For a more well-formatted blog post, see this link: https://messier87star.github.io/blog5

Welcome back to the blog! In the last post, I promised that we would finally get to the calculation of the critical exponents. Before introducing the critical exponents, however, I think it’s important to go over the actual metric that we will be calculating all of our critical exponents within. In this post, I’ll be introducing that particular metric that, the Kerr-Newman metric, as well as the first of the critical exponents we will be deriving in this project—the Lyapunov exponent.

The Kerr-Newman Metric

My intention with this project is to, at the very least, calculate the critical exponents for the most general black hole describable within general relativity. For most kinds of objects in the universe, this would be an extremely difficult task—after all, there are so many trivial ways one might perturb an object to add “extraneous” information to its description. However, thanks to the no-hair theorem, to which we have alluded previously in the blog, this is actually possible for black holes. This is because black holes are uniquely parameterized by three properties: mass, charge, and angular momentum. Because of this, by considering a black hole with a given mass $M$, a charge $q$, and an angular momentum $a$, we are able to solve Einstein’s field equation in order to derive the metric tensor gμν​ of the black hole, and this highly general metric is known as the Kerr-Newman (KN) metric. This derivation was first performed by Ezra Newman in 1965; I’m not going to present the entire derivation here, because it is highly nontrivial (in fact, it is incredibly complicated!), but the end result, a generalization of the more famous Kerr metric derived only a few years prior, describes all black holes that can be found in nature (at least, should general relativity and the no-hair theorem be absolutely correct). The metric, written out fully, is as follows:

ds2=gμν​dxμdxν=​−ΣΔ​(dt−asin2θdϕ)2+ΔΣ​dr2+Σdθ2+Σsin2​[(r2+a2)dϕ−adt]2​

where, for the sake of brevity, we have abbreviated the following functions:

Δ=r2−2Mr+a2+q2Σ=r2+a2cos2θ

Don’t worry if you don’t understand this entire function! It’s fairly complicated, and its behavior is fairly complicated; the important takeaway is that it provides some notion of “distance” that depends on the curvature of space, and this particular notion of distance depends on the mass, charge, and angular momentum of the black hole. This metric is central to our entire project, and we will use the behavior of this metric to derive the three critical exponents.

It’s worth discussing why the Kerr-Newman metric is generally less well-studied as compared to its less general cousin, the Kerr metric. The Kerr metric is simply the limit of the Kerr-Newman metric as the charge $q$ approaches 0; if you analyze the above metric closely, you may notice that this is equivalent to the limit Δ→r2−2Mr+a2. The extent to which this behavior affects the behavior of a black hole is still not totally well-understood; all black holes that have been observed seem to obey the Kerr metric fairly well, but this is likely because black holes are not easily imparted with charge. By contrast, they are very easily imparted with angular momentum (consider that the initial cloud of material that will eventually become a black hole almost certainly, at least statistically, possessed some angular momentum—thus, by conservation of angular momentum, the final black hole must be spinning). However, in contrast, one might imagine that highly positively charged black holes would tend to attract negative charge and repel positive charge, or vice versa for highly negatively charged black holes; thus the effect of charge on black holes is very difficult to gauge observationally. Another one of the goals of this project, then, will be to discuss the extent to which the charge of a black hole has a visible effect on its photon rings.

Now, we can begin discussing the first of the critical exponents by which we will gauge this effect.

The Lyapunov Exponent

The Lyapunov Exponent, often denoted $\gamma ,$ quantifies the extent to which subsequent photon rings are demagnified. It can also be thought of as relating to the relative width of subsequent photon rings, which is what governs the demagnification phenomenon. Recall that the photons that we observe on our screen are those associated with nearly bound geodesics; bound geodesics, after all, can never be observed. Consider a photon, then on a bound orbit of radius rB​. A nearly bound orbit might deviate by some factor δr0​<<rB​, and after $n$ half-orbits this deviation from this deviation the orbit will have deviated more from its nearly bound trajectory to some δrn​. We then conjecture that the relationship between δr0​ and δrn​ is

δrn​=(eγ)nδr0​

Thus to calculate the Lyapunov exponent $\gamma$ we can rearrange this to obtain

γ=n1​ln(δr0​δrn​​)

The reason we calculate $\gamma$ instead of eγ is partly a matter of convention (Lyapunov exponents appear more generally than in the study of photon rings; they are fairly central to chaos theory), and partly because the resulting expressions are in fact a little cleaner. Also note that we have not actually proved that the bound orbit deviations grow exponentially in $n$, simply assumed it; after making this ansatz, should we get an answer that does not involve $n$, we have proved it; otherwise, we have disproved it.

How might we compute this expression involving both the initial deviation of radius and a subsequent one? Stay tuned for the next blog post, in which we’ll be doing exactly that! (In particular, we’ll be following a derivation presented in the literature by the master’s thesis of Seppe Staelens, although explicitly keeping in mind both spin and charge.)

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.
• Misner, Thorne, and Wheeler. Gravitation. Princeton University Press. 1973.