Extensions at Quadrupolar Order

Abstract

In this last chapter, the aim is to extend the results of Chap. 5, obtained at dipolar order, to account for several finite-size effects. In particular, we detail two quadrupolar models, that account for the deformation of the physical body described by the particle: a spin-induced quadrupole, which encodes the information on the response of a body to its proper rotation, and a tidally-induced quadrupole, which accounts for the tidal effects.

Dear Mr Poincaré, I am sorry to say it is too true that there are, as you tell me in your letter which I have received this morning, several mistakes in respect to magnitude and to sign in my statements regarding the nutation which would exist if the earth consisted of a rigid ellipsoidal shell filled with frictionless liquid.

W. Thomson

Letter to H. Poincaré (1901).

Interlude

for most physics students learning about general relativity (GR), it is fair to say that having followed an introductory course on special relativity (SR) is recommended at the least. Most of these students (myself included) might have felt that SR is less “complicated” than GR. Without doubts, this is primarily because of the flatness of the Minkowski spacetime, which allows one to stay within the realm of affine geometry, a world already familiar to Euclideanly-trained students. This misconception may also be due to the way SR is usually taught, with more emphasis on a matrix formulation and Lorentz transformations than on the tensorial, operational nature of the theory. But as was beautifully showed by Éric Gourgoulhon in his treatise on SR in non-inertial frames [22], the geometry of Minkowski spacetime is nothing short of complicated and beautiful. It may as well be curved if one chooses arbitrary coordinates and/or studies accelerated, rotating motions. Of course, GR forces us to leave affine spaces for manifolds, but the effort to make the SR $\rightarrow$ GR step is, in my opinion, drastically reduced when learning about SR chronogeometrically, as Gourgoulhon teaches.

In any case, these relativistic theories showed how to think of space, time and gravitation geometrically. Einstein himself talked about this matter several times. His thoughts on the geometrization of physics are beautifully summarized in the article Geometrie und Erfahrung (Geometry and Experience) [549], which he had prepared for his “particularly nice lecture” given in 1921, at the Prussian Academy’s in honor of Frederick the Great. During this lecture, Einstein makes the distinction between the “practical” geometry, which he refers to as the oldest branch of physics; and “purely axiomatic geometry”, which he does not give any credit for his discovery of relativity. But by distinguishing between the two, Einstein only emphasizes of the geometry of space (and time) in which physical processes take place, and not on the inherent geometry of the laws themselves. Yet, this is where the geometrization of physics started, long before Einstein and his revolutionary ideas, at a time when the frontier between mathematics and physics was very much nuanced.

Indeed, it was classical mechanics that became the first geometrized branch of physics during the eighteenth century, through an ironical situation often found in books on physics history. While it is often said that Lagrange is the father of analytical mechanics (which would become the foundation of symplectic geometry), we find in his 1788 masterwork on analytical mechanics the following warning:

On ne trouvera point de Figures dans cet Ouvrage. Les méthodes que j’y expose ne demandent ni construction, ni raisonnemens géométriques ou méchaniques, mais seulement des opérations algébriques, assujetties à une marche réguliere et uniforme. ⁴

In English: The reader will find no figures in this work. The methods which I set forth require no constructions nor geometrical or mechanical reasoning, but only algebraic operations, subject to a regular and uniform rule of procedure. Here, Lagrange speaks, in this particular paragraph, of the “synthetic” geometry, which he will not be using, even though it was the method of choice among geometers and physicists of the time. Instead, Lagrange warns that he will be using new tools, that are now part of classical “analysis”. Nevertheless, it is ironic that by inventing his new analysis tools, in particular, the Lagrange parentheses, Lagrange did invent a new geometry: symplectic geometry. Indeed, we may rightfully set the birthdate of symplectic geometry in 1808, when Lagrange’s work on celestial mechanics showed that the equations of planetary motion could be cast in a greatly simplified form [550], using the tools he invented. The form of these equations were then extended to any mechanical problem by Hamilton, and these ideas were all simplified, extended and understood in details by Poisson, Jacobi and Liouville, to name but a few. An illustration of Lagrange’s early insight can be found in the “Lagrange parentheses”, equivalent to the more commonly known Poisson brackets, which are nothing but the components of the symplectic form [551].

The word “symplectic” was introduced by Hermann Weyl in his treatise on group theory [552] (see the first footnote in chapter VI there) as a replacement for (and Greek-equivalent of) complex, also introduced by himself in the context of group theory. Symplectic geometry, at the basis of Mechanics, is more demanding, in some sense, than Lorentzian geometry, on which GR is founded. While pretty much every manifold can be endowed with a Riemannian metric, regardless of its dimension, not all of them can carry a symplectic form. Even excluding odd-dimensional manifolds is not enough. For example, the 4D and 6D sphere cannot be endowed with a symplectic structure (not even a nondegenerate antisymmetric bilinear form for the former!) The main reason behind this fundamental difference between the two is encoded in Jacobi’s identity, or, to paraphrase Arnold [553], because the three altitudes of a triangle are concurrent [554]. We leave for the reader the pleasure of discovering why Arnold would have said such a thing.

The first part of this thesis was based on a relativistic problem, where a “symmetry” was provided by the existence of a helical Killing vector field. In the second part, we will be dealing with a non-relativistic problem, with a symplectic symmetry, hidden at the level of the Hamiltonian of the system. Since the new problem at hand s be rather independent of that studied in the first, let us, at least, keep with the tradition of letting Galileo open the introductory chapter, like in the first part of this thesis.