What all of these explanations get "wrong" is that they try to explain what a tensor is. Yet tensors themselves are neither fundamental nor important.

The fundamental/important thing is the tensor *product*. A tensor is just an object assembled using tensor products... (1/9)

https://twitter.com/vikhyatk/status/1969151806501487096

The tensor product ⊗ is, conceptually, the most general (binary) operation that behaves "how a product should behave". In practice, this means that the order of brackets shouldn't matter, i.e. X⊗(Y⊗Z) should be the same as (X⊗Y)⊗Z, for any objects X, Y and Z... (2/9)

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) 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 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)

https://twitter.com/EliotJacobson/status/1942937846484177239

"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)

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)

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) 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)

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)

https://twitter.com/ThePhysicsMemes/status/1927662467481469428

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)

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)
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

My recent "dunk" about encoding functions and the algorithmic/Kolmogorov complexity of the laws of physics may have seemed flippant, but it actually goes back to an old 17th century philosophical conundrum: the dichotomy between idealism and materialism.

Let me explain. (1/14)

https://twitter.com/getjonwithit/status/1911933388895756371

When attempting to model the world computationally, there are typically *three* computations that one needs to consider: the computation that the system itself (e.g. the universe) performs, the computation that the observer performs, and the "encoding function". (2/14)

New paper alert!

We developed the first automated theorem-proving framework for (hyperbolic) PDE solvers: now you can build *formally verified* physics simulations, with provable mathematical and physical correctness properties.

arXiv link and explanation in thread... (1/10)
Link:
Hyperbolic PDEs are the foundation of most simulations in hydrodynamics, electromagnetism, general relativity, etc. But solvers often become unstable, fail to preserve hyperbolicity, or introduce new extrema, unless one is very careful. (2/10)arxiv.org/abs/2503.13877

"General relativity doesn't admit black hole solutions. It only admits *wormhole* solutions."

I have previously made this statement and had people get confused by it. So let me try to clarify precisely what this means, using a neat analogy to real/complex analysis. (1/17)
When you first encounter the sine function, you probably see it defined in terms of triangles, which means that it really only makes sense for x between -pi and pi (or -180 and 180 degrees). But later on, you learn that the "true" sine function is much larger. (2/17)

Energy, momentum, pressure, stress, etc. are all just different ways of quantifying the same basic thing: how our perceptions of space and time get distorted over time.

And once you internalize this, it allows you to think about these concepts in a much more general way. (1/12) Suppose that the room you're in defines a 3-dimensional coordinate system (x, y, z). If the walls in the x-direction get pulled apart in the x-direction, and those in the y-direction get pulled apart in the y-direction at the same rate, etc., then the room is expanding. (2/12)

"Oh, I'm a *pure* mathematician, I don't write code/do calculations/etc.."
"Oh, I'm a *theoretical* physicist, I don't do experiments/analyze data/etc.."
Etc.

These kinds of statements are typically uttered with an air of intellectual smugness. But what are they really? (1/6) In actuality, this classification of certain tasks as "worthy" (e.g. proving theorems, developing models) and others as "beneath oneself" (e.g. doing calculations, writing code) is a signal of a fundamental absence of intellectual curiosity. (2/6)

To illustrate just how different neutron star and black hole metrics truly are, I simulated perfect fluid accretion onto a black hole vs. onto a neutron star of identical mass and spin.

Source code and simulation details in thread below. (1/4)

https://twitter.com/getjonwithit/status/1878094729809723568

To produce this, I simulated supersonic accretion of a perfect fluid, obeying an ideal gas equation of state, initially onto a standard black hole metric in Cartesian Kerr-Schild coordinates, and then onto a physically realistic rotating neutron star metric due to Pappas. (2/4)

There's something which almost every major neutron star simulation gets wrong, and it's related to a widespread confusion about the behavior of rotating objects in general relativity.

The confusion goes back to one of the most beautiful results in GR: Birkhoff's theorem. (1/14)
Here's one way to think about Birkhoff's theorem: suppose that you have a perfectly spherically-symmetric object that is uncharged and not rotating. Clearly, by measuring its gravitational field (i.e. the extent to which it curves spacetime), you can determine its mass. (2/14)

I'm often described (due to my work on computable physics) as being an advocate for the Simulation Hypothesis.
But I'm not.
In fact, I think the Hypothesis is nonsensical, and that the most famous argument in its favor actually proves the opposite of what it purports. (1/10) First, it is crucial to distinguish two things: a computation (which we can formalize as the operation of a Turing machine), and a simulation (which is a specific type of computation, whose output states are interpreted as approximate states of some *external* system). (2/10)

The apparent "philosophical problems" of quantum mechanics are not unique to QM at all: they are in fact the same problems that arise whenever one attempts to construct an abstract model of reality. We can see these problems already in high school-level mechanics. (1/14)
When you throw an object into the air and try to calculate when it will hit the ground, you can do this (neglecting air resistance) using a SUVAT equation: by solving for t in s = ut + 1/2 at^2. But this equation is a quadratic, and so generically has 2 solutions. (2/14)

Lord Kelvin famously believed that atoms were "knots in the aether".

And he was onto something: perhaps not atoms per se, but knots can indeed tell us something quite deep about the nature of the universe, and the relationship between classical and quantum field theories. (1/16)
A "knot" is an embedding of a circle (S^1) in 3D space (R^3). A "link" is a collection of knots (potentially linked together, but not intersecting). And whenever you smoothly (topologically) deform a knot or link, some properties will always remain the same. (2/16)

Much more work is needed before this is really a robust result (paper forthcoming, hopefully by the end of the year), but the initial findings are clear:

Spacetime discreteness may be observationally detectable in things like quasar luminosities. (1/6)
In previous work (), I showed that black holes in discrete spacetime accrete matter more slowly than their continuous spacetime counterparts, for essentially ergodic reasons: with only countable degrees of freedom,... (2/6)arxiv.org/abs/2402.02331

This is a rock.

More precisely, it's a sample of my favorite mineral: Zn Cu3 (OH)6 Cl2, otherwise known as anarakite or Herbertsmithite.

Here's what it may be telling us about the fundamental structure of space, time, and the universe. (1/17) Suppose that you have a big collection of matrices, representing elements of some symmetry group. For instance, the elements of the 2D rotation groups U(1) or SO(2) can be represented by 2x2 rotation matrices. Call this collection a "representation" of the group. (2/17)

New paper alert!

Numerical general relativity is hard, complicated and computationally expensive. Here, we develop a radically new approach to doing it: cheat, and do special relativity instead...

arXiv link and explanation in thread... (1/11)
Link: 
In GR, a deep consequence of the equivalence principle is that, within any curved spacetime, there must always exist an observer (a worldline) that perceives it to be flat. A free-falling observer feels no gravity. (2/11)arxiv.org/abs/2410.02549

Functions like sine and cosine have some pretty nice properties: they're always differentiable and well-behaved (i.e. don't ever blow up to infinity). Apparently.

Yet it turns out that such functions are actually impossible, at least over the complex numbers. (1/9) As soon as we start extending sine and cosine to the complex plane, we see that they start blowing up - they don't remain nicely bounded like they do over the reals. We could try imposing bounds on them, but that would quickly break their differentiability. (2/9)