If you have read my previous blogs, you must realize by now that I am deeply fascinated by the parallels between acoustic vibrations and the universe. Recent discoveries keep fueling my curiosity about the interconnectedness of the realms of sound and the cosmos. My previous post explored how a neutron star might ultimately collapse into a black hole, focusing on the resonant tones it then produces. Now, here’s another intriguing discovery about the behavior of black holes themselves.
All these findings are directly linked to the rapidly advancing field of gravitational wave detection (see my post on these waves). Interestingly, even before we had the technology to detect gravitational waves, mathematics had already enabled us to explore these phenomena theoretically. The possibility of detecting them is directly tied to Einstein’s theory of relativity, developed in 1917. For much of the period leading up to their discovery, there was considerable doubt about whether gravitational waves would ever be observed. But that didn’t stop astrophysicists from delving into this field in theory. And, indeed in a recent article “ in 1997 a graduate student, Hisashi Onozawa from the Tokyo Institute of Technology (now Institute of Science Tokyo), discovered a curious irregularity in what these waves should look like. In other words, some of the vibrations were dissonant in relation to the expected partial series from the fundamental.” (source: https://www.eurekalert.org/news-releases/1081428)
Here is a short description of the resonance found in bells:
Common partials in a bell include:
Hum: A low-pitched partial, one octave below the strike note (fundamental).
Prime (or strike): The main note you hear, typically the fundamental.
Tierce: A minor or major third above the prime (this depends on how the bell is tuned)
Quint: A perfect fifth above the prime.
Nominal (or octave): One octave above the prime.
Although bells differ in a number of subtle and debatable ways, there seems to be some common features. The sound of a bell is characterized by a large number of partials. Some of the upper partials harmonize (approximately) with the fundamental, but most do not. Assuming the lowest mode (hum tune) to be C0, the traditional bell is designed to have higher partials at C1, E1flat, G1, C2, E2, and C3 The minor third, C, - Eb, is desirable to give the bell its characteristic plaintive tone. In practice, the various partials do not match the harmonic series perfectly. The pitch (strike tone) will not necessarily coincide with the partial around C, but is apparently determined by some sort of an average fit to the nearly harmonic series. 3, 4 The clarity of the pitch seems to depend upon how closely the partials match the harmonic series. (source: Physics and Music, The Science of Musical Sound)
The sound of the bell is characterized by a large number of partials, and all these modes of vibration are interacting - which result in random partials that are unrelated to the fundamental. This is how these irregularities also form in black holes.
The article goes on “Now, after thirty years, Associate Professor Hayato Motohashi from Tokyo Metropolitan University, formerly affiliated with Kogakuin University, has resolved this problem using precision calculations and the nascent theoretical field of non-Hermitian* physics. Looking closely at the behavior of modes, he found that the dissonance was not isolated to one mode but was a result of two modes interacting with each other in a resonant fashion. In fact, by examining many modes, it turns out this kind of interaction between modes is not a rare incident, but one which turns up universally in a range of modes. Through a series of theoretical analyses, this resonance between modes in black holes was shown to be one example of a whole host of physical phenomena in not only astrophysics, but optical physics as well, looking at electromagnetic waves instead of gravitational ones. The interdisciplinary application of a new theoretical approach heralds the birth of the new field of non-Hermitian gravitational physics, unlocking the true potential of global-scale experiments like the LIGO-Virgo-KAGRA Collaboration which are coming online." https://www.eurekalert.org/news-releases/1081428)
Here’s another analogy I find interesting: sound (acoustic waves) travels faster in denser materials. For example, sound travels faster in water than in air—because water is 800 times denser than air. Sound moves at about 1500 meters per second in water (roughly 4.3 times faster than in air, where it’s 343 meters per second). Another example is the trick (often used by train robbers in the Wild West—please don’t try this!) of putting your ear to the rail of a train track to hear an approaching train much earlier than by listening in the air. That’s because sound travels at 5120 meters per second in iron—almost 15 times faster than in air! (see formulas a the bottom of this post)
This makes me wonder: what about the speed of sound in a black hole, with its infinite density? Would it travel at the speed of light? Perhaps. But we’ll probably never know—since not even light can escape a black hole, we’ll never be able to hear it, except for gravity’s effects on the fabric of spacetime.
All these analogies between acoustics and astrophysics underline the universality of the laws that govern everything, from the everyday objects around us to the most remote objects in the cosmos—even the bells that ring from our church towers. Imagine all the stars in the universe as resonating bells. I can hardly imagine what a cosmic carillon it would create!
* In standard quantum mechanics, the mathematical tools we use to describe things like energy and momentum are called Hermitian operators. These make sure that the measured values (called eigenvalues) are real numbers. If we use non-Hermitian operators instead, the measured values can be complex numbers.
If you want to figure it out by yourself, here are the formulas:
Solids:
v = √(Y/ρ).
where:
Y is Young's modulus (a measure of the stiffness of the material, in Pascals, Pa).
ρ is the density of the solid (in kg/m³).
Liquids:
v = √(B/ρ).
where:
B is the bulk modulus (a measure of the incompressibility of the liquid, in Pascals, Pa).
ρ is the density of the liquid (in kg/m³).
[? something to add here?]