A FASCINATING JOURNEY TO THE FIFTH DIMENSION

Human beings are capable of achieving things that at first seem completely beyond their reach. While extraordinary people like Magellan circumnavigated the globe in rudimentary sailing ships, or others like Neil Armstrong landed on the moon, other, no less extraordinary people made other, even more incredible journeys. In 1921, when humanity had barely begun to make the first air journeys, a physicist named Theodor Kaluza, with the simple help of paper and a pencil, was able to "travel" to the fifth dimension of space-time. Such a feat is only possible thanks to the universal reach of mathematics and the extraordinary development of 20th-century physics. Five years later, another physicist named Oskar Klein expanded on this journey by providing an explanation of the hidden phenomena in the fifth dimension.

What these physicists discovered on this journey was so impressive that Einstein himself was completely amazed. In this article, we will explore, using simple concepts and expressions, the hidden "treasures" that Kaluza and Klein discovered in the fifth dimension and how these discoveries continue to be studied by modern physics today.

Welcome to the new dimension of fundamental physics!

The metric tensor of General Relativity and Electromagnetism

Before describing the Kaluza-Klein theory (KK theory from now on)

We must briefly review some key concepts of general relativity and electromagnetism.

General relativity unifies the three spatial dimensions and the one time dimension into a single four-dimensional entity called space-time. The metric specifies how distances are measured in a given space-time. For relativistic flat Minkowski space-time, the metric is:

The metric tensor contains all the information needed to perform a coordinate change from one reference frame to another while preserving distances. The diagonal of the metric tensor contains the metric in all four coordinates; thus, the general relativity metric tensor for relativistic flat spacetime is:

If we want to describe curved space-times, the values of the metric tensor must measure curvilinear paths, which is why polar coordinates are often used.

For example, to transform the three-dimensional Euclidean plane metric given by:

In the curved metric of a sphere we use the spherical coordinates given by:

To find the metric in spherical coordinates we simply use the equivalence between Cartesian and polar coordinates and derive:

In this way:

And doing calculations we get that the metric is:

Since, considering the cross components the metric can be written as:

The metric tensor will be:

This would be the metric tensor for a spherical surface like the Earth. Another example would be the 4D metric for the spacetime created by a Schwarzschild black hole:

In this space-time it can be seen that at the point r=2GM (on the surface of the black hole) the metric tensor diverges and the distances become infinite!

To calculate the dynamics of a body in a curved spacetime, we must calculate the partial derivatives of the metric tensor with respect to all dimensions of spacetime. To do this, we use the so-called Christoffel symbols, whose notation is:

Where i, j, k vary from 1 to 4. This simply means taking the partial derivatives of the components of the metric tensor gi j with respect to k as follows:

This is the first-degree Christoffel symbol. To obtain the second-degree symbols, the notation is:

We simply multiply the already calculated first degree symbol by the metric:

As we will see later, this last expression will be crucial to understanding the so-called Kaluza-Klein mechanism.

To conclude this section, let us recall Maxwell's equations that describe electromagnetism:

All these equations can be derived by considering a potential vector A:

which satisfies the following expression (gauge invariance):

Where F is the force field of electromagnetism. With these ingredients, we're ready to enter the fifth dimension.

Adding the fifth dimension

Each row or column of the metric tensor can be considered as a vector of n components. In the 4D space-time of general relativity, we have 4-component vectors: V = (Xo,X1,X2,X3). If we want to add a new dimension, we must include a new spatial coordinate to the four-dimensional metric tensor of general relativity. To do this, we must add the coordinates shown in brown:

Since the metric tensor is generally symmetric, the coordinates g04, g14, g24, and g34 in the last column and row have the same values. Therefore, it is sufficient to define only a new five-component vector A for the new dimension. We will describe the new vector associated with the fifth dimension as follows: A=(A0, A1, A2, A3, psi). Where psi is a field that can take any value. The five-dimensional metric tensor will then be:

To describe the dynamics in the new components, we must calculate the associated Christoffel symbols. These "symbols" imply cross-partial derivatives between all dimensions of the metric. As we saw in the previous section, in general relativity, the way to calculate them is:

Where the values of n,v,p vary from 1 to 4. Similarly, the Christoffel symbols for five dimensions will be:

Where M, N, R now vary from 1 to 5. In this calculation, there are "cross" components that include both partial derivatives of the fifth dimension with respect to the other dimensions and partial derivatives of the other dimensions with respect to the fifth dimension . We are interested in these latter components. The Christoffel symbols for these "cross" components associated with the fifth dimension will be:

This is where we reach the crucial point of the so-called "Kaluza-Klein mechanism." Since there is no experimental evidence for the existence of the fifth dimension, it must be very small and periodic, so it must be "rolled up" into a cylinder. Furthermore, the influence of this new dimension on the rest of the larger dimensions must be negligible, which means that the value of these "cross" derivatives with respect to the fifth dimension must be zero:

Therefore, Christofell's symbols associated with the fifth dimension are reduced to:

Applying this to our vector A associated with the fifth dimension of the metric tensor we obtain:

But this is precisely the definition of the electromagnetic field force tensor! The tensor F that describes the electromagnetic force is defined as:

This is the first "treasure" that Kaluza and Klein found on their journey: by allowing the metric tensor to access a compactified fifth dimension, we have succeeded in "creating" the electromagnetic field from empty space-time . But we have also achieved another, even more impressive feat: we have managed to unify gravity and electromagnetism, since both fundamental forces can be derived from the same source: the five-dimensional metric tensor.

You can imagine Einstein's face when Kaluza sent him the article for review in 1919! But what does this fifth dimension mean? Is this new dimension real? How can we interpret this fascinating result?

From left to right: Theodor Kaluza, Oskar Klein and Albert Einstein

Unveiling the mystery of the fifth dimension

Five years after Kaluza's surprising work, physicist Oskar Klein published an article that clarified many aspects of this new and mysterious fifth dimension.

Klein identified the coordinate system that preserved the cylindrical condition associated with the fifth dimension. The coordinate system associated with five-dimensional geometry that preserves the cylindrical condition is precisely the one that contains a U(1) gauge symmetry and leaves the electromagnetic force field invariant.

The g44 coordinate of the metric tensor represents the value of a scalar field (sometimes called a radion) that "lives" in the fifth dimension. This scalar would propagate through the fifth dimension following the usual wave function. Due to the circular periodic geometry of the fifth dimension, this new field would have excited states n1, n2, ... whose increasing mass/energy would be proportional to n/R where R is the radius of the circular dimension. This "tower" of particles of increasing mass would be an indicator of the existence of new compactified dimensions and is being actively searched for by high-energy experiments such as the LHC accelerator.

Klein observed an analogy between optics and mechanics and commented: "The Hamilton-Jacobi wave equation has to be interpreted as an equation in five dimensions instead of four." With this interpretation we obtain that the electromagnetic charge that we feel in our four-dimensional Universe is due to the momentum of a scalar field moving in the fifth dimension . To see this more clearly, it should be noted that, in KK theory, the tensor equation that describes the acceleration of a charged particle in a magnetic field is:

While the equation for a freely moving particle in KK spacetime is:

Comparing the two previous expressions makes it clearer what we explained earlier: electric charge e constitutes the fifth component of momentum in five dimensions. That is, the charge detected in four dimensions is a manifestation of motion in the fifth dimension! This is the second "treasure" hidden in the fifth dimension.

Intuitively, the new compacted dimension can be visualized by assigning a circle to each point in 4D space-time. More specifically, viewed at very high energies, each point in our space-time is actually a circle.

Recreation of a very high-energy particle moving through 5-dimensional space-time

The third "treasure" hidden in the fifth dimension

To discover the last hidden treasure in the new dimension, we'll follow a wave function moving through 5-dimensional space-time. The wave function will be:

The invariance of this wave function with respect to phase changes implies:

Therefore:

What this entails:

Since k has units of action we can assign it the following constant value:

Which finally brings us to:

This expression is Heisenberg's uncertainty principle! All the principles of quantum mechanics can be derived from this expression!

By allowing access to a compacted fifth dimension, we have discovered the fundamental principle of quantum mechanics! A new compacted dimension would explain, in geometric terms, the appearance of the quantum effects we see in our 4D Universe. The interpretation of this fact is controversial since, as we shall see, a single extra dimension is not enough to explain our Universe.

The unification of fundamental forces

We know that in our Universe there are more forces than just gravity and electromagnetism: the weak nuclear force and the strong nuclear force. The obvious question is: Can we unify these forces by accessing new dimensions? We know from the Standard Model of particle physics that the symmetry responsible for the weak force is the SU(2) gauge symmetry and that for the strong force is the SU(3) gauge symmetry. Because of this, it seems natural to try to include these new symmetries in an extended KK theory. This has led physicists to include larger symmetry groups and to use a larger number of dimensions. This process has led fundamental physics to the only theory capable of unifying all the fundamental forces:

Superstring theory . Superstring theory incorporates six new compactified dimensions and uses higher symmetry groups such as the E8 group.

We can conclude that although the KK theory does not describe our real Universe, it does provide a mechanism capable of unifying the fundamental forces: the Kaluza-Klein mechanism lays the foundation for unifying gravity and the rest of the fundamental forces in a single 10-dimensional geometric framework.

Sources:

Kitcher Explanatory Unification, Kaluza-Klein Theories , Quantization as a consequence of Kaluza-Klein projection