1. Spacetime itself is self-imaging. An informal way of showing this is to take Maxwell’s equations and add to the Maxwell *F _{μ ν}* and its vector source

Already in **Post 5** of this series we noted that associated with the propagating R-C curvature tensor is a new particle, the massless spin-1 *riemann*, replacing the non-existent graviton. Also, buried deep within the homogeneous R-C field equations are the Einstein gravitational field equations. The gravitational force thus may be said ultimately to come from the R-C equations. But there is an additional force as well, one we call the R-C force—a *fifth force *of nature overall—analogous to the Lorentz force of electrodynamics. Between massive bodies separated by macroscopic distances *r*, the R-C force is entirely negligible; its strength in that context is ~ (*ℓ */*r *)^{2 }that of gravity, where the constant length *ℓ *is of the order of* ℓ *_{P}—Planck’s constant (1.6 x 10^{−35} m).

However, the R-C force acting as a *self-force* is anything but negligible. In geometrodynamics, self-force is the force that the field generated by a given mass—whether R-C or gravitational—exerts on the mass itself. As it turns out, the R-C self-force is *repulsive*, whereas the gravitational self-force is of course attractive. Consider a massive sphere of constant density and radius *a*. While for such a body the gravitational self-force is proportional to *a *^{2}, the R-C self-force is proportional to* ℓ*_{ }^{2}, the same constant value for all spheres irrespective of radius. Thus in gravitational collapse, the R-C self-force precludes the formation of the black-hole singularity predicted by general relativity, providing instead a volume of mass of finite radius *ℓ* ~* ℓ *_{P} . Alternately, should a massive point singularity spontaneously arise, the essentially infinite mismatch between the R-C and gravitational self-forces can result in an explosion of spacetime itself—the Big Bang. So the R-C fifth force, far from being unobservable, may well have brought about the expanding universe we live in.

2. But what happens to the universe after the Big Bang?** **Why, since relative to the gravitational force the R-C force is getting weaker and weaker, does it continue to expand? Suppose, instead of adding two indices to the Maxwell field and its source, we add just one. We obtain yet another set of equations of the Maxwell form, with *F ^{μ ν}* replaced by the rank-3 tensor

** ** Now look at the structure of the tensor source. Formally it is given by

*J *_{α }* ^{ν }*=

where *j* ^{ν} = *ρ **v*^{ ν} is the convection current density, *v *^{ν }the four-velocity, and *ρ* the mass density. (Here *ℓ *is the same constant appearing in the R-C equations: we are still doing geometrodynamics.) The key here is to observe that *ρ* cannot be ordinary gravitating matter; for gravitation is already described by the Einstein equations, and those in turn are embedded in the R-C equations as described above. But the only candidate for *ρ * is the quantum-theoretic density * ρ *_{Q F T , }which, as it is ~ 124 orders of magnitude larger than the observed vacuum mass density of the universe, obviously cannot be gravitating, i.e., causing either catastrophic collapse or runaway expansion. Thus we assume a non-gravitating (NG) mass density

*ρ ≡ ρ* _{N G} ~ *ρ *_{Q F T }, (2)

with a constant of proportionality to be determined.

Finally, with respect to the tensor current (1), we note that it is proportional to the *gradient* of the convection current *j* ^{ν} . Thus all fields and forces in this theory are essentially edge effects. If *ρ* _{N G} is assumed spatially uniform, then the only place where a force could be felt is at the edge of a *bounded* universe. This new force, if it pushes outwardly on the boundary against gravity, can account for the expansion of the physical universe. We call it—a potential *sixth *force of nature—the cosmic Casimir (C-C) force, as it arises from a step in the mass density at the edge of the cosmos.

The C-C force, like the R-C force, is formally analogous to the Lorentz force of electrodynamics. From that familiar formalism one readily calculates the C-C self-force of the universe—our putative force of cosmic expansion. The dynamics of the expansion in a Robertson-Walker metric are captured in an expression for the rate of increase, *d a */ *d t *, of the dimensionless scale factor, *a *( *t *) . This expression reproduces the fundamental Friedmann equation for the time evolution of *a*, provided *ρ* _{N G} varies in accordance to a specific function of *a*. For the present time ( *a * = 1 ), that function yields for the current non-gravitating mass density

*ρ* _{N G , 0 }= 1.048 (*D *_{U 0 }/* ℓ *) ^{2} *ρ *_{C R I T} , (3)

where *ρ *_{C R I T }~ 10 ^{−29} g / cm ^{3} is the critical mass density of the universe (below which the universe expands, and above which it contracts), and *D *_{U 0} is its present radius. Now suppose that *ρ* _{N G , 0 }~ *ρ *_{Q F T} as suggested above. The stupendous ratio ~ 10 ^{124} between *ρ *_{Q F T} and the vacuum mass density *ρ *_{Λ} associated with Einstein’s cosmological constant (comprising 68% of *ρ *_{C R I T} ) is then explained: it is equal to the square of the ratio of the largest and smallest measurable distances in the universe, *D *_{U 0} and * ℓ *_{P }.

There seems now general agreement that the accelerating cosmic expansion is due to negative pressure induced by Einstein’s cosmological constant. The Law of Laws, however, suggests that the true cause of expansion may instead be the C-C self-force applied at the edge of a bounded universe. Even so, that does not mean that the Einstein formulation is wrong. For after all, both theories—those of Einstein and C-C—yield the same Friedmann equation governing the expansion of the universe. It means only that, in the Einstein equations, the cosmological constant is a stand-in for the true actor, the formerly misunderstood quantum-theoretic mass density, *ρ *_{Q F T }.