Metric Engineering Investigations 1.2
Metric Engineering Investigations 1.2
On Dec 27, 2004, at 2:45 PM, firstname.lastname@example.org wrote:
OK, then can you explain your understanding of their distinction between "physical equivalence of all these reference frames", on the one hand, and general covariance of the laws, on the other? In your own words?
I have many times now. The distributions of detectors connected by GCTs - different regions of extended phase space of the detector distributions connected by the GCTs in same region of configuration space (within a given neighborhood of event P)
A set of Alice detectors Ai (i = 1 to N) and a set of Bob Bj (j = 1 to N) detectors pepper a neighborhood of even P in the overlap region that is the domain of the GCT transition function. Each set of detectors is a point in extended phase space including the (LC) at the position of each detector. Although the A & B detectors occupy the same point in configuration space, they occupy different points in phase space. Indeed, let the physical event be P with coincident (Ai,Bi) detectors be located at common events Pi close to P. We want to plot (LCA)i and (LCB)i as degrees of freedom, we can also plot the "velocities" relative to the micro-wave background in principle. This gives an extended "phase space" for the two detector configurations in a small neighborhood of the event/process P that is being simultaneously measured. Note that N = 1 is good enough. The GCT connects physically distinct reference frames in this sense.
Obviously the two configurations of detectors measuring the same event P are physically distinct. The GCT connecting these different points in phase space that share a common point in configuration space enable the objective comparison of raw data taken independently by both of them. Of course, this does not extend to quantum measurements on the same micro-quantum system. One can work with ensembles of course.
Back to 1.1 Newton's theory, L&L explicitly say that the idea of a uniform gravitational field is only an approximation in a limited region of space.
"The field of gravity of the Earth (over small regions, where the field can be considered uniform). Thus a uniformly accelerated system of reference is equivalent to a constant, uniform external field." p. 226 The sentence following that is garbled and incorrect, i.e. the incorrect sentence in my 4th revised English Edition is
"a nonuniformly accelerated linear motion of the reference system is clearly equivalent to a uniform but gravitational field" should be
"a nonuniformly accelerated linear motion of the reference system is clearly equivalent to a nonuniform gravitational field"
Limitations of the Galilean-Newtonian equivalence principle
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames."
Note that this is only true in the global inertial frames of Newton's theory. It is not at all true in the local inertial frames of Einstein's theory where the "gravitational field" as "Levi-Civita" connection vanishes. This is analogous to an electromagnetic gauge transformation at FIXED x
A^i(x) -> Ai'(x) = A^i(x) - Grad^if(x) = 0
Note that the "curvature" Curl A^i(x) is not affected since CurlGradf(x) = 0.
However, the global Wilson loop phase factor e^[i(e/hc)Closed Loop Integral A.dl] is the Bohm-Aharonov-Josephson nonlocal quantum observable, e.g. fringe shift at crossing point of an electron interferometer where the electrons only travel through regions where CurlA = 0. There should be a gravity analog to this in Einstein's general relativity where Closed Loop integral (LC)ds is a dimensionless quantum phase difference over two alternative indistinguishable Feynman micro-quantum histories. There may also be a GIANT QUANTUM EFFECT since the local vacuum coherence order parameter must also be single-valued, i.e. path independent state function mod winding number N, i.e. 2piN. That is, Lp^2(Macro-Quantum Coherent Goldstone Phase),u is the distortion of the fabric of spacetime whose strain tensor is the curved part of the Einstein metric tensor in a given local coordinate patch, i.e.
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
A GCT comes from the generating function of a canonical transformation overlap transition function
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)'(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?