Precessing conic sections:
black hole orbits and spin in general relativity
come alive with Maple

by bob jantzen, Dept of Math and Stat,Villanova University

March 31, 2017, 2pm Mendel 154
math major informal talk

Conic sections are fun curves to play with for those who like mathematical geometry, their properties known since Apollonius studied them 2000 years ago. Unfortunately, perhaps like most US math departments, we skip the section on conics in our Villanova calculus sequence [Stewart Calculus 8e, 10.5 Conic Sections, 10.6 Conic Sections in Polar Coordinates], but perhaps somewhere along the line our majors become sort of aware of these curves. They are well known as the orbits of planets around the sun in Newtonian mechanics, with the Sun (more properly the center of mass of the Sun-planet two body system) at one focus of the circular, elliptical, parabolic, and hyperbolic orbits of the planets. [These conic section orbits are intimately tied to the behavior of particles in inverse square force fields.]

Einstein's general relativity adds an additional element to these conic section orbits in the field of a rotating mass or black hole, namely a precession (rotation!)  of the classical Newtonian conic section orbits (in appropriate coordinates) for orbits confined to the equatorial plane of the Kerr rotating black hole spacetime. Furthermore, the spin vector of a gyroscope carried along such orbits, whose direction with respect to the distant starts remains fixed  in Newtonian theory, also precesses due both to the curved geometry in which the orbit is tracing out its "geodesic" path, as well as to the rotation of the source of the gravitational field. [Stewart Calculus 8e, 3.11 Hyperbolic Functions is another skipped section, describing the geometry of special relativity needed to get to general relativity.]

When a body is already rotating, and the axis of the rotation is itself rotating, the latter is said to precess. A the tip of a spinning top with its axis tilted will  precess around that axis in a circle. The Earth's rotation axis precesses ("axial precession") with a period of about 26,000 years. Who knew? "Apsidal precession" is a more elastic use of the term to describe the progressive rotation of elliptical orbits in their plane

To visualize the unseen geometry which puts these orbits and moving spin vectors into context, similar to the Apollonius geometry of ellipses and their evolutes, Maple proves again to be the perfect tool. [hype from bob]

A slowly precessing elliptical orbit is shown here.

For orbits, the conic sections are expressed in polar coordinates (r,θ) with a focus at the origin, parametrized by the eccentricity e and semilatus p as follows.

\begin{equation} r = \frac{p}{1+e \cos(\theta)} \end{equation}

Here on the left is a family of conics with unit minimum radius rmin = p/(1+e) = 1. The circle has e = 0, ellipses 0 < e < 1, the parabola e = 1, and the hyperbolas e > 1.

In the context of  Newtonian gravitational field orbits, the circle and ellipses are "bound" orbits with negative energy, the parabola (zero energy) and the hyperbolas (positive energy) are "unbound" orbits. At the point of minimum radius where the velocity is in the azimuthal direction, the speed at that moment determines which of these orbits results.

Prerequites for this talk:
MAT2500 Multivariable Calculus, MAT2705 Differential Equations with Linear Algebra

Maple worksheets will be used to deliver the content. This may be a bit ambitious, so the details can be consulted later if found interesting and worthy of any more time than sitting in this lecture.
This is meant to be inspiring, not delivery of testable math content.

  1. Introduction
  2. Conic sections  []
  3. Planetary orbits and central forces in Newtonian theory []
  4. The GR (general relativity) correction (additional inverse cubic force term): precessing conics
  5. Rotations and spin []
  6. Spin precession in precessing GR orbits []

For more details and the final animations which bring these ideas to life, see the precessing orbits and spin web page.

There was insufficient time to show the example of relative rotation of the TNB axes with respect to Cartesian axes along a helix: [ok, bob forgot to include that worksheet after spending a lot of time on it.]

Note also that any central force can have its orbits described by precessing conic sections, but for GR it is particularly useful since in normal circumstances the precession is small and slightly corrects our Newtonian picture, while in strong fields we see how the idea bridges the gap to the fast precessing case.