World Hologram and Conformal Group
On Nov 11, 2005, at 7:49 AM, art wagner wrote:
This paper has intuitive elegance & elucidates an important part of the teleparallel picture:

(http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0505/0505025.pdf)

Note my Heuristic Mathematically Non-Rigorous Metric Engineer's Rule of Thumb #1

Locally gauge every continuous symmetry group both internal and space-time.
Spontaneously break the internal symmetry groups required by standard particle theory that end up with U(1)em UNBROKEN. This guarantees S2 vacuum manifold with point defect for the geometrodynamical monopole with quantized Hawking-Bekenstein BITs ~ Area/Lp^2 from single-valuedness of the abstract 2-vector order parameter in S2 G/H space. Gauss's theorem implies the basic world hologram picture that all the dynamical degrees of freedom for quantum gravity are on the surrounding closed surfaces of a volume that maybe multiply-connected with wormholes. The geometrodynamic monopoles responsible for quantized area bits and world holography are not to be confused with the U(1)em magnetic monopoles at the end of the Dirac string that give quantized electric charge.


The above paper looks good from my POV of local gauge theory & spontaneous vacuum internal symmetry breakdown G ---> H = U(1)em implying from general theorems in topological quantum mechanics that the

Degenerate vacuum manifold coset space G/U(1)em ~ S2

using the adjoint 3x3 irrep of SU(2) weak for the local order parameter with second homotopy group = Z implying only POINT DEFECTS where the coherent vacuum Higgs field vanishes and its conjugat Goldstone phase is undetermined.

It looks like the non-metricity connection corresponds to locally gauging the dilaton and the special conformal boosts to uniform hyperbolic notion, i.e. Tony Smith conformal gravity?

That is, locally gauge entire 15-parameter conformal group of Penrose's twistors to get the exterior covariant derivative

D" = d + W(Spin Connection from T4)/\ + S(Torsion Connection from O(1,3)) + Q(non-metricity connection from remaining 5 generators of conformal group). These are all Cartan 1-forms where W, S, Q all have 2 indices in the tangent fiber space that are raised and lowered with the constant flat Minkowski metric nab.

Acknowledging Waldyr Rodrigues's true remark that this is not mathematically rigorous at present stage, using a quick and dirty short-hand AS IF notation,

Tetrad local frame-invariant e = 1 + B

B = 0 means globally flat Minkowski space-time with

W = S = Q = 0 identically globally

i.e. B = 0 means GLOBAL 1905 Special Relativity without gravity and without inertia. Plugging in rest masses m is strictly speaking inconsistent and incomplete in violation of the equivalence principle. The only consistent flat quantum field theory must have all rest masses strictly equal to zero.

The equivalence principle that all forms of {Pa} = Lie algebra of T4 are sources of curvature including virtual zero point stress-energy density (String Tension)/\zpfguv enforces the indirect relationship

W^a^b = -*[dB^a/\(1-B)^b] = 1-form

where * is Hodge dual

from

(d + W/\)(1 + B) = 0

i.e.

dB^a + W^a^b/\(1 + B)^b = 0

Note that

W ~ {Pa} Lie algebra of T4 operating on coherent vacuum Goldstone phase of the World Hologram

S ~ {Sab} Lie algebra of O(1,3) operating on Goldstone phase ...

Q ~ {dilaton, special conformal boosts} operating on Goldstone phase ...



On Nov 12, 2005, at 10:30 AM, Jack Sarfatti wrote:

There is obviously total disagreement among all the pundits on how to use the basic terms to begin with and then what starting premises to make.

Rovelli

Shipov

Kiehn

Poltorek

myself (local gauge invariance & spontaneous broken symmetry of vacuum POV with gravity as an emergent phenomenon not given apriori.

All have somewhat different POVs and physical motivations. I am closest to Rovelli on the basic standard definitions. I have basically been using Rovelli's Ch. 2 to connect my quantum substratum standard model of leptons & quarks to the classical geometrodynamics using Cartan's forms notation in at least a heuristic way in the spirit of the Feynman diagrams. I am also trying to see if the 't Hooft-Susskind hologram idea is really needed i.e. 2 + 1 boundary gauge theory --> 3 + 1 gravity on the interior. I seem to get Hawking-Bekenstein quantization of area A/Lp^2 quite trivially from single-valuedness of the adjoint 3-irrep of SU(2)weak vacuum coherent order parameter POINT DEFECT for T4 -> Diff(4) gravity compared to Dirac string LINE DEFECT for quantized U(1)em charge.

So, the final test is comparison to observations.

On Nov 12, 2005, at 8:09 AM, Bill Page wrote:

Gennady,

Maybe the better question is what connection is Alex talking
about? In the papers he cites;

http://arxiv.org/abs/gr-qc/0407060
http://arxiv.org/abs/gr-qc/0403107
http://arxiv.org/abs/gr-qc/0403050

there is only a "arbitrary affine connection". In the first paper
he writes:

"In part I we show that in the presence of an arbitrary affine
connection, the Einstein field equations lend themselves
to a novel geometrical interpretation wherein the affine
deformation tensor of the Levi-Civita connection plays the
role of a gravitational field. Furthermore, in the case of an
affine connection with vanishing torsion, the gravitational
field becomes the nonmetricity of spacetime."

The idea to refer to the symmetric part of the Ricci 2-form
as "non-metricity" I think is based on eq. (1) of this paper:

(1) GammaBar = Gamma + S + Q

where GammaBar is the arbitrary affine connection, Gamma
is Levi-Civita connection, S is non-metricity, and Q is
torsion. But probably this is not complete. More generally

(1') GammaBar = Gamma + S + K

where K is contorsion which includes both a torsion Q and
a symetric part different from S.

Under some conditions (perhaps?) the symmetric part of K
can be absorbed into S. But as you say, this is not the
same geometry as in your theory.

Regards,
Bill Page.

On November 12, 2005 7:16 AM Gennady Shipov wrote:

Nonmetricity means, that covariant derivative with respect
to the metric tensor is not equal to zero. In a case of
Teleparallelism it is equal to zero, therefore Teleparallelism
is a metric geometry.
What nonmetricity you are talking about?

On Nov 10, 2005, at 12:33 PM, Alex Poltorak wrote:

For the most part I agree with Gennady. However, my analysis
(published in GR9 and, more recently, GR17) suggests that gravity
is not a metric curvature 2-form but rather, it is a tensor of
nonmetricity or, in Cartan terminology, the symmetric part of
the Ricci 2-form; while the inertia is described as affine
connection plus Ricci torsion. the Equivalence Principle,
gravity = inertia, is therefore expressed as

[Gravity=nonmetricity] = [Inertia=affine connection+Ricci torsion]