Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Spin 1 vector curvature & torsion fields in tetrad substratum
Posted:
Sep 19, 2007 11:17 PM


On Sep 19, 2007, at 1:33 PM, Paul Zielinski wrote:
"In which case I suppose your
e^a = I^a + B^a
is an *arbitrary* basis in the tangent space?"
Yes in sense of equivalence principle that the metric in the a,b,c Latin indices is Minkowski nab i.e.
ea = nabe^a
In a coordinate basis
e^a = e^audx^u
ds^2 = guvdx^udx^v = e^aea = nabe^aue^bvdx^udx^v
hence
guv = nabe^aue^bv
Z: "As I understand it this is not a "tetrad"; the "tetrads" are given by the 4 x 4 2index quantities e^a_u."
I call e^a_u the "tetrad components" and I call e^a the four tetrad 1forms
Also eu = eu^aea
so eu is a basis vector in u space where each eu has 4 components eu^a
a duality
Z: "Am I right?"
OK, except for some name difference. BTW Hawking and Ellis have an interesting discussion on part of this. More on that when I get a chance.
Z: "If so, how can e^a carry information about reference frames?"Z:
Why do you keep asking this? e^a is independent of uframes, i.e. e^audx^u is a scalar invariant under
x^u(P) > x^u'(P)
P = objective local coincidence physically = gauge orbit formally
Z: "? Or is it only the components I^a and B^a that individually carry such coordinatedependent information, which exactly cancels out in the sum e^a = I^a + B^a?"
Yes. Now this is original with me and I may be wrong, but it seems it has to be that way in order that B^a transform like the EM vector potential A in a U(1) gauge transformation where A is not a U(1) tensor but is a U(1) connection with the inhomogeneous term. e^auPa is the GCT local T4 analog of the EM U(1) gauge covariant partial derivative! Then everything works! That is, compare
U(1) id/dx^u  (e/hc)Au gauge covariant derivative on the NR electron quantum wave field to
local T4 i.e. iI^auPa  B^auPa > id/dx^u  Bu
where {Pa} is Lie algebra of T4 rigid group of 1905 Special Relativity in a MATRIX REPRESENTATION matching that of the source fields under the Poincare group P10. For Dirac spinors the Pa have Dirac gamma matrices in them  see Rovelli Ch 2.
In my world hologram theory Bu = N^1/3Au
N = Bekenstein BITS to IT of a dominating world horizon like the RETROCAUSAL dark energy future deSitter horizon.
where Au is the local T4 geometrodynamic spin 1 vector field analog to the U(1) EM 4potential
for a curvature field only.
Similarly, if you add a new dynamically independent torsion field (e.g. Gennady Shipov) the larger localized P10 gauge covariant derivative on the matter source fields is of the form
iI^auPa  B^auPa + S^a^buP[ab] > id/dx^u  Bu
where P[ab] are the 3 Lorentz boosts and 3 space rotations of Lie algebra of rigid O(1,3) of 1905 special relativity and S^a^b = S^a^budx^u is the spin connection.
Just as Bu = B^auPa = curvature tetrad spin 1 vector field
Su = S^a^buP[ab] is the torsion spin 1 vector field.
Jack Sarfatti wrote: I never said e^u_a were coordinate invariant, I said e^a were coordinate invariant On Sep 19, 2007, at 7:54 AM, Paul Zielinski wrote:
Jack Sarfatti wrote:
Z: If the difference vector between two spacetime points P and P' is dx, then
dx = dx^u e^u
S: No, that formula makes no sense.
I only meant that you should have written
dx = dx^ueu without the ^ in e^u  otherwise obvious since eu is a basis set
dx = dx^u e_u.
Z: "The point is that the vielbeins e^u_a are clearly not coordinateinvariant, since they depend on the coordinate basis {e_u}."
I never said otherwise



