willj.dev / geometric-universe /

The goal of these pages is to provide a “museum tour” of the state-of-the-art mathematical framework that describes the fundamental behavior of the universe, which I have not very humbly decided to refer to as the Geometric Universe Hypothesis. In general, I will not be going into proofs, but I will be sure to point out where they can be found; as a result, it may be helpful to treat this more like an art project than a peer-reviewed scientific publication, which it emphatically is not. I certainly do not claim to solve any of the currently unsolved problems in physics; the theory presented here is no more or less than the generally accepted Standard Model + General Relativity paradigm, just written (impractically) in the same mathematical language.

The content here should agree with the treatments given in university classes. In fact, there are already many great books that provide a narrative account of modern physics; I first discovered the subject through Steven Hawking’s A Brief(er) History of Time. I have always longed for a version of those books with a bit more math—not so much as to need a degree, but enough to feel a bit closer to the “actual machinery”. Now that I can actually claim to be familiar with the math in question, I might as well just write it myself! However, to my knowledge, no one has given a recent, full-blown, accessibly-technical account of the “physical axioms” that we can treat as fundamental to the universe, from the metaphysical boundary (what does that even mean, anyway?) out to where physics starts to bleed into other scientific disciplines.

An important point to remember as we move along is that all of our equations are “museum pieces” in the sense that they look nice, and the math checks out, but they are essentially impossible to use in practice. The hypothesis we’re building up to is basically an algorithm for writing down a large number of absolutely gnarly differential equations. We can hand wave away many of these by doing a similar number of completely preposterous integrals. There’s still a good chance that what you’re left with could only be solved numerically, and that could take long enough that the sun will have gone out, and humanity will be doing better things on different planets (if we managed not to fizzle out on Earth).

Understanding the math is not required to follow along, but it will certainly help if you are on cordial terms with algebra, and perhaps even have a passing acquaintance with calculus. If you are out of practice reading text with equations in it, fear not! When you encounter an equation (or, equivalently, unfamiliar technical jargon) in the wild, the trick is to use a particular reading strategy that usually isn’t needed outside academic journals and textbooks (collectively, The Literature). Don’t panic! Take a moment to stare at the equation/term; if it doesn’t mean anything to you, it might come up later, and these things often make more sense when you see it used in context a few times. Just put a mental sticky note on it and move on, keeping an eye out for anywhere else it shows up in case that yields hints. It will feel very slow at first, but I promise the payoff of knowing how to read technical literature without becoming overwhelmed is a pretty sweet skill to have in your back pocket if you ever find yourself wanting to read up on an unfamiliar topic (it happens, I swear!). With practice it becomes second nature—it is, in fact, one of the unspoken basic skills / secret arts of grad school.

The mathematical framework I use is also fairly well established; mathematicians call it the Clifford Algebra1, specialized for physics as the Geometric Algebra2. Almost all of the results here are quoted from Geometric Algebra for Physicists, by Chris Doran and Anthony Lasenby3. It is equivalent to the notation that can be found in The Literature, but has a number of subtle generalizations and bonus features that make it easy to write things aesthetically. It has not seen widespread adoption simply because people who do physics for their actual job don’t have time to learn a new notation which, after all, is still equivalent to the notation they’re used to! We can use it here without guilt because this is, after all, a “mathematical art project” and so this choice can be classified under “artistic license”.

Before getting to the math, though, we should take care of some metaphysical groundwork. The goal is to be as precise as possible; to that end, we want to make sure we have a really good idea of what we mean when we say things like “perfectly controlled conditions”, and for that matter, what even is “reality” anyway?

  1. Developed in the 1870s by W. K. Clifford, now a mature topic in algebra theory.↩︎

  2. Developed in the 1950s by David Hestenes.↩︎

  3. Cambridge, 2003. I will cite pages like “(D&L, pg 481)”, sections like “(D&L, sec 13.5.4)”, and so on.↩︎


The Geometric Universe Hypothesis
Geometric Algebra
Geometric Physics
Science Near and Far from the Geometric Regime