Jonathan Gorard Profile picture
Jul 28, 2024 11 tweets 3 min read Read on X
Moths are attracted to lights because of the same mathematics that underlies twistor theory and compactification in theoretical physics: projective geometry.

It all starts from a simple observation: translations are just rotations whose center is located "at infinity". (1/11) Image
So if we take an ordinary space (like the 2D plane) and we adjoin a hypothetical point "at infinity", then the distinction between translations and rotations disappears: as the center of rotation moves further and further away from the object being rotated,... (2/11)
...the "translation to rotation" ratio gets larger and larger, until, at infinity, it becomes infinite. So translations reduce to a special case of rotations. We call such a space a "projective space", and it represents a "compactification" of the space we started from. (3/11)
The 2D plane with a point adjoined at infinity becomes a sphere (namely the "Riemann sphere"), meaning that every point on this projective plane can be mapped uniquely to a point on this sphere, via "stereographic projection". Imagine suspending the sphere above the plane. (4/11)
Then, stand at the North Pole of the sphere. If we cast a ray down from the Pole to any point on the surface of the sphere, that ray will intersect some point on the plane below. Hence every point on the sphere can be associated to a point on the plane (and vice versa). (5/11)
The only point for which that is not true is the North Pole itself: that is associated with the point "at infinity". Since this mapping takes us from a 2D plane (which is not compact) to a sphere (which is compact), it is referred to as a "compactification" of the plane. (6/11)
Compactifications appear all over the place in physics, and I plan to talk about them more in future posts. But for now, I want to talk about moths.
Moths fly at night using celestial navigation: they use the stars to ensure that they continue flying in a straight line. (7/11)
They do this by first selecting a star, and then keeping that star at a fixed angle in their vision. Since the star is effectively at "optical infinity", the moth can then exploit the basic feature of projective geometry that translations=rotations. (8/11)
By keeping the star at a fixed angle in their sight, they are rotating themselves around a point at projective infinity, which is equivalent to a straight translation. But what if the "star" that they lock on to is in fact not a star at all, but rather an artificial light? (9/11)
Rotating around a point that is not at infinity is *not* equivalent to a translation, and so they will not maintain a fixed course. Rather, by fixing an artificial light at a constant angle in their vision, the moth will instead orbit the object in a logarithmic spiral. (10/11)
So moths end up inside your house at night because their brains evolved to navigate a world in which geometry is projective, where rotations=translations; yet human lighting breaks that expectation, and places them inside the much uglier world of non-projective spaces. (11/11)

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More from @getjonwithit

Jul 21
How many holes does a straw have?

Topologically, of course, it has 1: it's homeomorphic to a punctured disk. But intuitively it has 2: one at the top and one at the bottom. And this answer lies at the heart of the most rigorous axiomatization of quantum field theory. (1/20) Image
In this intuitive picture, the two "holes" of the straw are 1-dimensional circles, and they're connected by a 2-dimensional cylinder (the straw itself). Mathematically, this relationship is called a "cobordism". Two n-dimensional manifolds are "cobordant" if they form... (2/20)
...the boundary of some n+1-dimensional manifold (like the two circles forming the boundary of the cylinder). And cobordisms give one a natural framework for thinking about time evolution in physics. Suppose we have two moments in time: t1 and t2, with t1 < t2. (3/20)
Read 20 tweets
Jul 10
The desiccated "Theorem, Lemma, Proof, Corollary,..." presentational style is staggeringly counterproductive, if one's objective is actually communicating the underlying mathematical intuitions and thought processes behind a result. In reality, the process is more like... (1/4)
"First, I tried <standard method>, but it failed for <enlightening reason>, so I investigated whether I could exploit this fact to find <counterexample> with <property>, but all objects obtained through this technique ended up having <interesting property> in common.... (2/4)
...So I tried relaxing <axiom> to see whether <related property> could be removed, and this led me to realize that <intermediate lemma> is actually crucial to the structure of <related object>..." Etc. You occasionally get these insights from (very good) mathematical talks. (3/4)
Read 4 tweets
Jun 2
What's a gravitational wave? Anything that distorts the shape of spacetime, but preserves its volume.

What's matter/energy/momentum? Anything that distorts the volume of spacetime, but preserves its shape.

A 🧵 on the Ricci decomposition theorem, as applied to gravity. (1/13)
Classical gravity is a manifestation of the Riemann curvature of spacetime, which describes how your coordinate system distorts as you move from point to point. More precisely, the *connection* describes how the coordinate system distorts, and the Riemann curvature... (2/13)
...describes how the connection distorts. So the Riemann curvature is effectively a second derivative of your coordinate system. The Ricci decomposition theorem then says that the Riemann curvature can be decomposed into two pieces: a "trace" part and a "trace-free" part. (3/13)
Read 13 tweets
May 31
Calling c the "speed of light" completely misses the point. Rather, c is the "spacetime exchange rate": how many units of space you can exchange for one unit of time.

In actuality, everything travels at the "speed of light", just not necessarily through space alone... (1/4) Image
Rather, everything travels through both space *and* time, simultaneously, with a speed of c. If you're standing still, then all of your velocity is focused in the time direction (with none in the space directions), so you move through time with a speed of c. (2/4)
If you start moving, then now a little bit of your velocity vector points in one of the space directions, so a little bit less must point in the time direction. So you move through time slightly slower than c, such that your overall speed through space *and* time remains c. (3/4)
Read 4 tweets
May 29
Sure, I’ll give it a go…

Consider a rotating disk. What does it mean to say that the disk has angular momentum? Well, imagine assigning a momentum vector to every point on the surface of the disk, and then slicing through the middle of the disk with a flat surface. (1/14) Image
Image
The "net flux” of momentum vectors through the surface is zero, since every momentum vector poking through the surface in one direction is counteracted by a momentum vector poking through in the opposite direction. In other words, the disk has no *linear* momentum. (2/14)
But the "total flux" of momentum vectors (i.e. the total amount of momentum intersecting the surface, irrespective of direction) is clearly non-zero, because the disk is rotating. This discrepancy between the total flux vs. the net flux is what we call "angular momentum". (3/14)
Read 14 tweets
May 9
New paper alert!

Birkhoff's theorem tells us that the spacetime around a non-rotating black hole is indistinguishable from that around any other non-rotating compact object, like a neutron star.

But what if it's rotating? Turns out, the differences can be *huge*. (1/4) Image
Image
Link:

Though the spacetime around an uncharged black hole depends on two parameters (mass and spin) by the no-hair theorem(s), objects like neutron stars have "hair" in the form of many other multipole moments: mass quadrupole, spin octupole, etc. (2/4)arxiv.org/abs/2505.05299
Most calculations and simulations of neutron stars assume that such moments don't matter (i.e. that the geometry is well-described by the Kerr metric of a spinning black hole). We show that this isn't true, for physically realistic neutron stars spinning at moderate speeds. (3/4)
Read 4 tweets

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