Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Mathematics o Cartan's tetrads in Einstein's gravity 1
Posted:
Oct 2, 2007 11:49 PM


On Oct 2, 2007, at 5:10 PM, Paul Zielinski wrote:
Jack Sarfatti wrote: #1
OK  then does A^a > A^a  X^a under GCTs? Or under stretchsqueeze 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 gforces, 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)
ds^2 = e^aea = (Minkowski LIF)abe^a^eb = (Curvilinear LNIF)uvdx^udx^v = guv(LNIF)dx^udx^v
GCT's act on the uindices.
e^a = e^audx^u
e^au is a 1st rank GCT covariant tensor
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(noninertial)
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 uindices
A^a > A'^a = A^a + X^a
I^a > I'^a = I^a  X^a
such that
e^a > e'^a = e^a
X^a = 0 for 1905 SR GIF > GIF'
but
X^a =/= 0 for GIF > GNIF'  this is the beginning of 1916 GR, but still with zero curvature Cartan 2form field R^a^b = 0
I^a is a 1905 SR "affine invariant" but it is not a nonaffine invariant i.e. including GNIF's as in the Galilean relativity nonaffine 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 4momentum in the offgeodesic uspace using curvilinear guv metric, technically the "tangent/cotangent spaces" depending on position of u lower or upper. LIFs are approximately geodesic aspaces 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 offgeodesic "gforce" inertial fields in noninertial frames which for translational motions always need a nongravity 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 offgeodesic observers at fixed r without any circulating orbital angular momentum.
I understand why basis vectors are coordinateinvariant under LLTs, but why should they be coordinateinvariant 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 largescale 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 (cbit) information, i.e. Wheeler's IT FROM BIT from the future infinity (horizon) acting retrocausally in Fred Hoyle's teleology ("Intelligent Universe", 1983) "Lambda" = Einstein's "cosmological constant"
http://qedcorp.com/APS/ureye.gif
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 11 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 ~ 10^210^3 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 firstorder 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 nonlinear GCTs.
No not for physical deformations, only for the nonphysical GCT changes in local coordinates  on same gauge orbit. Analogy is to U(1) EM gauge transformations
APsi > (A + grad Chi)Psi
igrad(e^iChi Psi) = [(gradChi)Psi + igradPsi]e^iChi
OK.
*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 nontidal field of the 1915 theory.
That's correct. X^a is simply a nonphysical 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 nongeodesic intersecting it at same P
I^a > I'^a = I^a + X^a
0 > A'^a =  X^a
e^a > e'a = e^a



