Describing Physics with the Spacetime Manifold

Origins

One of the most common questions that people ask about Knot Physics is why we are not affiliated with an institution. There are many advantages to working in academia, especially when it comes to communicating new ideas. The credentials that come along with an academic position are very helpful for reaching an audience. That said, there are also many constraints, including numerous demands on the time and attention of the researcher. 

When this project began, it was just an idea. Developing the idea required long periods of talking to no one and staring at walls—uninterrupted blocks of time that an academic career would not afford. As the idea progressed, it became clear that it had substance. Eventually, it was a fully formed theory that had developed as independent research.

From idea to theory

The theory began as a question: Can all of physics be described using only the spacetime manifold? General relativity succeeded in describing gravity as curvature of spacetime. Could that description be extended to particles? If spacetime can bend, perhaps it can bend so much that it forms a knot. What if those knots are the elementary particles?

The theory began as a question: Can all of physics be described using only the spacetime manifold?

embedded-spacetime-manifold-knot
Spacetime can be knotted only if it is inside of a higher-dimensional space.

In developing that idea, the first issue was dimension. We know from the mathematical theory of knots that a knotted manifold must be inside of—or embedded in—a space that is larger by 2 dimensions. Because the spacetime manifold is 4-dimensional, a knotted spacetime manifold must be embedded in a 6-dimensional space.

Having determined the dimension, there were two immediate results. 

The first result applied to gravity. Einstein’s theory of general relativity says that gravity is the curvature of spacetime; however, general relativity describes that curvature using a metric—a mathematical object defined on every point of spacetime—and there is no further discussion about the meaning of spacetime curvature. An embedded spacetime manifold allows an extension of that description. For an embedded spacetime manifold, curvature can be considered the bending of spacetime inside of the higher-dimensional space. 

For an embedded spacetime manifold, curvature can be considered the bending of spacetime inside of the higher-dimensional space. 

The second result that came from embedding spacetime was about the quantum wave function. Particles, like electrons, have a quantum wave function ψ that describes their location. One of the perplexing aspects of the wave function is that it uses complex numbers. Why is a particle usefully described by a complex number? We can use the embedded spacetime manifold to provide an interpretation. If spacetime is in a 6-dimensional space, a knot on spacetime can rotate in the two additional dimensions. We can describe the knot’s size with a radius r and describe its rotation with an angle θ. We can combine those two numbers to get a single complex number k=re, a knot amplitude that describes both the size and rotation of the knot. Maybe complex numbers are useful for the quantum wave function because they are a good way of describing a knot’s size and rotation in two extra dimensions.

The next consideration was, naturally, quantum mechanics. Quantum mechanics requires quantum superposition for particles, but a particle is just a knot in the spacetime manifold. So the spacetime manifold itself must be in a superposition. This can be achieved if spacetime is a branched manifold. In fact, by using knots on a branched spacetime manifold, we can recreate the quantum wave function. In addition, if we allow the branches of spacetime to split and recombine, the knots on those branches also split and recombine. The recombination of knots produces quantum interference.

spacetime-manifold-branches-wave-function
A particle on branched spacetime has one knot on each branch. Each knot has a complex amplitude. The distribution of knot amplitudes is the wave function (shown in red).

By using knots on a branched spacetime manifold, we can recreate the quantum wave function.

A branched spacetime manifold was necessary for quantum mechanics, but branching also introduced a new problem. If spacetime is branched, what prevents the spacetime manifold from branching and branching and branching infinitely? That one question began a long process of refining the theory to develop a truly fundamental description of physics. 

To prevent infinite branching, we needed constraints on the branched manifold. It turns out that the right constraints are motivated by basic requirements: The constraints must prevent the branched spacetime manifold from being either too simple (e.g. one branch that is perfectly flat) or too crazy (e.g. infinite branching and infinite wrinkling).

Once those minimal constraints were introduced, the rest of the theory followed as a consequence. The constraints allow the spacetime manifold to spontaneously create pairs of knots. If those knots are elementary fermions, then this is creation of particle-antiparticle pairs. We also find that those knots interact with each other, and after a little (a lot of) work, we see that interactions between those knots correspond to the forces that we observe in nature. From a simple set of constraints on a branched embedded spacetime manifold, particles and forces follow naturally.

From a simple set of constraints on a branched embedded spacetime manifold, particles and forces follow naturally.


Today, Knot Physics is a well-developed geometric unification theory. Particles, forces, and quantum mechanics follow from a simple set of assumptions. These and other discoveries are discussed in papers on the Knot Physics website.

Lately, the research has been paused while we work to increase awareness of the theory. While it is certainly possible to produce more papers on the topic, it also seems clear that the pace of work will be dramatically increased if the theory is more widely known. For that purpose, there is an ongoing effort to reach out to other physicists. We believe that developing on these results will open new doors in theoretical physics.


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Published by

Cliff Ellgen

Cliff Ellgen graduated from Caltech in 1999 with a degree in mathematics. He has been contemplating Knot Physics, a geometric unification theory, for more than fifteen years. Since August of 2014, he has been collaborating with Garrett Biehle, who received his Ph.D. from Caltech in astrophysics under advisors Kip Thorne and Roger Blandford.

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