Weak field approximation only? No.
On Jun 6, 2005, at 9:04 AM, ... wrote:
It is good, I agree fully with these ideas.
Maybe I am wrong, but it seems to me that the equation
eu^a = Iu^a + Bu^a
has to be valid only for the weak field approximation.
No definitely not - it is strong field and completely non-perturbative in origin. There is nothing in the principle of local gauge invariance that restricts it to weak field. That would mean QED & Yang-Mills were also a weak field approximations since Au^a plays same role in QED & Yang-Mills as Bu^a does in GR.
All the gauge potentials emerge from simply localizing rigid global symmetries of the dynamical action of some initial action. In the case of QED it is the action for the Dirac electron spinor field. In the case of GR it would be the trivial pre-Einstein-Hilbert action symmetric under rigid T4, i.e. zero action density ~ R = 0 with a Lagrange multiplier forcing Ruvwl = 0 identically I suppose. The idea is that the Einstein-Hilbert action density ~ (c^4/G) R =/= 0 emerges from B =/= 0.
In addition from ODLRO B^a ~ LpP^a/ih i.e. spontaneous broken T4 vacuum symmetry on top of local gauging of T4 to Diff(4).
Pa generates T4
Please, decode also such terms as ODLRO.
Vast literature from the 1940s or so. Google: Off-Diagonal-Long-Range-Order Oliver Penrose, Lars Onsager, density matrices. Much discussion of it in "A Career in Theoretical Physics" P.W. Anderson (World Scientific).
e.g. let psi be a 2nd quantized destruction boson operator at x - say in NR quantum many body problem in Fock occupation number space for example.
<0|psi*(x')psi(x)|0> is first reduced density matrix in the ground state |0> of the many body system.
ODLRO is the c-number factorization
<0|psi*(x')psi(x)|0> ---> <0'|psi*(x')|0'><0'|psi(x)|0'> + <0'|psi*(x')psi(x)|0'>
In a ground state phase transition from |0> ---> |0'>
|0> is unstable micro-quantum ground state
|0'> is more stable macro-quantum ground state
|0'> is a SQUEEZED GLAUBER state with two complex parameters z & s.
z gives the displacement of the Gaussian in the complex plane (phase space of Wigner function) and s gives the squashing of the degenerate Gaussian circle to an ellipse in phase space of the two quadratures psi* + psi & i(psi* - psi). Squeeze the normal vacuum one way to get repulsive dark energy of negative zero point pressure and the other to get attractive dark matter of positive zero point pressure? (Half-baked speculation of mine). s has the Bogoliubov pair correlations between different I & IV sections of the Penrose conformal diagram (e.g. Schwarzschild case) outside event horizon related to Unruh-Hawking radiation seen by accelerated observers.
<0'|psi(x)|0'> is the LOCAL macro-quantum COHERENT order parameter that obeys a NONUNITARY Landau-Ginzburg equation with SIGNAL NONLOCALITY & no Born probability rule. Micro-quantum theory with SIGNAL LOCALITY is violated when we have ODLRO!
B^a ~ (LpP^a/ih)arg<0'|psi(x)|0'>
B^a = Bu^adx^u
Lp^2 = hG/c3
This is spontaneous breakdown of real quanta ground state ("vacuum" for virtual quanta) symmetry for some group G of the dynamical action of the many-quanta system.
In the superconductor G ---> U(1)em internal fiber group for the electron field.
Rest mass of photon inside the super-conductor is precursor of rest mass of lepto-quarks in the U(1)xSU(2)xSU(3)xT4 local gauging with spontaneous broken vacuum symmetry not only in INTERNAL SU(2)weak force, but EVEN MORE IMPORTANTLY, in the T4 space-time group!
In Einstein relativity G ---> T4 space-time translation group that requires zero tidal geodesic deviation curvature for it to be the symmetry group of the, in this case, trivial constrained geometrodynamic action ZERO for the Einstein-Cartan tetrad field.
In special relativity SR, the tetrad field is the trivial non-dynamical Iu^a identity Kronecker-delta. In general relativity GR, as a local gauge theory, the tetrad field acquires a DYNAMICAL WARP FIELD Bu^a, i.e. now it is
eu^a = Iu^a + Bu^a
eu^a is a Diff(4) tensor under Diff(4) localization of T4 in the u index (base space)
eu^a is a O(1,3) tensor in the a index (tangent fiber space that still admits P^a) generated by the Sab Lie algebra of 6 space-time rotations whose conjugate phases are the "Ricci rotation coefficients".
Shipov's torsion theory, IN ADDITION, locally gauges O(1,3) to get 10-dim "space" of the oriented "point test particle" rather than the 4-dim "space-time" of the non-oriented point test particle. This adds another field Su^a to the Einstein-Cartan tetrad
eu^a = Iu^a(Minkowski non-dynamical) + Bu^a(tidal curvature disclination) + Su^a(torsion dislocation)
Where the EEP (Einstein Equivalence Principle) generalizes to
guv(curvature + torsion) = eu^a(Minkowski)abev^b
Geodesics in the extended geometry are not geodesics in the smaller geometry.