On the other hand, if the initial Planck scale vacuum phase transition has a degenerate vacuum manifold
V(Planck) = G(False Vacuum)/H(GUT) ~ S2
then the GEOMETRODYNAMIC FIELD does not have 1-D string defects, but, rather has 2D point monopole defects where the quantized 1D winding numbers are replaced by 2D wrapping numbers.
That is, we have 2 alternative models for the initial emergence of Einstein's gravity at 10^-43 sec.
I. V(Planck) ~ S1 circle
Here there is only ONE Goldstone phase THETA with 1D string defects where the Higgs magnitude of the vacuum order parameter vanishes along the string and the Goldstone phase THETA is undefined. This is the usual easy to visualize Mexican Hat Potential for renormalizable quantum field theories that spontaneously break internal symmetries.
Here the local virtual vacuum order parameter is a single complex scalar field like in superfluid helium and the BCS superconductor for real bosons on-mass-shell.
Stable quantized circulation comes from the first homotopy group being non-trivial, i.e.
PI1(S1) = Z (integers)
B(curved) = Lp'd'THETA 1-form curved tetrad of Einstein's GR
The geometrodynamic vorticity flux dB(curved) is quantized through areas including the string (vortex) defect. This is an example of Bohm-Aharonov "Flux without flux" because dB = 0 on the loop L in physical space that surrounds the singularity with the Higgs magnitude vanishes and the Goldstone phase is undefined. Going around the loop one in physical space means going around the S1 circle that is the vacuum manifold N times. N is the number of trapped geometrodynamics flux quanta. This is not to be confounded with magnetic flux.
II. V(Planck) ~ S2 spherical surface
Here we obviously have TWO Goldstone phases THETA and PHI with point defects.
How do we construct B(curved)?
Here the local vacuum order parameter is 3 real scalar fields like the 3x3 adjoint irreducible representation of the SU(2) group corresponding to "spin 1" rather than "spin 1/2".
However, the real non-vanishing vacuum order parameter can be chosen to lie completely along the arbitrarily chosen polar axis of S2 = G(False Vacuum)/H(GUT) so that the complementary Goldstone phase is the azimuthal angle PHI in the equatorial plane in this fiber space of degenerate order parameters beyond space-time.
B(curved) = Lp'd'PHI 1-form
Here its the second homotopy group that is non-trivial
PI2(S2) = Z
The curvature 2-form with physical dimension 1/(area) is
R = DW
where W is the spin connection 1-form
W = -*[dB/\(1 - B)]
Therefore, the dimensionless DeRham integral of R over a surrounding S2 surface in physical space enclosing the point defect is quantized even though the Bianchi identity DR = 0 everywhere other than the point defect singularity.
That is closed surface integral of the curvature 2-form R ~ Z
The wrapping number around V = G(False Vacuum)/H(GUT) = S2 is the number of Bekenstein bits equal to the number of curvature quanta through the closed surface.
This argument implies that there are no Bekenstein bits in globally flat spacetime.
Note that the curvature 2-form maps to the 4th rank Riemann-Christoffel curvature tensor i.e.
R^ab = R^abuvdx^udx^v
R^wluv = R^abuvea^we^bl
That is, the Bekenstein BITs require B(curved) =/= 0.
Therefore, quantization of "area" has the same status as quantization of circulation in a superfluid. Both point to an underlying ground state order parameter. Geometrodynamics is an emergent local gauge theory of T4 ---> Diff(4) from the spontaneous breaking of an internal symmetry at the Planck era.
We only need ONE point defect geometrodynamic monopole in the universe to produce the Bekenstein BIT exactly like we only need 1 magnetic monopole in the universe at the end of the Dirac string to produce quantization of electric charge.
On Nov 13, 2005, at 8:18 PM, Jack Sarfatti wrote:
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
THE QUANTUM GRAVITY ERA
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
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
PSI is the INFLATION FIELD
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.]
THE INFLATION AND SYMMETRY BREAKING ERA
1x10^-38 1x10^29 KT 72 Limit of perturbative interaction
thermalization of universe
1x10^-35 1x10^28 KT 72 Grand unification spontaneous
[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
[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.]
THE QUARK-LEPTON ERA
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
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
THE RADIATION ERA
1x10^2 1x10^9 BS Typical photon energies drop below
the deuteron binding energy and
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
THE MATTER ERA
4x10^12 3.5x10^3 KT 78 Electrons and protons recombine into
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.