What is spin in a curved torsioned spacetime?
1. We want to directly couple the torsion field not only to spin S of matter but to J = L + S of matter. Most important is orbital angular momentum L-torsion field coupling and I know papers exist on this topic.
2. The association of the 4 intrinsically warped tetrad fields A(4)^a is to a kind of Yang-Mills field not the EM potential.
Note the EM potential is A(EM) = Au(EM)dx^u
no "internal Yang-Mills index" ^a which in our case transforms under space-time SO(1,3) rather than internal SU(2)weak or SU(3)strong with chiral breaking of mirror symmetry in the fundamental lepton-quark multiplets etc.
The 4 tetrad 1-forms A(4)^a transform as a SO(1,3) 4-vector and they are separately GCT scalar invariants, i.e.
A(4)^a = A(4)^audx^u
u-indices are GCT tensor indices.
The 6 torsion field spin-connection 1-forms S(1,3)^a^b = - S(1,3)^a^b are anti-symmetric second rank SO(1,3) tensors analogous to the spin 1 EM field tensor 2-form F(EM) = dA(EM).
Note that each individual S(1,3)^a^b component is a GCT scalar invariant.
So what spin should we associate with the spin connection? I suppose it is spin 2 since it is a second-rank SO(1,3) tensor?
Note in the globally flat EM case we do not have this problem since
F(EM) = F(EM)uvdx^u/\dx^v
and here the u,v indices are globally flat Poincare group tensor indices so no ambiguity in how to define spin in that limiting simple case.
On Mar 8, 2007, at 11:16 AM, Jack Sarfatti wrote:
Show the details please Ark. Cannot evaluate your too-short cryptic verbal-only remarks.
One equation is worth a jillion words. ;-)
Also "tensor" is defined relative to a group. In context they mean
GCT, i.e. localized T4.
Also, as I show in detail in last message to Becker
the relation of torsion & curvature to the local gauging of which group is
a bit subtle & counter-intuitive though the formal math is clear enough, i.e. Robert Becker wrote:
"Hehl brings out another important point: the relation of Torsion to translation. It is customary in physics to associate Torsion with spin one way or another. I believe both you and Shipov follow this interpretation. However, Vargas and others, like Pommaret, reject the association of Torsion spin or rotation because it is mathematically closely related to translation as Hehl highlights in his Letter. Vargas finds a geometrodynamical association of Torsion with the EM field, rather than spin."
I pointed out this is a tricky point that Rovelli and Kibble clarify.
The GCT gauge freedom of 1915 GR comes from localizing only the 4-parameter translation subgroup T(4) of the 10-parameter Poincare group P[T(4),SO(1,3)]. Clearly the local GCT's
x^u = x^u(x^u') are localized infinitesimal translations a^u(x^u'), i.e. 4-parameter
x^u -> x^u + a^u(x^u')
In global 1905 special relativity the infinitesimal "elastic deformation" a^u(x^u') of Hagen Kleinert's 4D world crystal Planck lattice is a global constant over the entire infinity of Minkowski spacetime. This is global action at a distance that violates local objective light cone limited causality, hence localization is absolutely necessary to maintain orthodox causality. Global special relativity without gravity is a half-way house first-approximation that is not internally consistent from this POV.
On the other hand there is no question that dislocation defects in the Kleinert world crystal lattice (Burger's vectors et-al) form the torsion field gaps and that the disclination defects for parallel transport around closed loops of the lattice where
V^u(finish) - V^u(start) ~ SO(1,3)^uu'V^u'(start) ~ R^uvwlA^v^wV^l(start)
A^v^w = - A^w^v is the sectional area element of the small loop
form the curvature.
Therefore there are two dual POV
on the one hand
1915 GR with curvature only and zero torsion emerges from the local gauging of 4-parameter T(4)
on the other hand
1915 GR with curvature only and zero torsion is associated with a local Lorentz transformation LLT of the 4-vector around the closed loop (no torsion gap).
Of course when one does the actual Cartan form algebra there is no problem. The problem is only one of the informal language (Bohm) not of the mathematics.
Thus the 4 GCT invariant tetrad 1-forms are the LLT 4-vector components
e(4)^a = I^a + A(4)^a
I^a is the trivial globally flat 1905 SR tetrad
A(4)^a is the intrinsically warped "gauge connection potential" from localizing 4-parameter T(4).
Einstein's 1915 fundamental local GCT & LLT scalar invariant is
ds^2 = e^aea = (Minkowski metric)abe^ae^b = guvdx^udx^v
Einstein's 1915 constraint of zero torsion is the vanishing 2-form
T(4)^a = de(4)^a + S(4)^ac/\e(4)^c = 0
This with metricity, see Rovelli's explicit formula Ch 2 of his online "Quantum Gravity" gives the 6 zero torsion field effective spin-connnection 1-forms
S(4)^a^b = - S(4)^b^a
only from localizing T(4) to get the spin 1 Yang-Mills potential A(4)^a
A(4)^a spin 1 because it's a Lorentz group 4-vector in the a-index.
This warped localized tetrad resembles the EM 4-potential with "internal index" "a" i.e. "Yang-Mills".
The pure disclination curvature defects without torsion gaps then come from the curvature 2-form
R(4)^a^b = dS(4)^a^b + S(4)^ac/\S^cb
And no TP of course at this stage as you say.
Next step is to locally gauge SO(1,3) giving the additional independent torsion gap field spin connection 1-forms
The full connection is then S(4)^a^b + S(1,3)^a^b
T^a(1,3) = S(1,3)^ac/\e^c =/= 0
R(10)^a^b = d[S(4)^a^b + S(1,3)^a^b] + [S(4)^ac + S(1,3)^ac]/\[S(4)^c^b + S(1,3)^c^b]
Where the TP Ansatz is obviously in my transparent notation using only local objective GCT invariants (coordinate independent)
R(10)^a^b = 0
R(10)^a^b = R(4)^a^b + R(1,3)^a^b + S(4)^ac/\S(1,3)^cb + S(1,3)^ac/\S(4)^cb
Hence two "diagonal" curvature 2-forms. Utiyama in 1960 computed R(1,3)^a^b in effect without R(4)^a^b.
This settles the above informal language confusion you mention!
Note also the two T(4) x SO(1,3) cross-coupling terms.
*I am going to completely rewrite my emergent gravity archive paper with these new results of course.
On Mar 8, 2007, at 1:54 AM, Arkadiusz Jadczyk wrote:
On 7 Mar 2007 at 16:40, Jack Sarfatti wrote:
> On Mar 7, 2007, at 4:16 PM, Dr. Eric Davis wrote:
> > Jack:
> > Attached is a letter to the editor of Physics Today (March 2007)
> > written by F. W. Hehl. In answering a previously published
> > complaint about the relevancy of the torsion tensor by Steven
> > Weinberg (UT-Austin), Hehl explains the physical nature and
> > relevance of the torsion tensor in GR.
Both Hehl and Weinberg are right and wrong. Weinberg is right
because in the standard presentation torsion is just a tensor. Weinberg
is wrong because there are other presentations possible where torsion is
NOT just a tensor. Hehl is right with the fact that torsion may be an important
factor when it comes from gauging of the translation group. Hehl is wrong when
he writes that torsion is "just not a tensor, but very specific tensor". Even
tensor is just a tensor.
When one is VERY careful (I mean "mathematically careful" - which is not what
is being advocated among the readers of Sarfatti Physics seminars), one can
find that torsion
is NOT a tensor at at all and that the fact that it does not have to be a
tensor may have physical implications. Explaining this would take us into
"knowing what we do", while
here "not knowing what we do" is being considered as the superior way :)
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein