Blog Post 6 - Nearly Bound Orbits

In the previous blog post, we established the definition of the Lyapunov in terms of the deviations of a nearly bound geodesic from its bound orbit; namely, that

$$\gamma =\frac{1}{n}\ln \left(\frac{\delta r_n}{\delta r_0}\right)$$

How can we do this? First, we must more carefully and quantitatively consider the nature of these nearly bound geodesics which the Lyapunov Exponent quantifies. Below I’ll present a derivation of the Lyapunov exponent for Kerr-Newman black holes; most of this proof follows that shown in the Staelens thesis cited below—the proof there is explicitly presented for the Kerr metric rather than for the Kerr-Newman metric, but due to the similarities between the two the proof requires only very small tweaks.

Orbits and Momentum

The first thing to consider is what we even mean when we say ”$n$ half-orbits”. What qualifies as an “orbit”? Imagine we parameterize the space around the black hole using the elements we gave the metric in terms of, spherical coordinates $(t,r,\theta ,\varphi )$, where $\varphi$ is the polar angle and $\theta$ the azimuthal angle. If you think of orbits in terms of the very classical orbits you may be used to visualizing (picture the earth moving around the sun) your first instinct may be to define an orbit in terms of one full oscillation in the $\varphi$ variable. It turns out that this is not a very good definition for our purposes, particular for photons coming from very far away; what we are interested in is the number of times a geodesic crosses the equatorial plane of our black hole (since, as you may recall, the intensity of the photon scales with this). Thus, it actually makes more sense to define our orbit as a full oscillation in the $\theta$ variable (and thus a half-orbit as half such an oscillation).

Now that we’ve established a notion of “orbit”, we can try getting equations of motion for geodesics near the black hole by writing down expressions for the momentum of the black hole. For brevity’s sake, I won’t perform the full derivation how to generate the momentum from the black hole, the basic outline begins with finding conserved quantities that might help us. The conserved quantities aren’t always obvious they way they are in classical mechanics; quantities that you might guess are conserved might not be, and there may be conserved quantitied admitted that you can’t quite guess. The general way to do this is to solve an equation called Killing’s equation:

$${\nabla }_{(\mu }{K}_{\nu )}=0$$

(Note: if you’re unfamiliar with the tensor notation being used here with regards to the parentheses, it essentially means you take the average of all the permutations of the parenthetical variables.) The vectors that solve this equation are called the Killing vectors, and they define the conserved quantities that are admitted by a system. (Specifically, if a vector $K_{\mu }$ solves the equation, the quantity $K_{\mu }{p}^{\mu }$ is conserved, where $p^{\mu }$ is a contravariant component of the momentum.)

If we solve Killing’s equation for the Kerr-Newman metric, we find two conserved quantities, which correspond to energy and angular momentum, respectively (which makes a lot of sense—energy is usually conserved, and the black hole can be thought of as a central potential, thus conserving angular momentum like in typical orbital mechanics). These two conserved quantities also correspond to the $t$ and $\varphi$ components of momentum, respectively. (Another detail I’ve omitted for brevity is the existence of solutions to this equation called Killing tensors—the KN metric has one of these as well, for a total of 3 conserved quantities. This last conserved quantity, first discovered by Brandon Carter in 1968, is often referred to as “Carter’s constant”.)

The Radial and Angular Potentials

We can write out the momentum of a geodesic

$$p=-Edt+\frac{\sqrt{\mathcal{R}(r)}}{\Delta }dr+\sqrt{\Theta (\theta )}d\theta +Ld\varphi$$

where we have written it in terms of the “radial and angular potentials”, which are respectively, for null geodesics

$$\mathcal{R}(r)={\left[E\left(r^2+a^2\right)-aL\right]}^2-k\Delta$$ $$\Theta (\theta )=k-(L\csc \theta -aE\sin \theta {)}^2$$

(omitting a lot of algebra that can be found in the papers below.) We’ve also abbreviated an expression $\Delta$ which comes up a lot in studying Kerr and Kerr-Newman black holes, which is merely

$$\Delta =r^2-2Mr+a^2+q^2$$

(In the derivation provided by Staelens, $q$ is taken to be zero for Kerr black holes in particular; however, restoring $q$ for an appropriate $\Delta$ in fact the only real tweak needed to make that proof general to Kerr-Newman balck holes as well.) Note that these expressions are valid for null geodesics only; I’ve omitted terms that would otherwise appear for geodesics with $m\ne 0$. An interesting fact about null geodesics, however: you may notice the potentials are written in terms of the three constants of motion we derived from the Killing equation. It turns out we can actually describe these potentials using only two constants of motion for null geodesics, those of $\lambda =L/E$ and $\chi =k/E^2$, sort of representing an energy-normalized angular momentum and Carter’s constant, respectively. (Some sources also use a modified version of this normalized Carter’s constant as $\eta =\chi -(\lambda -a{)}^2$ to make some algebra cleaner at times.)

We can therefore write normalized versions of our potentials as

$$\frac{\mathcal{R}(r)}{E^2}=\left(r^2+a^2-a\lambda \right)-\chi \Delta$$ $$\frac{\Theta (\theta )}{E^2}=\chi -(\lambda \csc \theta -a\sin \theta {)}^2$$

Raising the indices using the contravariant form of the metric, we are able to derive relationships between components of the momentum and these potentials as

$$\frac{\Sigma }{E}{p}^r={\pm }_r\sqrt{\mathcal{R}(r)}$$ $$\frac{\Sigma }{E}{p}^{\theta }={\pm }_{\theta }\sqrt{\Theta (\theta )}$$

The components of momentum quantify the rate of change of a coordinate with respect to some parameter. When explaining the geodesic equation, you may recall we used the proper time $\tau$ for this. However we are now dealing with null geodesics—which travel at the speed of light and do not experience proper time! Thus, we instead use what is known as the “affine parameter” $\sigma$, which can be shown to be a unique parameter that satisfies the geodesic equation (technically only unique up to an affine transformation, to be precise). Thus a particular component of momentum can be described as $p^{\mu }=\frac{d{x}^{\mu }}{d\sigma }$. By eliminating $\frac{\Sigma }{E}$ in the above set of equations, we find that

$${\pm }_r\frac{1}{\sqrt{\mathcal{R}(r)}}\frac{dr}{d\sigma }={\pm }_{\theta }\frac{1}{\sqrt{\Theta (\theta )}}\frac{d\theta }{d\sigma }$$

We now integrate this equation over $\sigma$ for a nearly bound geodesic. Recall that we want to derive the expression at the beginning of this blog post. Thus, we might consider integrating from a photon that is nearly bound; that is, if a bound orbit exists at $r_B$, then we might begin integrating at $r_B+\delta r_0$ for this small deviation from boundness $\delta r_0$. We would then integrate until the bound geodesic makes $n$ half-orbits around the black hole, reaching a radius of $r_B+\delta r_n$.

What, then, would be the bounds of the angular integral? Well, as we’ve discussed before, we consider a full orbit to be an oscillation in $\theta$; thus, a half oscillation would be movement from some upper angle ${\theta }_{+}$ to a lower angle ${\theta }_{-}$. There are still some issues to deal with with regards to sign. By symmetry, we can integrate from the equatorial angle of $\pi /2$ up to ${\theta }_{+}$ and then double this integral—this is allowed because we can simply define our coordinate plane to contain the equatorial plane at $\pi /2$. This it the integral for a single half-orbit, so we then multiply by $n$ for $n$ half-orbits. (For transparency, I’ve omitted a lot of rigorous steps here for brevity, but some things to consider are proving that there exist turning points on both sides of the equator and that they are equidistant from it, and that the radial integral does not change direction outside a bound orbit.) Considering all this, we can write that

$${\int }_{r_B+\delta r_0}^{r_B+\delta r_n}\frac{dr}{\sqrt{\mathcal{R}(r)}}=2n{\int }_{{\theta }_{+}}^{\frac{\pi }{2}}\frac{d\theta }{\sqrt{\Theta (\theta )}}$$

In the next blog post, I’ll go over how we can actually derive an analytical expression for this, and perhaps even generate some graphs, so stay tuned!

Sources

• Staelens, Seppe. “Black Hole Photon Rings Beyond General Relativity: Analytic and Ray-Tracing Approaches.” KU Leuven, Faculty of Science. 2022, https://www.scriptiebank.be/sites/default/files/thesis/2022-09/Master_Thesis_Seppe_Staelens.pdf.
• Gralla, Samuel E., and Alexandru Lupsasca. “Null Geodesics of the Kerr Exterior.” Physical Review D, vol. 101, no. 4, Feb 2020, p. 044032. arXiv.org, https://doi.org/10.48550/arXiv.1910.12881