Wednesday, September 19, 2007

Curvature and torsion spin 1 vector fields in geometrodynamic sqrt substratum

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.

eb = nabe^a

In a coordinate basis

e^a = e^audx^u

ds^2 = guvdx^udx^v = e^aea = nabe^aue^bvdx^udx^v


guv = nabe^aue^bv

Z: "As I understand it this is not a "tetrad"; the "tetrads" are given by the 4 x 4 2-index quantities e^a_u."

I call e^a_u the "tetrad components" and I call e^a the four tetrad 1-forms

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 u-frames, i.e. e^audx^u is a scalar invariant under

x^u(P) morphs to 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 coordinate-dependent 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 morphs to 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 4-potential

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] morphs to 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 coordinate-invariant, since they depend
on the coordinate basis {e_u}."

I never said otherwise

No comments: