Sunday, November 13, 2005

Planck Era Emergent Gravity, Lepton Quark Rest Masses & Bekenstein Area Quantization
Note two meanings of "inertia".

Gravity as a "force" is locally indistinguishable from an inertial force in a non-inertial local frame that is either rotating or on a time-like nongeodesic path or both with respect to the Levi-Civita connection.

"Inertia" as the rest mass of quarks, leptons and W-mesons.

Our universe is thought to have formed in a sequence of spontaneous symmetry breakings of the physical vacuum. Assuming the old Heisenberg uncertainty principle

Planck scale breaking ~ 10^19 Gev ~ 10^-33 cm

GUT scale breaking ~ 10^15 Gev ~ 10^-29 cm

Electroweak breaking ~ 250 Gev ~ 10^-16 cm

In more detail

Time (s) T (Kelvin) Reference Event

1x10^-43 1x10^32 KT 72 Quantum limit of classical general

[My comment 1:

I postulate, as there is no fundamental theory of this as yet, the simplest possibility

G(Planck)---> H(Planck)

V (Planck)= G(Planck)/H(Planck) ~ S1(Planck)

from first homotopy group PI1(S1) = Z (integer winding numbers)

i.e. line defects i.e. 1-dim strings

Vacuum coherent order parameter is a single complex scalar field

PSI(Planck) = |PSI|e^iargPSI

Planck-scale Higgs amplitude is |PSI|, Planck-scale Goldstone phase is argPSI.

argPSI is a 0-form

Curved space-time tetrad 1-form is

B = Lp'd'argPSI

ds^2 = (1^a + B^a)(Minkowski)ab(1^b + B^b) is EEP

B is from local gauging of SPACE-TIME SYMMETRY T4 to Diff(4) AND ALSO from spontaneous breaking of INTERNAL SYMMETRY G(Planck) -> H(Planck) to S1 Post-Planck Vacuum Manifold


Diff(4) covariant exterior derivative is

D = d + W/
W^ac = T4 spin-connection

The vanishing torsion 2-form in 1915 GR model is

T^a = De^a = de^a + W^ac/\(1 + B)^c = dB^a + W^ac/\(1 + B)^c = 0

Invert this equation to get the 1-form spin connection

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

* is Hodge dual based on Minkowski metric in tangent fiber space.

The geodesic deviation curvature 2-form is

R^ab = DW^ab = dW^ab + W^acW^cb

The Bianchi identity is the 3-form equation (analog to dF(EM) = 0 3-form)

DR^ab = 0

The Einstein-Hilbert vacuum Lagrangian density is the 0-form


The Einstein curvature 4-form is

G^ab = D*W^ab

The Einstein 4-form source equation (analog to d*F(EM) = *j(electric) 3-form) is

G^ab = *J(T4)^ab

Local conservation of stress-energy source current densities is the vanishing 5-form

DG^ab = D*J(T4)^ab = 0 5-form in 4-space is identically zero.

G^ab = *J(T4)^ab

is 4-form because of equivalence principle where even zero point energy with w = -1 gravitates absolutely and not relatively.]


1x10^-38 1x10^29 KT 72 Limit of perturbative interaction

thermalization of universe

1x10^-35 1x10^28 KT 72 Grand unification spontaneous

symmetry breaking

[My Comment 2

OK, here H(Planck) ---> H(GUT) = U(1)hyperchargeSU(2)weak

SU(3)quark coupling constant splits off from U(1)hyperchargeSU(2)weak with same coupling constant

We do not know the topology of the new vacuum manifold

H(Planck)/U(1)hyperchargeSU(2)weak = V(GUT) = ?

Note the sequence:

G(Planck) ---> H(Planck) ---> H(GUT) = U(1)hyperchargeSU(2)weak

G/(Planck)/H(Planck) ---> H(Planck)/H(GUT)


V(Planck) ---> V(GUT)

But what does this mean in terms of the vacuum order parameter?

V(Planck) does not disappear. It is robust from Goldstone phase rigidity and it is the Hercules supporting The World Geometrodynamic Field


Indeed the total vacuum manifold has grown more complex into the TENSOR PRODUCT


1x10^-34 1x10^27 KT 274 Approximate start of inflation

1x10^-32 1x10^27 KT 274 Approximate start of reheating and

end of inflation

1x10^-11 3x10^15 KT 72 Electroweak unification spontaneous

symmetry breaking

[My Comment 3

Here H(GUT) = U(1)hyperchargeSU(2)weak --3x3 adjoint irrep--> U(1)em

V(Electroweak) = U(1)hyperchargeSU(2)weak/U(1)em ~ S^2

with point defects. W+,W0, W- massive A(photon) massless, 1 Higgs & masses of leptons & quarks from Yukawa couplings.

Prior to that 250 Gev scale all quantum fields are massless. There is gravity already emergent from Planck era. It does not have to be put in by hand adhoc anymore. You can have curved spacetime without massive particles but not vice-versa, because to do so violates the equivalence principle.

Now the vacuum manifold is V(Planck)xV(GUT)xV(Electroweak)

Although the short-scale vacuum order parameter may be a single complex scalar field with an S1 vacuum manifold, there is reason to believe that the larger-scale vacuum order parameter after the electroweak splitting of the weak SU(2) charge from the U(1) electromagnetic charge is essentially of the character of isovector, i.e. 3x3 adjoint irrep of SU(2) with effective vacuum manifold S2 hence geometrodynamic point defects not to be confused with magnetic monopoles. Having geometrodynamic point defects instantly gives the Hawking-Bekenstein area quantization because of the second homotopy group PI2(S2) = Z giving stable 2D "wrapping numbers" instead of the stable 1D winding numbers. The 3-adjoint irrep of SU(2) is 3 real scalar fields of which in one Higgs model only one scalar field develops a non-vanishing VEV Higgs amplitude. We can imagine its conjugate Goldstone phase as the rotation about it as an axis in V(Electroweak) ~ S2 i.e. the azimuthal angle about the "z" axis chosen so that the order parameter is electrically neutral. We then must assume that this is the same Goldstone phase that survived from the Planck era. These speculations are of course very new.]


2x10^-7 2x10^13 HA 353 Tauon anti-tauon annihilation

1x10^-5 2x10^12 KT 72 Formation of hadrons from quarks

7x10^-5 1x10^12 HA 353 Muon anti-muon annihilation

5x10^-4 4x10^11 KT 159, 281 By this time the universe has a

baryon-antibaryon asymmetry

which results from post-inflationary

B,C,CP violating processes

1x10^-1 3x10^10 BS Neutral current weak interactions

become too slow and neutrinos


1x10^0 1x10^10 BS Charged current weak interactions

become too slow and neutron to

proton ratio freezes out

1x10^1 5x10^9 BS Electron positron annihilation


1x10^2 1x10^9 BS Typical photon energies drop below

the deuteron binding energy and

nucleosynthesis begins

1x10^3 4x10^8 BS Particle energies drop below coulomb

barrier energies and nucleosynthesis


4x10^10 6x10^4 KT 77 Matter density becomes equal to

radiation density


4x10^12 3.5x10^3 KT 78 Electrons and protons recombine into

hydrogen atoms

6x10^12 3.0x10^3 KT 80 Photon decoupling

2x10^16 20 HA 349 Formation of galaxies

(X-1.4x10^17) --- KT 12 Formation of the solar system

(X-9.9x10^16) --- HA 390 Emergence of life on Earth

X ~ 4.7x10^17 2.726 KT 12 Today

These values are rough estimates only. I have taken Omega_0=1, Lambda_0=0, and h=1.

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