn't use them to model anything physical.

What would it take to make the signature background independent and what would happen to GR if this were done?

I don't know. I don't think this is really a GR question, because GR only deals with the classical limit of whatever underlying quantum gravity theory turns out to be the right one. In the classical limit, a locally Minkowski metric signature is an observed fact, and as above, we pick which mathematical solutions we use for physical models in GR based on that observed fact. I'm not aware of any mathematical solutions with a signature that changes from point to point; I'm not sure you could even model that in a framework like GR. In an underlying quantum gravity theory, of course, "spacetime" is supposed to be an emergent feature arising from something else, but questions about that belong in the Quantum Physics forum, or perhaps the Beyond the Standard Model forum; they can't be answered in the context of GR.