Ernst Mach

How to use this course

Ernst Mach is known to engineers for a number, to psychologists for an illusion, to physicists for a principle he never wrote down, and to philosophers for the view that science describes sensations. These are less separate than they look. Mach was an experimental physicist who treated the observer as part of the experiment, so he studied the eye and the inner ear with the same apparatus and the same care he gave to sound waves and bullets.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations in modern notation and exercises with folded solutions. The order roughly follows his career: the Doppler effect (1860), vision (1865), the sense of motion (1873–75), shock waves (1875–1888), and the critique of mechanics (1883) that shaped Einstein’s thinking.

Timeline

Moravia and Vienna, 1838–1864

Ernst Mach was born on 18 February 1838 in Chirlitz, a village outside Brno in Moravia, then part of the Austrian Empire. The family spoke German; the surname suggests Czech roots. His father, Johann Nepomuk Mach, had studied at the university in Prague and worked as a private tutor. Ernst was taught at home by both parents until he was fourteen. He then spent three years at the gymnasium in Kremsier (Kroměříž).

Mach later told a story about his own intellectual awakening. As a boy of about fifteen he read Kant’s Prolegomena and, some time afterwards, came to feel that the “thing in itself” behind appearances was an idle addition. The world, he decided, was made of colours, sounds, pressures and their regular connections. This view stayed with him for life and shaped every part of his science.

The University of Vienna

In 1855 Mach entered the University of Vienna to study physics. He also took a semester of medical physiology, an unusual choice that prepared him for his later work on the senses. Vienna had a strong physics institute, recently directed by Christian Doppler, who had died in 1853. Mach’s doctoral advisor was Andreas von Ettingshausen, and his 1860 thesis dealt with electrical charges and induction.

The same year he turned to Doppler’s principle, still disputed in Vienna, and demonstrated the acoustic effect with a rotating apparatus (Lesson 04). In 1861 he obtained the habilitation and began lecturing as a Privatdozent, paid only by student fees. To earn a living he gave popular lectures on physics and on the physics of music, and he took up psychophysics after reading Gustav Fechner’s Elemente der Psychophysik (1860). By the time he left Vienna he had published on optics, acoustics and the perception of time intervals.

Graz, 1864–1867

In 1864 Mach declined a chair of surgery at the University of Salzburg and accepted the chair of mathematics at the University of Graz. In 1866 he moved to the chair of physics there. The Graz years were short but productive. Without a well-equipped laboratory, he turned his own eyes and ears into instruments.

In 1865 he reported a striking effect. When a surface is lit so that its brightness rises steadily from one level to another, the eye sees a thin dark line at the foot of the ramp and a bright line at its top, though no such lines exist in the light itself. These Mach bands led him to argue that the retina does not simply record brightness: neighbouring points influence one another (Lesson 05). Around the same time he studied hearing, and the way the ear separates tones, in dialogue with the work of Hermann von Helmholtz.

On 1 August 1867 Mach married Ludovica Marussig in Graz. They had five children: Ludwig (1868), Caroline (1873), Heinrich (1874), Felix (1879) and Viktor (1881). Ludwig became a physician and his father’s closest scientific collaborator.

Prague, 1867–1895

In 1867 Mach became professor of experimental physics at the Charles-Ferdinand University in Prague. He stayed twenty-eight years, the core of his scientific life. In Prague he published about ninety papers and six books. He built apparatus for stroboscopy and high-speed photography, and he continued experiments on sound, light and the senses, often with himself as the subject.

The work of these decades included the explanation of the sense of rotation (1873–75), the first soot-plate records of colliding shock waves (1875), and the photographs of supersonic bullets taken with Peter Salcher (1886). He was elected to the Royal Bohemian Society of Sciences in 1871 and to the Leopoldina in 1873.

Rector during the division

Mach served as rector of the university in 1879–80 and again in 1883–84. In these years the university was divided into a Czech and a German institution, with separate teaching from the winter semester of 1882–83. Mach objected to the way the division was carried out and resigned from the rectorship in protest. He then directed the physics institute of the German university, where many Czech students still attended his lectures. One of his students in Prague was the seismologist Andrija Mohorovičić.

The historical-critical books

Mach’s Prague teaching led him to write histories of physical ideas that were also critiques. Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit (1872) traced the conservation of energy. Die Mechanik in ihrer Entwicklung (1883) examined the foundations of Newtonian mechanics and rejected absolute space (Lessons 09 and 10). Beiträge zur Analyse der Empfindungen (1886) set out his view that physics and psychology study the same elements, sensations, in different connections. These books made him famous far beyond physics.

Testing Doppler’s principle

For the general reader. A siren sounds higher as it approaches and lower as it moves away. Christian Doppler predicted this in 1842, but in the Vienna of Mach’s student years the effect was still contested. Mach, aged 22, settled the matter in the lecture room with a sound source that went round in a circle.

Derivation

Let sound travel at speed \(c\) relative to still air. A source emits frequency \(f_0\) while moving with speed \(v_s\) along the line towards an observer, who moves towards the source with speed \(v_o\). In one period \(T_0=1/f_0\) the source advances \(v_sT_0\), so successive crests in front of it are separated by \(\lambda=(c-v_s)T_0\). The observer meets crests at relative speed \(c+v_o\).

The asymmetry between source and observer is real: the medium picks out a rest frame. For light there is no medium, and only the relative velocity matters, which is one root of special relativity.

The rotating source

Mach’s apparatus carried a sounding source around a circle. A listener in the plane of rotation, some distance away, sees the source’s velocity component along the line of sight swing between \(+v\) and \(-v\) once per revolution, so the pitch rises and falls. A listener on the axis of rotation sees a velocity always perpendicular to the line of sight, and hears a steady pitch. The contrast rules out explanations by loudness or by vibration of the apparatus: only the line-of-sight component of motion matters, exactly as Doppler’s theory says.

Mach then urged astronomers to look for the optical analogue in stellar spectra. His later papers on Doppler’s theory (1874, 1878) returned to the question.

Mach bands and lateral inhibition

For the general reader. Look at a shadow with a soft edge, and you may see a darker line at its inner border and a lighter line at its outer border. The lines are not in the light. They are made by the eye, which sharpens edges by letting each point of the retina suppress its neighbours. Mach described the effect in 1865 and guessed the mechanism nearly a century before it was measured.

The phenomenon

Let the illumination \(I(x)\) be constant at a low level for \(x<0\), rise linearly for \(0<x<a\), and stay constant at a higher level for \(x>a\). Observers see a dark band at \(x=0\) and a bright band at \(x=a\). Where \(I\) is constant or linear no bands appear. Mach concluded that perceived brightness depends on how the illumination curves, and he tied the bands to the second derivative of \(I\). He proposed that each retinal point is excited by its own light and inhibited by the light falling on its surroundings.

The linear term in the Taylor expansion vanishes by symmetry of \(w\), which is why a uniform gradient produces no band. The model is a spatial high-pass filter. In the 1950s Haldan Keffer Hartline and Floyd Ratliff measured exactly this mutual inhibition between receptors in the eye of the horseshoe crab, and Ratliff’s 1965 book Mach Bands tied the physiology back to Mach’s papers, which he translated.

The sense of rotation

For the general reader. Spin on an office chair with your eyes closed, then stop. You feel yourself turning the other way. Mach built a large rotating frame in his Prague laboratory to study such sensations. In 1873 he concluded, at the same time as the physician Josef Breuer and independently of him, that the semicircular canals of the inner ear sense rotation, and that they respond to changes in rotation rather than to rotation itself.

Mach’s findings

Seated in his apparatus, with vision excluded, Mach found that a subject feels the start of a rotation, but during uniform rotation the feeling fades and disappears. When the rotation stops, a rotation in the opposite sense is felt, which also fades. Only angular acceleration is sensed directly. He proposed that the fluid in the semicircular canals lags behind the bony canal when the head starts to turn, and that the relative flow stimulates the sensory cells. He published the full account in Grundlinien der Lehre von den Bewegungsempfindungen (1875). The Scottish chemist Alexander Crum Brown reached similar conclusions in the same period.

A modern model

The model is a descendant of the torsion-pendulum picture of the cupula developed in the twentieth century. It reproduces each of Mach’s observations: fading during uniform rotation, and a reversed after-sensation on stopping.

The apparent vertical

Mach also studied how a rotating subject judges the vertical. In a frame rotating at \(\omega\), a body at radius \(r\) feels gravity \(g\) downwards and centrifugal acceleration \(\omega^2r\) outwards. The otoliths register the resultant, so the apparent vertical tilts outwards by an angle \(\theta\) with \(\tan\theta=\omega^2r/g\).

Shock waves and the Mach cone

For the general reader. A bullet faster than sound outruns its own noise. The disturbances it makes pile up on a cone trailing behind it, a thin layer where pressure jumps abruptly. In 1886 Mach and Peter Salcher, a physicist at the Austro-Hungarian Naval Academy in Fiume (Rijeka), photographed this cone for the first time, using an electric spark triggered by the bullet itself.

The Mach angle

Let a point source move in a straight line at constant speed \(v>c\) through still air. The sound emitted at time \(-t\) is now a sphere of radius \(ct\) centred on a point a distance \(vt\) behind the source. All these spheres are tangent to a cone with its apex at the source.

Proof. The tangent from the apex to the sphere of radius \(ct\) whose centre lies at distance \(vt\) makes an angle \(\mu\) with the line of motion, with \(\sin\mu=ct/vt\), independent of \(t\). So every sphere touches the same cone. The hero illustration draws this construction for \(M=2\).

Photographing the invisible

A shock wave is a thin layer where the density of the air changes abruptly. Density changes the refractive index, so light passing through the layer is bent. Mach and Salcher used the schlieren method of August Toepler, in which a knife edge blocks part of the light focused by a lens, so that small deflections appear as bright or dark regions. The bullet passed between two wires and closed a circuit that fired a spark, giving an exposure short enough to freeze it. Their paper, Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge, appeared in the Sitzungsberichte of the Vienna Academy in 1887. It showed the bow wave ahead of the bullet, the conical wave behind it, and the turbulent wake. In 1888 they extended the work to artillery shells, including tests at Meppen. In the early 1890s Ludwig Mach improved the interferometer used for such studies; the Mach–Zehnder interferometer is named after him and Ludwig Zehnder.

Mach reflection

In 1875 Mach studied colliding shock waves from two electric sparks by letting them sweep across a glass plate coated with soot. Where two shocks met head-on, soot piled up in a thin line. Where they met at a glancing angle, the record showed a V-shaped pattern: the two incident waves and the reflected waves no longer met at the plate, but merged into a third, nearly normal shock, now called the Mach stem. This irregular or Mach reflection occurs whenever the angle of incidence of a shock on a surface exceeds a critical value. It matters for blast waves: an air burst at the right height produces a Mach stem near the ground with roughly double the overpressure.

The Mach number and compressible flow

For the general reader. Air behaves like an incompressible fluid when it moves slowly, but near and above the speed of sound its density changes, and the rules change with it. The single number that measures this is the ratio of the flow speed to the local speed of sound. The Swiss engineer Jakob Ackeret named it after Mach in 1929.

Speed of sound

For small disturbances in a fluid, \(c^2=(\partial p/\partial\rho)_s\), the derivative at constant entropy. For an ideal gas with \(p\propto\rho^\gamma\) along an adiabat and \(p=\rho RT\) (with \(R\) the specific gas constant),

For dry air, \(\gamma\approx1.4\) and \(R\approx287\ \text{J kg}^{-1}\text{K}^{-1}\).

The area relation explains the shape of a rocket nozzle. In subsonic flow (\(M<1\)) a narrowing duct speeds the gas up; in supersonic flow (\(M>1\)) a widening duct does. To go from subsonic to supersonic the duct must first converge, then diverge, with \(M=1\) at the throat. Gustaf de Laval used this in his steam turbines in the 1880s, the decade of Mach’s bullet photographs.

Normal shocks

Mass, force and the critique of Newton

For the general reader. Newton defined mass as quantity of matter, the product of density and volume, which is circular because density is mass per volume. He also assumed an absolute space against which all motion is measured. In Die Mechanik (1883) Mach showed how to define mass from observed accelerations alone, and argued that absolute space has no place in physics.

Mach’s definition of mass

Mach began from an experimental proposition: when two bodies \(A\) and \(B\) interact in isolation, they give each other accelerations in opposite directions, and the ratio of their magnitudes is fixed, independent of the kind of interaction and of the time.

Force is then defined as \(\mathbf F=m\mathbf a\), and the third law becomes a consequence of the definition: \(m_A\mathbf a_A=-m_B\mathbf a_B\). The physical content sits in the two experimental propositions, which could in principle fail. Mach also noted that accelerations must be measured relative to some frame, and he took the fixed stars as the reference.

Newton’s bucket

Newton hung a bucket of water by a twisted rope and let it spin. At first the bucket turns while the water is at rest and flat; later the water turns with the bucket and its surface becomes concave. The concavity tracks the water’s rotation, not its rotation relative to the bucket. Newton concluded that the water rotates relative to absolute space.

Mach drew a different lesson. The experiment shows only that rotation relative to the bucket walls produces no noticeable centrifugal force, while rotation relative to the Earth and the rest of the universe does. He remarked that nobody could say how the experiment would turn out if the walls of the bucket were several leagues thick. For Mach, the rotation that matters is rotation relative to the distant masses of the universe. He did not propose a theory of how those masses act, but he pointed to where one should look.

Mach’s principle and Einstein

For the general reader. If rotation means rotation relative to the distant stars, then perhaps the stars themselves cause the forces we feel when we spin. Einstein took this idea, which he called Mach’s principle, as a guide in building general relativity. General relativity turned out to contain part of it, but not all.

Einstein’s reading

Einstein read Die Mechanik as a student and later said it had a profound influence on him. On 25 June 1913 he wrote to Mach that in the theory he was developing, inertia would arise from an interaction between bodies, and that a heavy shell rotating relative to the fixed stars would drag the plane of a Foucault pendulum inside it. In 1918 Einstein listed three principles for a theory of gravitation and named one of them Mach’s principle: the metric field should be fully determined by the masses of bodies. Mach himself never stated a principle in this form.

Frame dragging inside a shell

This is the effect Einstein had predicted to Mach, now called frame dragging or the Lense–Thirring effect. It is tiny for ordinary bodies but grows with \(GM/Rc^2\). Satellite experiments, including Gravity Probe B, have detected frame dragging by the rotating Earth.

How Machian is general relativity?

Not fully. Einstein’s equations have solutions with no matter at all, such as flat spacetime, in which inertia is perfectly well defined; this violates version 3. Kurt Gödel found a rotating cosmological solution in 1949 in which matter rotates relative to the local inertial frames. On the other hand, in realistic cosmologies the local inertial frames do not rotate relative to the distant galaxies, to high precision, and frame dragging is real. Robert Dicke and Carl Brans proposed in 1961 a scalar–tensor theory in which the gravitational constant is set by the matter of the universe, partly to make gravity more Machian. Physicists still disagree about what exactly the principle should say.

Vienna, philosophy and the atom, 1895–1916

In 1895 Mach returned to Vienna to a chair created for him, in the history and philosophy of the inductive sciences. His lectures drew large audiences. He argued that science aims at the most economical description of the connections among phenomena, and that concepts which cannot be tied to possible experience, such as absolute space or the atom taken as a real thing, should be treated as provisional aids at best.

The dispute over atoms

This brought him into conflict with Ludwig Boltzmann, whose kinetic theory rested on atoms. After a lecture by Boltzmann at the Vienna Academy in 1897, Mach declared that he did not believe atoms exist. The disagreement was intellectual, and both men were respected figures in Vienna; Boltzmann took over lectures on natural philosophy at the university in 1903. In 1908 Max Planck attacked Mach’s philosophy in a lecture at Leiden, and Mach replied sharply in 1910. Within a few years the work of Einstein and Jean Perrin on Brownian motion had persuaded most physicists that atoms are real.

Illness and last years

In 1898 a stroke paralysed Mach’s right side. He continued to write, using his left hand and dictating. He retired from the university in 1901 and was appointed to the Austrian House of Lords, the upper chamber of parliament; he declined ennoblement as unsuitable for a scientist. Erkenntnis und Irrtum (1905) collected his thoughts on the psychology of research. In 1913 he moved to Vaterstetten near Munich, to the home of his son Ludwig. He died there on 19 February 1916, a day after his seventy-eighth birthday. His ashes, and later Ludwig’s, rest at a memorial in Munich’s Nordfriedhof.

Mach’s attitude to relativity at the end of his life is disputed. The preface to his posthumous Die Prinzipien der physikalischen Optik (1921) rejects Einstein’s theory, but some historians argue that Ludwig wrote or altered it.

Legacy

On Mach’s death Einstein wrote an obituary in the Physikalische Zeitschrift, and in later years he repeatedly named Mach, together with Hume, as an important influence on the road to relativity. In 1900 Mach became godfather to Wolfgang Pauli, whose father was a friend. In the 1920s the philosophers and scientists of the Vienna Circle took Mach as a founding figure and named their public society after him.

His philosophy also had a political afterlife. Some Russian Marxists, notably Alexander Bogdanov, adopted his ideas, and in 1908 Lenin attacked them in Materialism and Empirio-criticism. Psychologists remember him for anticipating Gestalt ideas about the perception of form, and the behaviourist B. F. Skinner cited him as a model of inductive science.

Names and honours

• Aerodynamics: the Mach number (named by Ackeret, 1929), Mach angle, Mach cone, Mach reflection and Mach stem, and the “Mach diamonds” in supersonic exhaust plumes.
• Vision: Mach bands, a textbook example of lateral inhibition.
• Optics: the Mach–Zehnder interferometer, after his son Ludwig and Ludwig Zehnder.
• Physiology: the Mach–Breuer theory of the semicircular canals.
• Cosmology: Mach’s principle.
• Other: the lunar crater Mach, the asteroid 3949 Mach, and the Fraunhofer Institute for High-Speed Dynamics in Freiburg, known as the Ernst-Mach-Institut, which holds part of his papers. He was nominated for the Nobel Prize in Physics several times.

Sources and further reading

Biography

1. Ernst Mach, Wikipedia, and Ernst Mach, German Wikipedia (family, honours, grave).
2. J. J. O’Connor and E. F. Robertson, Ernst Mach, MacTutor History of Mathematics.
3. Institute of Theoretical Physics, Charles University, Ernst Mach in Prague.
4. J. T. Blackmore, Ernst Mach: His Work, Life, and Influence, University of California Press, 1972.
5. A. Einstein, Ernst Mach, Physikalische Zeitschrift 17 (1916).
6. Bibliography of Ernst Mach’s writings, University of Washington.

Physics

1. E. Mach, Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt, Leipzig, 1883. English: The Science of Mechanics, Open Court.
2. E. Mach, Grundlinien der Lehre von den Bewegungsempfindungen, Leipzig, 1875.
3. E. Mach, Beiträge zur Analyse der Empfindungen, Jena, 1886.
4. E. Mach and P. Salcher, Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge, Sitzungsber. Akad. Wiss. Wien 95 (Abt. II), 764–780 (1887).
5. F. Ratliff, Mach Bands: Quantitative Studies on Neural Networks in the Retina, Holden-Day, 1965.
6. Mach bands; Mach number; Mach’s principle; Mach–Zehnder interferometer, Wikipedia.
7. The discovery of the Mach reflection effect and its demonstration in an auditorium (abstract, drawing on Mach’s papers at the Ernst-Mach-Institut).
8. J. D. Anderson, Modern Compressible Flow, McGraw-Hill (normal shocks and nozzle flow).
9. J. Barbour and H. Pfister (eds.), Mach’s Principle: From Newton’s Bucket to Quantum Gravity, Birkhäuser, 1995.