Dark matter from the fifth dimension

After years of tinkering with tensor calculus, my paper explaining dark matter with a 5th dimension has been published in the peer-reviewed journal Classical and Quantum Gravity.

Dark matter has had a five-dimensional explanation for a while actually but that explanation proposed a particle arising from a fifth dimension that is tightly rolled up. If you take a long sheet of paper and roll it into a small tube, the number of dimensions as seen from far away goes from 2 to 1. The tube has a lot of length but very little width. A being that is much bigger than the tube will not perceive the width.

This concept is used in string theory but need not be. In dark matter theories, extra compact dimensions produce light particles that do not interact electromagnetically when energy propagates in the small dimension. This type of dark matter is called the lightest Kaluza-Klein particle. Kaluza was the first person to propose a fifth dimension to unify gravity with electromagnetism. Kaluza didn’t know what to do with the extra dimension, however. He couldn’t explain why we didn’t perceive it. Klein proposed that this dimension was rolled up very tightly and thus was too small for our measurements to detect.

My paper shows how you can get even more out of Kaluza-Klein theory. The reason why you would want to is because particle-based dark matter doesn’t quite fit the observations we make that we attribute to dark matter, specifically the speed at which galaxies rotate.

The rotational speed of galaxies was the first hint that dark matter exists but since then we have discovered many more indications of dark matter at larger scales of galaxy clusters and in the Cosmic Microwave Background.

The problem is that galaxies should rotate sort of like the solar system does. Inner parts of a spiral galaxy, for example, should rotate quickly like the inner planets do while outer parts of the galaxy should rotate much more slowly like the outer planets do. This is not, however, what we observe. Instead, we see the outer parts rotating around the galactic cores nearly as fast as the inner ones. These galaxies should fly apart.

Our current dark matter model proposes that dark matter is some kind of massive particle and that fits most of these phenomena quite well. Galactic rotation curves, on the other hand, have stubbornly resisted fitting dark matter models well.

The best theory is that galaxies have what is called a dark matter halo, a kind of bubble of dark matter completely surrounding the galaxy. This halo allows matter to rotate at a much faster rate because it keeps the galactic density from falling off. In this model, the Milky Way is predicted to be 95% dark matter.

One of the leading modifications to gravity, MOdified Newtonian Dynamics or MOND, on the other hand, explains these curves better (but not perfectly). MOND suggests that rather than there being a mysterious particle causing the curves, it is simply how Newton’s gravitational pull changes as the acceleration due to gravity decreases.

In essence, when acceleration is relatively high, which is true for objects in the solar system and globular clusters, Newtonian gravity is the norm. When acceleration is very low as in the outer reaches of gravity, the induced velocity becomes nearly constant. That allows galaxies to rotate almost as rigid bodies.

MOND is simply a modification of Newton’s gravitational force so how to connect that to Einstein’s theory of general relativity has been an open question, but back in 2004 Bekenstein of Bekenstein-Hawking formula fame proposed a modification to a theory called Tensor-Vector-Scalar (TeVeS) theory that would explain dark matter.

Tensor-Vector-Scalar theory added two new gravitational fields to the existing Einstein theory which is just a Tensor theory. Bekenstein painstakingly showed how his TeVeS could produce results that behave like dark matter not only in terms of gravitational influence but also in terms of gravitational lensing, which MOND didn’t explain. Hence, it was a modification of Einstein’s theory that explained why modified gravity looks like dark matter.

Unfortunately, Bekenstein’s version of TeVeS predicted that gravitational waves travel more slowly than light, and this prediction was proved wrong when a pair of neutron stars colliding with one another (GW170817) produced both a gravitational wave and light show. Those signals arrived at the same time proving that, even over millions of light years, gravitational waves and light travel at the speed of light.

Bekenstein had passed away at this point, so another researcher demonstrated that a modified version of TeVeS called generalized TeVeS could save the theory. The modification to the theory was pretty ad hoc, however.

This was the starting point for my theory. I wanted to show that a 5-dimensional pure Einstein theory could, when decomposed into 4 dimensions and one compactified dimension, reproduce generalized TeVeS.

My personal goal for this was simply to create evidence for a fifth dimension. Quantum theory showed no promise in generating the desired predictions so I turned to the stars. I ended up with more than I bargained for.

The math was tricky and elaborate which is probably why nobody had even attempted it before.

Eventually, I succeeded in showing that I could indeed derive generalized TeVeS from a five-dimensional Einstein theory, which I called a “Geometric interpretation” of TeVeS.

To make the theory work, I had to use a concept called a reference fluid which is a kind of field that creates a preferred coordinate system. This reference fluid arises from the spontaneous symmetry breaking that causes compactification of the 5th dimension, I conjectured.

This led to the interesting development that the compactified 5th dimension could produce both generalized TeVeS as well as light Kaluza-Klein particles. Perhaps that was exactly what we needed to explain all Dark Matter attributed phenomena. It was not one thing but two arising from the same cause.

Hence, my theory became a hybrid theory, part dark matter, part MOND, all arising from this fifth dimension.

Even better, it predicted, indeed required, the modification that had been added ad hoc to fix Bekenstein’s theory.