lucas_ said:
"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".)
lucas_ said:
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.
lucas_ said:
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.
