Hendrik Antoon Lorentz

How to use this course

Every student of physics meets Lorentz’s name twice before the end of the second year: in the force \(q(\mathbf E+\mathbf v\times\mathbf B)\) on a moving charge, and in the transformation that relates the clocks and rulers of moving observers. Both came out of one programme. Lorentz set out to explain all of optics and electromagnetism by small charged particles moving through a motionless ether. The programme was largely right about electrons and wrong about the ether, and its failures led directly to relativity and the quantum.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Formulas use SI units. Lorentz himself worked in Gaussian-type units with the factor \(4\pi\) placed differently; the physics is the same. The technical lessons follow the order of his work: optics of matter (1878), the electron theory (1892–1895), the Zeeman effect (1896), moving bodies (1892–1904), the electron’s mass (1904), and metals and radiation (1903–1908).

Timeline

Arnhem, 1853–1870

Hendrik Antoon Lorentz was born in Arnhem, in the eastern Netherlands, on 18 July 1853. His father, Gerrit Frederik Lorentz, kept a nursery business; his mother, Geertruida van Ginkel, died in 1861, when he was a small boy. In 1862 his father married Luberta Hupkes, who raised the boy.

He went to Mr Timmer’s primary school and at thirteen entered Arnhem’s new Hogere Burgerschool. These schools, created by an 1863 education law, taught science and modern languages instead of the classics, and they were staffed by unusually able teachers. Lorentz excelled in physics and mathematics and read widely in English, French and German, languages he would later use with ease at international meetings. Because the Burgerschool did not teach Latin and Greek, he had to pass extra examinations in the classical languages before the university would admit him. He passed them in 1870.

Leiden and the thesis, 1870–1877

Lorentz entered Leiden University in 1870. The astronomer Frederik Kaiser influenced him strongly, and he took his candidate degree in mathematics and physics within a year. In 1872 he went back to Arnhem to teach evening classes at a secondary school, and he prepared his doctorate there, largely on his own, from the papers of James Clerk Maxwell, Hermann von Helmholtz and Augustin-Jean Fresnel.

The thesis

In 1875 he received his doctorate under the physicist Pieter Rijke with a dissertation Over de theorie der terugkaatsing en breking van het licht (On the theory of the reflection and refraction of light). Maxwell’s theory was then little accepted on the Continent. Lorentz used it, in a form adapted from Helmholtz, to derive the laws of reflection and refraction at a boundary. The thesis already showed the theme of his career: optics should follow from electromagnetism once one knows how matter responds to an electric field. He was 22.

The first chair of theoretical physics, 1878–1892

In 1877 Leiden created a chair of theoretical physics, among the first in Europe. It was offered first to Johannes Diderik van der Waals, who declined. Lorentz accepted and gave his inaugural lecture on 25 January 1878, aged 24. He would hold the chair for 34 years.

His first major paper as professor, in 1878, connected the refractive index of a substance to the polarizability of its molecules (Lesson 04). The Danish physicist Ludvig Lorenz had found the same relation in 1869, which is why it carries both names. In 1881 Lorentz married Aletta Catharina Kaiser. Their daughter Geertruida later became a physicist, wrote her doctorate under her father, and married the physicist Wander de Haas.

During the 1880s Lorentz published on kinetic theory and thermodynamics as well as optics, and in 1886 on the influence of the Earth’s motion on optical phenomena. The change came in 1892, when he published a long memoir in the Archives Néerlandaises that founded the electron theory. Part II follows that theory from its roots in 1878 to its limits in 1908.

Light in matter: the Lorentz–Lorenz formula

For the general reader. Light slows down in glass or water. Lorentz explained this by supposing that matter is full of tiny charges bound to molecules, which the light wave sets oscillating. Each oscillating charge sends out its own small wave, and the sum of all these waves is a wave that travels more slowly. The amount of slowing depends on how easily the charges are pushed and on how many there are.

The local field

In a dielectric with polarization \(\mathbf P\) (dipole moment per unit volume), the field acting on one molecule is not the average field \(\mathbf E\). Lorentz cut a small sphere around the molecule: outside it the medium is continuous, inside it the neighbouring dipoles are treated individually. For a cubic lattice or a disordered isotropic medium the near dipoles contribute nothing on average. Removing the uniformly polarized sphere from the continuum adds back the field \(\mathbf P/3\varepsilon_0\).

Derivation

We have \(\mathbf P=N\alpha\,(\mathbf E+\mathbf P/3\varepsilon_0)\). Write \(x=N\alpha/\varepsilon_0\). Solving, \(\mathbf P=\varepsilon_0\,\dfrac{x}{1-x/3}\,\mathbf E\), so \(\varepsilon_r-1=\dfrac{x}{1-x/3}\) and \(\varepsilon_r+2=\dfrac{3}{1-x/3}\). Dividing gives \((\varepsilon_r-1)/(\varepsilon_r+2)=x/3\). At optical frequencies \(\varepsilon_r=n^2\). The same relation for static fields is the Clausius–Mossotti formula. Its practical content is that the molar refraction \(\frac{n^2-1}{n^2+2}\,\frac{M}{\rho}\) is nearly independent of density, which Lorentz checked against data for liquids and their vapours.

The Lorentz oscillator

Let a charge \(-e\) of mass \(m\) be bound with natural frequency \(\omega_0\) and damping \(\gamma\). A field \(E_0e^{-i\omega t}\) drives \(m\ddot x=-m\omega_0^2x-m\gamma\dot x-eE\), giving \(p=-ex=\alpha(\omega)E\) with

For a dilute gas, \(n^2-1\approx N\alpha/\varepsilon_0\). The index rises toward an absorption line (normal dispersion), the imaginary part of \(\alpha\) gives absorption, and on the far side of the line the index falls (anomalous dispersion). This model, still taught under Lorentz’s name, gives the shape of resonance lines that physicists call Lorentzian.

The electron theory and the Lorentz force

For the general reader. Maxwell’s theory described electricity as a strained state of a medium and was vague about what charge actually is. Lorentz made a clean division. Empty space, the ether, carries electric and magnetic fields and never moves. Matter contains small charged particles, which he called ions and later electrons. Fields push on charges, and moving charges create fields. Every electrical and optical property of matter was to follow from this one picture.

The fundamental equations

In the 1892 memoir and the 1895 Versuch, Lorentz wrote the field equations for the ether containing charge of density \(\rho\) moving with velocity \(\mathbf v\). In modern SI form:

The magnetic part of the force on a moving charge had been obtained by J. J. Thomson in 1881, with an erroneous factor of one half, and corrected by Oliver Heaviside in 1889. Lorentz’s 1895 Versuch gave the complete law in the form above, as one of the postulates of a single theory. Macroscopic quantities such as \(\mathbf D\), \(\mathbf H\), the dielectric constant and the conductivity were to be obtained by averaging over the electrons. This is the procedure still used in textbooks to derive macroscopic electrodynamics.

Consistency

Taking the divergence of the last field equation gives \(0=\mu_0\nabla\cdot(\rho\mathbf v)+\mu_0\varepsilon_0\partial_t\nabla\cdot\mathbf E\), that is \(\partial_t\rho+\nabla\cdot(\rho\mathbf v)=0\): charge is conserved. Dotting the force density with \(\mathbf v\) gives the power delivered to the charges, \(\mathbf v\cdot\mathbf f=\rho\mathbf v\cdot\mathbf E\), and the field equations give Poynting’s theorem

The ether in Lorentz’s theory is absolutely at rest and has no mechanical properties of its own. One consequence troubled him: the force of the field on matter has no reaction force on the ether, so Newton’s third law fails for matter alone. The missing momentum is carried by the field, with density \(\varepsilon_0\mathbf E\times\mathbf B\).

The Zeeman effect

For the general reader. In 1896 Pieter Zeeman, a former student of Lorentz, placed a flame between the poles of an electromagnet and saw its spectral lines widen. Lorentz explained at once what should happen: a single line should split into three, with definite polarizations. Zeeman then saw the predicted pattern. The measured splitting showed that the oscillating particles inside atoms were negatively charged and very light, a year before J. J. Thomson identified the electron in cathode rays.

Derivation

The equation of motion is \(m\ddot{\mathbf r}=-m\omega_0^2\mathbf r-e\,\dot{\mathbf r}\times\mathbf B\). The \(z\)-motion is unaffected. In components, \(m\ddot x=-m\omega_0^2x-eB\dot y\) and \(m\ddot y=-m\omega_0^2y+eB\dot x\). With \(u=x+iy\),

Trying \(u=e^{-i\omega t}\) gives \(m\omega^2+eB\omega-m\omega_0^2=0\), whose roots are \(\omega\approx\pm\omega_0-eB/2m\). The two signs are the two senses of rotation, with angular frequencies \(\omega_0\mp eB/2m\). The quantity \(eB/2m\) is the Larmor frequency. A dipole does not radiate along its own axis, which explains why the central component disappears when the light is viewed along the field. The sense of circular polarization of the higher-frequency line tells the sign of the charge: Zeeman’s observation showed it was negative.

Zeeman’s measurement gave \(e/m\) of order \(10^{11}\ \mathrm{C\,kg^{-1}}\), more than a thousand times larger than for the hydrogen ion in electrolysis. Lorentz and Zeeman shared the 1902 Nobel Prize in Physics in recognition of their research into the influence of magnetism upon radiation phenomena. Many lines split into more than three components (the anomalous Zeeman effect); explaining those required electron spin and quantum mechanics, which arrived in 1925–1927.

Moving bodies, ether drift and contraction

For the general reader. If light is a wave in a motionless ether, then the Earth, moving around the Sun at 30 km/s, should feel an ether wind. Experiment after experiment failed to detect it. Lorentz showed that his theory explains why small effects of the wind cancel. When the very precise experiment of Albert Michelson and Edward Morley also found nothing, he proposed a bold addition: bodies moving through the ether shrink slightly along their motion.

The Michelson–Morley null result

An interferometer arm of length \(L\) parallel to the ether wind \(v\) takes light a round-trip time \(t_\parallel=\frac{L}{c-v}+\frac{L}{c+v}=\frac{2L}{c}\frac{1}{1-v^2/c^2}\). A perpendicular arm takes \(t_\perp=\frac{2L}{c}\frac{1}{\sqrt{1-v^2/c^2}}\). The difference, about \(\frac{L}{c}\frac{v^2}{c^2}\), is of second order in \(v/c\). In 1887 Michelson and Morley found no shift, to well below the expected value.

George FitzGerald had stated the idea qualitatively in a short letter of 1889; Lorentz reached it independently in 1892 and gave it a reason. If the forces that hold molecules together transform like electromagnetic forces when a body moves, the equilibrium shape of the body must change in exactly this way.

Local time and corresponding states (1895)

In the Versuch, Lorentz introduced, for a system moving with velocity \(\mathbf v\) through the ether, the coordinates \(\mathbf x_r=\mathbf x-\mathbf vt\) and the local time

For Lorentz, \(t'\) was a mathematical auxiliary. The true time was \(t\). In 1900 Poincaré pointed out that local time is exactly what moving observers would read on clocks synchronized by light signals, assuming light speed \(c\) in their own frame.

Fresnel drag

In 1818 Fresnel had proposed that a moving transparent body drags light only partially, with coefficient \(1-1/n^2\), and Hippolyte Fizeau confirmed this with flowing water in 1851. Lorentz derived the coefficient from the electron theory: the electrons in the moving body re-emit light in local time, not ether time. In modern terms, it follows from the relativistic addition of velocities (Exercise 7.2).

The Lorentz transformation

For the general reader. To explain why no experiment of any accuracy could detect the ether wind, Lorentz extended his first-order theorem to all orders. The result, in 1899 and 1904, was a set of formulas that relate the positions and times assigned to events by a moving system to those of the ether. Einstein derived the same formulas in 1905 from two simple principles and drew the conclusion Lorentz had avoided: there is no privileged rest frame, and time itself is relative.

Lorentz’s 1904 form

Lorentz worked from Galilean relative coordinates \(x_r=x-vt\). His 1904 paper used \(x'=\gamma l\,x_r\), \(y'=l\,y\), \(z'=l\,z\), \(t'=\frac{l}{\gamma}t-\gamma l\frac{v}{c^2}x_r\), with a factor \(l\) that he fixed to 1 by an argument about the electron. Substituting \(x_r=x-vt\) into \(t'\) with \(l=1\) gives \(\frac{t}{\gamma}+\frac{\gamma v^2t}{c^2}-\frac{\gamma vx}{c^2}\). Since \(\frac1\gamma+\frac{\gamma v^2}{c^2}=\gamma\left(1-\frac{v^2}{c^2}+\frac{v^2}{c^2}\right)=\gamma\), this is \(\gamma(t-vx/c^2)\). The two forms agree. Lorentz assumed that all forces, including molecular forces and those inside the electron, transform like electromagnetic ones, and that electrons contract.

Proof of the invariances

Write \(ct=\tau\) and \(\beta=v/c\). Then \(x'=\gamma(x-\beta\tau)\), \(\tau'=\gamma(\tau-\beta x)\), and

The transformation is a hyperbolic rotation: with \(\beta=\tanh\phi\), \(\gamma=\cosh\phi\). The chain rule gives \(\partial_\tau=\gamma(\partial_{\tau'}-\beta\partial_{x'})\) and \(\partial_x=\gamma(\partial_{x'}-\beta\partial_{\tau'})\), so \(\partial_\tau^2-\partial_x^2=\partial_{\tau'}^2-\partial_{x'}^2\) by the same algebra.

Three readings of one formula

In June 1905 Poincaré corrected Lorentz’s transformation of charge density and current, showed that the transformations form a group, and named them after Lorentz. Einstein’s paper of 1905 derived them from the relativity principle and the constancy of the speed of light, without reference to the ether or the structure of electrons. In 1908 Hermann Minkowski recast them as rotations of four-dimensional spacetime. Lorentz admired Einstein’s approach and taught it, but for years he kept a preference for a real ether and a true time, which he regarded as a matter of taste rather than of experiment.

The mass of the electron

For the general reader. A charged particle carries its electric field along when it moves, and that field stores energy and momentum. Part, or perhaps all, of the electron’s inertia might therefore be electromagnetic. Around 1900 this was a live and exciting hypothesis, and different models of the electron predicted different laws for how its mass should grow with speed. Lorentz’s contracting electron gave the law that experiment eventually confirmed, the same law that follows from Einstein’s relativity.

Electromagnetic mass

For a spherical shell of radius \(a\) and charge \(e\) moving slowly with velocity \(\mathbf v\), the field momentum \(\varepsilon_0\int\mathbf E\times\mathbf B\,dV\) equals

The factor \(4/3\), instead of the 1 suggested by \(E=mc^2\), signals that a purely electromagnetic electron is incomplete. Max Abraham noted in 1904 that a charged shell blows itself apart, and Poincaré in 1905 added non-electromagnetic cohesive stresses that restore both stability and the factor 1.

Derivation

Differentiate \(\mathbf p=m_0\gamma\mathbf v\) with \(d\gamma/dt=\gamma^3\,\mathbf v\cdot\dot{\mathbf v}/c^2\):

If \(\dot{\mathbf v}\parallel\mathbf v\), the bracket gives \(m_0\gamma(1+\gamma^2v^2/c^2)\dot{\mathbf v}=m_0\gamma^3\dot{\mathbf v}\). If \(\dot{\mathbf v}\perp\mathbf v\), only \(m_0\gamma\dot{\mathbf v}\) remains. Abraham’s rigid spherical electron gave different, more complicated functions of \(v\). Walter Kaufmann’s deflection experiments on fast beta-ray electrons (1901–1906) at first seemed to favour Abraham, and Lorentz was briefly ready to abandon his theory. Later experiments, beginning with Alfred Bucherer’s in 1908, supported the Lorentz–Einstein formula, and particle accelerators confirm it every day.

Electrons in metals and heat radiation

For the general reader. Metals conduct electricity and heat well, and both properties were explained around 1900 by a gas of free electrons moving among the atoms. Lorentz gave the first careful statistical treatment of this gas in 1905, using the methods of Ludwig Boltzmann. He then turned the same electrons into a tool for the most important open problem of the day, the spectrum of heat radiation, and found that classical physics gives an absurd answer at short wavelengths.

The Lorentz gas

Paul Drude had proposed in 1900 that conduction electrons behave like a classical gas, colliding with the metal ions. In 1905 Lorentz treated the problem rigorously. He took light point particles scattering elastically from fixed heavy spheres, with no collisions among the light particles, and solved the Boltzmann transport equation for their distribution in weak fields and temperature gradients. This model, now called the Lorentz gas, is a standard object in kinetic theory and in the mathematical theory of chaotic billiards.

The proportionality to \(T\) and the universality of the ratio among metals were strong evidence for the electron theory. The trouble was the heat capacity: a classical electron gas should add \(\tfrac32k_B\) per electron to the specific heat of a metal, which was not observed. The resolution waited for the Pauli principle and Arnold Sommerfeld’s theory of 1927.

Radiation: 1903 and 1908

In 1903 Lorentz calculated the long-wavelength radiation emitted and absorbed by the free electrons of a metal and obtained, by Kirchhoff’s law, the spectrum of black-body radiation in that limit. His result agreed with the long-wavelength limit of Max Planck’s 1900 law. In April 1908, at the International Congress of Mathematicians in Rome, he applied Gibbs’s statistical mechanics to matter and ether together and showed that equipartition forces the Rayleigh–Jeans law at all wavelengths. He concluded that this contradicts experiment. Soon after he accepted that Planck’s energy elements were required, a turning point in the acceptance of quantum theory.

Haarlem, Solvay and the Zuiderzee, 1911–1926

By 1910 Lorentz was the most respected theoretical physicist in Europe. He spoke the main languages fluently, listened carefully and summarized debates fairly, and younger physicists trusted his judgement. In autumn 1911 the industrialist Ernest Solvay convened the first Solvay Conference in Brussels on radiation and the quanta, and Lorentz chaired it. He went on to chair the later Solvay physics conferences up to the fifth, in October 1927, the meeting at which Niels Bohr and Einstein debated the meaning of quantum mechanics.

In 1912 he gave up his Leiden chair to become curator of the physical cabinet at Teylers Museum in Haarlem, which left him free for research. He first asked Einstein to succeed him at Leiden. Einstein declined, and Lorentz chose Paul Ehrenfest. Lorentz kept a position as extraordinary professor at Leiden and continued his weekly Monday-morning lectures there. His doctoral students over the years included Leonard Ornstein, Adriaan Fokker, Johannes Droste, Hendrika van Leeuwen and his daughter Geertruida de Haas-Lorentz. Droste found, independently of Karl Schwarzschild, the exact field of a point mass in general relativity, and van Leeuwen proved that classical statistical mechanics allows no magnetism. Lorentz himself wrote papers on the variational formulation of Einstein’s general relativity between 1915 and 1917.

The Zuiderzee committee

In 1918 the Dutch government decided to close off the Zuiderzee, a large inlet of the North Sea, with a dam. Engineers feared that the dam would raise storm-tide levels along the Wadden Sea coast. Lorentz chaired the state committee formed to predict the effect, from 1918 to 1926. He and his colleagues built a mathematical model of tidal flow in the network of channels and computed the changes in tide heights. Their predictions guided the design and were later broadly confirmed when the Afsluitdijk was completed in 1932. One set of its locks is named the Lorentzsluizen in his honour. The work is an early example of computational hydraulic engineering.

Statesman of science, death and legacy, 1919–1928

After the First World War, Lorentz worked to restore international scientific relations, which were divided by the exclusion of German scientists from the new international bodies. In the Netherlands he led the committee whose work led to the national applied research organization TNO. In 1925 he succeeded the philosopher Henri Bergson as chairman of the League of Nations International Committee on Intellectual Cooperation, a forerunner of UNESCO, of which Einstein and Marie Curie were also members.

He remained scientifically active. In the autumn of 1926 he lectured at Cornell University on the new quantum mechanics, including Erwin Schrödinger’s wave mechanics, which he had studied and discussed with Schrödinger by letter. In January 1928 he fell seriously ill, and he died in Haarlem on 4 February 1928, aged 74. His funeral took place at noon on Friday 10 February. The Dutch state telegraph and telephone services were suspended for three minutes as a mark of respect. Ernest Rutherford spoke at the graveside on behalf of the Royal Society, and Einstein attended.

Legacy

• Electrodynamics: the separation of field and charges, the Lorentz force and the averaging procedure from microscopic to macroscopic equations remain the backbone of classical electromagnetism.
• Relativity: the Lorentz transformation, Lorentz group and Lorentz invariance are the structure of special relativity and of every relativistic quantum field theory.
• Atomic physics: the Zeeman effect and the oscillator model of dispersion gave the first quantitative picture of electrons in atoms.
• Statistical physics: the Lorentz gas and his radiation analysis were central to the crisis that produced the quantum theory.
• Honours: Nobel Prize (1902), Rumford Medal (1908), Franklin Medal (1917), Copley Medal (1918). The Royal Netherlands Academy awards the Lorentz Medal in theoretical physics, first given to Planck in 1927. Leiden’s Lorentz Institute for theoretical physics bears his name.

Connections

People linked to Hendrik Antoon Lorentz across this collection. A name without a link has no page here yet. See Hendrik Antoon Lorentz in the network of all 337 lives →

Teachers and advisors

• Pieter Rijke 1875

Students

• Adriaan Fokker
• Hendrika van Leeuwen 1919
• Johannes Droste
• Leonard Ornstein

Influenced by

• Christian Doppler
• Heinrich Hertz
• Hippolyte Fizeau
• James Clerk Maxwell
• Ludwig Boltzmann
• Oliver Heaviside
• Frederik Kaiser 1870-1872

Influenced

• Albert Einstein
• Hermann Minkowski
• Paul Langevin

Colleagues and collaborators

• Albert A. Michelson
• Heike Kamerlingh Onnes 1882-1912
• Henri Poincaré
• Johannes Diderik van der Waals
• Max Planck
• Paul Ehrenfest 1912-1928
• Wilhelm Wien
• George FitzGerald
• Ludvig Lorenz
• Pieter Zeeman 1896-1902

Rivals

• Max Abraham

Friends

• Albert Einstein 1909-1928

Family

• Geertruida de Haas-Lorentz
• Wander de Haas

Also mentioned in this life

• André-Marie Ampère
• Niels Bohr
• Rudolf Clausius
• Enrico Fermi
• Benjamin Franklin
• Augustin-Jean Fresnel
• Josiah Willard Gibbs
• Hermann von Helmholtz
• J. J. Thomson
• Gustav Kirchhoff
• Marie Curie
• Isaac Newton
• Wolfgang Pauli
• Lord Rayleigh
• Count Rumford
• Ernest Rutherford
• Erwin Schrödinger
• Arnold Sommerfeld

Fields

• theoretical physics 11 lives
• electromagnetism 12 lives
• optics 38 lives
• special relativity 3 lives
• statistical physics 3 lives

On CoursesHub

• Lorentz Transformations · Special Relativity
• Classical Electrodynamics
• Optics & Photonics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Hendrik Antoon Lorentz, MacTutor History of Mathematics.
2. Hendrik Lorentz, Wikipedia (with references).
3. H. A. Lorentz, biographical, NobelPrize.org.
4. G. L. de Haas-Lorentz (ed.), H. A. Lorentz: Impressions of his Life and Work, North-Holland, 1957.
5. A. J. Kox, Hendrik Antoon Lorentz’s struggle with quantum theory, Archive for History of Exact Sciences 67 (2013). PDF.

Physics

1. H. A. Lorentz, La théorie électromagnétique de Maxwell et son application aux corps mouvants, Archives Néerlandaises 25 (1892) 363–552.
2. H. A. Lorentz, Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern, Brill, Leiden, 1895.
3. H. A. Lorentz, Electromagnetic phenomena in a system moving with any velocity smaller than that of light, Proc. Royal Netherlands Academy 6 (1904) 809–831. Wikisource text.
4. H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat, Teubner, Leipzig, 1909.
5. Lorentz ether theory, Wikipedia, for the sequence 1892–1905.
6. O. Darrigol, Electrodynamics from Ampère to Einstein, Oxford University Press, 2000.
7. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999, chapters 4, 7, 11 and 16.
8. N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976, chapters 1–2 (Drude model and Wiedemann–Franz law).