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...

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Compare (8) to my

e^Iu = ||^Iu + A^Iu

 ||^Iu ---> first two terms on RHS of (8)

A^Iu ---> third term on RHS of (8)

this generalizes my Minkowski to any curved background metric field.

<pastedGraphic.jpg>

Compare (9) to my

guv = nuv + nIJ||u^{IA^J}v + nIJA^IuA^J

similar idea I got intuitively.

I also have written it is not a weak field expansion (e.g. to Carlos Castro quite a while ago)

(9) is more general and more rigorous of course.

On Jul 17, 2009, at 6:43 PM, JACK SARFATTI wrote:

http://en.wikipedia.org/wiki/Coset

http://en.wikipedia.org/wiki/Quotient_group

http://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking

http://en.wikipedia.org/wiki/Goldstone_boson

<pastedGraphic.jpg>

<pastedGraphic.jpg>

<pastedGraphic.jpg>

On Jul 17, 2009, at 6:14 PM, JACK SARFATTI wrote:

Using Feynman’s words “gravity is that field which corresponds to a gauge

invariance with respect to displacement transformations”.

On Jul 17, 2009, at 7:21 AM, art wagner wrote:

http://xxx.lanl.gov/PS_cache/arxiv/pdf/0907/0907.2871v1.pdf

Gravity as a gauge theory of translations

J. Mart´ın–Mart´ın ∗ A. Tiemblo†

Abstract

"The Poincar´e group can be interpreted as the group of isometries

of a minkowskian space. This point of view suggests to consider the

group of isometries of a given space as the suitable group to construct

a gauge theory of gravity. We extend these ideas to the case of maximally

symmetric spaces to reach a realistic theory including the presence

of a cosmological constant. Introducing the concept of “minimal

tetrads” we deduce Einstein gravity in the vacuum as a gauge theory

of translations."

Exactly my point about globally rigid T4 to locally variable T4(x).

"In the absence of fermions the theory can be established in a purely

geometrical framework. Nevertheless, a relevant question arises when we

attempt to couple fermions with gravity, a program which implies the necessity

to enlarge the framework with the introduction of an internal symmetry

group suitable to include spinorial fields in the theory.

There is, in fact, a very rich literature [1] [2] [3] [4] [5] [6] [7] [8] [9]

suggesting different extensions of the gauge principle to space–time groups,

an attempt that exploits the properties of the Yang–Mills Theories in the

hope to express gravity as mediated by gauge potentials only, as it happens

with the other interactions.

Special mention is deserved to the occurrence in these kind of theories of

the tetrad a typical structure which is neither a purely geometrical object

nor an internal tensor, but both things at the same time, as deduced from

the double behavior with respect to space-time coordinates and internal

degrees of freedom."

Equivalence principle: covariant tensor acceleration of the detector is locally equivalent to standing still (static LNIF) in Newton's gravitational field. (Einstein's happiest moment of illumination from reading about a workman falling off his ladder in Zurich)

e^Iu = (Minkowski)^Iu + (Acceleration)^Iu

Locally coincident

<Transition.gif>

Each formal coordinate patch (circle) of abstract point set topology to differential geometry via the Klein Erlanger partially ordered lattice is physically a Cartan "frame mobile" whose material incarnation for experimental physicists and space warp metric engineers are actual space platforms loaded with detectors. Damn the torpedoes from the rigor morticians, the philofawzers and the metatheoreticians, engage warp drive! ;-)

<Pioneer-10-Einstein16may08.gif> 

Local Frame of Reference (LIF when unpowered and not rotating about its center of mass).

"covariant" before "acceleration" understood unless otherwise noted.

I = "internal" unaccelerated LIF

u = "spacetime" accelerated LNIF

"To explain tetrad’s properties several proposals have

been considered and to this purpose Hehl’s Poincar´e gauge theory as well as

metric affine gravity [10] [11] [12] must be mentioned.

It must be stressed that tetrads can be considered as the footprint of

the unavoidable presence of translations in a true gauge theory of gravity.

Using Feynman’s words “gravity is that field which corresponds to a gauge

invariance with respect to displacement transformations”. We claim that

the cornerstone to include translations in such a gauge theory is given by

non linear realizations of space-time symmetry groups containing translations

[13] [14] [15] [16] [17] [18] [19]. On these grounds we are going to

show that the formal structure of tetrads can be expressed in terms of well

known gauge objects, connections and Goldstone–like bosons associated to

the displacement transformations that, as we will see, can be locally chosen

as coordinates that, in this way, can be interpreted as well as dynamical

objects.

...

[Given that] the quantum vacuum [is] the basic ingredient of the physical reality,

any realistic attempt to construct a dynamical description of space–time's

nature, must include the role of the cosmological constant as an initial requirement.

In this aim the extension of the gauge principle to the [deSitter dark energy] group of

isometries of a space of constant curvature, appears as the natural generalization

of the Poincar´e group that can be considered as a limiting case when

the curvature tends to zero.

to be continued

On Jul 17, 2009, at 4:08 PM, JACK SARFATTI wrote:

The Good:

There are some common ideas in Waldyr's theory and my own starting in my 2006 archive paper culminating (though by no means finished nor free of errors) in

http://www.iop.org/EJ/article/1742-6596/174/1/012045/jpconf9_174_012045.pdf?request-id=36d11a5e-8b4c-4544-888b-9fb4fc6344f7

e.g.

1)  the use of four spin 1 tetrad 1-form fields as the basic observables

2) attempt to use non-dynamical global Minkowski spacetime whilst still obeying background independence - good trick if you can do it. 

If that can be done one can use momentum space Feynman diagrams in the Dyson book keeping sense if not the literal Feynman sense in renormalizable quantum gravity.

This means spin 0, spin 1 and spin 2 quantum gravity corrections.

More Good later on.

The Bad: The physical meaning of the non-metricity tensor is obscure. Maybe Waldyr can fix that. There may be a contradiction with the equivalence principle that the naked formalism does not show - a matter of connecting the empty symbols to operational definitions (P.W. Bridgman).

The Ugly - the formalism is difficult and the model is contrived Rube Goldberg - a mathematical mule, but this is a matter of taste and maybe I am being unfair and will change my mind in due course.

Starting point is

<pastedGraphic.jpg>

Now there is an ambiguity to begin with. One can add more GCT tensors to the connection and get new physics to be tested against experiment. The connection is the compensating gauge potential from localizing a symmetry group of the actions of the matter fields. For gravity the group must be universal applying equally to all matter source fields whose stress-energy density tensors generate gravity.

That is, the pattern of timelike and lightlike geodesics may change. Waldyr does not want to do that. He proposes to only look at Einstein's 1915 GR from a new POV. The jury is still out on whether he succeeds.

These quotes are from Waldyr's recent note. The idea is to keep the geodesic pattern fixed. The physics is in the invariant geodesic pattern. Thus Waldyr presents us with a bizarre Coney Island funhouse distorted mirror image of plain vanilla Einstein 1915 GR, no torsion as in Shipov's theory and no Weyl conformal dilation as in Castro's theory. We will not find any explanation of dark energy, dark matter or the Pioneer Anomaly in Waldyr's non-metricity model nor in his different teleparallel model that I will not further discuss here.

<pastedGraphic.jpg>

Note the strange look of the SSS vacuum metric field in Cartesian coordinates instead of the more usual spherical polar coordinates.

<pastedGraphic.jpg>

The new second ghostly shadow connection is <pastedGraphic.tiff>its physical meaning is obscure. 

I will use D instead of <pastedGraphic.tiff>and D' for the curved LC connection (which may be confusing)

i.e.

D = flat LC connection 

e.g. x --> x' = x + (1/2)gt^2

dx' = dx + gtdt

       t --> t'

gt/c << 1

ds^2 = (cdt')^2 - dx'^2 = (cdt)^2 - (dx + gtdt)^2 = (cdt)^2(1 - (gt/c)^2) - 2gtdxdt - dx^2

therefore there are non-vanishing first order partial derivatives of the metric tensor in the accelerating non-inertial (x,t) frame. So this is an example of what Waldyr means with his abstract <pastedGraphic.tiff>.

The key is (7) - definition of the non-metricity tensor A. Look at the bottom of (6).

Subtract it from (7) and use (3.15). This is justified only because we have ad_hoc demanded that the geodesic pattern Z is invariant.

Waldyr wrote:

"In GRT a frame in free fall is a unity time like vector field Z such that

 D'_Z Z = 0 "

This is the curved spacetime geodesic equation.

 

"where D' is the Levi-Civita connection of g on a manifold M.

Of course, in Minkowski spacetime structure over the same manifold M you have a connection D and a Minkowski metric n.

Now, it is trivial to verify that

 D_ZZ = A =//= 0."

Using Sean Carroll's (3.15) above

"So, ( D_ZZ - D'_Z Z)=S=A=//=0."

i.e.

D' = D - A

Curved space connection = Flat space connection + Non-metricity tensor

slight notation change

i.e. <pastedGraphic.tiff>

Therefore, it's plausible to identify A as the intrinsic non-tidal gravity field.

A can be calculated formally, but can it be measured physically?

<pastedGraphic.tiff>

<pastedGraphic.jpg>

Note that (10) is simply the component of Newton's gravity force in the x1 direction. But here we have a clue that Waldyr's formalism may violate the equivalence principle even though it looks airtight formally. Along the timelike geodesic Z the total Newtonian gravity force must vanish. This is "inertial compensation" because the argument is then that the Minkowski connection contribution cancels (10) exactly. Well this is very ugly - a mathematical mule, a Rube Goldberg contraption because it suggests that there is no way of measuring the non-metricity tensor. It's a purely formal device. Waldyr must come up with a way to measure this tensor or the theory is not good physics - not a serious contender to replace Einstein 1915.