Tuesday, July 05, 2005

Lectures in Macro-Quantum Gravity 1 & 2

Part 1


The big questions are

1. How does inflation work in the creation of the universe?

2. Why is the entropy low in the early universe?

3. What is dark energy and dark matter?

4. Why is the electron stable? (implication for Ken Shoulders EVOs)

5. What is the Galactic Halo? Why is the stellar rotation curve flat in a wide region?

6. What is causing the gravity anomaly in the two NASA Pioneer space probes?

7. What makes the gamma ray bursts?

8. Why the universal slope of the Regge paths of the hadronic resonances?

I suppress indices as much as possible for brevity in e-mail. When I am forced to use them, a,b,c in tangent fiber, u,v,w in base space, with I = Kronecker delta &u^a in the global aligned frame in flat Minkowski space-time. This frame is physical i.e. all non-rotating inertial detectors are on globally flat timelike geodesics where the geodesic deviation tidal curvature tensor field is identically zero in that limiting case.

The Einstein-Cartan tetrad is

e = I + B

I = identity

When the substratum warp field B = 0 identically, the tangent fiber is aligned with the base space in the "convenient" global frame. Of course Diff(4) transformations in this globally flat B = 0 everywhere-when limit will misalign the fiber with the base describing inertial forces on globally flat non-geodesic detectors. These inertial forces, that are locally equivalent, to gravity cannot exist without non-gravity electrical forces. In this B = 0 limit the geodesic deviation tidal tensor field AKA Cartan-Einstein curvature 2-form is identically zero.

B ~ Lp^2GradargPSI

Goldstone phase of vacuum ODLRO is argPSI, Grad is 4D.

In terms of Cartan's exterior derivative d with d^2 = 0

argPsi is a 0-form

B ~ Lp^2dargPSI

dB =/= 0 only because of multiple connectivity, i.e. singularities in argPSI that make B closed but not exact despite the notation dargPSI. See John Baez's book, on Gravity and Knots for example. The cohomology is non-trivial like vortex lines in superfluid helium which is, like the Higgs Ocean post-inflationary vacuum, a macro-quantum condensed system of real quanta rather than, as in our problem of emergent gravity, virtual zero-point quanta.

Technically H1, the first cohomology group of the local macro-quantum ODLRO order parameter PSI is larger than the trivial identity group that describes a simply-connected manifold.

PSI = LOCAL post-inflation order parameter (a complex-numbered scalar field in a 4D real manifold)

This explains why macro-spacetime physics is local and why the early universe has a small entropy.

Lp^2 = hG/c^3 = Loop quantum of area


1. if c -> infinity a real geodesic deviation tidal curvature warp field is impossible. That is, no gravity possible in Galilean relativity. You need special relativity as the local limit AKA the Einstein Equivalence Principle (EEP).

2. if h -> 0 a real geodesic deviation tidal curvature warp field is impossible. That is, you need a finite quantum of action to get Einstein gravity as a "More is different" (PW Anderson) Andrei Sakharov emergent macro-quantum phenomenon.

These two conditions are non-trivial and are only explained clearly in my theory. The third condition is trivial, i.e.

4. if G -> 0 a real geodesic deviation tidal curvature warp field is impossible.

In essence I show here how Loop Quantum Gravity gives classical Einstein theory.

Part 2

EEP (Einstein's Equivalence Principle) in this formalism is symbolically

g(curved) = (I + B)(flat)(I + B)

= In(flat)I + I(flat)B + B(flat)I + B(flat)B

The terms of the curved metric field g linear in B are "elastic" terms and the nonlinear terms quadratic in B are the "plastic" terms causing the "cracking" of the world crystal lattice whose defects appear as curvature and torsion.

Under X in Diff(4), which is the locally gauged T4, with B as the substratum's compensating warp Yang-Mills gauge force potential of "spin 1" not spin 2, which appears only at the bi-linear level.

e' = Xe

That is total tetrad e is a Diff(4) first rank tensor.

e is also a Cartan 1-form in the substratum

Under L in tangent fiber Lorentz group SO(1,3)

e'' = Le

Note that XI = I' =/= I

What is physical meaning of X?

X is the field mapping of possible coincident detectors in arbitrary relative motion to each other in the neighborhood of the same physical event P. Such a mapping corresponds to a transition function connecting overlapping local coordinate charts. However, there are more such transition functions then physically significant Diff(4) X. Therefore X is a quotient set mod the equivalence relation ~ that leaves the relative motion of coincident detector sets invariant. That is

X = {transition functions}/~

i.e. X is a non-overlapping "coset" equivalence class mod ~ in the quotient group with the unphysical gauge freedom factored out.

Note that

e' = Xe = XI + XB = I' + XB = I + B'


B' = (I'-I) + XB = XI - I + XB

Therefore, B' has an inhomogeneous term under Diff(4) if we wish to make the split into a globally flat part and a warped part post the X transition. On the other hand, if we are content to use I' = XI, then B is a tensor under X. So it depends on how we want to make the split. Obviously, when B = 0 identically

g' = XI(flat)XI

is the apparent curved metric from the non-geodesic motion of the detectors. However, locally there is no way to distinguish an apparent gravity force from an actual gravity force. Globally we can, of course tell the difference. EEP is only a local principle. Geodesic deviation stretch-squeeze measurements of B in the relative coordinates are forbidden in this statement since they do not affect the actual force on the center of mass of an extended test object. You need non-gravity forces to create such non-geodesic motions in the detectors. Here we are talking about globally flat timelike geodesics when B is identically zero.

To proceed further we need to define the spin-connection W that is more fundamental physically than the Levi-Civita connection. W is a Cartan 1 form with values in the Lorentz Lie Algebra.

The gauge covariant exterior derivative at the bi-linear level of the geometrodynamic field is

D = d + W/
where d^2 = 0 ONLY in SIMPLY-CONNECTED manifolds without holes. /\ is the Cartan exterior product.

Do not confuse D with

D* = d + B/\ in the substratum of the fabric of spacetime!

D & D* are qualitatively physically different. I will use * on right of a symbol for substratum quantities, not to be confused with same * to left for the Hodge dual using the fully anti-symmetric tensors of different ranks 2 to 4. The Yang-Mills spin 1 substratum field is the 2-form

F* = D*B = dB + B/\B

The Bianchi identities in the substratum give

DF* = 0

i.e. the 3-form equation

dF* + B/\F* = d^2B + d(B/\B) + B/\dB + B/\B/\B = 0

The 1-form Yang-Mills source equation is

D*F* = *J

Remember B is from the local gauging of T4 a 4-parameter commutative group unless you deform it to a non-commutative space-time geometry where it may look like U(1)xSU(2)?

Next, I go to the bi-linear geometrodynamic level. The Cartan-Shipov torsion 2-form is defined as

T = De = de + W/\e

Since the tetrad is a 1-form.

Einstein's 1915 GR has T = 0 identically. This is what you get when you do NOT locally gauge SO(1,3) the 6-parameter Lorentz Lie Group.


de + W/\e = 0

implicitly determines the spin-connection 1-form W in terms of the tetrad e = I + B

That is

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

Since I is a constant, dI = 0

dB = - W/\(I + B)

Note, we cannot assume dX = 0 in a Diff(4) transformation.

When B = 0 identically, obviously W = 0 because

0 = - W/\I

The Levi-Civita connection is

(LC) = eWe = (I + B)W(I + B) = IWI + BWI + IWB + BWB

EEP here then implies that

(LC) = 0 in local curved spacetime geodesic coordinates only at a point event P - not identically. That is, one must make a different choice as one moves about in curved spacetime.

The stretch-squeeze geodesic deviation tidal curvature Cartan 2-form is

R = DW = dW + W/\W

Globally-flat Minkowski space-time has R = 0 identically.

The 1915 zero torsion Einstein-Hilbert classical action density is the 0-form

&S/&V^4 = *(e/\e/\R) + /\zpfe/\e/\e/\e)

Note that the Hodge star dual of a 4-form is a 0-scalar form.

where /\zpf is the dark zero point energy term shown now by observation to be 96% of the large-scale limit of the universe.

However, it is not required to form the nonlocal unitary Feynman micro-quantum gravity amplitudes


B ~ Lp^2GradargPSI

Instead one must use the NON-UNITARY local macro-quantum coherent Landau-Ginzburg equation with a micro-quantum dark energy noise term against a coherent vacuum ODLRO dynamical background.

No comments: