## Thursday, December 23, 2004

Meaning of Einstein's Gravity

Paul
Wake up and smell the coffee. General relativity is a mature well understood theory. There are important issues today with interesting questions and you are not asking them. I started out on this with you very open-minded. Basically you are garbling Newton's force idea with Einstein's geometrical ideas. Your idea is "not even wrong". You cannot compute anything. You cannot explain anything that needs explaining. You have not asked the right question.

"The Question is: What is The Question?" (Wheeler)

There are 3 analogies of some interest (electromagnetic, fluid & elastic) none of which you have even thought of.

The GR (LC) connection is analogous to the EM vector potential connection.

The curvature is the NONLINEAR NON-ABELIAN (LC) curl of itself analogous to the ABELIAN magnetic field or the circulation of a fluid.

Any 3D-vector field A has a gradient (divergence) part and a curl (circulation part).

Roughly and non-rigorously to start (Kiehn can say it better more rigorously)

DivCurl = 0

In the language of Cartan forms

1-form = exact 1-form + non-exact 1-form

Here we are in 3-space.

The Cartan exterior derivative on a 0-form is a one-form.

Neglecting anholonomic singularities

d^2 = 0

Exact 1-form = dF

F = 0 form scalar

B is a 1 form

curlB = dB is a 2-form

e converts the 2-form to a dual 1 form in 3-space or a dual 2-form in 4-space of GR depending on the context.

The point is that CURVATURE is a 2-form like EM field tensor, and like VORTICITY in a fluid, and like disclination topological defect densities in a crystal lattice (H. Kleinert).

Your COORDINATE PART is the analog to the DIVERGENCE or GRAD exact 1-form part.

The INTRINSIC PART is the analog to the Curl or non-EXACT 1-form part.

However, your ERROR is to think that the non-EXACT 1-form part is a GCT tensor T ALL BY ITSELF.

What we have in fact is

(LC) 1-form = exact 1-form + non-exact 1-form = N NOT A GCT TENSOR in the metric space.

Curvature 2-form = d(LC) = d(non-exact 1-form)

Now in Minkowski space

(LC) = exact 1-form

Note that even in this case, you can still get the Vilenken-Taub "curvature without curvature" effect from global non-trivial topology like the quantized vortices in the superfluid irrotational flow and in Type II superconductors. The dark energy is the "normal fluid" of the macro-quantum coherent holographic vacuum.

Bottom line, the only GCT tensor you can get in 1916 GR (no torsion) from the metric tensor is its (LC) covariant derivative which is the ZERO nonmetricity tensor. (LC) is always a non-GCT tensor that under the GCT X transforms as

(LC) = N -> (LC)' = XXX(LC) + XY = N'

The only possible tensor T here is guv;w = 0 where ;w is the (LC) covariant derivative.

Y is a derivative of X so (LC) is a quasi-tensor under the limited group of linear transformations, i.e. a subgroup of GCT.

The pure gravity energy pseudo-tensor has this same kind of property as (LC). We want gravity energy to be nonlocal. That is a good thing not a bad thing to be eliminated with Rube Goldberg devices.

Curvature is like the breakdown of irrotational flow in a superfluid. There is an analogy to Bohm-Aharonov effect because of the local quantum vacuum coherence that is a giant single-valued wave function. Indeed, this explains the Pioneer 10-11 anomaly as a hedgehog exotic vacuum zero point dark energy topological defect centered at the Sun and maybe ALL stars, the Galactic Halo and other interesting observations that appear mysterious.

PS

Note in the Newtonian mechanics of rotating non-inertial frames

,t' (non inertial) = ,t(inertial) + Wx = Galilean relativity analog to the (LC) covariant derivative of GR

W = instantaneous rotation axial vector of the non-inertial frame (common origin with inertial frame)

For a test particle at displacement r

r' = r

v' = v + Wxr

a' = a + Wxv + W,txr + Wxv + WxWxr

= a + 2Wxv + WxWxr + W,txr

2Wxv = INERTIAL CORIOLIS acceleration of the non-inertial rotating frame

WxWxr = INERTIAL CENTRIFUGAL acceleration ...

W,txr = INERTIAL TORQUE acceleration

In Gennady's Shipov's torsion theory extension of Einstein's 1916 GR this is part of a TORSION FIELD .

Note that there are no-translational inertial accelerations in this particular problem.

If P is fixed to the rotating frame S' then in this REST rotating non-inertial frame, obviously

v' = 0

therefore

v = -Wxr

a' = 0

0 = a + WxWxr + W,txr

When W,t = 0, conservation of angular momentum L

a + WxWxr = 0

But

F = ma

F/m + WxWxr = 0

is compensation of the centrifugal inertial force in the rest rotating non-inertial force by the applied force F measured in the inertial frame.

In Newton's theory, unlike Einstein's, gravity is an external force in the inertial frame

F = GM/r^2

Giving, for a circular orbit Kepler's law

GM = r^3/T^2

T = period of orbit

in flat Euclidean space where the geodesics are traced out by point test particles at constant speed in straight lines.

In contrast, the Newtonian non-inertial non-geodesic motion of this test particle is an inertial geodesic motion in Einstein's curved spacetime in which there is no gravity force.

That is, in Einstein's theory

F = 0 and W = 0 in the above problem.

What we have instead is the geodesic equation

D^2x^u/ds^2 = 0

In a non-inertial LNIF this becomes

d^2x^u/ds^2 + (LC)^uvw(dx^v/ds)(dx^w/ds) = 0

Where (LC) is computed from guv

e.g.

gtt = (1 - 2GM/c^2r) = - grr^-1, gtheta,theta = gphi,phi = -1

in local orthogonal basis (cdt)et, drer, rdthetaetheta, rsinthetadphiephi

r >> GM/c^2 gives same answers as Newton's theory.

This particular REPRESENTATION guv of a CURVED SPACE-TIME is only for REST LNIF observers at fixed r because of some non-gravity force!

In a coincident LIF the EEP gives (LC) = 0 and d^2x^u/dt^2 = 0.

In general for COINCIDENT LNIF & LNIF'

gu'v' = Xu'^uXv'^vguv

Where X is the Jacobian matrix for an element of the GCT symmetry group.

EEP mean the existence of tetrads Eu^a and their inverses where

guv(LNIF) = Eu^anabEv^b

nab = Minkowski metric

Where LNIF and LIF are COINCIDENT at same physical event E.

"Physics is simple, when it is local." Wheeler

* Curved space-time physics is local because Einstein's gravity is "More is different" emergence of the LOCAL macro-quantum cohering of the nonlocal micro-quantum zero point pre-inflationary false vacuum fluctuations. The cohering is only partial so there is some dark energy/matter remnants in our post-inflationary universe.

Cartan Tetrad 1-form ~ (1 + Lp^2(Goldstone Phase),u) dx^u

Vacuum Coherence = (Higgs Field)e^i(Goldstone Phase)

G/c^4 comes from "Goldstone Phase Rigidity" (P.W. Anderson)

X Jacobian matrix of GCT comes from canonical transformation generating function THETA(x^u,x^u')

Where Goldstone Phase(x^u) -> Goldstone Phase(x^u) + THETA(x^u,x^u')

That is Einstein's GR is really a local gauge theory from a post-inflationary macro-quantum LOCAL vacuum coherence ODLRO parameter of zero entropy setting the direction of the Arrow of Time of The Second Law of Thermodynamics in the same direction as the accelerating expansion of the universe.

Note that

Curvature (tidal stretch-squeeze) = (LC) Covariant Curl of (LC) = Ricci Part + Conformal Part

Guv = Ruv - (1/2)Rguv = (8piG/c^4)Tuv(non-gravity source)

When Tuv = 0 Ricci Part of curvature = 0

In "classical vacuum" there is only conformal curvature with 10 independent parameters at each point.

Obviously if Cuvwl =/= 0 in any LNIF, then Cabcd =/= 0 in any COINCIDENT LIF.