Torsion Math 1
D’ = d + W/\ + S/\ = D + S/
W is the (tangent fiber) 2-index spin connection 1-form from T4, i.e. W^ab
W ~ Lie algebra of T4 on the macro-quantum vacuum coherent Goldstone phase field with point defects from vacuum manifold G/U(1)em ~ S2 and 3-vector local order parameter whose single-valuedness implies Bekenstein-Hawking (Area/Lp^2) is quantized geometrodynamic area flux.
S is the 2-index torsion connection 1-form from O(1,3), i.e. S^ab
S ~ Lie algebra of O(1,3) on Goldstone phase.
E’ = 1 + B + T’
is the Einstein-Cartan tetrad field with a one-index torsion 1-form T', i.e. T'^a
D = d + W/
is the 1915 GR covariant exterior derivative with zero torsion i.e. only T4 is locally gauged.
T = DE = 0
W = -*[dB/\(1 - B)] 1-form with 2 indices.
E = 1 + B
is 1915 GR tetrad in invariant SYMBOLIC short-hand notation.
T’ = D’E’ = (d + W/\ + S/\)(1 + B + T’) = (d + W/\)T’ + S/\(1 + B + T’)
T’^a = dT^a + W^ab/\T^b + S^ab/\(1 + B + T’)^b
Note that this is a nonlinear differential equation for T'^a.
T' is to O(1,3) as B is to T4.
T' & S are both 1-forms, but T' has one index and S has two indices.
We can get 2-forms from each of them.
D'S and D'T' are both 2-forms.
Therefore, the possibilities are larger than I first suspected and are even larger if we throw in non-metricity.
B is what I call a substratum quantity whilst W is the geometrodynamical quantity.
Similarly T' is in substratum and S is geometrodynamical.
Generalized curvature 2-form with 2 indices
R' = D'W = DW + S/\W = R + S/\W
with Bianchi identity
D'R' = 0 3-form
Geometrodynamic curvature source equation is
D'*W = *J(T4) 4-form
Current conservation is
D'*J(T4) = 0 5-form
Generalized torsion 2-form with 2-indices is
TORSION = D'S = DS + S/\S
Bianchi identity is
D'TORSION = 0 3-form
Geometrodynamic torsion source equation is
D'*S = *J(O(1,3)) 4-form
D'**J(O(1,3)) = 0 5-form
In contrast INSIDE THE SUBSTRATUM where the 1-forms B & T' have ONE index
F = dB 2-form
dF = 0 3-form
d*F = *j(T4) 3-form
d*j(T4) = 0 4-form
Just like U(1) EM theory
Same story for T' one-index torsion 1-form.