Remember the Future
On Mar 15, 2005, at 6:17 PM, firstname.lastname@example.org wrote:
After all, the observer's world line is itself a geometric object, and thus from a purely mathematical standpoint itself has a coordinate-independent definition. Yet we find that in GR each observer world line is at the same time associated with a particular choice of coordinates -- which seems somewhat paradoxical. A kind of duplicity.
Not at all. Fix on the Schwarzschild
ds^2 = (1 - 2rs/r)(cdt)^2 - (1 - 2rs/r)^-1dr^2 + r^2(dtheta^2 + sin^2thetadphi^2)
There are the trivial GCT's like going to isotropic coordinates, or to Cartesian.
They all correspond to a kind of gauge freedom in which the LNIF observers are still hovering at fixed r (Schwarzschild).
So what I mean is NON-TRIVIAL GCT's with TRIVIAL GCT's FACTORED OUT, i.e. a QUOTIENT SPACE. See Einstein's "HOLE PARADOX" before the gauge freedom was understood. Same thing happens in Yang-Mills (Faddeev).
OK, so you do make the distinction. But notice that every "non-trivial" GCT also has a parallel "trivial" definition.
I do not understand your sentence.
This is exactly what I mean by "duplicity". It's not simply that a subset of mathematically possible CSs is marked out as non-mathematical,
so much that a certain subset of mathematical CSs at the same time *also* have a physical interpretation as characterizing observer frames
I understand this one less than I understood first, which was not at all! ;-)
I.e. NON-TRIVIAL GCTs connect physically distinct OBSERVER MATRICES
= ALL GCTs/TRIVIAL GCTs
I am asking if, in your gauge field model, these dual aspects of "general covariance" -- one mathematical, and one not just mathematical but also physical -- can be separated?
Yes, it's the gauge freedom.
So your "gauge freedom" is due purely to coordinate generality? It is a trivial consequence of formal general covariance?
It's worked out quite rigorously in a lot of books. As I recall John Baez has a pretty good discussion of all this on his website. Look and see what Penrose says in Road. I am on the road to LA and have almost no books with me. I am using a direct thought-computer interface given to me by ET Black Ops as I drive looking like a Borg. :-)
As I understand it, in Feynman's non-Abelian spin-2 model, not just covariance but also equivalence was explained in terms of an underlying
Of course he was working in a flat background spacetime.
Yes, basic idea is in Feynman. He did original stuff for Yang-Mills I think.
A GCT is to ANOTHER MATRIX of LNIF guys firing their rockets in space in a different way.
Sure, of course I know this is how 1916 GR actually works.
That is the PHYSICAL MEANING of LOCAL COORDINATE CHARTS! Physics is more than math. It's MATH PLUS.
Yes, I agree.
But at the same time, how then do you separate mere coordinate-generality from the extra-mathematical question of physical relativity? Or do you think these are not distinguishable in 1916 GR?
Gauge freedom. The physics is a quotient structure. The trivial GCTs are an equivalence relation.
OK, so you do seem to be saying that your "gauge freedom" is entirely due to formal covariance, as opposed to EEP equivalence.
Local gauging of T4 IS Diff(4)! The compensating B field that gives REAL TIDAL WARPS is as real at the EM field!
OK. But what part of your "gauge freedom" is due purely to coordinate generality, and what part is due to "general relativity" (equivalence)?
Can you separate them? Can you even distinguish them?
Easy. B/gauge ~ -> Non-trivial GCT
B/gauge ~ done by Fadeev et-al in Yang-Mills theory.
In quantum field theory it's done with Feynman histories path integrals.
See A. Zee "Quantum Field Theory in a Nutshell" that must have it. I am on road.
OK, this looks good.
Hey what are those strange lights up ahead? :-)
But aren't there two things going on here? (1) coordinate generality; and (2) inertial-gravitational equivalence?
I don't know. That's too vague. Spell it out using Norton quotes.
In this context, "coordinate generality" is the purely mathematical requirement that the *form* of the laws of physics in a formally covariant physical theory be the same in every mathematically well-behaved spacetime CS.
~ equivalence relation expressing gauge freedom, which are the mathematically distinct forms that correspond to INVARIANT OBSERVER MATRICES M, i.e.
~ is the set of TRIVIAL GCTs U such that
M -> M' = UMU^-1 = M
i.e. [M,U] = 0
"Inertial-gravitational equivalence", on the other hand, is a *physical* stipulation that entails that physical laws governing gravitation-inertia are actually the same in every *observer frame of reference* ("general relativity").
This means that the local laws must be tensor/spinor equations relative to the relevant symmetry group G,
Physical ("active") symmetry group, yes.
What do you mean by "active" exactly? Everything is LOCAL, i.e. at FIXED PHYSICAL EVENT P, which is not same as MANIFOLD POINT p. Everything "real" (in physics) is LOCALLY COINCIDENT - that's in Norton citing Einstein. I use that idea!
"Physics is simple only when it's local." Wheeler
Keep it simple, but not too simple. (Einstein's Rule violated in Hal Puthoff' PV).
which in the special case of 1916 GR is simply
Diff(4)xO(1,3) in the tangent bundle.
In Shipov's theory it's something like
But formal covariance alone demands that physical quantities be represented by coordinate-invariant objects -- and all such invariant objects must be tensors or spinors of various ranks.
There is sloppiness here - in the literature. If you mean LOCAL tensor/spinors like Fuv EM field intensities as representations of OBJECTIVE F = dA, then OK. But some of these guys want everything to be GLOBAL integrals and that's obviously stupid contradicting COINCIDENCE idea cited above. Also GR measurements are always local! One finds nonlocal correlations after the fact. Local relative to scale of curvature that is.
Any nonlinear GCT when B = 0 is simply SIMULATED GRAVITY in sense of Landau & Lifshitz Ch 10 Classical Theory of Fields.
But such GCT's even in flat space-time need not be dynamically trivial, e.g. Sagnac effect in flat space-time with g0i.
OK. Although I understand this involves rotating *sources*, as opposed to frames. There's a distinction in GR (according to Pauli p 5).
NO, ROTATING SOURCES AS IN Tuv, or as in vacuum Kerr Metric is TORNADO SWIRL OF SPACE-TIME, i.e. REAL GRAVIMAGNETIC FRAME DRAG that even drags along LIFs! Nothing escapes. Resistance is futile.
Forget all that. In Sagnac mount the interferometer, ring lasers what have you on your 78 RPM turntable, and, in principle you will get an effect out in far space in free float. I mean base of turn table is freely floating and the table is turning.
So call this an INERTIAL GRAVIMAGNETIC FIELD, a FAKE ONE, like in Landau & Lifshitz.
Of course LOCALLY if you are a bit sloppy you cannot tell to first order. If you work hard enough you can tell, like with tidal curvature. Remember Einstein was a "Lazy Dog" - Minkowski said so.
Now, do you (or can you) distinguish here between mere mathematical "coordinate generality" and physical "EEP" equivalence?
I showed you above. The coordinate REPRESENTATION choice is a DYNAMICAL MATRIX of possible LNIF (and LIF) observers no holds barred - every possible PHYSICAL timelike motion.
Well, I suppose I'll have to take this as a tentative "no".
No, it's a YES, via moding out the trivial GCT's by the Faddeev type equivalence relation in the Yang-Mills B-representation.
OK. You now seem to have this resolved in your own mind.
Yeah, I knew it all the time in the Implicate Order. Plato's remembering FROM THE FUTURE.