Parallel volume forms

In the previous post on Newton-Cartan theory and Galilean geometry, I showed that the Galilean manifolds admit a preferred volume form. After some discussion with my old officemate Pin Yu, and a bit of digging on the internet, I found that this notion of a parallel volume form is a rather well-developed one in classical differential and affine geometry.

Newton-Cartan, Part 2

After writing up the previous post on Newton-Cartan theory, I came to realize that it is actually a very nice exercise for myself to dig into the geometry more. So here goes a bit more on the implications of the Galilean geometry and the Newton-Cartan theory.

Newton-Cartan Gravity

A few weeks ago I discussed Einstein-Cartan geometry, with a focus on relaxing the “torsion-free” condition on a Levi-Civita connection. In this post I will talk about the Newton-Cartan theory of gravity, which is in some sense the Newtonian limit of general relativity, and which relaxes the metric-compatibility of a Levi-Civita connection (while keeping the torsion-free condition).

Newtonian gravity and the naive formulation
(Note: the material in this section is re-hashed from Section 12.1 of Misner, Thorne, and Wheeler’s black-covered bible.)

Consider first Newtonian theory of gravity. The space-time is with Galilean symmetry, and gravitational interaction is represented by the gravitation potential . In Newtonian theory, the gravitational field is given by minus the gradient of the potential (I will put the arrows over symbols to denote the fact that they are three-dimensional vectors, and the derivative symbols should be interpreted in the sense of three-dimensional vector calculus). The force on a particle is given by the product of the gravitational field and the gravitational charge of the particle . By Newton’s second law, the force is also equal to the product of the inertial mass and the acceleration of the particle . Now, by the principle of equivalence (or the observation that the gravitational charge is equal to the inertial mass), we have that the gravitational field is equal to the acceleration of the particle.

Now consider a particle traveling in the gravitational field in free fall. Write its trajectory in as . Lifting to the space-time the world line of the particle is given by . (For people familiar with General Relativity already: in GR the world-line is usually given as a geodesic with unit speed. Under the 3+1 split in Newtonian theory, “proper time” is not defined, so the natural parametrization is by the global/invariant time.) The velocity vector in the space-time is and the acceleration vector is . The Newtonian equation of motion then is described by

Equation 1

Now, observe that if we do an affine change of variables (affine means here $d^2t/ds^2 = 0$), and notice that the chain rule gives , (and by abuse of notation we write )

Equation 1′

Digital computing and catastrophic failures

I just read a wonderful article on Discover magazine. The article centers around Kwabena Boahen (and other members of the school of Carver Mead) in creating electronic circuitry modeled more after the human brain. The main claim is that these types of neurocircuits have the potential in significantly lowering the power consumption for computing. If the claim were correct, though, it will imply there are certain nontrivial relationship between the voltage applied to a transistor and the noise experienced.

The idea, I think, if I understood correctly just from the lay explanation, is a trade-off between error rates versus power. Let us consider the completely simplified and idealized model given by the following. A signal is sent in at voltage . The line introduces thermal noise in the form of a Gaussian distribution. So the signal that comes out at the other end has a distribution , where the Gaussian family is defined as

Definition 1 (Noisy signal)

(Note: our definition is not the standard definition, in particular our Gaussian is centered at ! This definition makes calculations later simpler, as we shall see.)

Dos and Don’ts of giving talks

(I am putting this on my blog mostly so I can find it again later.)

Jordan Ellenberg stirred up a discussion of bad practices that happens in talks. Then he decided (after a commenter) that positive reinforcement of good practices may work better.

Cartan’s Spiral Staircase

I’ve just spent much too long puzzling over the geometric formalism of a paper of Lazar and Hehl, so I figure I’ll write a little something about Riemann-Cartan geometry here. Note: I will only discuss geometry, and from a very limited perspective at that. For physical applications please see the paper cited above.

(pseudo/semi-)Riemannian geometry
First we recall what a semi-Riemannian geometry is (see also Chapter 3 of B.O’Neill, Semi-Riemannian Geometry). To begin, let us recall some definitions.

Definition 1 (Affine connection)
An affine connection on a smooth manifold is a map taking two smooth vector-fields to a third smooth vector-field denoted satisfying the following three properties:

• (-linearity in ) for any real number ,
• (-linearity in ) for any smooth, real-valued function on , and
• (Leibniz rule for -multiplication in ) for any smooth, real-valued function .

Intuitively an affine connection affords us a way of identifying the tangent space at two points , but the identification depends on the path taken from . This is the notion of a parallel transport (perhaps someday I will write more on these fundamental things about geometry; but I’ll just assume that the reader is familiar with it for now).

Aharonov-Bohm effect’s analogue in water waves

Rather indirectly through Claude (a not-so-short story there) I learned of a paper by Michael Berry and collaborators titled “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue” (a copy can be found on Berry’s website at number 96). It is among the most pleasurable papers I have read in physics to date.

To understand why this Chandrasekhar-esque paper tickles me so, I need to explain a little bit of the physics.

The Aharonov-Bohm effect
The Aharonov-Bohm effect is one of the classic illustrations of the difference between classical and quantum mechanics. More importantly though, it also demonstrated that the use of the vector magnetic potential is not a mere computational convenience, but that the potential also manifests itself in physical effects.

“No Hair” theorems

Hum, now I am a bit confused. I doubt any “prominent mathematician” will read this blog and comment, so I guess I’ll ask them in person next time I go to a conference.

The question is about the term “no hair theorem”.

Healthy skepticism

Besicovitch once said that “A mathematician’s reputation rests on the number of bad proofs he has given.” Of course, originating from someone educated in the Russian school, the word “bad” in the quote should probably be taken to mean “inelegant”. However, lack of beauty is certainly not the only possible deficiency in the quality of a published result. Cases abound where a proof is bad not in the aesthetics, but in something more fundamental.

Irrationality of a common constant

It is rather embarrassing that, as a professional mathematician, that I didn’t know how to prove the irrationality of . The most common proof of this fact is this argument which PlanetMath cites to Hardy and Wright, but which I’ve seen also to referred to in Bourbaki. I will discuss a slightly different proof here. The version that I read is from Zhou and Markov, though the idea is due originally to Lambert in 1761.

Theorem. If is a non-zero rational number, then the value is irrational.

Before we continue to give the proof of the theorem, observe that the contrapositive of the theorem gives that if is a rational number, then is either 0 or irrational. Since , an immediate corollary is that , and hence , is irrational.