Harvey Mudd Senior Publishes Two Physics Papers in Major Journals

Math-physics major Matthew Fox ’20 has published two single-author papers in high-impact journals. Physical Review D published “Palatini f(R,Lm,RμνTμν) gravity and its Born-Infeld semblance” in June, and the Journal of Mathematical Physics published “Multipole hair of Schwarzschild-Tangherlini black holes” in October. In December, in a seminar slot usually reserved for graduate students, postdocs and faculty, Fox gave a talk on his black hole research to the Particle Theory Group at University of California, Riverside. Here, Fox describes his research.

“Palatini f(R,Lm,RμνTμν) gravity and its Born-Infeld semblance”

There are many outstanding problems in physics. One is the dark energy problem: A judicious look at the sky will reveal that the universe is expanding at an ever-increasing rate. We say the universe expands because of “dark energy.” To me, the phrase “dark energy” comes off as some supremely technical term that only people with advanced degrees really understand. But there is no there there. Dark energy is just a label we give to the thing that causes the universe to expand. You can call it whatever you like! We simply don’t know what dark energy is. We only know what it does.

A particle physicist or cosmologist might encourage a particular way of thinking about dark energy. They might say it is some fantastical fluid-like substance that permeates all of space and that also pushes space apart.

Indeed, this is how it is often modeled. But there is another view, a view in which dark energy is not some substance, some thing, but instead it is a trait, a feature, of the gravitational force. In this view, dark energy (i.e., the expansion of the universe) is effectively encoded in the DNA of gravity. Thus, from this perspective, where there is gravity, there is dark energy forcing the universe to spread out.

My paper concerns a very specific model that belongs to this latter perspective. The model is called Palatini f(R, Lm, RT) gravity. Basically, it is a generalization of Einstein’s theory of general relativity, which is the contemporary theory of how gravity works. The details of my theory aren’t all that important. What is important is its relation to an interesting anomaly that arises in Einstein’s great theory, having to do with the notion of infinity.

In physics, we measure and calculate numbers. We say a theory is successful if the theory predicts a number that agrees with the numerical output of an experiment testing that theory. In Einstein’s theory, there are some numbers which do not make sense, but which the theory nonetheless predicts. One example is infinity.

Infinity can be hard to come by. But at the center of resplendent objects known as black holes, you will find an infinity. Einstein’s theory predicts that the curvature of space and time is infinite there, which is nonsensical in many respects. Interestingly, one can modify Einstein’s original theory to cure these so-called “curvature singularities.” In essence, these theories modify Einstein’s theory such that the infinities no longer appear at the center of black holes.

In my paper, I seek to unify two fundamentally different modified theories of gravity. One theory, the Palatini f(R, Lm, RT) theory I mentioned above, can account for the dark energy problem under certain circumstances. The other, called Eddington-inspired Born-Infeld (EiBI) gravity, cures the curvature singularities at the nuclei of black holes. This is interesting since the unified theory might engender models that explain dark energy and remove infinities at the centers of black holes. Two birds with one stone!

“Multipole hair of Schwarzschild-Tangherlini black holes”

Another major problem in physics (perhaps the greatest unsolved problem in physics today) concerns reconciling gravity and quantum mechanics. We don’t understand how quantum mechanics (the theory of small things) relates to gravity (which is a force concerning large things like planets and stars).

Many theories that try and understand this invoke something wholly unfamiliar: extra dimensions of space. Never mind the reasons why we don’t see these extra dimensions. Physicists are clever, and there are bona fide ways to keep these extra dimensions hidden from us in our daily lives. The point is that, in principle, there is no reason for extra dimensions to not exist. It makes sense, therefore, to study physical phenomena (using mathematics) assuming extra dimensions of space, in order to gain a better understanding about how things would work in a world with more dimensions. (If things don’t work like we see them assuming more dimensions, then there can’t be extra dimensions, since things do work in our day-to-day lives!)

The topic of my paper is black holes. A black hole is what you get when you have a lot of material stuff packed so tightly that the mutual gravity of everything overwhelms all the interparticle forces pushing out. When this happens, the thing collapses in on itself, producing a region of space from which not even light can escape (hence black hole).

In three dimensions, which is the canonical setting of Einstein’s theory of gravity, these black holes are suspiciously simple creatures. It doesn’t matter what the original stuff was that was later packed up tightly. A black hole is related to the original stuff from which it is made in a very simple manner: It only cares about the net mass, the net charge and the net angular momentum of the original stuff. All the other properties of the original stuff is history.

That’s really remarkable, since there is a whole periodic table out there describing all these different materials with varying properties. Yet black holes don’t care. They know nothing of this table! They only care about the mass, charge and angular momentum of the stuff that makes them up! For this reason, we say black holes have “no hair,” meaning they have no other features to them.

Let me stress one final, ostensibly unrelated point: In three dimensions black holes are spheres. Or at least they are related to a sphere in the same way an egg-shaped piece of cookie dough is related to a sphere: You can very easily mold the egg-shaped cookie dough into a sphere without tearing it. Just keep rolling it between your hands. The same is true for all three-dimensional black holes: They can be molded into a sphere without tearing. For this reason, we say all three-dimensional black holes are topologically spherical. This is one of Stephen Hawking’s many famous results.

OK, back to my paper. When you go to higher dimensions, black holes are not so simple anymore. In higher dimensions, not all black holes are topologically spherical. Rather, you can get black holes that resemble a higher dimensional version of a torus (a surface with a hole in it, like a donut), among other things. This suggested to me that perhaps there is some physical way to change the topology of a higher dimensional black hole.

I studied the effect of electrically charging up a specific higher dimensional black hole, known as a Schwarzschild-Tangherlini black hole (in many respects, this is the simplest type of black hole you can have in higher dimensions). The process itself is quite straightforward: I have some uncharged thing (the Schwarzschild-Tangherlini black hole), and I want to charge it. So all I do is deposit some charge on to it. Then the thing is charged! What is so interesting about this is what actually happens to the topology of the original black hole.

In three dimensions, nothing interesting really happens. I start with an uncharged black hole, which is topologically a sphere, that turns into a charged black hole that retains its spherical topology. The topology doesn’t change because Hawking’s theorem says it can’t!

In higher dimensions, things get weird. Hawking’s work doesn’t extend to higher dimensions. So, when charge is deposited on to the higher dimensional black hole, it could lose its spherical topology. I showed it does lose its spherical topology! This has at least two fascinating technical implications. One is that higher dimensional black holes can sport some hair, they are not characterized by just their mass, charge and angular momentum. They have additional features! The second is that the topology of the original black hole is no longer a sphere. In other words, infalling electric charges change the topology of Schwarzschild-Tangherlini black holes.

I don’t know what the topology of the charged-up black hole is, but I do know it is not a sphere. In this sense, the electric charges act a bit like a nail and the black hole a bit like a piece of wood. You start out with some surface with no holes in it (the wood), but a sufficiently driven nail will puncture a hole right through the surface. My results suggest that electric charges do a similar thing to the surfaces of higher dimensional black holes!