Note also, the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields.

Look at the exterior covariant derivative 1-form

D = (d + S)/\

note that d is usually written without the /\ that is tacitly understood since

d(p-form) = (p + 1)-form

dual to boundary operator

&(p + 1) dim manifold = p-dim manifold so that Stokes theorem generalizes to

Integral of d(p-form) on p + 1 dim manifold = Integral of (p-form) on the boundary &(p + 1)manifold

for example when p = 0 in 3D space i.e. fundamental definite integral formula of calculus

d(0-form) = gradient of a function

p = 1 e.g. loop integral of vector (1-form) field is interior flux (2-form) integral

d(1-form) - curl of a vector (2-form)

p = 2 Gauss's divergence theorem - end of short digression

d(2-form) = divergence of a 1-form vector field in 3D space.

can generalize to Minkowski spacetime of 1905 SR

Back to main point

d in a sense is e

i.e. in a flat spacetime in a geodesic GIF coordinate basis

d(0-form) is "4-gradient" d/dx^a on a 0-form function since ea^u = Kronecker delta

ea = ea^u(d/dx^u)

e^a = e^audx^u

That is we can think of d/\ as e/\

or

D/\ = (e + S)/\

e = I + A


i.e.

D/\ = I/\ + A/\ + S/\

where

I/\ is in flat Minkowski spacetime of 1905 SR

A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.

Note that A induces S that is not independent when the torsion T field vanishes globally.

In a GNIF

R = D/\S = 0

i.e.

R = (I/\ + A/\ + S/\)/\S = 0

and also

T = De = 0

i.e.

(I/\ + A/\ + S/\)(I + A) = 0

when rigid T4 is localized to T4(x) then we have possibility that

R = (I/\ + A/\ + S/\)/\S =/= 0

This notation makes the Yang-Mills field structure more apparent.

In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89

S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged.

That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli (2.89). Hence GR is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field that may give Salam's strong short-range f-gravity in addition to zero mass geometrodynamic field quanta. Indeed the composite quanta are spin 0, spin 1 and spin 2 from pairs of the fundamental spin 1 geometrodynamic quanta i.e. zero point fluctuations of the uncondensed part of the post-inflation vacuum ODLRO field. Think of the geometrodynamic field random "dark energy" quanta like the zero point motions of helium 4 atoms in the T = 0 ground state that is only 10% coherent condensate even though the effective superfluid density is 100%.