According to InspireHEP.net, our quasinormal paper with Andy Neitzke joined our "famous" articles according to their terminology (250+ cits; more inclusive Google Scholar shows 302) because a two-year-old Indian thesis was finally scanned.
It was a fun calculation with a rather exciting pre-history and post-history – with lots of interesting technicalities but also some very general conceptual insights about the scientific method as a general approach to the truth. I described much of the context in A Quasinormal Story On Quasinormal Modes but let me add a few more comments about this specific paper.
Intrinsically, it could have been another calculation that no one had ordered i.e. that no one was really interested in, like a recent proof of an eigenvalue identity by Terence Tao. What made this thinking exciting and what mostly motivated us, or at least me, was the claims that the frequencies of these quasinormal modes had an almost direct relationship to the black hole entropy.
Highly heuristic but attractive features of the quasinormal frequencies were speculated upon by Hod in Dec 1998 and Dreyer in Nov 2002. The former paper tried to interpret the quasinormal modes by a not quite coherent but very diverse collection of musings about Bohr's complementarity and the black hole thermodynamics while the latter paper was framed in a more focused way as a rationalization of loop quantum gravity.
So I learned about these papers exactly because I was interested in the crazy claims by all those Jeffrey-Epstein-funded scientists who ludicrously claimed that they had a competition for string theory and it could perhaps even count the black hole states microscopically, too.
Normal modes are some solutions to a linear equation – for oscillations around a static state – that depend on time as \(\cos(\omega t)\). This cosine may be identified as the real part of \(\exp(i\omega t)\) and if you allow complexified solutions, this complex exponential may be viewed as the "object of interest" which is more well-behaved. Normal modes oscillate (or breathe or "ring") normally. What do quasinormal modes do? They oscillate quasinormally. Well, this really means that \(\omega\) is allowed to be complex. The real part of \(\omega\) still encodes some oscillation while the imaginary part of \(\omega\) adds some exponential increase or damping. It's the damped solutions that are close to the "ringing" that may occur in the real world.
In this context, we were interested in the Klein-Gordon equation for a scalar field that lives on the background of a Schwarzschild black hole in \(d=4\) – and then other black holes in general spacetime dimensions. And the focus was on the "highly damped" modes with frequencies whose imaginary part is much larger than the real part. So they're much more damped than they oscillate! This makes them "seemingly" much less physical than the normal modes and "slightly damped" quasinormal ones.
But in the highly damped limit, it was shown and understood why the imaginary part of the frequency had many possible values that were equally spaced (and what the spacing was in the limit). The real part was shown numerically to be \(\log 3\) times the Hawking temperature. The value of the coefficient \(\log 3\) was known to be this correct number with the precision of "billionths" or so. By this experimental argument, there was no real doubt that it was \(\log 3\) exactly.
Now, to an irrational person, the appearance of \(\log 3\) (it is a natural logarithm! Only natural logarithms naturally appear in natural sciences, and that's why \(\log\) is used to mean the natural logarithm by physicists, a point that the high school math and physics teachers were completely unaware of, I was assured during a talk I gave last summer!) may look miraculous, religious, stunning, and I think that it was the situation of Shahar Hod who had probably tried many other crazy "fits" for the numerical constant.
To a rational person with some knowledge of complex calculus, it's common sense that the real part is the "logarithm of a simpler number". Why? Imagine that I tell you that the frequencies \(\omega_j\) behave as \[
\omega_j = M+2\pi i k
\] for \(k\in\ZZ\), a countable number of such frequencies are allowed. What can you say about the value of \(M\)? Well, a nice way to get rid of the \(k\)-degeneracy is to exponentiate the equation\[
\exp(\omega_j) = \exp(M)
\] which is a simpler equation than before in which the \(k\in\ZZ\) mess was eliminated. So fundamentally, the "grid" of solutions is obtained from an equation such as one above. And \(\exp(M)\) has no reason to be this explicit exponential of something else. It's much more likely that it's some "simpler something else", like three, so you get \(\exp(M)=3\) and \(M=\log 3\). Or the logarithm of something that is simpler than its exponential.
(Note that above, you only have \(\omega_j\) in the exponent without \(i\). That seems different from the usual dependence on time, \(\exp(i\omega t)\). Why is there the exponential of \(\omega_j\) without the imaginary unit anywhere? Well, it's because you effectively substitute \(t=i\beta\) where \(\beta\) is the inverse temperature: the thermal calculations are those with imaginary time and the black hole setup effectively produces a thermal calculation based on the Hawking temperature.)
In this way, I guessed the form of the equation – the last step of the analytic derivation of the real part of the quasinormal frequency. And indeed, this guess worked well. First, I derived that logarithm of three using continued fractions. It's a pretty unusual, algebraic, but cool calculation that resembles some "evolution in discrete time". Again, there are some parameters labeled by \(k\in\ZZ\) that govern an expansion of the solution and they are constrained by some recursive equations. Recursive equations are just like "discretized differential equations". The most violent behavior of the numbers in takes place for \(k\approx 0\), i.e. when \(k\) is zero or a small positive or negative integer. The numbers have the capacity to jump all over the place but there is a condition that a nice given asymptotic behavior for \(k\to-\infty\) evolves to another nice behavior for \(k\to +\infty\). In the highly damped limit, everything is well-behaved, solvable, and the behavior near \(k\approx 0\) is exactly computable in terms of the Gamma function.
This method was nice and I think it may be generalized to other black holes and black holes in other dimensions. But you need recursive relations that involve a greater number of neighbors – i.e. discretizations of higher-order differential equations. Before I ever completed any nontrivial calculation for non-Schwarzschild black holes or black holes in other dimensions, I switched to the "geometric" calculation that we completed within some 3 weeks with Andy. We had the advantage of knowing the right result but it was still hard. We had to try to combine and recombine the phrases such as "Bessel's function", "monodromies", "boundary conditions" in many ways before we got the right combination that made sense and that implied the right result. But we were sure of the existence of such a solution with the same words and equations in advance LOL.
In the geometric solution, the quasinormal function is written as a function of time – that's the damped-oscillating complex exponential behavior – and as a function of \(r\), some cleverly chosen radial coordinate in general relativity that is usually named after a Ninja turtle. In that coordinate, the relevant solutions to the differential equation arising from GR are Bessel's functions. Those have monodromies and Stokes' phenomenon that we really had to intuitively master because although we had heard that phrase before, we had no "real" understanding for how it worked before.
This Bessel-based calculation seemed easier for the calculation of quasinormal modes on top of other black holes than the continued-fraction computation. And we have obtained many solutions that had looked very hard. We immediately falsified the general prediction that the asymptotic real part of the quasinormal frequencies of highly damped modes were \(\log 3\) times the Hawking temperature. It didn't work for charged black holes or black holes in other dimensions. Indeed, it didn't even work for fields of higher spin in the \(d=4\) Schwarzschild.
So we got celebrated by the anti-string "quantum gravity" people who study theories that obviously can't be consistent theories of quantum gravity. But a somewhat more general application of our results also showed that they had no reason to celebrate. The body of such results is another detailed and wonderful proof of the trivial statement that the people who claim to be finding a "quantum theory of gravity without strings" are full of šit and they're just producing illusions and delusions that may always be proven wrong or trivial if you try just a little bit harder.
Also, I think that someone should actually try to extend my first "algebraic" method using the continued fractions. As I said, it becomes more complex because the recursive relations become higher-order ones. That differs from our approach with Neitzke where the more complex black holes lead to the same Bessel-like equations in a complex plane where you have to draw a more complicated contour. Many examples are our figures – created with TkPaint which runs under ActiveTcl and is still my preferred tool to make physics-like diagrams (in EPS).
I believe that the preference for simpler geometric pictures is just a "human idiosyncrasy". With a neutral enough brain, the calculation using the continued fraction could be considered "equally hard" even for other black holes.
Also, I think that the behavior of the amplitudes that is very far from the real axis (real values of frequencies, wave numbers – i.e. energies and momenta) is "physical" in some fundamental, mathematically heavy sense. It's ironic that the discussion of the highly damped quasinormal frequencies had been linked to loop quantum gravity because the loop quantum gravity practitioners are childishly naive, anti-mathematical, and down-to-Earth. They think in terms of "real pictures" and the precise values of highly damped frequencies are nothing like a real picture of anything. Instead, these frequencies are some abstract mathematical traits associated with the analytic behavior of the amplitudes. The naive people with their "real pictures" – and that includes all critics of quantum mechanics, loop quantum gravity would-be scientists, and tons of other folks – don't even appreciate the importance of the facts that in credible theoretical physics, amplitudes are analytic functions of energies and momenta. The mathematical notion of analytic or meromorphic functions plays no role in their big picture of the world. This absence of meromorphic functions is a part of their innumeracy and naivite.
And be sure that the amplitudes are analytic or meromorphic functions of energies and momenta. Here you have a random 1966 paper by Lehmann that shows that good physicists have understood the universality of the analytic behavior of amplitudes for more than half a century. (I think that the first insights about the natural character of the analytic continuation of amplitudes were obtained already in the 1920s.) Needless to say, the analytic behavior of amplitudes was crucial especially at the beginning of string theory as well as in the twistor/amplituhedron minirevolution.
There have been some interesting applications of similar monodromy methods, e.g. Lapan-Maloney in 2013. Statements about monodromies may be not only clever ways to "solve a problem". They may also be important to "define a fundamental law of physics" or at least to explain why two seemingly different aspects of the laws of physics are actually tightly connected.
I also want to say that this paper, like many other papers, could serve as the example of the fact that the proper physics research doesn't really respect any "politically protected boundaries between subcommunities". Physics is either correct or incorrect, good or bad. If you claim to have some new theory to explain some phenomena and evidence that it is a good theory, it must be understood by everyone who is smart enough and who tries hard enough. If it's not the case and if you want to build the community of fans of your theory as if you were building a political party, then you a charlatan, not a scientist.
So of course we weren't ashamed of following some ideas by people who "belonged elsewhere" and who had clearly written tons of things that were no good. But they had also found something intriguing – which also opened many new questions and challenges – that should be clarified and we tried to do so. While doing so, we also arrived at stronger evidence that other claims by those people were totally wrong. That's how it should be. Scientists should generally listen to each other, build on the ideas of others from all other places, and also confirm or falsify the hypotheses by others, whether the others are "in the same camp or not", depending on where the evidence points to.
When someone tries to define a subfield that should be protected against thinkers from outside, and that demands to be "protected" in order to keep "diversity" of theories, then it's a pseudoscientific subfield that is on par with the grievance studies. Real science doesn't respect any identity politics, doesn't consider "diversity" a goal and by definition, scientific theories simply can't get any guarantees to survive no matter what. If someone wants to declare "diversity" a universal protective shield against falsification, then he or (in this case equally likely) she is firmly outside the realm of science.