OK -- then does A^a -> A^a - X^a under GCTs? Or under stretch-squeeze deformations of the manifold? Critical point.
A real physical deformation always comes from either some dynamical change in the Tuv tensor. I am not talking about that at all. We are only talking about changes in POV or perspective of locally coincident observers in arbitrary relative motion, some feel g-forces, some do not - for a fixed geometrodynamical field configuration. Their momentary separation is small compared to the local radii of curvature. That's what "locally coincident" means operationally empirically in sense of Einstein's "gedankenexperiment."
The 2 always go together to keep e^a invariant - same as ihd/dx^u - (e/c)Au gauge invariant derivative on charged source fields
What do this mean? Keep the components fixed? Or that the e^a transform as vectors under GCTs?
Remember Einstein's basic physical local frame invariant is on some "gauge orbit" ("local coincidence") physical event P (defined by the web of relations of geometrodynamic field to EM field etc as in Rovelli Ch 2)
therefore e^a is a GCT invariant scalar zero rank GCT tensor
but it is a single Lorentz group vector (contravariant first rank Lorentz tensor).
e^a = I^a(Minkowski) + A^a(non-inertial)
such that A^a = 0 in both GIFs and LIFs, but not in GNIFs nor LNIFs (curvature does not matter here)
under a GCT on the u-indices
A^a -> A'^a = A^a + X^a
I^a -> I'^a = I^a - X^a
e^a -> e'^a = e^a
X^a = 0 for 1905 SR GIF -> GIF'
X^a =/= 0 for GIF -> GNIF' - this is the beginning of 1916 GR, but still with zero curvature Cartan 2-form field R^a^b = 0
I^a is a 1905 SR "affine invariant" but it is not a non-affine invariant i.e. including GNIF's as in the Galilean relativity non-affine GIF to GNIF' transformation
t -> t'
z -> z' = z - (1/2)gt^2
gt/c << 1
I^a is the geometrodynamic field analog of ih(d/dx^a) in D^a = ihd/dx^a - (e/c)A^a in U(1) EM this U(1) gauge covariant derivative on a the charged complex number Psi field gives under a gauge transformation with phase chi
D^aPsi -> D^aPsi' = e^ichiD^aPsi
That is minimal coupling (equivalence principle) gives
Pa -> Pu = e^auPa
where the Pa generate the globally flat translation group T4, so Pu is the 4-momentum in the off-geodesic u-space using curvilinear guv metric, technically the "tangent/co-tangent spaces" depending on position of u lower or upper. LIFs are approximately geodesic a-spaces using constant Minkowski metric - I mean their origins in a region small compared to the radii of local curvature, i.e. 1905 SR works locally to a good approximation. Note that the local curvature tensor Rabcd =/= 0 in a LIF if Ruvwl =/= 0 in a coincident LNIF. This is basically from John A. Wheeler.
So A^a (and also X^a - you were right) are basically the off-geodesic "g-force" inertial fields in non-inertial frames which for translational motions always need a non-gravity force to maintain them. For example in the text book SSS vacuum solution
g00 = - 1/grr = 1 - rs/r
rs/r < 1
area of closed 2D spherical surface surrounding point source = 4pir^2
is only for special "static" or "shell" (Wheeler) LNIF off-geodesic observers at fixed r without any circulating orbital angular momentum.
I understand why basis vectors are coordinate-invariant under LLTs, but why should they be coordinate-invariant under GCTs?
? e^a transform as
e^a' = [SO(1,3]^a'ae^a
and each e^a is a GCT invariant (scalar).
in a sense ds^a = e^a
ds^2 = ds^adsa
ds connects P with P' where their separation is small compared to local radii of curvature. In 1905 SR the radii of curvature are infinite. In our universe - the constant large-scale de Sitter radius of curvature is (Lambda)^-1/2 where the positive dark zero point energy density is (c^4/8piG)(Lambda) ~ hc/NLp^4 ~ 10^-29 gm/cc where Lp^2 = hG/c^3 ~ 10^-66 cm^2 & N ~ 10^122 BITS of Shannon classical (c-bit) information, i.e. Wheeler's IT FROM BIT from the future infinity (horizon) acting retro-causally in Fred Hoyle's teleology ("Intelligent Universe", 1983) "Lambda" = Einstein's "cosmological constant"
The future de Sitter horizon is the "world hologram" and our 3D space "volume without volume" is its retrocausal hologram image with exactly N volume quanta in 1-1 correspondence with the N future area quanta. That is
&L ~ (Lp^2L)^1/3 = size of quantum foam virtual bubble from Wigner
L ~ N^1/2Lp world hologram from t'Hooft
&L ~ N^1/6Lp ~ 10^-13 cm for cosmological L
Note that the world hologram Hubble scale is N^1/2Lp ~ 10^61 10^-33 ~ 10^28 cm, the geometric mean between smallest and largest lengths in the universe is simply N^1/4Lp ~ (10^28 10^-33)^1/2 ~ 10^-2-10^-3 cm and the dark energy density is ~ hc/(geometric mean)^4. Note also that c/(geometric mean) ~ 10^12 Hertz.
If A^a -> A^a - X^a under *deformations* (holding the local coordinates invariant) then the -X^a term would directly correspond to the first-order effect of the "actual" gravitational field in the 1915 theory. Then *intrinsic* -X^a cancels the effect of "curved coordinate" X^a appearing under the action of the non-linear GCTs.
No not for physical deformations, only for the non-physical GCT changes in local coordinates - on same gauge orbit. Analogy is to U(1) EM gauge transformations
*I suppose you can say that all the different points p on the same GCT gauge orbit P represent all possible subjective POVs of the infinity of possible locally coincident observers looking at the same physical P. The p's are the Shadows and P is the Platonic Idea (Allegory of the Cave, "The Republic") Observables are GCT invariant.
If, on the other hand, you get A^a -> A^a - X^a under GCTs alone, without deformation of the manifold, then there is nothing I can see in this -X^a term that corresponds to the actual non-tidal field of the 1915 theory.
That's correct. X^a is simply a non-physical gauge artifact that is cancelled out - using entire gauge orbit as the physical object - deformed by changes in the energy density etc of the source fields.
OK. But at the same time, the components e^a do nonetheless change under GCTs,
according to I^a -> I^a + A^a?
Even in a flat spacetime?
What happens is that e^a is GCT invariant, however, in general if you jump from a geodesic to a non-geodesic intersecting it at same P