The unification of classical electrodynamics and general relativity within the context of five-dimensional general relativity (Kaluza, 1921, and Thiry, 1948) contains a scalar field which may be identified with the gravitational constant,

General relativity and classical electrodynamics are elegantly unified in the classical five-dimensional (5D) theory introduced by Kaluza [

The essence of the classical 5D theory is to posit a 15-component 5D metric comprising the usual 10-component four-dimensional (4D) metric, the electromagnetic 4-vector potential, and a scalar field. Applying the 5D vacuum Einstein equations to the 5D metric yields the 4D Einstein equations with electromagnetic sources, plus the vacuum Maxwell equations. A fifteenth equation describes the scalar field. Applying the 5D geodesic hypothesis to the same metric yields the 4D geodesic equation with the Lorentz force term. To this framework is added the constraint, known as the cylinder condition, that none of the fields are observed to depend on the fifth coordinate.

It is unorthodox to seek fundamental unification in a theory which predates the quantum revolution. And it was the quantum discoveries which spurred abandonment of the purely classical 5D theory. But the perspective of 85 years has revealed to us what could not have been known when work on the classical theory was dropped: that a quantum theory of general relativity may be impossible, thereby blocking any attempt at a unification with quantum electrodynamics. If we are to unify general relativity with the other forces, we are obligated to look again at the common symmetries of general relativity and classical electrodynamics.

An intriguing result of 5D relativity is that the coupling of electromagnetic stress energy in the 4D Einstein equations is variable, depending on the scalar field. In other words, the gravitational “constant" will vary in a radiation-dominated universe. A radiation-dominated universe is an appropriate model of the early history of our own universe, from the Big Bang until the time of radiation-matter equality when the universe was around 50,000 years old [

The fully self-consistent field equations of 5D relativity [

In the limit that the scalar field

The 5D unification leads to couplings between

Now we apply these equations to the standard results for a radiation-dominated universe, for example, [

For (

We turn now to the modified Friedmann equation, which is obtained from the

The electromagnetic energy density will scale as

Equations (

In these equations,

When

The reference time

Note that the effective gravitational constant

Of course, the variation of the gravitational constant,

Inferences about

The result (

Let us now modify the Dirac hypothesis by adjusting the LHS of (

The theory does not provide us with the normalizing constant for (

When Brandenburg's expression (

Of course, if Brandenburg's expression (

The point is that one may expect quantum effects to provide the normalization of this classical theory, as in (

The classical unification of electrodynamics and general relativity predicts that the electromagnetically induced deformation of spacetime is mediated by a scalar field. This scalar field will manifest as a varying gravitational constant,