###### By Dan Hooper, Ph.D., University of Chicago

## Einstein wasn’t alone in the search of a Unified Field Theory. From Weyl’s version of metric tensors to Kaluza’s Fifth Dimension to Eddington’s Affine connection, there were many attempting to build a Unified Theory.

### Weyl’s Metric Tensor and Unified Field Theory

The first attempt at a unified field theory wasn’t made directly by Einstein himself. Instead, it was by the German physicist and mathematician Hermann Weyl. However, Einstein and Weyl were in communication during this time, and they discussed some of the aspects of this problem together. So, at least to some extent, Einstein was involved.

From Weyl’s perspective, there was one central challenge that made it so hard to combine general relativity and electromagnetism into one unified field theory. This challenge was that general relativity is a theory of geometry, while electromagnetism is not. Maxwell’s equations described the forces that act on electrically charged particles. They don’t involve any changes to the geometry of space or time.

Weyl felt that if he wanted to merge these two theories together into a common framework, he would need to find a new geometrical way to formulate the theory of electromagnetism. In general relativity, the geometry of space and time is described by a mathematical object called the metric tensor. A tensor is essentially a special kind of matrix or array of numbers.

In general relativity, the metric tensor is a 4×4 array of numbers, so it contains a total of sixteen entries. But of these sixteen quantities, six are redundant, so there are really only 10 independent numbers described by the metric tensor. And we need all 10 of these numbers just to describe the effects of gravity.

The problem in combining general relativity with electromagnetism is that when we incorporate electromagnetism we need at least four more numbers at every point in space. This made it hard to see how one could explain both gravity and electromagnetism in terms of geometry. There just aren’t enough numbers in the metric tensor to describe both gravity and electromagnetism at the same time.

To try to get around this problem, Weyl proposed a version of non-Euclidean geometry. In doing so, he argued that it was possible to construct a geometrical system that wasn’t limited to the 10 independent numbers. In addition to those 10 numbers, Weyl’s version of the metric tensor contained other additional quantities. And Weyl hoped that these additional numbers could somehow encode the effects of electromagnetism.

The theory that Weyl ultimately came up with was very complicated. Although it was mathematically sound, physically, it just didn’t make much sense. After a series of exchanges with Einstein, even Weyl became convinced that his work hadn’t gotten them any closer to viable unified field theory.

This is a transcript from the video seriesWhat Einstein Got Wrong.Watch it now, on The Great Courses Plus.

### Kaluza’s Fifth Dimension and Unified Field Theory

Only a year later or so, another idea in this direction was proposed. This time by the mathematician Theodor Kaluza. Most people find Kaluza’s idea to be pretty strange and surprising. What he proposed was a unified field theory in which the space and time of our universe aren’t limited to four, but five dimensions.

To see why a fifth dimension might be helpful in building a unified field theory, we need to remember metric tensor. A tensor is a 4×4 array of numbers, for a total of sixteen entries—10 of which are independent of each other. But tensor is a 4×4 array of numbers only because it was formulated in four-dimensional spacetime. If spacetime is five-dimensional, then the metric tensor will be a 5×5 array of numbers, for a total of twenty-five entries.

After removing all of the redundant entries, the five-dimensional metric tensor contains fifteen independent quantities. 10 of these fifteen numbers are needed to describe gravity. And this leaves us with five others, which is more than enough to potentially encode the phenomena of electromagnetism.

There is, though, one immediate and obvious objection that one might raise to Kaluza’s five-dimensional theory. As far as we can tell, our universe doesn’t have a fifth dimension.

Fortunately, there is a way that a fifth dimension might be able to remain hidden in a system like Kaluza’s. In this geometrical system, the fifth dimension isn’t like the others. The three dimensions of space that we are familiar with are large, and as far as we know, they go on forever in any given direction. If there were an extra dimension like this, it would be impossible for us not to notice it.

But the fifth dimension being imagined by Kaluza doesn’t go on forever. Instead, it’s wrapped up, or curled up, into a tiny circle. If something moved even a short distance along the direction of this fifth dimension, it would simply return to where it started. If the circumference of the fifth dimension is small enough, it would be almost impossible for us to perceive it.

It was in 1919 that Kaluza described his idea to Einstein for the first time. And despite the fact that there were significant problems with the 5-dimensional theory, Einstein liked it a great deal.

With Einstein’s help, Kaluza managed to publish his theory a couple of years later, in 1921. And only a few weeks after that, Einstein himself wrote and published an article that investigated some of the aspects of similar five-dimensional unified field theories. But, despite the enthusiasm, it was pretty clear that there were serious problems with Kaluza’s theory. Einstein, though, continued to work on this theory not because he thought it was a viable unified field theory, but because he thought it might lead to something more promising.

After all, while Einstein was developing general relativity, he went through several incorrect versions of the gravitational field equations before he found the right answer.

Learn more about Quantum Entanglement.

### Arthur Eddington’s Affine Connection and Unified Field Theory

Another scientist who worked on unified field theories during this period of time was the famous astronomer and physicist Arthur Eddington. However, Eddington didn’t focus on expanding the metric tensor. In fact, he didn’t focus on the metric tensor at all. Instead, he focused on a different mathematical structure, known as the ‘affine connection’. In the end, Eddington didn’t really get any closer than Weyl or Kaluza to building a viable unified field theory. But Eddington’s work was important because his approach was quite different, and along with Kaluza, Eddington probably had the most influence on Einstein’s later efforts to develop such a theory.

Learn more about what Einstein got right: Special Relativity.

### Einstein’s Early Work on Unified Field Theory

Einstein himself began to focus on unified field theories in the early 1920s. During this period of time, he remained enthusiastic about the work that had been earlier done by both Kaluza and Eddington. In fact, a lot of Einstein’s early work in this area consisted of extending and building upon these earlier ideas.

Einstein was deeply enthusiastic about this program of exploration. Although in this respect, he was relatively isolated since most physicists didn’t share his excitement. Quantum physics was developing rapidly, and that was occupying the bulk of the field’s attention during this time.

Einstein was deeply unhappy with the developments occurring in quantum theory as it moved away from the predictive determinism. Einstein’s views about quantum mechanics also served to bolster his interest in unified field theories.

In addition to unifying general relativity with electromagnetism, Einstein hoped that a unified field theory might also somehow be able to restore determinism and scientific realism to the quantum world.

### Common Questions about the Early Works on Unified Field Theory

**Q: Is unified field theory possible?**

Yes, it’s possible to have a unified field theory similar to that of James Clerk Maxwell who successfully combined electric and magnetic fields into Electromagnetic theory.

**Q: What does the unified theory mean?**

Unified field theory is an attempt to unify different fundamental forces and the relationships into a single theoretical framework. There have been many attempts at unified theories, some were successful, some failed.

**Q: Who discovered field theory?**

James Clerk Maxwell was the first one to create a unified field theory. He also combined electric and magnetic fields into Electromagnetic theory.

**Q: Who gave the quantum theory?**

The founding fathers of quantum theory are Niels Bohr, Max Planck, and, to a certain extent, Albert Einstein.