Curvature, Torsion, Nonmetricity

bcc
On Nov 11, 2005, at 7:49 AM, art wagner wrote:

This paper has intuitive elegance & elucidates an important part of the teleparallel picture:

(http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0505/0505025.pdf)


OK

T4 locally gauged gives tidal curvature spin connection with compensating 1-form potential B

B = Bu^adx^u(Pa/ih)(Goldstone Phase of Higgs Field)

{Pa} = Lie algebra of T4

Note the electro-weak breaking to give the inertia of leptons and quarks happens relatively late as the universe forms, yet there is gravity before that. This is an issue for my model to clarify. I will need a larger group G valid at the inflation on scale of 10^19Gev that breaks in a cascade ending with H = U(1)em. There is a theorem that the vacuum manifold is still G/U(1)em ~ S2 with a point defect for a 3-component "vector" vacuum order parameter that we can point along "z" in abstract 3D parameter space (not physical space).

O(1,3) locally gauged gives the torsion 1-form

S = Su^a^bdx^uSab(Goldstone Phase)

{Sab} = Lie algebra of O(1,3)

What group do we need to locally gauge to get nonmetricity Qu^a^b?

We need to do some work to get the 1-form spin connection W from B.

W = -*[dB/\(1 - B)]

W = (1/ih)^2Wu^a^bdx^uPaPb(Goldstone Phase) ?

Assuming also the non-metricity connection 1-form Q, using the symbolic short-hand notation. I will call it the "AS IF" notation - not to be taken literally, but in the spirit of the rules for Feynman diagrams that can be made rigorous later i.e. I give here a "Windows" high-level language that mathematicians can later translate into more rigorous "machine code" as it were.

s is a 0-form. It is essentially the Goldstone phase conjugate to the important component of the vacuum order parameter that gives emergent c-number gravity. d is the Cartan exterior derivative.

I define the 1-form ds = 1 + B = differential invariant line element

Remember Newton's infinitesimals were not made rigorous until non-standard analysis. They were "Ghosts of departed quantities" and I am trying to avoid premature rigor mortis in order to get the physical idea set forth throwing caution to the winds.

The spin-connection 1-form is then

W = *[-dB/\(1 - B)]

from De = 0

e = 1 + B

D = d + W/
Therefore using the connection W + S + Q

D" = d + W/\ + S/\ + Q/
The coupled field equations with curvature, torsion & non-metricity are

R" = D"W curvature 2-form

D"R" = 0 3-form Bianchi identity

D"*W = *J(T4 source) 4-form from equivalence principle

D"*J(T4 source) = 0 5-form in 4-space current conservation

Similarly

T" = D"S torsion 2-form

D"T" = 0 3-form

D"*S = *J(O(1,3) source) 4-form

D"*J(O(1,3) = 0

And

N" = D"Q = non-metricity 2-form

D"N" = 0

D"*Q = *J(non-metricity sources)

D"*J(non-metricity) = 0

Maxwell's EM field equations are where A is from locally gauging U(1)em

F(em) = D"A(em)

D"F(em) = 0

D"*F(em) = *j(em) 3-form for internal symmetries compared to 4-form for space-time symmetries. Difference is from the equivalence principle.

D"*j(em) = 0

Similarly for Yang-Mills SU(2) & SU(3) with 1-forms Y

and for the Dirac spinor lepto-quark equations use for connection

D"" = D" + A + Y

i.e. Homework problem for the Yang-Mills & Spinor source fields.