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3.4. Gravity

Spin(5) has 10 infinitesimal generators. How Spin(5) gives produces gravity by the MacDowell-Mansouri mechanism 5,6,7 is shown in Chapter 4.

Spin(5) acts naturally on the quaternionic manifold S4 = Spin(5)/Spin(4),

giving the 4-dimensional Lagrangian, with curvature Fg4 and Dirac operator g¶g,

º ( - Fg4/\*Fg4 + `S8± g¶g S8± )

where g¶g acts locally on Q5+ = RP1 x S4 = ¶Silov(D5+ Å

Å Spin(7)/Spin(5) x U(1)).

V(M) = V(S4) and V(Q)/(V(D)^1/m) = V(RP1 x S4)/(V(D5))^1/4.

The resulting volume for full Spin(5) anti-de Sitter gravitation is, using volumes from Hua11:

V(S4) V(Q5+)/(V(D5+))^1/4 = (8¹^2/3)(8¹^3/3)/(¹^5/2^4 5!)^1/4 =

= 28¹^4(15/2¹)^1/4 / 9 = 3444.0924 .

The relative geometric force strength of the full Spin(5) anti-de Sitter gravitational force is taken to be 1, because it is the force with the greatest volume: aG = 1 .

[ Note that, about 10 years after 1992, I think that the physical interpretation for the value of unity for the geometric strength of gravity is that gravity is what holds spacetime together, and the amplitude for a graviton to be emitted/absorbed from one point of spacetime to a neighboring (with respect to Planck-length lattice structure) point of spacetime is unity, so that, with probability 1, each spacetime point is connected to its neighboring points, and spacetime holds together. ]

Full Spin(5) anti-de Sitter gravitation has a mass factor of me^2/mPlanck^2, where me is the electron mass and mPlanck is the Planck mass.

The value of the Planck mass is estimated in the F4 model, as discussed in

Section 6.5., to be mPlanck Å 1.0-1.6 x 10^19 GeV.

The electron mass me is the one assumed mass in the F4 model,

me = 0.5110 MeV, so the mass factor in terms of the electron mass is me^2/mPlanck^2 Å 1.0-2.6 x 10^-45.

That gives an effective gravitational force strength, including mass factor, of GGmproton^2 Å 3.4-8.8 x 10^-39 .

The characteristic distance of full Spin(5) anti-de Sitter gravitation is the Planck length, which can be considered to be the gravitational

Bohr radius 1/ aG mPlanck Å 1.6 x 10^-33 cm.

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