Curvature-Torsion Substratum

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

implies

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.