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/\
D/\ = (e + S)/\
e = I + A
D/\ = I/\ + A/\ + S/\
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
R = (I/\ + A/\ + S/\)/\S = 0
T = De = 0
(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%.