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, 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.