This is the third in the series of posts on Physics and the Origins of the Universe, and contains a discussion of the Kaluza-Klein hypothesis, which is a hypothesis which brings Einstein's field equations together with Maxwell's equations for Electrodynamics. Its proposition, that there are five dimensions of space and time, not four, and that it is hard to characterise the fifth spatial dimension in terms of size, may point to the presence of a plenum beneath physical reality.

The Plenum and its Properties (II)

How many dimensions are there? It depends on what you are
doing and how many you need to describe and understand physical reality. The
number will change according to what phenomenon is being explored.

We are accustomed to thinking that, for most of recorded
history, the human race has made do with three dimensions of space, plus time,
in order to make sense of the world. If we treat time as a dimension, that
gives us four dimensions. Three separate dimensions in which things can exist,
and in which they can move. And whether these things are motionless or moving,
they are also moving through time, at the same rate, irrespective of whether
they are travelling left or right, backwards or forwards, or up or down.

In fact four dimensions is an inadequate number to explain
how the world of physical reality actually works, and how its component parts
fit together. For the most part, our understanding of what physically exists is
described in scalar and vector terms within four dimensions, because that seems
to work. Most of the time. Unless you are working in Fermilab or CERN, where
their descriptions of physical reality concern the quantum level of physical
reality, and things are more complicated there. But even at the anthropic
scale, sometimes four dimensions can be found wanting. A description is not an
explanation, and some things are therefore not explained.

I will borrow a little from the Wikipedia article on the
Kaluza-Klein theory, which has been described as a unified theory of
gravitation and electromagnetism built around the idea of a fifth dimension
beyond the standard four of space and time. The article describes how this five
dimensional theory was developed, and breaks it down into three steps.

The original hypothesis came
from Theodor Kaluza, who sent his results to Einstein in 1919,

^{[1]}and published them in 1921. Kaluza's theory was a purely classical extension of general relativity to five dimensions. The five-dimensional metric has 15 components. Ten components are identified with the four-dimensional spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field.
This extension to five dimensions for Einstein’s equations yields

…the four-dimensional Einstein field
equations, the Maxwell equations for the electromagnetic field, and
an equation for the scalar field.

Now that is pretty interesting, and it is surprising that
physicists were not falling over themselves at the time to understand the
mechanics of this mathematical description which tied electromagnetic vector
potential with Space and Time.

It would be intriguing to know what was in Theodor Kaluza’s
mind when he came up with this five dimensional model, which involved an
unidentified scalar field. His model
involved a hypothesis which is described as the ‘cylinder condition’, which
means that

that no component of the five-dimensional
metric depends on the fifth dimension. Without this assumption, the field
equations of five-dimensional relativity are enormously more complex.

In other words, the five dimensional model of physical
reality has two potential states – one of which results in impossibly difficult
field equations. The cylinder condition is the other state for the model, which
results in relatively simple and intelligible equations.

The article suggests that

^{}
standard four-dimensional physics seems to
manifest the cylinder condition.

Which may be the case, or it may be that there is a
fundamental problem with the five dimensional model, and that it is a
conception which doesn’t reflect physical reality. Kaluza

… set the scalar field equal to a
constant, in which case standard general relativity and electrodynamics are
recovered identically.

Meaning that Kaluza knew how the equations should turn out,
and he knew how to characterise the unknown scalar field in order for the
equations to turn out correctly.

The second step in the development of Kaluza’s theory was a
quantum interpretation, supplied by Oskar Klein in 1926. This to enable the
theory to

accord with the then-recent
discoveries of Heisenberg and Schrödinger. Klein introduced the hypothesis that
the fifth dimension was

*curled up and microscopic,*[my emphasis] to explain the cylinder condition. Klein also calculated a scale for the fifth dimension based on the quantum of charge.
The third step happened in the 1940s, when the classical
theory was completed, and the

full field equations including the
scalar field were obtained by three independent research groups

The introduction to the article concludes by saying that
even under the cylinder condition the full Kaluza equations are quite complex,
and that

The complete Kaluza equations were
evaluated using tensor algebra software in 2015

Here is a section from the main body of the Wikipedia page.
I can understand the concepts involved, but I’m not remotely competent to
evaluate the equations. I’ve left the
links to the Wikipedia references intact. [this version of the document presents the Wikipedia text as graphics, so the links are not functional. The original article is at: https://en.wikipedia.org/wiki/Kaluza-Klein_theory]

I’ll pick out some parts of the text -

So far, this decomposition is quite
general and all terms are dimensionless. Kaluza then applies the machinery of
standard general relativity to this metric. The field equations are
obtained from five-dimensional Einstein equations, and the equations of
motion are obtained from the five-dimensional geodesic hypothesis. The
resulting field equations provide both the equations of general relativity and
of electrodynamics; the equations of motion provide the four-dimensional geodesic
equation and the Lorentz force law, and one finds that electric
charge is identified with motion in the fifth dimension.

It has been an objection to the
original Kaluza hypothesis to invoke the fifth dimension only to negate its
dynamics. But Thiry argued that the interpretation of the Lorentz force
law in terms of a 5-dimensional geodesic mitigates strongly for a fifth
dimension irrespective of the cylinder condition. Most authors have therefore
employed the cylinder condition in deriving the field equations.

…It shows that the electromagnetic
field is a source for the scalar field. Note that the scalar field cannot be
set to a constant without constraining the electromagnetic field. The earlier
treatments by Kaluza and Klein did not have an adequate description of the
scalar field, and did not realize the implied constraint on the electromagnetic
field by assuming the scalar field to be constant.

In the Kaluza theory, the
gravitational constant can be understood as an electromagnetic coupling
constant in the metric. There is also a stress-energy tensor for the scalar
field. The scalar field behaves like a variable gravitational constant, in
terms of modulating the coupling of electromagnetic stress energy to spacetime
curvature. The sign of phi in the metric
is fixed by correspondence with 4D theory so that electromagnetic energy
densities are positive. This turns out to imply that the 5th coordinate is
spacelike in its signature in the metric.

***

So what are we actually looking at with this five
dimensional model of physical reality? The theory unifies Einstein’s field
equations, and Maxwell’s equations, and does appear to be describing physical
reality, even if there are problematic aspects to it.

The fifth dimension is invoked, and then dismissed as a
functional component in the model, which reduces it to a dimension on a par
with the other four.

*But it is very small*, and may be rolled up. It is practical to treat it as if it is on a par with the other dimensions, which is what physicists generally do. But ignoring the complexity the model may have beyond the cylinder condition – one of the two states of the model - doesn’t mean that it does not have this complexity, and that the fifth dimension does not have an important role in how the physical world functions.
It is possible that what has been identified in the course
of the development of the theory is in fact the plenum, and not simply another
dimension on a par with the other four.

Thomas Yaeger, 22 February 2016.