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.

Conformal compactification of space-time

I’ve been reading the article “Relativistic Symmetry Groups” by Roger Penrose, which appeared in Group Theory in Non-Linear Problems: Lectures Presented at the NATO Advanced Study Institute on Mathematical Physics ed. A.O.Barut (1974). In the article he talked a bit about conformal compactification of space-time. The notion of a conformal infinity is, by now, a pretty standard topic in introductory courses in General Relativity. I mention this article of Penrose because it contains a few small facts which is often not discussed in said courses or textbooks.

Questions and Answers: my progress through problems

I’ve put up a new page at this blog called Questions and Answers. It documents my progress through research problems that I am thinking about (and helps me keep track of ideas that I have). It is mostly for my own benefit.

What is mathematical research?

Recently I sat on a panel of “research scientists” for Career Day at the New Jersey Governor’s School in the Sciences. During the discussion, I was pointedly asked, by a young woman who aspires to be a research mathematician, what “mathematicians do for research?” Much to my own dismay, I fumbled around for a short answer, but ended up giving a shpiel that is much too long for the occassion and which ended up mostly sound and fury, and probably confused everyone in the room except for myself.

Contrary to the impression I have just given, I have thought quite a bit on how to answer the question “what is it that you do?” I am even quite good at tailoring the answer to the audience. If that were the question asked, I would have told the audience of high school students that I study the long term behaviour of the universe using general relativity, that this falls in the realm of mathematics and not physics because it is patently impossible to recreate large scale structures of the universe, such as nebulae or black holes, in a laboratory on Earth, and so I study the mathematical consequences of the assumptions underlying general relativity. If time allowed or if I were further prompted, I would have told them about the propagation of waves and other fancy things from partial differential equations.

Straightedge and compass constructions

Classical Euclidean geometry is based on the familiar five postulates. The first two are equivalent to the assumption of the existence of a straightedge; the third gives the existence of a compass. The fourth essentially states that space is locally Euclidean, while the fifth, the infamous parallel postulate, assumes that space is globally Euclidean.

A quick digression: by now, most people are aware of the concept of non-Euclidean geometry as described by Bolyai and Lobachevsky, and independently by Gauss. These types of geometries make also the first four postulates. By simply abandoning the fifth postulate, we can arrive a geometries in which every pair of “lines” intersect (in other words, geometries in which “parallel lines” do not exist), or in which parallel lines are non-unique. (As a side note, one of the primary preoccupations of mathematicians is the existence-and-uniqueness of objects. With our obsession of classifying things, we ask often the question, “Does an object exist with properties X and Y?” and follow it up with, “Is there only one object that possesses properties X and Y?” One may have been subjected to this analysis as early as high school algebra when one is asked to classify whether a system of linear, algebraic equations have solutions, and whether the solution is a point or a line.) With even this relaxation, the space is still locally Euclidean (by the fourth postulate): a small portion of the surface will look “flat” when you blow it up (the same way how from our point of view, the Earth often looks flat). It is, however, possible to also relax the fourth postulate. The easiest example to imagine is by taking a piece of paper and rolling it up into a cone. At the vertex of the cone, going around one full turn is no longer going around 360 degrees. The “total angle” at the vertex will be smaller. So if we define a “right angle” as an angle that is exactly one quarter of the “total angle” at the point, the right angle at the vertex will be smaller! You can experiment with it yourself by drawing four rays from the vertex and then unfurling the piece of paper. Geometries with this kind of structures are often studied under the name of orbifolds.

In any case, let us return to the topic at hand.

Classical Euclidean geometry is filled with constructive proofs (the modern methods of mathematics–proof by contradiction, principle of strong induction, and existence proofs without construction–are only really popular after the 18th century).

How to derive the Kerr metric by cheating quite a bit. Part 3

( … continued from Part 2)

3. Deriving the Kerr metric

In this section we show how the Kerr metric may be (in a large part) derived by studying Problem 2.23.

The main result that we rely on is a lemma given in Mars’ 1999 paper (If you actually look at the paper, you’d see that there are some factors of 2 differences in a lot of the statements. These are related to the fact that our definitions of anti-self-dual forms differs by a factor of 2, and that our definitions of the Ernst two-form and the Ernst potential also differ by a factor of 2).

Lemma 3.1
We can define the real-valued function and by . Then there exists a non-negative real number such that , and

How to derive the Kerr metric by cheating quite a bit. Part 2

(… continued from Part 1)

The anti-self-dual fields and complexification

For ease of algebraic manipulations, often we consider the anti-self-dual versions of two-forms. Observe that on a four-dimensional Lorentzian manifold, the Hodge star operator takes two-forms to two-forms, and squares to -1. This implies that its eigenvalues can only be . So we complexify our geometry by linearly (so in particular and not the Hermitian product). It is clear that (via a little bit of linear algebra) that the space of two-forms splits after complexification

Equation 2.10

where are spaces of complex-valued two-forms that have eigenvalues under * respectively. It is also clear that there is a natural isomorphism from to each of (they all have real dimension 6).

How to derive the Kerr metric by cheating quite a bit. Part 1

This came from a lecture I gave to MAT 451 at Princeton University on April 23, 2009. I have originally written this up as a LaTeX document; since I won’t be publishing this in any conventional way (and indeed, the material covered is rather unconventional), I figure I’ll use this as an experiment for the first posts on this blog. The original is a 10 page paper, which is why I am splitting this into several installments.

MAT 451 is a senior level mathematics course in which the instructor has great leeway in deciding what to teach. This year my thesis advisor was in charge, and focused the discussion on mathematical aspects of general relativity. This first post will, therefore, be rather on the technical side: the reader is assumed to have familiarity with basic pseudo-Riemannian geometry and with various aspects of general relativity. I will, however, be happy to answer any questions left in the comments.

The Kerr metric I refer to is, of course, the rotating black hole solution in Einstein’s theory of general relativity. For a brief history surrounding its discovery, see Dautcourt’s survey “Race for the Kerr field”.

1. Introduction and the first ansatz

In this note we give a heuristic derivation of the Kerr metric, in a way quite significantly different from the classical methods. This is in no way a formal write-up, so for a more rigorous derivation, and for references, please see the wonderful article by Roberto Bergamini and Stefano Viaggiu, “A novel derivation for Kerr metric in Papapetrou gauge,” Class. Quantum Grav. 21 4567–4573 (2004). The method described herein is inspired by Marc Mars’ paper “A spacetime characterization of the Kerr metric,” Class. Quantum Grav. 16 2507–2523 (1999), and also by my 2009 PhD dissertation.