On Oct 2, 2007, at 5:31 PM, Paul Zielinski wrote:
Jack Sarfatti wrote:
On Oct 2, 2007, at 1:31 PM, Paul Zielinski wrote:
In other words, linear coordinate substitutions?
I^u is a homogeneous affine tensor under the above i.e.
I^au to I^au' = C^uu'I^au
Which is analogous to a "Cartesian tensor" under orthogonal transformations in ordinary 3D space?
I^audx^u is an affine invariant (scalar) under I.
In addition, A^a is a Lorentz 4-vector under rigid O(1,3).
This could either mean that the components indexed by a are the components of a Lorentz vector, or
that there is a separate Lorentz vector for each value of a.
also "a" labels each GCT u-index tensor
e.g. e^au means 4 distinct first-rank GCT tensors whose components labeled by "u"
i.e. "a" are O(1,3) indices & u are GCT indices.
So the a-indexed qualtities are components of vectors under LLTs, and the u-indexed quantities
are components of vectors under GCTs?
yes, or more generally tensors (or non-tensor connections as the case may be) relative the respective groups if multiple indices of same type..
e^au means 4 Lorentz vectors in u-space and 4 u-vectors in a-space.
OK. And this is what they call the "flat" vs. "curved" indices?
Yes, u (curved) are also technically "tangent or co-tangent" and a(flat) Minkowski, I prefer "u" as non-geodesic LNIF and "a" as geodesic LIF to connect the formalism directly to the equivalence principle - using Wheeler's operational approach closest to actual experimental physics. In past I called "u" base space and "a" fiber" but I guess technically that's not what the mathematicians like. So basically "a,b ..." means zero g-force geodesic local observer-detector and "u,v ..." means non-zero g-force off-geodesic local observer detectors and all the transformations are "local" in the sense of almost colliding observers looking at the same processes - each observer is a point p on the same GCT gauge orbit P, i.e. p(Alice) ~ p'(Bob) belong to the same P - as in Rovelli's pictures in Ch 2 of "Quantum Gravity."
e^a' = O(1,3)^a'ae^a
eu' = (GCT)^uu'eu
e^a = e^audx^u
eu = e^au(d/dx^a)
eu = e^au(d/dx^a)?
yes, that was a typo
Then e^a = I^a + A^a remains scalar invariant under the larger NON-AFFINE transformations that include those conformal boosts to constant accelerating GNIFs as well as the full GCTs of 1916 GR
i.e. I^a to I^a + X^a under GCTs
A^a to A^a - X^a
just like gauge invariant P + eA in U1(x) electromagnetism.
So A^a to A^a - X^a cancels or "compensates" I^a to I^a + X^a under the "non-affine" transformations?
Such that I^a + A^a to I^a + A^a?
And is that how you model an LIF in gauge gravity theory? Cancellation of the two spoiler non-homogeneous
terms X^a and -X^a in I^a and A^a?
No, A^a = 0 in a LIF, A^a =/= 0 in a LNIF.
OK. So A^a represents the net field observed in the LIF?
No, A^a = 0 in both a Minkowski GIF (R^a^b = 0) and a curved LIF (R^a^b =/= 0) where I^au = Kronecker delta (this is in Rovelli explicitly)
A^a =/= 0 in both a Minkowski GNIF (R^a^b = 0) and a curved LNIF (R^a^b =/= 0), therefore A^a =/= 0 is Shipov's "inertial geometrodynamic field" - even though it is zero in a LIF its gradients need not be zero so that R^a^b =/= 0. Remember this curvature field 2-form
R^a^b = dS^a^b + S^ac/\S^c^b
where S^a^b = spin connection 1-form
is already of Yang-Mills form and it is a GCT local scalar invariant (zero rank GCT tensor) like ds^2 = e^aea is.
S^a^b is the compensating Yang-Mills type gauge potential from localizing the rigid 6-parameter Lorentz group SO(1,3)
R^a^b = - R^b^a is ALSO an antisymmetric second rank 6-parameter Lorentz group SO(1,3) tensor just like the EM field tensor is!
I was also trying to force the torsion field 2-form T^a into the Yang-Mills mold
e^a = I^a + A^a
T^a = d(I^a + A^a) + S^ac/\(I^a + A^a)
in 1905 SR only allowing GIF -> GIF' ("affine" global frame transformations)
A^a = 0 and S^a^b = 0
dI^a = 0
half-way to 1916 GR allowing GIF -> GNIF'
A^a =/= 0 and S^a^b =/= 0
R^a^b = 0
T^a = 0
Going all the way to 1916 GR, i.e. replace rigid T4 by local T4(x) (defines "GCT), i.e. LIF -> LNIF (tetrads) & LNIF -> LNIF' as well as LIF -> LIF' all at same P, i.e. on fixed GCT gauge orbit
R^a^b =/= 0 allowed
T^a = 0 enforced.
This gives a dummy redundant S^a^b(T4) (see Rovelli eq. 2.89) that I cannot put into Yang-Mills form
S^a^b(T4) = w^a^bce^c
Might be able to do it for the actual torsion field, but I am not sure - maybe not.
Einstein-Cartan theory is locally gauging full Poincare P10 to P10(x) so that T^a =/= 0
Note if T^a =/= 0 from localizing only SO(1,3) subgroup of P10 you do get a torsion induced curvature. This is what Utiyama did in 1956, but he did not have GCTs as a set of gauge transformations. He stuck them in ad-hoc because he did not localize T4.
e^a(LIF) = I^a
e^a(LNIF) = I'^a + A'^a
I'^a = I^a + X^a
A'^a = -X^a
The intrinsic curvature is in the gradients of A^a =/= 0 even when A^a = 0 in a LIF - same as Levi-Civita.
OK. So then it is A^a that should split up, in my model, into "curved-coordinate" and "intrinsic" parts -- as I thought.
Maybe, but why bother?
Consider the space of local frames on a fixed GCT gauge orbit (see pictures in Rovelli Ch 2) - each local frame is a point p on the orbit. A^a is a functional on this gauge orbit "space". LIFs are critical points with horizontal tangents (analogy) - intrinsic curvature is in the second order partial derivatives of the A^a inertial field.
Do a Taylor series. In a LIF starting point (remember "p" here is in the abstract space of local frames on a gauge orbit not physical spacetime)
A^a(LNIF at p') ~ 0 + 0(p' - p) + (1/2)(d^A^a(LIF at p)/dp^2)(p' - p)^2 + ...
If we expand around a LNIF at p to a LIF at p' then
0 ~ A^a(LNIF at p) + (dA^a(LNIF at p)/dp)(p' - p) + (1/2)(d^A^a(LNIF at p)/dp^2)(p' - p)^2 + ...
Then in this model the cancellation of the non-tidal g-field in an LIF is entirely a function of the coordinate
transformation properties of the tetrad basis? With no reference to the intrinsic spacetime geometry?
Exactly - g-forces are simply non-geodesic artifacts of those non-affine transformations to accelerating frames, curvature is completely irrelevant.
If by this you mean "...to local frames accelerating with respect to LIFs", then OK. That is certainly consistent with the position you've taken previously.
Yes. The LIFs are determined by the geodesics that depend on the curvature GCT invariants R^a^b =/= 0 i.e. relative tilts of neighboring light cones (e.g. R. Penrose, "The Road to Reality"). By definition LIFs defined relative to R^a^b =/= 0 are "nonaccelerating" (zero g-forces). If you force this into Minkowski spacetime like Puthoff & Yilmaz want to do then you change the meaning of "nonaccelerating" causing a lot of confusion! A curved (R^a^b =/= 0) geodesic LIF looks approximately like a Minkowski (R^a^b = 0 globally) GNIF! Hence a lot of useless arguments.
There are g-forces in Minkowski S-T GNIFs - what curvature does is to change what is meant by "geodesic".
If by "curvature" you mean "intrinsic stretch-squeeze deformation of the manifold", then I agree.
Yes, that is the Weyl vacuum curvature there is also the Ricci compression/expansion in the presence of real matter quanta and also virtual matter quanta, i.e. the gravitating dark matter and the anti-gravitating dark energy both!
Note that in 1905 SR X^a = 0 and A^a = 0 i.e. only GIF -> GIF'
Right. That's the easy part.
So A^a is a kind of "inertial field" in G. Shipov's sense.
I don't understand this. Why doesn't the term X^a (in I^a -> I^a + X^a) represent an inertial field?
OK I see what you mean as in my above
e^a(LIF) = I^a
e^a(LNIF) = I'^a + A'^a
I'^a = I^a + X^a
A'^a = -X^a
Therefore, you can think of X^a as a purely contingent "inertial field", because of the equivalence principle absent in internal Yang-Mills of weak flavor and strong color forces, the GCT (local T4(x)) gauge transformations on fixed gauge orbits have direct physical meaning in terms of what actual detectors register.