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Theodor Kaluza (University of Konigsberg)
Oskar Klein
Kaluza-Klein theory - 1919, 1921, 1926
Theodor Kaluza, a Polish mathematician, sent a proposal to Einstein on April 21, 1919. It was a short article, only a few pages long. This was an astounding suggestion that left Albert Einstein speechless. In just a few lines, Kaluza had shown that it was possible to unify Albert Einstein's theory of gravity with Maxwell's theory of electromagnetism. To achieve this unification, Kaluza proposed that there were actually more than 3 dimensions of space.
Theodor Kaluza, extended general relativity, to include and introduce a 5th dimension. In this picture, there is 4 dimensions of space and 1 of time. Light would be a ripple in the fabric of this 5th dimension. For Kaluza, light was the warping of a higher dimensional geometry.
Kaluza wrote the field equations of Albert Einstein for gravitation in 5 dimensions, instead of 4. The metric tensor of Bernhard Riemann can be formulated in any number of dimensions. Kaluza showed that these 5-dimensional equations included the 4-dimensional equations of Einstein (which was to be expected). However, it also included an additional piece. This additional piece was Maxwell's equations for light. Kaluza's idea was that the two greatest fields of science, Einstein's and Maxwell's could be harmonized in the 5th dimension. This was a purely geometrical theory. Albert Einstein, initially was intrigued with the idea. This was a framework that could potentially weave together Albert Einstien's general theory of relativity with James Clerk Maxwell's theory of electromagnetism. Kaluza found that by including an extra spatial dimension, he found additional equations to Einstein's. Kaluza found that these equations, were the very same equations that Maxwell had written for electromagnetism in the 1880s! By adding another dimension, Kaluza had unified gravity with electromagnetism. Prior to this, there was no indication of an underlying symmetry between these two forces. Light and gravity appear, at first glance to have nothing in common. They are like oil and water, even mathematically. Maxwell's theory requires 4 fields and Einstein's metric theory of gravity requires 10 fields. However, even Einstein had to admit that the theory of Kaluza was elegant. Gravity and electromagnetism were both associated with ripples in the fabric of spacetime in this new curled-up 5th dimension of Kaluza.
As stated earlier, Maxwell's theory requires 4 fields and Einstein's metric theory of gravity requires 10 fields. In order to proceed, we need to understand the concept of a metric tensor in general relativity. This is a collection of numbers that assigns a value to every point in space that can define its curvature at that particular point. The metric tensor, in 4 spacetime dimensions, requires 10 numbers at each point of the manifold to describe how distorted it is. This is the basis of the work of Albert Einstein and Bernhard Riemann. This is a collection of 10 numbers defined at each point in space. These 10 numbers can be arranged in a checkerboard pattern of 4 x 4. Thus, for Maxwell, his field will be a collection of 4 numbers defined at each point in space. The idea of Kaluza, thus, was to re-write the metric tensor in 5 dimensions, in the pattern: 5 x 5. The 5th row was for the Maxwell field, while the remaining 4 x 4 block was for the 4-dimensional metric tensor of Einstein. Kaluza was able to combine the fields of Einstein and Maxwell simply by adding another dimension. However, there was one last component to this 5-dimensional metric, creating a total of 15 components:
There were 10 components for the four-dimensional spacetime in Einstein's field for gravity.
There were 4 components for the electromagnetic field. Maxwell's field is nicely included into the Riemannian metric tensor in Kaluza's work.
The last and 15th component of Kaluza's 5-dimensional metric was for a scalar particle. This particle is not important for our conversation here.
For Einstein, the idea that unification could be achieved through a 5-dimensional universe (4 of space and 1 of time) had never occurred to him. However, about a week later, Einstein will write to Kaluza again. This time Einstein is a little bit more skeptical. Einstein found Kaluza's theory interesting and didn't see any real impossibilities in the framework. However, he was not fully convinced. Nevertheless, Kaluza's idea was published 2 years later on October 14, 1921. Einstein decided to present Kaluza's idea to the academy after all. He even stated that he regretted his suspicion and having restrained publication of Kaluza's idea. The name of the paper was: "On the Unity Problem of Physics." Einstein recognized the originality of Kaluza's theory, despite having delayed it's publication.
However, the question still remains: where is this extra 5th dimension? The answer is that our universe may have both extended and curled-up dimensions of space. The 4 extended dimensions of our universe are the 3 of space and 1 of time that are easily visible, large and accessible to common experience. However, the universe could have dimensions that are invisible and tightly curled up into a tiny space. These are spaces so tiny that they have evaded our detection by even our most advance experimental apparatuses. Our current cutting-edge equipment can detect scales that are about a billionth of a billionth of a meter. Distance scales smaller than this cannot be detected, this includes curled-up and microscopic dimensions.
Oskar Klein, in 1926, gave a quantum physical interpretation for the extra 5th dimension. Klein's idea was that quantum mechanics could explain the geometric nature of this curled up dimension. Quantum mechanics was emerging as a serious field during this period of time in the 1920s. For Klein, the extra dimension was a curled up, microscopic and undetectable circle, with the radius 10^-33 cm, that would exist at each point in space. Klein calculated that this extra curled up dimension might be as small as the Planck length! This is the only length scale that would naturally arise in a quantum theory of gravity.
The analogy for compactification that physicists love to use: "A garden hose, viewed from a sufficient distance, appears one-dimensional. However, when it is magnified, another dimension becomes visible: the circular circumference of the hose. In the same way, our universe could have tiny curled up dimensions that we just can't observe!"
This extra dimension would be so small that atoms cannot even fit inside. The Planck length is 24 orders of magnitude smaller than an atom and 19 orders of magnitude smaller than a proton. This small circular dimension is too small to measure and is far shorter than what is accessible to even our most high energy experiments. This is an additional curled up dimension of space. This is a new dimension that would exist at every point in the extended three dimensions of space! There is also interesting topological implications for our universe. This extra dimension, would take on the geometry of a circle, while the universe would take on the geometry of a cylinder.
This Kaluza-Klein theory was an attempt at unification of gravitation and electromagnetism by extending general relativity to a 5th dimension. Sadly, despite the theory being a beautiful idea, the theory proved unsuccessful, since:
It proved the existence of a particle that has never been shown to exist.
It was not able to correctly predict the ratio of an electron’s mass to its charge. The attempts that were made to bring the electron into Kaluza-Klein theory predicted a relationship between the mass and charge that were vastly contradictory from it's measured values.
Kaluza's theory raised more questions than it answered. Physicists were not fully convinced that the 5th dimension actually existed. Perhaps, worst of all, the proposal that there is a curled-up circular dimension on the scale of the Planck length could not be tested. The energy necessary to do this is called the Planck energy and it is 10^19 billion electron volts. This amount of energy is beyond comprehension. At this point, many physicists lost interest in Kaluza-Klein theory and in curled-up extra dimensions. The attention of the mainstream theoretical physics community would turn over to quantum mechanics. Nevertheless, Kaluza-Klein theory will prove to be an important precursor to string theory, as a potential means of making sense of the extra dimensions. As we will see, Kaluza and Klein were ahead of their time.
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