Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: Mathematics o Cartan's tetrads in Einstein's gravity 1
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List  
Jack Sarfatti

Posts: 1,942
Registered: 12/13/04
Mathematics o Cartan's tetrads in Einstein's gravity 1
Posted: Oct 2, 2007 11:49 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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 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)

ds^2 = e^aea = (Minkowski LIF)abe^a^eb = (Curvilinear LNIF)uvdx^udx^v =
guv(LNIF)dx^udx^v

GCT's act on the u-indices.

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(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

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 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"

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 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

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 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

I^a -> I'^a = I^a + X^a

0 -> A'^a = - X^a

e^a -> e'a = e^a




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.