**THURSDAY, 29 APRIL 2021**

The Pythagoreans discovered that there was a simple mathematical relationship between the length of the string that was plucked and the pitch (or frequency) of the note that is produced. This can be visualised in physical terms. The violin string vibrates in a smooth, regular wave. Since both ends of the string are fixed in place, a half-integer number of wavelengths must fit into the length of the string, so the longest wavelength possible is the length of the string itself. Frequency is inversely proportional to wavelength, so the lowest, or fundamental, frequency is inversely proportional to the length of the string. The string can also vibrate at higher frequencies, corresponding to two, three or more wavelengths fitting into the length of the string. These are called the pure tones of the string.

It took many centuries, from the Pythagoreans to the Napoleonic Wars, for the next chapter in this story. In 1801 Joseph Fourier started to attack the problem of the conduction of heat in a solid. This work culminated in 1822 with the publication of The Analytical Theory of Heat.

Although the subject of the book is the study of heat, it also introduces Fourier Analysis. The central assertion of Fourier Analysis is that essentially any mathematical function can be decomposed as a sum of sinusoidal waves. Although the functions that Fourier was interested in represented the heat of a metal rod, the theory applies to the function representing the displacement of our violin string. Interpreted in this context, Fourier says that any vibration of the violin string can be understood as a superposition of pure tones; all of the varied complexity of notes produced by a violin can be understood in terms of this discrete family of pure tones.

If we plucked our violin string in a vacuum chamber, with no friction or air resistance, then the resulting waves would carry on indefinitely, neither increasing nor decreasing in amplitude. Each pure tone is described by a single number, the frequency of oscillation, which is called the normal frequency. The corresponding mathematical function representing the pure tone is called a normal mode. Of course, this situation is very theoretical.

In reality, there will always be some friction which will cause the oscillations to lose energy, causing the amplitudes to decay exponentially in time. Each pure tone has a specific rate of decay associated with it. In this case, the fundamental mathematical objects are called quasi-normal modes. Quasi-normal modes are described by two numbers, the frequency of oscillation and the rate of exponential decay. These two numbers describe a single complex number called the quasi-normal frequency.

Moving forwards in history again by another hundred years, we reach Einstein’s discovery of the general theory of relativity. General relativity was formulated by Einstein and his contemporaries as an attempt to resolve conflicts between the earlier special theory of relativity and Newton’s theory of universal gravitation. The end result was the hypothesis that four dimensional space-time is curved by the presence of matter. The curvature of space-time in

turn affects the motion of matter. It is this interplay which explains the force of gravity.

General relativity accounted for some hitherto unexplained astronomical observations, most notably in the orbit of Mercury. However, the most profound and unexpected prediction of the theory was the existence of black holes: regions of space-time which are so warped that nothing, not even light, can escape. The first indication of black holes in general relativity came from Karl Schwarzschild in 1915, months after Einstein published his theory.

For many years, black holes were a purely theoretical prediction, and many experts, notably Cambridge astronomer Arthur Eddington, refused to believe that they were anything other than a mathematical oddity. As such, black holes were neglected by physicists for decades.

However, a renaissance began in the 1960s with a flurry of exciting work from the likes of John Archibald Wheeler (who first coined the term black hole), Roger Penrose, and many others. In particular, Penrose provided the first convincing evidence of the physical existence of black holes by showing mathematically that, under very general conditions, black holes would form as the result of gravitational collapse. It was for this work that Penrose won the 2020 Nobel Prize in Physics. Observational evidence of black holes had to wait until 1971, when astronomers Louise Webster and Paul Murdin at the Royal Observatory in Greenwich and Charles Thomas Bolton in Toronto found a binary system consisting of a star and a black hole orbiting each other in our galaxy.

Whenever a black hole is formed, whether by gravitational collapse, or by the merger of two other black holes, the final result will always be either perfectly spherical, or spheroidal if the black hole is spinning. However, in the early stages of formation, the black hole takes on a different shape. As the differences from the final

state die down, the black hole will emit gravitational waves, akin to ripples in space-time itself. Although the existence of gravitational waves had been predicted by Einstein, it was not until the renaissance of the 1960s that serious effort was put into understanding how to observe them.

In the early 1970s, Zerilli derived the equations which govern black hole perturbations. There is a very strong mathematical analogy with the equations which govern the vibrations of a string. Thus it was discovered that black holes also emit gravitational waves at certain pure tones, oscillating at one of a discrete family of frequencies which depends only on the final state, completely independent of the process by which the black hole is formed. Although there is no friction or air resistance to damp gravitational waves, the presence of the black hole, into which energy can fall never to return, causes the same exponentially decaying behaviour that we saw with the violin string.

Astronomers began their first serious attempts to understand how to detect gravitational waves in the 1960s. However, due to technological difficulties and shortage of government funding, it wasn’t until 1994 that the construction of the Laser Interferometer Gravitational-Wave Observatory (LIGO) began. The first observations from LIGO between 2002 and 2010 made no discoveries, but after 2010, upgrades led to an advanced LIGO in 2015. On 11th February 2016, LIGO made the first direct detection of gravitational waves. This was a striking confirmation of the theory which Einstein had first proposed almost exactly a hundred years earlier, one which fundamentally changed the way we understand the universe. In the process, humanity heard the beautiful orchestra of black holes for the very first time. This symphony is one that astronomers will enjoy for years to come

Owain Salter Fitz-Gibbon is a 4th year PhD student in Mathematics at St. Johns College. Artwork by Debbie Ho.