The Music of Black Holes Owain Salter Fitz-Gibbon discusses how sound waves from a violin resemble gravitational waves emitted from black holes P icture a violin string . When it is plucked, it vibrates and disturbs the air around it, causing sound waves to travel to your ears. The result may be a rich and pleasant sound, maybe even the start of your favourite symphony. The story of how a few vibrating strings can lead to such a rich diversity of phenomena as Bach’s violin concertos and how this is connected to the modern science of gravitational waves is a long one, going back at least as far as the ancient Greeks. 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
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The Music of Black Holes
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
Lent 2021
