Is there a possibility that general covariance be an approximation or violated ?
to me that seems like an interesting question. I hope other people will give some discussion.
just to be as inclusive as possible, would you talk a bit about what "general covariance" MEANS
I know what a smooth or differentiable manifold, a continuum, is.
I know what a smooth invertible map is, a DIFFEOMORPHISM.
Any time you give me a smooth manifold I can tell you what the diffeomorphisms are that map that manifold onto itself.
So touqra, would you say in those terms what "general covariance" means?
What kind of setups or objects can be general-covariant?
and then let's think about what it would look like for that kind of covariance to fail.
but meopemuk, that gets us into Independent Research territory! We can't discuss that here, we'd all have to go over into the Indy section.
Just for starters, somebody tell us what GenCov means and let's think what it would be like if it were violated. We don't have to get into anyone's private ideas in order to do that.
I think that for a mathematician, GenCov means "DIFFEOMORPHISM INVARIANT"
and it is talking about the relation of Matter to Geometry
Please any knowledgeable person (touqra, other..) correct me if I am wrong, but what it says is that
If the Geometry fits the Matter, then you can stir things around with a smooth (even nonlinear, quite general) map, so that the Matter is now in a different place and the geometrical features are reorganized and the Geometry will STILL fit the Matter.
the disposition of the matter determines the SHAPE of the geometry, and you can stir things around anyway you please with a diffeomorphism (aka smooth map) and the NEW disposition of the matter will still be determining the NEW shape of the geometry by exactly the same equation (Einstein's equation) as it did before.
Therefore one says that Einstein's equation is "diffeomorphism invariant".
does anyone want to elaborate? Correct what I said? Add some illuminating detail?
This topic has been discussed for years starting with Kretschmann and Einstein in 1917, and continuing up to Smolin and Polchinski this year. Closely related topics include Lorentz invariance, which Bee investigates, active versus passive gauge invariance, coordinate independence and background independence. If you Google “general covariance” and look at the first six or seven entries, you will get a lot of good information.
I particularly like the appendix to this one:
At least one paper on the arXiv is titled “Violating General Covariance” and tries to relate it to Dark Energy and Dark Matter.
The question that goes back to Kretschmann is whether general covariance is merely a mathematical coordinate requirement which is physically vacuous. The same issue arises with regard to passive gauge invariance. On the other hand, active gauge invariance and background independence are claimed to be physically meaningful and hence able to be experimentally violated. But IIRC, Smolin and Polchinski could not agree on the meaning of background independence.
If you can find someone to shed more light on this, I would be most eager to try to understand it.
Thanks for the comment! Can you suggest to me how to picture an experimental violation of background independence?
As you recall it, how did Smolin and Polchinski's concepts of background independence differ?
I'd rather hear your answer but for what it's worth I will interject what I recall.
I think it turned on what you say a THEORY is. You can say that a theory is a human artifact that is for practical purposes identical to its FORMULATION. (the mathematical formulation is what one derives predictions from and what one has notions about like what is the domain of applicability)
And so one can see in a straightforward manner whether the formulation of a theory uses a PRIOR FIXED GEOMETRY at any point. If it doesnt, the formulation is independent of any background geometry. And the typical example is GR.
That is just one POV. But at least with that POV I don't understand how one can say Nature is or is not B.I. because B.I. is a property of human artifacts, namely mathematical models.
You can't transfer that property to Nature except in a vague sense that it is or is not adequately describable by B.I. theories.
It strikes me that this is too simple to be what you are talking about. Too trivial or something. So I would really like to hear what you remember about the issue and how you picture an experimental violation of either B.I. or of Diffeomorphism Invariance.
Violating Background Independence
My memory is not that good, so I googled. The last directly relevant quote I found was this one by Joe Polchinski, which came from this Cosmic Variance thread:
Background independence. I think people are a bit tired of the who-is-more-background-independent argument, since it seems to come down to definitions. Let me put things in physical terms. As you say, suppose that the strong form of Maldacena duality is true. This would mean that we can consider a box as large as we want - a light-year, 106 light-years, with an arbitrarily small negative cosmological constant, and AdS/CFT provides a complete construction of quantum gravity within that space. This would include: the formation and decay of (nonsupersymmetric) black holes; graviton scattering at hyper-Planckian energies; physically continuous transitions from one topology, through a quantum state with no geometric interpretation, to a different topology; states where a submanifold of spacetime has a noncommutative geometry; states with a variety of apparent geometric singularities, where the physics is nonsingular. All of these, and many others with a variety of geometries and topologies (you can put a lot in an AdS box), and arbitrary quantum superpositions of them, can be identified in the gauge theory, and so are described algorithmically by the duality. It may not include spaces with interesting cosmologies, or with an effective positive cosmological constant. You call this a very weak and limited form of background independence.
Even here you are blowing things out of proportion: your reply refers five times to the “global symmetry algebra,” but almost immediately after the original work of Maldacena, the duality was extended to systems with reduced symmetry, or none. Your own PI colleagues, Alex Buchel and Rob Myers, have made important contributions to this subject, and I note also the series of papers by Hertog and Horowitz on strongly time-dependent boundary conditions.
A second physics point concerns the constraints. It is not that I am ignorant of the conventional wisdom here, I am challenging it. You believe that the large Hilbert space in which the constraints act is necessary in order to describe all possible backgrounds of quantum gravity. No, only the much smaller set of states that satisfy the constraints is needed. The larger space may play a useful auxiliary role, but it is not physical: the universe cannot be in such a state, and observables must keep the system within the physical subspace. So what are these larger spaces for? One thing we have learned, from emergent gauge theory, is that they are not necessary: one can start from a system with no constraints, only physical variables, and the constraints are needed only to describe the classical limit efficiently. We have learned a similar lesson from dualities such as AdS/CFT: these larger spaces are very different in different classical limits, they are not intrinsic to the quantum theory. Thus, all this focus on constraints is putting effort into something that is unphysical and actually intrinsic to a certain classical limit.
Me again. Both Joe and Lee Smolin posted several more times in this thread, but never got back to BI explicitly. The conversation veered off to cover the KKLT paper and the landscape, physics sociology and the importance of rigor in physics, with Lee championing rigor and Joe holding out for physical insight. This rigor vs. insight issue seems to also apply in some measure to the paragraphs above.
Let me first say that this is very much not my area of expertise, and if you can see a strong reason for preferring one or the other of these two viewpoints, you’re a better man than I am, Gunga Din! But plunging on, to me it appears that the remaining differences hinge on the importance of (getting rid of) global symmetries, and the size of the set of constraints.
Let me also say I have heard both Joe and Lee speak several times, (Lee more than Joe) and have asked them both questions, either after their talks or in hallways at APS meetings (or book tours) or similar places. I am very impressed with both of them, and consider myself also to be pretty neutral between Strings and Loops, basically hoping that at least one of them succeeds. So I don’t have an opinion on this point, other than that it is a very difficult point, not yet fully resolved. I don’t have the technical chops to go any deeper.
Coming back to something like the original question, violating general covariance (GC), now modified to violating BI, it appears that a global symmetry, as opposed to a local symmetry, might be said to be such a violation. Or just possibly some unjustified constraint.
I can think of one simple example to propose: a manditorily flat universe. This is a very old idea, and one very much out of favor right now. As I understand it, a flat universe is not enough to violate GC or BI because the universe could be flat by accident or only locally flat due to inflation, as is now part of the standard model of cosmology. In order to violate BI, you must also have some kind of preferred frame effect or some other form of experimentally visible form of rigidity not allowing the flatness to be merely accidental. Of course preferred frame effects are ruled out to quite high precision, so you would have to look for very small effects, or else some subtler form of violation. So perhaps this is an unsatisfying example of BI violation.
I am definitely trying to come up with a physical, experimentally observable violation, not merely a mathematical one. With regard to GC, this may be either impossible or a matter of definitions. With regard to BI, it is generally agreed that special relativity and perturbative string theory are BD, ie not BI. GR on the other hand is generally agreed to be BI. (String field theory is at least claimed to be BI. That is part of the point of the argument between Joe and Lee, I think.) So an observation that violated GR, but supported perturbative string theory would be a good candidate for an observational violation of BI. Fifth forces, moduli fields (and associated quanta), and violations of the principle of equivalence are possibilities that have been discussed. In particular, I think this has revived interest in much more precise equivalence principle tests.
Take all of the above with a great big grain of salt, or check it with someone who knows more than I do.
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