If you are a fan of the Doppler effect, then time trials at the Indy 500 Speedway will floor you. Even if you have experienced the fall in pitch of a passing train whistle while stopped in your car at a railroad crossing, or heard the falling whine of a jet passing overhead, I can guarantee that you have never heard anything like an Indy car passing you by at 225 miles an hour.
Indy 500 Time Trials and the Doppler Effect
The Indy 500 time trials are the best way to experience the effect, rather than on race day when there is so much crowd noise and the overlapping sounds of all the cars. During the week before the race, the cars go out on the track, one by one, in time trials to decide the starting order in the pack on race day. Fans are allowed to wander around the entire complex, so you can get right up to the fence at track level on the straight-away. The cars go by only thirty feet away, so they are coming almost straight at you as they approach and straight away from you as they leave. The whine of the car as it approaches is 43% higher than when it is standing still, and it drops to 33% lower than the standing frequency—a ratio almost approaching a factor of two. And they go past so fast, it is almost a step function, going from a steady high note to a steady low note in less than a second. That is the Doppler effect!
But as obvious as the acoustic Doppler effect is to us today, it was far from obvious when it was proposed in 1842 by Christian Doppler at a time when trains, the fastest mode of transport at the time, ran at 20 miles per hour or less. In fact, Doppler’s theory generated so much controversy that the Academy of Sciences of Vienna held a trial in 1853 to decide its merit—and Doppler lost! For the surprising story of Doppler and the fate of his discovery, see my Physics Today article.
From that fraught beginning, the effect has expanded in such importance, that today it is a daily part of our lives. From Doppler weather radar, to speed traps on the highway, to ultrasound images of babies—Doppler is everywhere.
Development of the Doppler-Fizeau Effect
When Doppler proposed the shift in color of the light from stars in 1842 , depending on their motion towards or away from us, he may have been inspired by his walk to work every morning, watching the ripples on the surface of the Vltava River in Prague as the water slipped by the bridge piers. The drawings in his early papers look reminiscently like the patterns you see with compressed ripples on the upstream side of the pier and stretched out on the downstream side. Taking this principle to the night sky, Doppler envisioned that binary stars, where one companion was blue and the other was red, was caused by their relative motion. He could not have known at that time that typical binary star speeds were too small to cause this effect, but his principle was far more general, applying to all wave phenomena.
Six years later in 1848 , the French physicist Armand Hippolyte Fizeau, soon to be famous for making the first direct measurement of the speed of light, proposed the same principle, unaware of Doppler’s publications in German. As Fizeau was preparing his famous measurement, he originally worked with a spinning mirror (he would ultimately use a toothed wheel instead) and was thinking about what effect the moving mirror might have on the reflected light. He considered the effect of star motion on starlight, just as Doppler had, but realized that it was more likely that the speed of the star would affect the locations of the spectral lines rather than change the color. This is in fact the correct argument, because a Doppler shift on the black-body spectrum of a white or yellow star shifts a bit of the infrared into the visible red portion, while shifting a bit of the ultraviolet out of the visible, so that the overall color of the star remains the same, but Fraunhofer lines would shift in the process. Because of the independent development of the phenomenon by both Doppler and Fizeau, and because Fizeau was a bit clearer in the consequences, the effect is more accurately called the Doppler-Fizeau Effect, and in France sometimes only as the Fizeau Effect. Here in the US, we tend to forget the contributions of Fizeau, and it is all Doppler.
Doppler and Exoplanet Discovery
It is fitting that many of today’s applications of the Doppler effect are in astronomy. His original idea on binary star colors was wrong, but his idea that relative motion changes frequencies was right, and it has become one of the most powerful astrometric techniques in astronomy today. One of its important recent applications was in the discovery of extrasolar planets orbiting distant stars.
When a large planet like Jupiter orbits a star, the center of mass of the two-body system remains at a constant point, but the individual centers of mass of the planet and the star both orbit the common point. This makes it look like the star has a wobble, first moving towards our viewpoint on Earth, then moving away. Because of this relative motion of the star, the light can appear blueshifted caused by the Doppler effect, then redshifted with a set periodicity. This was observed by Queloz and Mayer in 1995 for the star 51 Pegasi, which represented the first detection of an exoplanet . The duo won the Nobel Prize in 2019 for the discovery.
Doppler and Vera Rubins’ Galaxy Velocity Curves
In the late 1960’s and early 1970’s Vera Rubin at the Carnegie Institution of Washington used newly developed spectrographs to use the Doppler effect to study the speeds of ionized hydrogen gas surrounding massive stars in individual galaxies . From simple Newtonian dynamics it is well understood that the speed of stars as a function of distance from the galactic center should increase with increasing distance up to the average radius of the galaxy, and then should decrease at larger distances. This trend in speed as a function of radius is called a rotation curve. As Rubin constructed the rotation curves for many galaxies, the increase of speed with increasing radius at small radii emerged as a clear trend, but the stars farther out in the galaxies were all moving far too fast. In fact, they are moving so fast that they exceeded escape velocity and should have flown off into space long ago. This disturbing pattern was repeated consistently in one rotation curve after another for many galaxies.
A simple fix to the problem of the rotation curves is to assume that there is significant mass present in every galaxy that is not observable either as luminous matter or as interstellar dust. In other words, there is unobserved matter, dark matter, in all galaxies that keeps all their stars gravitationally bound. Estimates of the amount of dark matter needed to fix the velocity curves is about five times as much dark matter as observable matter. In short, 80% of the mass of a galaxy is not normal. It is neither a perturbation nor an artifact, but something fundamental and large. The discovery of the rotation curve anomaly by Rubin using the Doppler effect stands as one of the strongest evidence for the existence of dark matter.
There is so much dark matter in the Universe that it must have a major effect on the overall curvature of space-time according to Einstein’s field equations. One of the best probes of the large-scale structure of the Universe is the afterglow of the Big Bang, known as the cosmic microwave background (CMB).
Doppler and the Big Bang
The Big Bang was astronomically hot, but as the Universe expanded it cooled. About 380,000 years after the Big Bang, the Universe cooled sufficiently that the electron-proton plasma that filled space at that time condensed into hydrogen. Plasma is charged and opaque to photons, while hydrogen is neutral and transparent. Therefore, when the hydrogen condensed, the thermal photons suddenly flew free and have continued unimpeded, continuing to cool. Today the thermal glow has reached about three degrees above absolute zero. Photons in thermal equilibrium with this low temperature have an average wavelength of a few millimeters corresponding to microwave frequencies, which is why the afterglow of the Big Bang got its name: the Cosmic Microwave Background (CMB).
Not surprisingly, the CMB has no preferred reference frame, because every point in space is expanding relative to every other point in space. In other words, space itself is expanding. Yet soon after the CMB was discovered by Arno Penzias and Robert Wilson (for which they were awarded the Nobel Prize in Physics in 1978), an anisotropy was discovered in the background that had a dipole symmetry caused by the Doppler effect as the Solar System moves at 368±2 km/sec relative to the rest frame of the CMB. Our direction is towards galactic longitude 263.85o and latitude 48.25o, or a bit southwest of Virgo. Interestingly, the local group of about 100 galaxies, of which the Milky Way and Andromeda are the largest members, is moving at 627±22 km/sec in the direction of galactic longitude 276o and latitude 30o. Therefore, it seems like we are a bit slack in our speed compared to the rest of the local group. This is in part because we are being pulled towards Andromeda in roughly the opposite direction, but also because of the speed of the solar system in our Galaxy.
Aside from the dipole anisotropy, the CMB is amazingly uniform when viewed from any direction in space, but not perfectly uniform. At the level of 0.005 percent, there are variations in the temperature depending on the location on the sky. These fluctuations in background temperature are called the CMB anisotropy, and they help interpret current models of the Universe. For instance, the average angular size of the fluctuations is related to the overall curvature of the Universe. This is because, in the early Universe, not all parts of it were in communication with each other. This set an original spatial size to thermal discrepancies. As the Universe continued to expand, the size of the regional variations expanded with it, and the sizes observed today would appear larger or smaller, depending on how the universe is curved. Therefore, to measure the energy density of the Universe, and hence to find its curvature, required measurements of the CMB temperature that were accurate to better than a part in 10,000.
Equivalently, parts of the early universe had greater mass density than others, causing the gravitational infall of matter towards these regions. Then, through the Doppler effect, light emitted (or scattered) by matter moving towards these regions contributes to the anisotropy. They contribute what are known as “Doppler peaks” in the spatial frequency spectrum of the CMB anisotropy.
The examples discussed in this blog (exoplanet discovery, galaxy rotation curves, and cosmic background) are just a small sampling of the many ways that the Doppler effect is used in Astronomy. But clearly, Doppler has played a key role in the long history of the universe.
 C. A. DOPPLER, “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (About the coloured light of the binary stars and some other stars of the heavens),” Proceedings of the Royal Bohemian Society of Sciences, vol. V, no. 2, pp. 465–482, (Reissued 1903) (1842)
 H. Fizeau, “Acoustique et optique,” presented at the Société Philomathique de Paris, Paris, 1848.
 M. Mayor and D. Queloz, “A JUPITER-MASS COMPANION TO A SOLAR-TYPE STAR,” Nature, vol. 378, no. 6555, pp. 355-359, Nov (1995)
 Rubin, Vera; Ford, Jr., W. Kent (1970). “Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions”. The Astrophysical Journal. 159: 379
One of the most important conclusions from chaos theory is that not all random-looking processes are actually random. In deterministic chaos, structures such as strange attractors are not random at all but are fractal structures determined uniquely by the dynamics. But sometimes, in nature, processes really are random, or at least have to be treated as such because of their complexity. Brownian motion is a perfect example of this. At the microscopic level, the jostling of the Brownian particle can be understood in terms of deterministic momentum transfers from liquid atoms to the particle. But there are so many liquid particles that their individual influences cannot be directly predicted. In this situation, it is more fruitful to view the atomic collisions as a stochastic process with well-defined physical parameters and then study the problem statistically. This is what Einstein did in his famous 1905 paper that explained the statistical physics of Brownian motion.
Then there is the middle ground between deterministic mechanics and stochastic mechanics, where complex dynamics gains a stochastic component. This is what Paul Langevin did in 1908 when he generalized Einstein.
Paul Langevin (1872 – 1946) had the fortune to stand at the cross-roads of modern physics, making key contributions, while serving as a commentator expanding on the works of the giants like Einstein and Lorentz and Bohr. He was educated at the École Normale Supérieure and at the Sorbonne with a year in Cambridge studying with J. J. Thompson. At the Sorbonne he worked in the laboratory of Jean Perrin (1870 – 1942) who received the Nobel Prize in 1926 for the experimental work on Brownian motion that had set the stage for Einstein’s crucial analysis of the problem confirming the atomic nature of matter.
Langevin received his PhD in 1902 on the topic of x-ray ionization of gases and was appointed as a lecturer at the College de France to substitute in for Éleuthère Mascart (who was an influential French physicist in optics). In 1905 Langevin published several papers that delved into the problems of Lorentz contraction, coming very close to expressing the principles of relativity. This work later led Einstein to say that, had he delayed publishing his own 1905 paper on the principles of relativity, then Langevin might have gotten there first .
Also in 1905, Langevin published his most influential work, providing the theoretical foundations for the physics of paramagnetism and diamagnetism. He was working closely with Pierre Curie whose experimental work on magnetism had established the central temperature dependence of the phenomena. Langevin used the new molecular model of matter to derive the temperature dependence as well as the functional dependence on magnetic field. One surprising result was that only the valence electrons, moving relativistically, were needed to contribute to the molecular magnetic moment. This later became one of the motivations for Bohr’s model of multi-electron atoms.
Langevin suffered personal tragedy during World War II when the Vichy government arrested him because of his outspoken opposition to fascism. He was imprisoned and eventually released to house arrest. In 1942, his son-in-law was executed by the Nazis, and in 1943 his daughter was sent to Auschwitz. Fearing for his own life, Langevin escaped to Switzerland. He returned shortly after the liberation of Paris and was joined after the end of the war by his daughter who had survived Auschwitz and later served in the Assemblée Consultative as a communist member. Langevin passed away in 1946 and received a national funeral. His remains lie today in the Pantheon.
The Langevin Equation
In 1908, Langevin realized that Einstein’s 1905 theory on Brownian motion could be simplified while at the same time generalized. Langevin introduced a new quantity into theoretical physics—the stochastic force . With this new theoretical tool, he was able to work with diffusing particles in momentum space as dynamical objects with inertia buffeted by random forces, providing a Newtonian formulation for short-time effects that were averaged out and lost in Einstein’s approach.
Stochastic processes are understood by considering a dynamical flow that includes a random function. The resulting set of equations are called the Langevin equation, namely
where fa is a set of N regular functions, and σa is the standard deviation of the a-th process out of N. The stochastic functions ξa are in general non-differentiable but are integrable. They have zero mean, and no temporal correlations. The solution is an N-dimensional trajectory that has properties of a random walk superposed on the dynamics of the underlying mathematical flow.
As an example, take the case of a particle moving in a one-dimensional potential, subject to drag and to an additional stochastic force
where γ is the drag coefficient, U is a potential function and B is the velocity diffusion coefficient. The second term in the bottom equation is the classical force from a potential function, while the third term is the stochastic force. The crucial point is that the stochastic force causes jumps in velocity that integrate into displacements, creating a random walk superposed on the deterministic mechanics.
Random Walk in a Harmonic Potential
Diffusion of a particle in a weak harmonic potential is equivalent to a mass on a weak spring in a thermal bath. For short times, the particle motion looks like a random walk, but for long times, the mean-squared displacement must satisfy the equipartition relation
The Langevin equation is the starting point of motion under a stochastic force F’
where the second equation has been multiplied through by x. For a spherical particle of radius a, the viscous drag factor is
and η is the viscosity. The term on the left of the dynamical equation can be rewritten to give
It is then necessary to take averages. The last term on the right vanishes because of the random signs of xF’. However, the buffeting from the random force can be viewed as arising from an effective temperature. Then from equipartition on the velocity
Making the substitution y = <x2> gives
which is the dynamical equation for a particle in a harmonic potential subject to a constant effective force kBT. For small objects in viscous fluids, the inertial terms are negligible relative to the other terms (see Life at small Reynolds Number ), so the dynamic equation is
This solution at short times describes a diffusing particle (Fickian behavior) with a diffusion coefficient D. However, for long times the solution asymptotes to an equipartition value of <x2> = kBT/k. In the intermediate time regime, the particle is walking randomly, but the mean-squared displacement is no longer growing linearly with time.
Constrained motion shows clear saturation to the size set by the physical constraints (equipartition for an oscillator or compartment size for a freely diffusing particle ). However, if the experimental data do not clearly extend into the saturation time regime, then the fit to anomalous diffusion can lead to exponents that do not equal unity. This is illustrated in Fig. 3 with asymptotic MSD compared with the anomalous diffusion equation fit for the exponent β. Care must be exercised in the interpretation of the exponents obtained from anomalous diffusion experiments. In particular, all constrained motion leads to subdiffusive interpretations if measured at intermediate times.
Random Walk in a Double Potential
The harmonic potential has well-known asymptotic dynamics which makes the analytic treatment straightforward. However, the Langevin equation is general and can be applied to any potential function. Take a double-well potential as another example
The resulting Langevin equation can be solved numerically in the presence of random velocity jumps. A specific stochastic trajectory is shown in Fig. 4 that applies discrete velocity jumps using a normal distribution of jumps of variance 2B. The notable character of this trajectory, besides the random-walk character, is the ability of the particle to jump the barrier between the wells. In the deterministic system, the initial condition dictates which stable fixed point would be approached. In the stochastic system, there are random fluctuations that take the particle from one basin of attraction to the other.
The stochastic long-time probability distribution p(x,v) in Fig. 5 introduces an interesting new view of trajectories in state space that have a different character than typical state-space flows. If we think about starting a large number of systems with the same initial conditions, and then letting the stochastic dynamics take over, we can define a time-dependent probability distribution p(x,v,t) that describes the likely end-positions of an ensemble of trajectories on the state plane as a function of time. This introduces the idea of the trajectory of a probability cloud in state space, which has a strong analogy to time-dependent quantum mechanics. The Schrödinger equation can be viewed as a diffusion equation in complex time, which is the basis of a technique known as quantum Monte Carlo that solves for ground state wave functions using concepts of random walks. This goes beyond the topics of classical mechanics, and it shows how such diverse fields as econophysics, diffusion, and quantum mechanics can share common tools and language.
“Stochastic Chaos” sounds like an oxymoron. “Chaos” is usually synonymous with “deterministic chaos”, meaning that every next point on a trajectory is determined uniquely by its previous location–there is nothing random about the evolution of the dynamical system. It is only when one looks at long times, or at two nearby trajectories, that non-repeatable and non-predictable behavior emerges, so there is nothing stochastic about it.
On the other hand, there is nothing wrong with adding a stochastic function to the right-hand side of a deterministic flow–just as in the Langevin equation. One question immediately arises: if chaos has sensitivity to initial conditions (SIC), wouldn’t it be highly susceptible to constant buffeting by a stochastic force? Let’s take a look!
To the well-known Rössler model, add a stochastic function to one of the three equations,
in this case to the y-dot equation. This is just like the stochastic term in the random walks in the harmonic and double-well potentials. The solution is shown in Fig. 6. In addition to the familiar time-series of the Rössler model, there are stochastic jumps in the y-variable. An x-y projection similarly shows the familiar signature of the model, and the density of trajectory points is shown in the density plot on the right. The rms jump size for this simulation is approximately 10%.
Now for the supposition that because chaos has sensitivity to initial conditions that it should be highly susceptible to stochastic contributions–the answer can be seen in Fig. 7 in the state-space densities. Other than a slightly more fuzzy density for the stochastic case, the general behavior of the Rössler strange attractor is retained. The attractor is highly stable against the stochastic fluctuations. This demonstrates just how robust deterministic chaos is.
On the other hand, there is a saddle point in the Rössler dynamics a bit below the lowest part of the strange attractor in the figure, and if the stochastic jumps are too large, then the dynamics become unstable and diverge. A hint at this is already seen in the time series in Fig. 6 that shows the nearly closed orbit that occurs transiently at large negative y values. This is near the saddle point, and this trajectory is dangerously close to going unstable. Therefore, while the attractor itself is stable, anything that drives a dynamical system to a saddle point will destabilize it, so too much stochasticity can cause a sudden destruction of the attractor.
 E. M. Purcell, “Life at Low Reynolds-Number,” American Journal of Physics, vol. 45, no. 1, pp. 3-11, (1977)
 Ritchie, K., Shan, X.Y., Kondo, J., Iwasawa, K., Fujiwara, T., Kusumi, A.: Detection of non- Brownian diffusion in the cell membrane in single molecule tracking. Biophys. J. 88(3), 2266–2277 (2005)
“Society is founded on hero worship”, wrote Thomas Carlyle (1795 – 1881) in his 1840 lecture on “Hero as Divinity”—and the society of physicists is no different. Among physicists, the hero is the genius—the monomyth who journeys into the supernatural realm of high mathematics, engages in single combat against chaos and confusion, gains enlightenment in the mysteries of the universe, and returns home to share the new understanding. If the hero is endowed with unusual talent and achieves greatness, then mythologies are woven, creating shadows that can grow and eclipse the truth and the work of others, bestowing upon the hero recognitions that are not entirely deserved.
“Gentlemen! The views of space and time which I wish to lay before you … They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
Herman Minkowski (1908)
The greatest hero of physics of the twentieth century, without question, is Albert Einstein. He is the person most responsible for the development of “Modern Physics” that encompasses:
Relativity theory (both special and general),
Quantum theory (he invented the quantum in 1905—see my blog),
Astrophysics (his field equations of general relativity were solved by Schwarzschild in 1916 to predict event horizons of black holes, and he solved his own equations to predict gravitational waves that were discovered in 2015),
Cosmology (his cosmological constant is now recognized as the mysterious dark energy that was discovered in 2000), and
Solid state physics (his explanation of the specific heat of crystals inaugurated the field of quantum matter).
Einstein made so many seminal contributions to so many sub-fields of physics that it defies comprehension—hence he is mythologized as genius, able to see into the depths of reality with unique insight. He deserves his reputation as the greatest physicist of the twentieth century—he has my vote, and he was chosen by Time magazine in 2000 as the Man of the Century. But as his shadow has grown, it has eclipsed and even assimilated the work of others—work that he initially criticized and dismissed, yet later embraced so whole-heartedly that he is mistakenly given credit for its discovery.
For instance, when we think of Einstein, the first thing that pops into our minds is probably “spacetime”. He himself wrote several popular accounts of relativity that incorporated the view that spacetime is the natural geometry within which so many of the non-intuitive properties of relativity can be understood. When we think of time being mixed with space, making it seem that position coordinates and time coordinates share an equal place in the description of relativistic physics, it is common to attribute this understanding to Einstein. Yet Einstein initially resisted this viewpoint and even disparaged it when he first heard it!
Spacetime was the brain-child of Hermann Minkowski.
Minkowski in Königsberg
Hermann Minkowski was born in 1864 in Russia to German parents who moved to the city of Königsberg (King’s Mountain) in East Prussia when he was eight years old. He entered the university in Königsberg in 1880 when he was sixteen. Within a year, when he was only seventeen years old, and while he was still a student at the University, Minkowski responded to an announcement of the Mathematics Prize of the French Academy of Sciences in 1881. When he submitted is prize-winning memoire, he could have had no idea that it was starting him down a path that would lead him years later to revolutionary views.
The specific Prize challenge of 1881 was to find the number of representations of an integer as a sum of five squares of integers. For instance, every integer n > 33 can be expressed as the sum of five nonzero squares. As an example, 42 = 22 + 22 + 32 + 32 + 42, which is the only representation for that number. However, there are five representation for n = 53
The task of enumerating these representations draws from the theory of quadratic forms. A quadratic form is a function of products of numbers with integer coefficients, such as ax2 + bxy + cy2 and ax2 + by2 + cz2 + dxy + exz + fyz. In number theory, one seeks to find integer solutions for which the quadratic form equals an integer. For instance, the Pythagorean theorem x2 + y2 = n2 for integers is a quadratic form for which there are many integer solutions (x,y,n), known as Pythagorean triplets, such as
The topic of quadratic forms gained special significance after the work of Bernhard Riemann who established the properties of metric spaces based on the metric expression
for infinitesimal distance in a D-dimensional metric space. This is a generalization of Euclidean distance to more general non-Euclidean spaces that may have curvature. Minkowski would later use this expression to great advantage, developing a “Geometry of Numbers”  as he delved ever deeper into quadratic forms and their uses in number theory.
Minkowski in Göttingen
After graduating with a doctoral degree in 1885 from Königsberg, Minkowski did his habilitation at the university of Bonn and began teaching, moving back to Königsberg in 1892 and then to Zurich in 1894 (where one of his students was a somewhat lazy and unimpressive Albert Einstein). A few years later he was given an offer that he could not refuse.
At the turn of the 20th century, the place to be in mathematics was at the University of Göttingen. It had a long tradition of mathematical giants that included Carl Friedrich Gauss, Bernhard Riemann, Peter Dirichlet, and Felix Klein. Under the guidance of Felix Klein, Göttingen mathematics had undergone a renaissance. For instance, Klein had attracted Hilbert from the University of Königsberg in 1895. David Hilbert had known Minkowski when they were both students in Königsberg, and Hilbert extended an invitation to Minkowski to join him in Göttingen, which Minkowski accepted in 1902.
A few years after Minkowski arrived at Göttingen, the relativity revolution broke, and both Minkowski and Hilbert began working on mathematical aspects of the new physics. They organized a colloquium dedicated to relativity and related topics, and on Nov. 5, 1907 Minkowski gave his first tentative address on the geometry of relativity.
Because Minkowski’s specialty was quadratic forms, and given his understanding of Riemann’s work, he was perfectly situated to apply his theory of quadratic forms and invariants to the Lorentz transformations derived by Poincaré and Einstein. Although Poincaré had published a paper in 1906 that showed that the Lorentz transformation was a generalized rotation in four-dimensional space , Poincaré continued to discuss space and time as separate phenomena, as did Einstein. For them, simultaneity was no longer an invariant, but events in time were still events in time and not somehow mixed with space-like properties. Minkowski recognized that Poincaré had missed an opportunity to define a four-dimensional vector space filled by four-vectors that captured all possible events in a single coordinate description without the need to separate out time and space.
Minkowski’s first attempt, presented in his 1907 colloquium, at constructing velocity four-vectors was flawed because (like so many of my mechanics students when they first take a time derivative of the four-position) he had not yet understood the correct use of proper time. But the research program he outlined paved the way for the great work that was to follow.
On Feb. 21, 1908, only 3 months after his first halting steps, Minkowski delivered a thick manuscript to the printers for an article to appear in the Göttinger Nachrichten. The title “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern” (The Basic Equations for Electromagnetic Processes of Moving Bodies) belies the impact and importance of this very dense article . In its 60 pages (with no figures), Minkowski presents the correct form for four-velocity by taking derivatives relative to proper time, and he formalizes his four-dimensional approach to relativity that became the standard afterwards. He introduces the terms spacelikevector, timelike vector, light cone and world line. He also presents the complete four-tensor form for the electromagnetic fields. The foundational work of Levi Cevita and Ricci-Curbastro on tensors was not yet well known, so Minkowski invents his own terminology of Traktor to describe it. Most importantly, he invents the terms spacetime (Raum-Zeit) and events (Erignisse) .
Minkowski’s four-dimensional formalism of relativistic electromagnetics was more than a mathematical trick—it uncovered the presence of a multitude of invariants that were obscured by the conventional mathematics of Einstein and Lorentz and Poincaré. In Minkowski’s approach, whenever a proper four-vector is contracted with itself (its inner product), an invariant emerges. Because there are many fundamental four-vectors, there are many invariants. These invariants provide the anchors from which to understand the complex relative properties amongst relatively moving frames.
Minkowski’s master work appeared in the Nachrichten on April 5, 1908. If he had thought that physicists would embrace his visionary perspective, he was about to be woefully disabused of that notion.
Despite his impressive ability to see into the foundational depths of the physical world, Einstein did not view mathematics as the root of reality. Mathematics for him was a tool to reduce physical intuition into quantitative form. In 1908 his fame was rising as the acknowledged leader in relativistic physics, and he was not impressed or pleased with the abstract mathematical form that Minkowski was trying to stuff the physics into. Einstein called it “superfluous erudition” , and complained “since the mathematics pounced on the relativity theory, I no longer understand it myself! ”
With his collaborator Jakob Laub (also a former student of Minkowski’s), Einstein objected to more than the hard-to-follow mathematics—they believed that Minkowski’s form of the pondermotive force was incorrect. They then proceeded to re-translate Minkowski’s elegant four-vector derivations back into ordinary vector analysis, publishing two papers in Annalen der Physik in the summer of 1908 that were politely critical of Minkowski’s approach [7-8]. Yet another of Minkowski’s students from Zurich, Gunnar Nordström, showed how to derive Minkowski’s field equations without any of the four-vector formalism.
One can only wonder why so many of his former students so easily dismissed Minkowski’s revolutionary work. Einstein had actually avoided Minkowski’s mathematics classes as a student at ETH , which may say something about Minkowski’s reputation among the students, although Einstein did appreciate the class on mechanics that he took from Minkowski. Nonetheless, Einstein missed the point! Rather than realizing the power and universality of the four-dimensional spacetime formulation, he dismissed it as obscure and irrelevant—perhaps prejudiced by his earlier dim view of his former teacher.
Raum und Zeit
It is clear that Minkowski was stung by the poor reception of his spacetime theory. It is also clear that he truly believed that he had uncovered an essential new approach to physical reality. While mathematicians were generally receptive of his work, he knew that if physicists were to adopt his new viewpoint, he needed to win them over with the elegant results.
In 1908, Minkowski presented a now-famous paper Raum und Zeit at the 80thAssembly of German Natural Scientists and Physicians (21 September 1908). In his opening address, he stated :
To illustrate his arguments Minkowski constructed the most recognizable visual icon of relativity theory—the space-time diagram in which the trajectories of particles appear as “world lines”, as in Fig. 1. On this diagram, one spatial dimension is plotted along the horizontal-axis, and the value ct (speed of light times time) is plotted along the vertical-axis. In these units, a photon travels along a line oriented at 45 degrees, and the world-line (the name Minkowski gave to trajectories) of all massive particles must have slopes steeper than this. For instance, a stationary particle, that appears to have no trajectory at all, executes a vertical trajectory on the space-time diagram as it travels forward through time. Within this new formulation by Minkowski, space and time were mixed together in a single manifold—spacetime—and were no longer separate entities.
In addition to the spacetime construct, Minkowski’s great discovery was the plethora of invariants that followed from his geometry. For instance, the spacetime hyperbola
is invariant to Lorentz transformation in coordinates. This is just a simple statement that a vector is an entity of reality that is independent of how it is described. The length of a vector in our normal three-space does not change if we flip the coordinates around or rotate them, and the same is true for four-vectors in Minkowski space subject to Lorentz transformations.
In relativity theory, this property of invariance becomes especially useful because part of the mental challenge of relativity is that everything looks different when viewed from different frames. How do you get a good grip on a phenomenon if it is always changing, always relative to one frame or another? The invariants become the anchors that we can hold on to as reference frames shift and morph about us.
As an example of a fundamental invariant, the mass of a particle in its rest frame becomes an invariant mass, always with the same value. In earlier relativity theory, even in Einstein’s papers, the mass of an object was a function of its speed. How is the mass of an electron a fundamental property of physics if it is a function of how fast it is traveling? The construction of invariant mass removes this problem, and the mass of the electron becomes an immutable property of physics, independent of the frame. Invariant mass is just one of many invariants that emerge from Minkowski’s space-time description. The study of relativity, where all things seem relative, became a study of invariants, where many things never change. In this sense, the theory of relativity is a misnomer. Ironically, relativity theory became the motivation of post-modern relativism that denies the existence of absolutes, even as relativity theory, as practiced by physicists, is all about absolutes.
Despite his audacious gambit to win over the physicists, Minkowski would not live to see the fruits of his effort. He died suddenly of a burst gall bladder on Jan. 12, 1909 at the age of 44.
Arnold Sommerfeld (who went on to play a central role in the development of quantum theory) took up Minkowski’s four vectors, and he systematized it in a way that was palatable to physicists. Then Max von Laue extended it while he was working with Sommerfeld in Munich, publishing the first physics textbook on relativity theory in 1911, establishing the space-time formalism for future generations of German physicists. Further support for Minkowski’s work came from his distinguished colleagues at Göttingen (Hilbert, Klein, Wiechert, Schwarzschild) as well as his former students (Born, Laue, Kaluza, Frank, Noether). With such champions, Minkowski’s work was immortalized in the methodology (and mythology) of physics, representing one of the crowning achievements of the Göttingen mathematical community.
Already in 1907 Einstein was beginning to grapple with the role of gravity in the context of relativity theory, and he knew that the special theory was just a beginning. Yet between 1908 and 1910 Einstein’s focus was on the quantum of light as he defended and extended his unique view of the photon and prepared for the first Solvay Congress of 1911. As he returned his attention to the problem of gravitation after 1910, he began to realize that Minkowski’s formalism provided a framework from which to understand the role of accelerating frames. In 1912 Einstein wrote to Sommerfeld to say 
I occupy myself now exclusively with the problem of gravitation . One thing is certain that I have never before had to toil anywhere near as much, and that I have been infused with great respect for mathematics, which I had up until now in my naivety looked upon as a pure luxury in its more subtle parts. Compared to this problem. the original theory of relativity is child’s play.
By the time Einstein had finished his general theory of relativity and gravitation in 1915, he fully acknowledge his indebtedness to Minkowski’s spacetime formalism without which his general theory may never have appeared.
Einstein is the alpha of the quantum. Einstein is also the omega. Although he was the one who established the quantum of energy and matter (see my Blog Einstein vs Planck), Einstein pitted himself in a running debate against Niels Bohr’s emerging interpretation of quantum physics that had, in Einstein’s opinion, severe deficiencies. Between sessions during a series of conferences known as the Solvay Congresses over a period of eight years from 1927 to 1935, Einstein constructed a challenges of increasing sophistication to confront Bohr and his quasi-voodoo attitudes about wave-function collapse. To meet the challenge, Bohr sharpened his arguments and bested Einstein, who ultimately withdrew from the field of battle. Einstein, as quantum physics’ harshest critic, played a pivotal role, almost against his will, establishing the Copenhagen interpretation of quantum physics that rules to this day, and also inventing the principle of entanglement which lies at the core of almost all quantum information technology today.
Fifth Solvay Congress: 1927 October Brussels: Debate Round 1
Einstein and ensembles
Sixth Solvay Congress: 1930 Debate Round 2
Photon in a box
Seventh Solvay Congress: 1933
Einstein absent (visiting the US when Hitler takes power…decides not to return to Germany.)
Physical Review 1935: Debate Round 3
EPR paper and Bohr’s response
Notable Nobel Prizes
1933 Dirac and Schrödinger
The Solvay Conferences
The Solvay congresses were unparalleled scientific meetings of their day. They were attended by invitation only, and invitations were offered only to the top physicists concerned with the selected topic of each meeting. The Solvay congresses were held about every three years always in Belgium, supported by the Belgian chemical industrialist Ernest Solvay. The first meeting, held in 1911, was on the topic of radiation and quanta.
The fifth meeting, held in 1927, was on electrons and photons and focused on the recent rapid advances in quantum theory. The old quantum guard was invited—Planck, Bohr and Einstein. The new quantum guard was invited as well—Heisenberg, de Broglie, Schrödinger, Born, Pauli, and Dirac. Heisenberg and Bohr joined forces to present a united front meant to solidify what later became known as the Copenhagen interpretation of quantum physics. The basic principles of the interpretation include the wavefunction of Schrödinger, the probabilistic interpretation of Born, the uncertainty principle of Heisenberg, the complementarity principle of Bohr and the collapse of the wavefunction during measurement. The chief conclusion that Heisenberg and Bohr sought to impress on the assembled attendees was that the theory of quantum processes was complete, meaning that unknown or uncertain characteristics of measurements could not be attributed to lack of knowledge or understanding, but were fundamental and permanently inaccessible.
Einstein was not convinced with that argument, and he rose to his feet to object after Bohr’s informal presentation of his complementarity principle. Einstein insisted that uncertainties in measurement were not fundamental, but were caused by incomplete information, that , if known, would accurately account for the measurement results. Bohr was not prepared for Einstein’s critique and brushed it off, but what ensued in the dining hall and the hallways of the Hotel Metropole in Brussels over the next several days has become one of the most famous scientific debates of the modern era, known as the Bohr-Einstein debate on the meaning of quantum theory. The debate gently raged night and day through the fifth congress, and was renewed three years later at the 1930 congress. It finished, in a final flurry of published papers in 1935 that launched some of the central concepts of quantum theory, including the idea of quantum entanglement and, of course, Schrödinger’s cat.
Einstein’s strategy, to refute Bohr, was to construct careful thought experiments that envisioned perfect experiments, without errors, that measured properties of ideal quantum systems. His aim was to paint Bohr into a corner from which he could not escape, caught by what Einstein assumed was the inconsistency of complementarity. Einstein’s “thought experiments” used electrons passing through slits, diffracting as required by Schrödinger’s theory, but being detected by classical measurements. Einstein would present a thought experiment to Bohr, who would then retreat to consider the way around Einstein’s arguments, returning the next hour or the next day with his answer, only to be confronted by yet another clever device of Einstein’s clever imagination that would force Bohr to retreat again. The spirit of this back and forth encounter between Bohr and Einstein is caught dramatically in the words of Paul Ehrenfest who witnessed the debate first hand, partially mediating between Bohr and Einstein, both of whom he respected deeply.
“Brussels-Solvay was fine!… BOHR towering over everybody. At first not understood at all … , then step by step defeating everybody. Naturally, once again the awful Bohr incantation terminology. Impossible for anyone else to summarise … (Every night at 1 a.m., Bohr came into my room just to say ONE SINGLE WORD to me, until three a.m.) It was delightful for me to be present during the conversation between Bohr and Einstein. Like a game of chess, Einstein all the time with new examples. In a certain sense a sort of Perpetuum Mobile of the second kind to break the UNCERTAINTY RELATION. Bohr from out of philosophical smoke clouds constantly searching for the tools to crush one example after the other. Einstein like a jack-in-the-box; jumping out fresh every morning. Oh, that was priceless. But I am almost without reservation pro Bohr and contra Einstein. His attitude to Bohr is now exacly like the attitude of the defenders of absolute simultaneity towards him …” 
The most difficult example that Einstein constructed during the fifth Solvary Congress involved an electron double-slit apparatus that could measure, in principle, the momentum imparted to the slit by the passing electron, as shown in Fig.3. The electron gun is a point source that emits the electrons in a range of angles that illuminates the two slits. The slits are small relative to a de Broglie wavelength, so the electron wavefunctions diffract according to Schrödinger’s wave mechanics to illuminate the detection plate. Because of the interference of the electron waves from the two slits, electrons are detected clustered in intense fringes separated by dark fringes.
So far, everyone was in agreement with these suggested results. The key next step is the assumption that the electron gun emits only a single electron at a time, so that only one electron is present in the system at any given time. Furthermore, the screen with the double slit is suspended on a spring, and the position of the screen is measured with complete accuracy by a displacement meter. When the single electron passes through the entire system, it imparts a momentum kick to the screen, which is measured by the meter. It is also detected at a specific location on the detection plate. Knowing the position of the electron detection, and the momentum kick to the screen, provides information about which slit the electron passed through, and gives simultaneous position and momentum values to the electron that have no uncertainty, apparently rebutting the uncertainty principle.
This challenge by Einstein was the culmination of successively more sophisticated examples that he had to pose to combat Bohr, and Bohr was not going to let it pass unanswered. With ingenious insight, Bohr recognized that the key element in the apparatus was the fact that the screen with the slits must have finite mass if the momentum kick by the electron were to produce a measurable displacement. But if the screen has finite mass, and hence a finite momentum kick from the electron, then there must be an uncertainty in the position of the slits. This uncertainty immediately translates into a washout of the interference fringes. In fact the more information that is obtained about which slit the electron passed through, the more the interference is washed out. It was a perfect example of Bohr’s own complementarity principle. The more the apparatus measures particle properties, the less it measures wave properties, and vice versa, in a perfect balance between waves and particles.
Einstein grudgingly admitted defeat at the end of the first round, but he was not defeated. Three years later he came back armed with more clever thought experiments, ready for the second round in the debate.
The Sixth Solvay Conference: 1930
At the Solvay Congress of 1930, Einstein was ready with even more difficult challenges. His ultimate idea was to construct a box containing photons, just like the original black bodies that launched Planck’s quantum hypothesis thirty years before. The box is attached to a weighing scale so that the weight of the box plus the photons inside can be measured with arbitrarily accuracy. A shutter over a hole in the box is opened for a time T, and a photon is emitted. Because the photon has energy, it has an equivalent weight (Einstein’s own famous E = mc2), and the mass of the box changes by an amount equal to the photon energy divided by the speed of light squared: m = E/c2. If the scale has arbitrary accuracy, then the energy of the photon has no uncertainty. In addition, because the shutter was open for only a time T, the time of emission similarly has no uncertainty. Therefore, the product of the energy uncertainty and the time uncertainty is much smaller than Planck’s constant, apparently violating Heisenberg’s precious uncertainty principle.
Bohr was stopped in his tracks with this challenge. Although he sensed immediately that Einstein had missed something (because Bohr had complete confidence in the uncertainty principle), he could not put his finger immediately on what it was. That evening he wandered from one attendee to another, very unhappy, trying to persuade them and saying that Einstein could not be right because it would be the end of physics. At the end of the evening, Bohr was no closer to a solution, and Einstein was looking smug. However, by the next morning Bohr reappeared tired but in high spirits, and he delivered a master stroke. Where Einstein had used special relaitivity against Bohr, Bohr now used Einstein’s own general relativity against him.
The key insight was that the weight of the box must be measured, and the process of measurement was just as important as the quantum process being measured—this was one of the cornerstones of the Copenhagen interpretation. So Bohr envisioned a measuring apparatus composed of a spring and a scale with the box suspended in gravity from the spring. As the photon leaves the box, the weight of the box changes, and so does the deflection of the spring, changing the height of the box. This change in height, in a gravitational potential, causes the timing of the shutter to change according to the law of gravitational time dilation in general relativity. By calculating the the general relativistic uncertainty in the time, coupled with the special relativistic uncertainty in the weight of the box, produced a product that was at least as big as Planck’s constant—Heisenberg’s uncertainty principle was saved!
Entanglement and Schrödinger’s Cat
Einstein ceded the point to Bohr but was not convinced. He still believed that quantum mechanics was not a “complete” theory of quantum physics and he continued to search for the perfect thought experiment that Bohr could not escape. Even today when we have become so familiar with quantum phenomena, the Copenhagen interpretation of quantum mechanics has weird consequences that seem to defy common sense, so it is understandable that Einstein had his reservations.
After the sixth Solvay congress Einstein and Schrödinger exchanged many letters complaining to each other about Bohr’s increasing strangle-hold on the interpretation of quantum mechanics. Egging each other on, they both constructed their own final assault on Bohr. The irony is that the concepts they devised to throw down quantum mechanics have today become cornerstones of the theory. For Einstein, his final salvo was “Entanglement”. For Schrödinger, his final salvo was his “cat”. Today, Entanglement and Schrödinger’s Cat have become enshrined on the alter of quantum interpretation even though their original function was to thwart that interpretation.
The final round of the debate was carried out, not at a Solvay congress, but in the Physical review journal by Einstein  and Bohr , and in the Naturwissenshaften by Schrödinger .
In 1969, Heisenberg looked back on these years and said,
To those of us who participated in the development of atomic theory, the five years following the Solvay Conference in Brussels in 1927 looked so wonderful that we often spoke of them as the golden age of atomic physics. The great obstacles that had occupied all our efforts in the preceding years had been cleared out of the way, the gate to an entirely new field, the quantum mechanics of the atomic shells stood wide open, and fresh fruits seemed ready for the picking. 
 A. Whitaker, Einstein, Bohr, and the quantum dilemma : from quantum theory to quantum information, 2nd ed. Cambridge University Press, 2006. (pg. 210)
 A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 47, no. 10, pp. 0777-0780, May (1935)
Christian Andreas Doppler (1803 – 1853) was born in Salzburg, Austria, to a longstanding family of stonemasons. As a second son, he was expected to help his older brother run the business, so his Father had him tested in his 18th year for his suitability for a career in business. The examiner Simon Stampfer (1790 – 1864), an Austrian mathematician and inventor teaching at the Lyceum in Salzburg, discovered that Doppler had a gift for mathematics and was better suited for a scientific career. Stampfer’s enthusiasm convinced Doppler’s father to enroll him in the Polytechnik Institute in Vienna (founded only a few years earlier in 1815) where he took classes in mathematics, mechanics and physics  from 1822 to 1825. Doppler excelled in his courses, but was dissatisfied with the narrowness of the education, yearning for more breadth and depth in his studies and for more significance in his positions, feelings he would struggle with for his entire short life. He left Vienna, returning to the Lyceum in Salzburg to round out his education with philosophy, languages and poetry. Unfortunately, this four-year detour away from technical studies impeded his ability to gain a permanent technical position, so he began a temporary assistantship with a mathematics professor at Vienna. As he approached his 30th birthday this term expired without prospects. He was about to emigrate to America when he finally received an offer to teach at a secondary school in Prague.
To read about the attack by Joseph Petzval on Doppler’s effect and the effect it had on Doppler, see my feature article “The Fall and Rise of the Doppler Effect“ in Physics Today, 73(3) 30, March (2020).
Doppler in Prague
Prague gave Doppler new life. He was a professor with a position that allowed him to marry the daughter of a sliver and goldsmith from Salzburg. He began to publish scholarly papers, and in 1837 was appointed supplementary professor of Higher Mathematics and Geometry at the Prague Technical Institute, promoted to full professor in 1841. It was here that he met the unusual genius Bernard Bolzano (1781 – 1848), recently returned from political exile in the countryside. Bolzano was a philosopher and mathematician who developed rigorous concepts of mathematical limits and is famous today for his part in the Bolzano-Weierstrass theorem in functional analysis, but he had been too liberal and too outspoken for the conservative Austrian regime and had been dismissed from the University in Prague in 1819. He was forbidden to publish his work in Austrian journals, which is one reason why much of Bolzano’s groundbreaking work in functional analysis remained unknown during his lifetime. However, he participated in the Bohemian Society for Science from a distance, recognizing the inventive tendencies in the newcomer Doppler and supporting him for membership in the Bohemian Society. When Bolzano was allowed to return in 1842 to the Polytechnic Institute in Prague, he and Doppler became close friends as kindred spirits.
Prague, Czech Republic
On May 25, 1842, Bolzano presided as chairman over a meeting of the Bohemian Society for Science on the day that Doppler read a landmark paper on the color of stars to a meagre assembly of only five regular members of the Society . The turn-out was so small that the meeting may have been held in the robing room of the Society rather than in the meeting hall itself. Leading up to this famous moment, Doppler’s interests were peripatetic, ranging widely over mathematical and physical topics, but he had lately become fascinated by astronomy and by the phenomenon of stellar aberration. Stellar aberration was discovered by James Bradley in 1729 and explained as the result of the Earth’s yearly motion around the Sun, causing the apparent location of a distant star to change slightly depending on the direction of the Earth’s motion. Bradley explained this in terms of the finite speed of light and was able to estimate it to within several percent . As Doppler studied Bradley aberration, he wondered how the relative motion of the Earth would affect the color of the star. By making a simple analogy of a ship traveling with, or against, a series of ocean waves, he concluded that the frequency of impact of the peaks and troughs of waves on the ship was no different than the arrival of peaks and troughs of the light waves impinging on the eye. Because perceived color was related to the frequency of excitation in the eye, he concluded that the color of light would be slightly shifted to the blue if approaching, and to the red if receding from, the light source.
Doppler wave fronts from a source emitting spherical waves moving with speeds β relative to the speed of the wave in the medium.
Doppler calculated the magnitude of the effect by taking a simple ratio of the speed of the observer relative to the speed of light. What he found was that the speed of the Earth, though sufficient to cause the detectable aberration in the position of stars, was insufficient to produce a noticeable change in color. However, his interest in astronomy had made him familiar with binary stars where the relative motion of the light source might be high enough to cause color shifts. In fact, in the star catalogs there were examples of binary stars that had complementary red and blue colors. Therefore, the title of his paper, published in the Proceedings of the Royal Bohemian Society of Sciences a few months after he read it to the society, was “On the Coloured Light of the Double Stars and Certain Other Stars of the Heavens: Attempt at a General Theory which Incorporates Bradley’s Theorem of Aberration as an Integral Part” .
Title page of Doppler’s 1842 paper introducing the Doppler Effect.
Doppler’s analogy was correct, but like all analogies not founded on physical law, it differed in detail from the true nature of the phenomenon. By 1842 the transverse character of light waves had been thoroughly proven through the work of Fresnel and Arago several decades earlier, yet Doppler held onto the old-fashioned notion that light was composed of longitudinal waves. Bolzano, fully versed in the transverse nature of light, kindly published a commentary shortly afterwards  showing how the transverse effect for light, and a longitudinal effect for sound, were both supported by Doppler’s idea. Yet Doppler also did not know that speeds in visual binaries were too small to produce noticeable color effects to the unaided eye. Finally, (and perhaps the greatest flaw in his argument on the color of stars) a continuous spectrum that extends from the visible into the infrared and ultraviolet would not change color because all the frequencies would shift together preserving the flat (white) spectrum.
The simple algebraic derivation of the Doppler Effect in the 1842 publication..
Doppler’s twelve years in Prague were intense. He was consumed by his Society responsibilities and by an extremely heavy teaching load that included personal exams of hundreds of students. The only time he could be creative was during the night while his wife and children slept. Overworked and running on too little rest, his health already frail with the onset of tuberculosis, Doppler collapsed, and he was unable to continue at the Polytechnic. In 1847 he transferred to the School of Mines and Forrestry in Schemnitz (modern Banská Štiavnica in Slovakia) with more pay and less work. Yet the revolutions of 1848 swept across Europe, with student uprisings, barricades in the streets, and Hungarian liberation armies occupying the cities and universities, giving him no peace. Providentially, his former mentor Stampfer retired from the Polytechnic in Vienna, and Doppler was called to fill the vacancy.
Although Doppler was named the Director of Austria’s first Institute of Physics and was elected to the National Academy, he ran afoul of one of the other Academy members, Joseph Petzval (1807 – 1891), who persecuted Doppler and his effect. To read a detailed description of the attack by Petzval on Doppler’s effect and the effect it had on Doppler, see my feature article “The Fall and Rise of the Doppler Effect” in Physics Today, March issue (2020).
It is difficult today to appreciate just how deeply engrained the reality of the luminiferous ether was in the psyche of the 19th century physicist. The last of the classical physicists were reluctant even to adopt Maxwell’s electromagnetic theory for the explanation of optical phenomena, and as physicists inevitably were compelled to do so, some of their colleagues looked on with dismay and disappointment. This was the situation for Woldemar Voigt (1850 – 1919) at the University of Göttingen, who was appointed as one of the first professors of physics there in 1883, to be succeeded in later years by Peter Debye and Max Born. Voigt received his doctorate at the University of Königsberg under Franz Neumann, exploring the elastic properties of rock salt, and at Göttingen he spent a quarter century pursuing experimental and theoretical research into crystalline properties. Voigt’s research, with students like Paul Drude, laid the foundation for the modern field of solid state physics. His textbook Lehrbuch der Kristallphysik published in 1910 remained influential well into the 20th century because it adopted mathematical symmetry as a guiding principle of physics. It was in the context of his studies of crystal elasticity that he introduced the word “tensor” into the language of physics.
At the January 1887 meeting of the Royal Society of Science at Göttingen, three months before Michelson and Morely began their reality-altering experiments at the Case Western Reserve University in Cleveland Ohio, Voit submitted a paper deriving the longitudinal optical Doppler effect in an incompressible medium. He was responding to results published in 1886 by Michelson and Morely on their measurements of the Fresnel drag coefficient, which was the precursor to their later results on the absolute motion of the Earth through the ether.
Fresnel drag is the effect of light propagating through a medium that is in motion. The French physicist Francois Arago (1786 – 1853) in 1810 had attempted to observe the effects of corpuscles of light emitted from stars propagating with different speeds through the ether as the Earth spun on its axis and traveled around the sun. He succeeded only in observing ordinary stellar aberration. The absence of the effects of motion through the ether motivated Augustin-Jean Fresnel (1788 – 1827) to apply his newly-developed wave theory of light to explain the null results. In 1818 Fresnel derived an expression for the dragging of light by a moving medium that explained the absence of effects in Arago’s observations. For light propagating through a medium of refractive index n that is moving at a speed v, the resultant velocity of light is
where the last term in parenthesis is the Fresnel drag coefficient. The Fresnel drag effect supported the idea of the ether by explaining why its effects could not be observed—a kind of Catch-22—but it also applied to light moving through a moving dielectric medium. In 1851, Fizeau used an interferometer to measure the Fresnel drag coefficient for light moving through moving water, arriving at conclusions that directly confirmed the Fresnel drag effect. The positive experiments of Fizeau, as well as the phenomenon of stellar aberration, would be extremely influential on the thoughts of Einstein as he developed his approach to special relativity in 1905. They were also extremely influential to Michelson, Morley and Voigt.
In his paper on the absence of the Fresnel drag effect in the first Michelson-Morley experiment, Voigt pointed out that an equation of the form
is invariant under the transformation
From our modern vantage point, we immediately recognize (to within a scale factor) the Lorentz transformation of relativity theory. The first equation is common Galilean relativity, but the last equation was something new, introducing a position-dependent time as an observer moved with speed relative to the speed of light . Using these equations, Voigt was the first to derive the longitudinal (conventional) Doppler effect from relativistic effects.
Voigt’s derivation of the longitudinal Doppler effect used a classical approach that is still used today in Modern Physics textbooks to derive the Doppler effect. The argument proceeds by considering a moving source that emits a continuous wave in the direction of motion. Because the wave propagates at a finite speed, the moving source chases the leading edge of the wave front, catching up by a small amount by the time a single cycle of the wave has been emitted. The resulting compressed oscillation represents a blue shift of the emitted light. By using his transformations, Voigt arrived at the first relativistic expression for the shift in light frequency. At low speeds, Voigt’s derivation reverted to Doppler’s original expression.
A few months after Voigt delivered his paper, Michelson and Morley announced the results of their interferometric measurements of the motion of the Earth through the ether—with their null results. In retrospect, the Michelson-Morley experiment is viewed as one of the monumental assaults on the old classical physics, helping to launch the relativity revolution. However, in its own day, it was little more than just another null result on the ether. It did incite Fitzgerald and Lorentz to suggest that length of the arms of the interferometer contracted in the direction of motion, with the eventual emergence of the full Lorentz transformations by 1904—seventeen years after the Michelson results.
In 1904 Einstein, working in relative isolation at the Swiss patent office, was surprisingly unaware of the latest advances in the physics of the ether. He did not know about Voigt’s derivation of the relativistic Doppler effect (1887) as he had not heard of Lorentz’s final version of relativistic coordinate transformations (1904). His thinking about relativistic effects focused much farther into the past, to Bradley’s stellar aberration (1725) and Fizeau’s experiment of light propagating through moving water (1851). Einstein proceeded on simple principles, unencumbered by the mental baggage of the day, and delivered his beautifully minimalist theory of special relativity in his famous paper of 1905 “On the Electrodynamics of Moving Bodies”, independently deriving the Lorentz coordinate transformations .
One of Einstein’s talents in theoretical physics was to predict new phenomena as a way to provide direct confirmation of a new theory. This was how he later famously predicted the deflection of light by the Sun and the gravitational frequency shift of light. In 1905 he used his new theory of special relativity to predict observable consequences that included a general treatment of the relativistic Doppler effect. This included the effects of time dilation in addition to the longitudinal effect of the source chasing the wave. Time dilation produced a correction to Doppler’s original expression for the longitudinal effect that became significant at speeds approaching the speed of light. More significantly, it predicted a transverse Doppler effect for a source moving along a line perpendicular to the line of sight to an observer. This effect had not been predicted either by Doppler or by Voigt. The equation for the general Doppler effect for any observation angle is
Just as Doppler had been motivated by Bradley’s aberration of starlight when he conceived of his original principle for the longitudinal Doppler effect, Einstein combined the general Doppler effect with his results for the relativistic addition of velocities (also in his 1905 Annalen paper) as the conclusive treatment of stellar aberration nearly 200 years after Bradley first observed the effect.
Despite the generally positive reception of Einstein’s theory of special relativity, some of its consequences were anathema to many physicists at the time. A key stumbling block was the question whether relativistic effects, like moving clocks running slowly, were only apparent, or were actually real, and Einstein had to fight to convince others of its reality. When Johannes Stark (1874 – 1957) observed Doppler line shifts in ion beams called “canal rays” in 1906 (Stark received the 1919 Nobel prize in part for this discovery) , Einstein promptly published a paper suggesting how the canal rays could be used in a transverse geometry to directly detect time dilation through the transverse Doppler effect . Thirty years passed before the experiment was performed with sufficient accuracy by Herbert Ives and G. R. Stilwell in 1938 to measure the transverse Doppler effect . Ironically, even at this late date, Ives and Stilwell were convinced that their experiment had disproved Einstein’s time dilation by supporting Lorentz’ contraction theory of the electron. The Ives-Stilwell experiment was the first direct test of time dilation, followed in 1940 by muon lifetime measurements .
 Bradley, J (1729). “Account of a new discoved Motion of the Fix’d Stars”. Phil Trans. 35: 637–660.
 C. A. DOPPLER, “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels (About the coloured light of the binary stars and some other stars of the heavens),” Proceedings of the Royal Bohemian Society of Sciences, vol. V, no. 2, pp. 465–482, (Reissued 1903) (1842).
 B. Bolzano, “Ein Paar Bemerkunen über die Neu Theorie in Herrn Professor Ch. Doppler’s Schrift “Über das farbige Licht der Doppersterne und eineger anderer Gestirnedes Himmels”,” Pogg. Anal. der Physik und Chemie, vol. 60, p. 83, 1843; B. Bolzano, “Christian Doppler’s neuste Leistunen af dem Gebiet der physikalischen Apparatenlehre, Akoustik, Optik and optische Astronomie,” Pogg. Anal. der Physik und Chemie, vol. 72, pp. 530-555, 1847.
 W. Voigt, “Uber das Doppler’sche Princip,” Göttinger Nachrichten, vol. 7, pp. 41–51, (1887). The common use of c to express the speed of light came later from Voigt’s student Paul Drude.
 A. Einstein, “On the electrodynamics of moving bodies,” Annalen Der Physik, vol. 17, pp. 891-921, 1905.
 J. Stark, W. Hermann, and S. Kinoshita, “The Doppler effect in the spectrum of mercury,” Annalen Der Physik, vol. 21, pp. 462-469, Nov 1906.
 A. Einstein, “”Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips”,” vol. 328, pp. 197–198, 1907.
 H. E. Ives and G. R. Stilwell, “An experimental study of the rate of a moving atomic clock,” Journal of the Optical Society of America, vol. 28, p. 215, 1938.
 B. Rossi and D. B. Hall, “Variation of the Rate of Decay of Mesotrons with Momentum,” Physical Review, vol. 59, pp. 223–228, 1941.
Albert Einstein defies condensation—it is impossible to condense his approach, his insight, his motivation—into a single word like “genius”. He was complex, multifaceted, contradictory, revolutionary as well as conservative. Some of his work was so simple that it is hard to understand why no-one else did it first, even when they were right in the middle of it. Lorentz and Poincaré spring to mind—they had been circling the ideas of spacetime for decades—but never stepped back to see what the simplest explanation could be. Einstein did, and his special relativity was simple and beautiful, and the math is just high-school algebra. On the other hand, parts of his work—like gravitation—are so embroiled in mathematics and the religion of general covariance that it remains opaque to physics neophytes 100 years later and is usually reserved for graduate study.
Yet there is a third thread in Einstein’s work that relies on pure intuition—neither simple nor complicated—but almost impossible to grasp how he made his leap. This is the case when he proposed the real existence of the photon—the quantum particle of light. For ten years after this proposal, it was considered by almost everyone to be his greatest blunder. It even came up when Planck was nominating Einstein for membership in the German Academy of Science. Planck said
That he may sometimes have missed the target of his speculations, as for example, in his hypothesis of light quanta, cannot really be held against him.
In this single statement, we have the father of the quantum being criticized by the father of the quantum discontinuity.
Max Planck’s Discontinuity
In histories of the development of quantum theory, the German physicist Max Planck (1858—1947) is characterized as an unlikely revolutionary. He was an establishment man, in the stolid German tradition, who was already embedded in his career, in his forties, holding a coveted faculty position at the University of Berlin. In his research, he was responding to a theoretical challenge issued by Kirchhoff many years ago in 1860 to find the function of temperature and wavelength that described and explained the observed spectrum of radiating bodies. Planck was not looking for a revolution. In fact, he was looking for the opposite. One of his motivations in studying the thermodynamics of electromagnetic radiation was to rebut the statistical theories of Boltzmann. Planck had never been convinced by the atomistic and discrete approach Boltzmann had used to explain entropy and the second law of thermodynamics. With the continuum of light radiation he thought he had the perfect system that would show how entropy behaved in a continuous manner, without the need for discrete quantities.
Therefore, Planck’s original intentions were to use blackbody radiation to argue against Boltzmann—to set back the clock. For this reason, not only was Planck an unlikely revolutionary, he was a counter-revolutionary. But Planck was a revolutionary because that is what he did, whatever his original intentions were, and he accepted his role as a revolutionary when he had the courage to stand in front of his scientific peers and propose a quantum hypothesis that lay at the heart of physics.
Blackbody radiation, at the end of the nineteenth century, was a topic of keen interest and had been measured with high precision. This was in part because it was such a “clean” system, having fundamental thermodynamic properties independent of any of the material properties of the black body, unlike the so-called ideal gases, which always showed some dependence on the molecular properties of the gas. The high-precision measurements of blackbody radiation were made possible by new developments in spectrometers at the end of the century, as well as infrared detectors that allowed very precise and repeatable measurements to be made of the spectrum across broad ranges of wavelengths.
In 1893 the German physicist Wilhelm Wien (1864—1928) had used adiabatic expansion arguments to derive what became known as Wien’s Displacement Law that showed a simple linear relationship between the temperature of the blackbody and the peak wavelength. Later, in 1896, he showed that the high-frequency behavior could be described by an exponential function of temperature and wavelength that required no other properties of the blackbody. This was approaching the solution of Kirchhoff’s challenge of 1860 seeking a universal function. However, at lower frequencies Wien’s approximation failed to match the measured spectrum. In mid-year 1900, Planck was able to define a single functional expression that described the experimentally observed spectrum. Planck had succeeded in describing black-body radiation, but he had not satisfied Kirchhoff’s second condition—to explain it.
Therefore, to describe the blackbody spectrum, Planck modeled the emitting body as a set of ideal oscillators. As an expert in the Second Law, Planck derived the functional form for the radiation spectrum, from which he found the entropy of the oscillators that produced the spectrum. However, once he had the form for the entropy, he needed to explain why it took that specific form. In this sense, he was working backwards from a known solution rather than forwards from first principles. Planck was at an impasse. He struggled but failed to find any continuum theory that could work.
Then Planck turned to Boltzmann’s statistical theory of entropy, the same theory that he had previously avoided and had hoped to discredit. He described this as “an act of despair … I was ready to sacrifice any of my previous convictions about physics.” In Boltzmann’s expression for entropy, it was necessary to “count” possible configurations of states. But counting can only be done if the states are discrete. Therefore, he lumped the energies of the oscillators into discrete ranges, or bins, that he called “quanta”. The size of the bins was proportional to the frequency of the oscillator, and the proportionality constant had the units of Maupertuis’ quantity of action, so Planck called it the “quantum of action”. Finally, based on this quantum hypothesis, Planck derived the functional form of black-body radiation.
Planck presented his findings at a meeting of the German Physical Society in Berlin on November 15, 1900, introducing the word quantum (plural quanta) into physics from the Latin word that means quantity . It was a casual meeting, and while the attendees knew they were seeing an intriguing new physical theory, there was no sense of a revolution. But Planck himself was aware that he had created something fundamentally new. The radiation law of cavities depended on only two physical properties—the temperature and the wavelength—and on two constants—Boltzmann’s constant kB and a new constant that later became known as Planck’s constant h = ΔE/f = 6.6×10-34 J-sec. By combining these two constants with other fundamental constants, such as the speed of light, Planck was able to establish accurate values for long-sought constants of nature, like Avogadro’s number and the charge of the electron.
Although Planck’s quantum hypothesis in 1900 explained the blackbody radiation spectrum, his specific hypothesis was that it was the interaction of the atoms and the light field that was somehow quantized. He certainly was not thinking in terms of individual quanta of the light field.
When Einstein analyzed the properties of the blackbody radiation in 1905, using his deep insight into statistical mechanics, he was led to the inescapable conclusion that light itself must be quantized in amounts E = hf, where h is Planck’s constant and f is the frequency of the light field. Although this equation is exactly the same as Planck’s from 1900, the meaning was completely different. For Planck, this was the discreteness of the interaction of light with matter. For Einstein, this was the quantum of light energy—whole and indivisible—just as if the light quantum were a particle with particle properties. For this reason, we can answer the question posed in the title of this Blog—Einstein takes the honor of being the inventor of the quantum.
Einstein’s clarity of vision is a marvel to behold even to this day. His special talent was to take simple principles, ones that are almost trivial and beyond reproach, and to derive something profound. In Special Relativity, he simply assumed the constancy of the speed of light and derived Lorentz’s transformations that had originally been based on obtuse electromagnetic arguments about the electron. In General Relativity, he assumed that free fall represented an inertial frame, and he concluded that gravity must bend light. In quantum theory, he assumed that the low-density limit of Planck’s theory had to be consistent with light in thermal equilibrium in thermal equilibrium with the black body container, and he concluded that light itself must be quantized into packets of indivisible energy quanta . One immediate consequence of this conclusion was his simple explanation of the photoelectric effect for which the energy of an electron ejected from a metal by ultraviolet irradiation is a linear function of the frequency of the radiation. Einstein published his theory of the quanta of light  as one of his four famous 1905 articles in Annalen der Physik in his Annus Mirabilis.
Einstein’s theory of light quanta was controversial and was slow to be accepted. It is ironic that in 1914 when Einstein was being considered for a position at the University in Berlin, Planck himself, as he championed Einstein’s case to the faculty, implored his colleagues to accept Einstein despite his ill-conceived theory of light quanta . This comment by Planck goes far to show how Planck, father of the quantum revolution, did not fully grasp, even by 1914, the fundamental nature and consequences of his original quantum hypothesis. That same year, the American physicist Robert Millikan (1868—1953) performed a precise experimental measurement of the photoelectric effect, with the ostensible intention of proving Einstein wrong, but he accomplished just the opposite—providing clean experimental evidence confirming Einstein’s theory of the photoelectric effect.
The Stimulated Emission of Light
About a year after Millikan proved that the quantum of energy associated with light absorption was absorbed as a whole quantum of energy that was not divisible, Einstein took a step further in his theory of the light quantum. In 1916 he published a paper in the proceedings of the German Physical Society that explored how light would be in a state of thermodynamic equilibrium when interacting with atoms that had discrete energy levels. Once again he used simple arguments, this time using the principle of detailed balance, to derive a new and unanticipated property of light—stimulated emission!
The stimulated emission of light occurs when an electron is in an excited state of a quantum system, like an atom, and an incident photon stimulates the emission of a second photon that has the same energy and phase as the first photon. If there are many atoms in the excited state, then this process leads to a chain reaction as 1 photon produces 2, and 2 produce 4, and 4 produce 8, etc. This exponential gain in photons with the same energy and phase is the origin of laser radiation. At the time that Einstein proposed this mechanism, lasers were half a century in the future, but he was led to this conclusion by extremely simple arguments about transition rates.
Detailed balance is a principle that states that in thermal equilibrium all fluxes are balanced. In the case of atoms with ground states and excited states, this principle requires that as many transitions occur from the ground state to the excited state as from the excited state to the ground state. The crucial new element that Einstein introduced was to distinguish spontaneous emission from stimulated emission. Just as the probability to absorb a photon must be proportional to the photon density, there must be an equivalent process that de-excites the atom that also must be proportional the photon density. In addition, an electron must be able to spontaneously emit a photon with a rate that is independent of photon density. This leads to distinct coefficients in the transition rate equations that are today called the “Einstein A and B coefficients”. The B coefficients relate to the photon density, while the A coefficient relates to spontaneous emission.
Using the principle of detailed balance together with his A and B coefficients as well as Boltzmann factors describing the number of excited states relative to ground state atoms in equilibrium at a given temperature, Einstein was able to derive an early form of what is today called the Bose-Einstein occupancy function for photons.
Derivation of the Einstein A and B Coefficients
Detailed balance requires the rate from m to n to be the same as the rate from n to m
where the first term is the spontaneous emission rate from the excited state m to the ground state n, the second term is the stimulated emission rate, and the third term (on the right) is the absorption rate from n to m. The numbers in each state are Nm and Nn, and the density of photons is ρ. The relative numbers in the excited state relative to the ground state is given by the Boltzmann factor
By assuming that the stimulated transition coefficient from n to m is the same as m to n, and inserting the Boltzmann factor yields
The Planck density of photons for ΔE = hf is
which yields the final relation between the spontaneous emission coefficient and the stimulated emission coefficient
The total emission rate is
where the p-bar is the average photon number in the cavity. One of the striking aspects of this derivation is that no assumptions are made about the physical mechanisms that determine the coefficient B. Only arguments of detailed balance are required to arrive at these results.
Einstein’s Quantum Legacy
Einstein was awarded the Nobel Prize in 1921 for the photoelectric effect, not for the photon nor for any of Einstein’s other theoretical accomplishments. Even in 1921, the quantum nature of light remained controversial. It was only in 1923, after the American physicist Arthur Compton (1892—1962) showed that energy and momentum were conserved in the scattering of photons from electrons, that the quantum nature of light began to be accepted. The very next year, in 1924, the quantum of light was named the “photon” by the American American chemical physicist Gilbert Lewis (1875—1946).
A blog article like this, that attributes the invention of the quantum to Einstein rather than Planck, must say something about the irony of this attribution. If Einstein is the father of the quantum, he ultimately was led to disinherit his own brain child. His final and strongest argument against the quantum properties inherent in the Copenhagen Interpretation was his famous EPR paper which, against his expectations, launched the concept of entanglement that underlies the coming generation of quantum computers.
Einstein’s Quantum Timeline
1900 – Planck’s quantum discontinuity for the calculation of the entropy of blackbody radiation.
1905 – Einstein’s “Miracle Year”. Proposes the light quantum.
1911 – First Solvay Conference on the theory of radiation and quanta.
1913 – Bohr’s quantum theory of hydrogen.
1914 – Einstein becomes a member of the German Academy of Science.
1915 – Millikan measurement of the photoelectric effect.
1916 – Einstein proposes stimulated emission.
1921 – Einstein receives Nobel Prize for photoelectric effect and the light quantum. Third Solvay Conference on atoms and electrons.
1927 – Heisenberg’s uncertainty relation. Fifth Solvay International Conference on Electrons and Photons in Brussels. “First” Bohr-Einstein debate on indeterminancy in quantum theory.
1930 – Sixth Solvay Conference on magnetism. “Second” Bohr-Einstein debate.
1935 – Einstein-Podolsky-Rosen (EPR) paper on the completeness of quantum mechanics.
Selected Einstein Quantum Papers
Einstein, A. (1905). “Generation and conversion of light with regard to a heuristic point of view.” Annalen Der Physik 17(6): 132-148.
Einstein, A. (1907). “Die Plancksche Theorie der Strahlung und die Theorie der spezifischen W ̈arme.” Annalen der Physik 22: 180–190.
Einstein, A. (1909). “On the current state of radiation problems.” Physikalische Zeitschrift 10: 185-193.
Einstein, A. and O. Stern (1913). “An argument for the acceptance of molecular agitation at absolute zero.” Annalen Der Physik 40(3): 551-560.
Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.
Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.
Einstein, A., B. Podolsky and N. Rosen (1935). “Can quantum-mechanical description of physical reality be considered complete?” Physical Review 47(10): 0777-0780.
 M. Planck, “Elementary quanta of matter and electricity,” Annalen Der Physik, vol. 4, pp. 564-566, Mar 1901.
 Klein, M. J. (1964). Einstein’s First Paper on Quanta. The natural philosopher. D. A. Greenberg and D. E. Gershenson. New York, Blaidsdell. 3.
 A. Einstein, “Generation and conversion of light with regard to a heuristic point of view,” Annalen Der Physik, vol. 17, pp. 132-148, Jun 1905.
It is surprising how much of modern dynamics boils down to an extremely simple formula
This innocuous-looking equation carries such riddles, such surprises, such unintuitive behavior that it can become the object of study for life. This equation is called a vector flow equation, and it can be used to capture the essential physics of economies, neurons, ecosystems, networks, and even orbits of photons around black holes. This equation is to modern dynamics what F = ma was to classical mechanics. It is the starting point for understanding complex systems.
The Magic of Phase Space
The apparent simplicity of the “flow equation” masks the complexity it contains. It is a vector equation because each “dimension” is a variable of a complex system. Many systems of interest may have only a few variables, but ecosystems and economies and social networks may have hundreds or thousands of variables. Expressed in component format, the flow equation is
where the superscript spans the number of variables. But even this masks all that can happen with such an equation. Each of the functions fa can be entirely different from each other, and can be any type of function, whether polynomial, rational, algebraic, transcendental or composite, although they must be single-valued. They are generally nonlinear, and the limitless ways that functions can be nonlinear is where the richness of the flow equation comes from.
The vector flow equation is an ordinary differential equation (ODE) that can be solved for specific trajectories as initial value problems. A single set of initial conditions defines a unique trajectory. For instance, the trajectory for a 4-dimensional example is described as the column vector
which is the single-parameter position vector to a point in phase space, also called state space. The point sweeps through successive configurations as a function of its single parameter—time. This trajectory is also called an orbit. In classical mechanics, the focus has tended to be on the behavior of specific orbits that arise from a specific set of initial conditions. This is the classic “rock thrown from a cliff” problem of introductory physics courses. However, in modern dynamics, the focus shifts away from individual trajectories to encompass the set of all possible trajectories.
Why is Modern Dynamics part of Physics?
If finding the solutions to the “x-dot equals f” vector flow equation is all there is to do, then this would just be a math problem—the solution of ODE’s. There are plenty of gems for mathematicians to look for, and there is an entire of field of study in mathematics called “dynamical systems“, but this would not be “physics”. Physics as a profession is separate and distinct from mathematics, although the two are sometimes confused. Physics uses mathematics as its language and as its toolbox, but physics is not mathematics. Physics is done best when it is done qualitatively—this means with scribbles done on napkins in restaurants or on the back of envelopes while waiting in line. Physics is about recognizing relationships and patterns. Physics is about identifying the limits to scaling properties where the physics changes when scales change. Physics is about the mapping of the simplest possible mathematics onto behavior in the physical world, and recognizing when the simplest possible mathematics is a universal that applies broadly to diverse systems that seem different, but that share the same underlying principles.
So, granted solving ODE’s is not physics, there is still a tremendous amount of good physics that can be done by solving ODE’s. ODE solvers become the modern physicist’s experimental workbench, providing data output from numerical experiments that can test the dependence on parameters in ways that real-world experiments might not be able to access. Physical intuition can be built based on such simulations as the engaged physicist begins to “understand” how the system behaves, able to explain what will happen as the values of parameters are changed.
In the follow sections, three examples of modern dynamics are introduced with a preliminary study, including Python code. These examples are: Galactic dynamics, synchronized networks and ecosystems. Despite their very different natures, their description using dynamical flows share features in common and illustrate the beauty and depth of behavior that can be explored with simple equations.
One example of the power and beauty of the vector flow equation and its set of all solutions in phase space is called the Henon-Heiles model of the motion of a star within a galaxy. Of course, this is a terribly complicated problem that involves tens of billions of stars, but if you average over the gravitational potential of all the other stars, and throw in a couple of conservation laws, the resulting potential can look surprisingly simple. The motion in the plane of this galactic potential takes two configuration coordinates (x, y) with two associated momenta (px, py) for a total of four dimensions. The flow equations in four-dimensional phase space are simply
where the terms in the light blue box describe a two-dimensional simple harmonic oscillator (SHO), which is a linear oscillator, modified by the terms in the magenta box that represent the nonlinear galactic potential. The orbits of this Hamiltonian system are chaotic, and because there is no dissipation in the model, a single orbit will continue forever within certain ranges of phase space governed by energy conservation, but never quite repeating.
# -*- coding: utf-8 -*-
Created on Wed Apr 18 06:03:32 2018
D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd ed. (Oxford,2019)
import numpy as np
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
from scipy import integrate
from matplotlib import pyplot as plt
from matplotlib import cm
# model_case 1 = Heiles
# model_case 2 = Crescent
print('Case: 1 = Heiles')
print('Case: 2 = Crescent')
model_case = int(input('Enter the Model Case (1-2)'))
if model_case == 1:
E = 1 # Heiles: 1, 0.3411 Crescent: 0.05, 1
epsE = 0.3411 # 3411
x, y, z, w = x_y_z_w
a = z
b = w
c = -x - epsE*(2*x*y)
d = -y - epsE*(x**2 - y**2)
E = .1 # Crescent: 0.1, 1
epsE = 1
x, y, z, w = x_y_z_w
a = z
b = w
c = -(epsE*(y-2*x**2)*(-4*x) + x)
d = -(y-epsE*2*x**2)
prms = np.sqrt(E)
pmax = np.sqrt(2*E)
# Potential Function
if model_case == 1:
V = np.zeros(shape=(100,100))
for xloop in range(100):
x = -2 + 4*xloop/100
for yloop in range(100):
y = -2 + 4*yloop/100
V[yloop,xloop] = 0.5*x**2 + 0.5*y**2 + epsE*(x**2*y - 0.33333*y**3)
V = np.zeros(shape=(100,100))
for xloop in range(100):
x = -2 + 4*xloop/100
for yloop in range(100):
y = -2 + 4*yloop/100
V[yloop,xloop] = 0.5*x**2 + 0.5*y**2 + epsE*(2*x**4 - 2*x**2*y)
fig = plt.figure(1)
contr = plt.contourf(V,100, cmap=cm.coolwarm, vmin = 0, vmax = 10)
fig.colorbar(contr, shrink=0.5, aspect=5)
fig = plt.show()
repnum = 250
mulnum = 64/repnum
for reploop in range(repnum):
px1 = 2*(np.random.random((1))-0.499)*pmax
py1 = np.sign(np.random.random((1))-0.499)*np.real(np.sqrt(2*(E-px1**2/2)))
xp1 = 0
yp1 = 0
x_y_z_w0 = [xp1, yp1, px1, py1]
tspan = np.linspace(1,1000,10000)
x_t = integrate.odeint(flow_deriv, x_y_z_w0, tspan)
siztmp = np.shape(x_t)
siz = siztmp
if reploop % 50 == 0:
lines = plt.plot(x_t[:,0],x_t[:,1])
y1 = x_t[:,0]
y2 = x_t[:,1]
y3 = x_t[:,2]
y4 = x_t[:,3]
py = np.zeros(shape=(2*repnum,))
yvar = np.zeros(shape=(2*repnum,))
cnt = -1
last = y1
for loop in range(2,siz):
if (last < 0)and(y1[loop] > 0):
cnt = cnt+1
del1 = -y1[loop-1]/(y1[loop] - y1[loop-1])
py[cnt] = y4[loop-1] + del1*(y4[loop]-y4[loop-1])
yvar[cnt] = y2[loop-1] + del1*(y2[loop]-y2[loop-1])
last = y1[loop]
last = y1[loop]
lines = plt.plot(yvar,py,'o',ms=1)
if model_case == 1:
Networks, Synchronization and Emergence
A central paradigm of nonlinear science is the emergence of patterns and organized behavior from seemingly random interactions among underlying constituents. Emergent phenomena are among the most awe inspiring topics in science. Crystals are emergent, forming slowly from solutions of reagents. Life is emergent, arising out of the chaotic soup of organic molecules on Earth (or on some distant planet). Intelligence is emergent, and so is consciousness, arising from the interactions among billions of neurons. Ecosystems are emergent, based on competition and symbiosis among species. Economies are emergent, based on the transfer of goods and money spanning scales from the local bodega to the global economy.
One of the common underlying properties of emergence is the existence of networks of interactions. Networks and network science are topics of great current interest driven by the rise of the World Wide Web and social networks. But networks are ubiquitous and have long been the topic of research into complex and nonlinear systems. Networks provide a scaffold for understanding many of the emergent systems. It allows one to think of isolated elements, like molecules or neurons, that interact with many others, like the neighbors in a crystal or distant synaptic connections.
From the point of view of modern dynamics, the state of a node can be a variable or a “dimension” and the interactions among links define the functions of the vector flow equation. Emergence is then something that “emerges” from the dynamical flow as many elements interact through complex networks to produce simple or emergent patterns.
Synchronization is a form of emergence that happens when lots of independent oscillators, each vibrating at their own personal frequency, are coupled together to push and pull on each other, entraining all the individual frequencies into one common global oscillation of the entire system. Synchronization plays an important role in the solar system, explaining why the Moon always shows one face to the Earth, why Saturn’s rings have gaps, and why asteroids are mainly kept away from colliding with the Earth. Synchronization plays an even more important function in biology where it coordinates the beating of the heart and the functioning of the brain.
One of the most dramatic examples of synchronization is the Kuramoto synchronization phase transition. This occurs when a large set of individual oscillators with differing natural frequencies interact with each other through a weak nonlinear coupling. For small coupling, all the individual nodes oscillate at their own frequency. But as the coupling increases, there is a sudden coalescence of all the frequencies into a single common frequency. This mechanical phase transition, called the Kuramoto transition, has many of the properties of a thermodynamic phase transition, including a solution that utilizes mean field theory.
The simulation of 20 Poncaré phase oscillators with global coupling is shown in Fig. 4 as a function of increasing coupling coefficient g. The original individual frequencies are spread randomly. The oscillators with similar frequencies are the first to synchronize, forming small clumps that then synchronize with other clumps of oscillators, until all oscillators are entrained to a single compromise frequency. The Kuramoto phase transition is not sharp in this case because the value of N = 20 is too small. If the simulation is run for 200 oscillators, there is a sudden transition from unsynchronized to synchronized oscillation at a threshold value of g.
The Kuramoto phase transition is one of the most important fundamental examples of modern dynamics because it illustrates many facets of nonlinear dynamics in a very simple way. It highlights the importance of nonlinearity, the simplification of phase oscillators, the use of mean field theory, the underlying structure of the network, and the example of a mechanical analog to a thermodynamic phase transition. It also has analytical solutions because of its simplicity, while still capturing the intrinsic complexity of nonlinear systems.
# -*- coding: utf-8 -*-
Created on Sat May 11 08:56:41 2019
D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd ed. (Oxford,2019)
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
import networkx as nx
from UserFunction import linfit
tstart = time.time()
Nfac = 25 # 25
N = 50 # 50
width = 0.2
# model_case 1 = complete graph (Kuramoto transition)
# model_case 2 = Erdos-Renyi
model_case = int(input('Input Model Case (1-2)'))
if model_case == 1:
facoef = 0.2
nodecouple = nx.complete_graph(N)
elif model_case == 2:
facoef = 5
nodecouple = nx.erdos_renyi_graph(N,0.1)
# function: omegout, yout = coupleN(G)
# function: yd = flow_deriv(x_y)
yp = np.zeros(shape=(N,))
for omloop in range(N):
temp = omega[omloop]
linksz = G.node[omloop]['numlink']
for cloop in range(linksz):
cindex = G.node[omloop]['link'][cloop]
g = G.node[omloop]['coupling'][cloop]
temp = temp + g*np.sin(y[cindex]-y[omloop])
yp[omloop] = temp
yd = np.zeros(shape=(N,))
for omloop in range(N):
yd[omloop] = yp[omloop]
# end of function flow_deriv(x_y)
mnomega = 1.0
for nodeloop in range(N):
omega[nodeloop] = G.node[nodeloop]['element']
x_y_z = omega
# Settle-down Solve for the trajectories
tsettle = 100
t = np.linspace(0, tsettle, tsettle)
x_t = integrate.odeint(flow_deriv, x_y_z, t)
x0 = x_t[tsettle-1,0:N]
t = np.linspace(1,1000,1000)
y = integrate.odeint(flow_deriv, x0, t)
siztmp = np.shape(y)
sy = siztmp
# Fit the frequency
m = np.zeros(shape = (N,))
w = np.zeros(shape = (N,))
mtmp = np.zeros(shape=(4,))
btmp = np.zeros(shape=(4,))
for omloop in range(N):
if np.remainder(sy,4) == 0:
mtmp,btmp = linfit(t[0:sy//2],y[0:sy//2,omloop]);
mtmp,btmp = linfit(t[sy//2+1:sy],y[sy//2+1:sy,omloop]);
mtmp,btmp = linfit(t[sy//4+1:3*sy//4],y[sy//4+1:3*sy//4,omloop]);
mtmp,btmp = linfit(t,y[:,omloop]);
sytmp = 4*np.floor(sy/4);
mtmp,btmp = linfit(t[0:sytmp//2],y[0:sytmp//2,omloop]);
mtmp,btmp = linfit(t[sytmp//2+1:sytmp],y[sytmp//2+1:sytmp,omloop]);
mtmp,btmp = linfit(t[sytmp//4+1:3*sytmp/4],y[sytmp//4+1:3*sytmp//4,omloop]);
mtmp,btmp = linfit(t[0:sytmp],y[0:sytmp,omloop]);
#m[omloop] = np.median(mtmp)
m[omloop] = np.mean(mtmp)
w[omloop] = mnomega + m[omloop]
omegout = m
yout = y
return omegout, yout
# end of function: omegout, yout = coupleN(G)
Nlink = N*(N-1)//2
omega = np.zeros(shape=(N,))
omegatemp = width*(np.random.rand(N)-1)
meanomega = np.mean(omegatemp)
omega = omegatemp - meanomega
sto = np.std(omega)
lnk = np.zeros(shape = (N,), dtype=int)
for loop in range(N):
nodecouple.node[loop]['element'] = omega[loop]
nodecouple.node[loop]['link'] = list(nx.neighbors(nodecouple,loop))
nodecouple.node[loop]['numlink'] = np.size(list(nx.neighbors(nodecouple,loop)))
lnk[loop] = np.size(list(nx.neighbors(nodecouple,loop)))
avgdegree = np.mean(lnk)
mnomega = 1
facval = np.zeros(shape = (Nfac,))
yy = np.zeros(shape=(Nfac,N))
xx = np.zeros(shape=(Nfac,))
for facloop in range(Nfac):
fac = facoef*(16*facloop/(Nfac))*(1/(N-1))*sto/mnomega
for nodeloop in range(N):
nodecouple.node[nodeloop]['coupling'] = np.zeros(shape=(lnk[nodeloop],))
for linkloop in range (lnk[nodeloop]):
nodecouple.node[nodeloop]['coupling'][linkloop] = fac
facval[facloop] = fac*avgdegree
omegout, yout = coupleN(nodecouple) # Here is the subfunction call for the flow
for omloop in range(N):
yy[facloop,omloop] = omegout[omloop]
xx[facloop] = facval[facloop]
lines = plt.plot(xx,yy)
elapsed_time = time.time() - tstart
print('elapsed time = ',format(elapsed_time,'.2f'),'secs')
The Web of Life
Ecosystems are among the most complex systems on Earth. The complex interactions among hundreds or thousands of species may lead to steady homeostasis in some cases, to growth and collapse in other cases, and to oscillations or chaos in yet others. But the definition of species can be broad and abstract, referring to businesses and markets in economic ecosystems, or to cliches and acquaintances in social ecosystems, among many other examples. These systems are governed by the laws of evolutionary dynamics that include fitness and survival as well as adaptation.
The dimensionality of the dynamical spaces for these systems extends to hundreds or thousands of dimensions—far too complex to visualize when thinking in four dimensions is already challenging. Yet there are shared principles and common behaviors that emerge even here. Many of these can be illustrated in a simple three-dimensional system that is represented by a triangular simplex that can be easily visualized, and then generalized back to ultra-high dimensions once they are understood.
A simplex is a closed (N-1)-dimensional geometric figure that describes a zero-sum game (game theory is an integral part of evolutionary dynamics) among N competing species. For instance, a two-simplex is a triangle that captures the dynamics among three species. Each vertex of the triangle represents the situation when the entire ecosystem is composed of a single species. Anywhere inside the triangle represents the situation when all three species are present and interacting.
A classic model of interacting species is the replicator equation. It allows for a fitness-based proliferation and for trade-offs among the individual species. The replicator dynamics equations are shown in Fig. 5.
The population dynamics on the 2D simplex are shown in Fig. 6 for several different pay-off matrices. The matrix values are shown in color and help interpret the trajectories. For instance the simplex on the upper-right shows a fixed point center. This reflects the antisymmetric character of the pay-off matrix around the diagonal. The stable spiral on the lower-left has a nearly asymmetric pay-off matrix, but with unequal off-diagonal magnitudes. The other two cases show central saddle points with stable fixed points on the boundary. A very large variety of behaviors are possible for this very simple system. The Python program is shown in Trirep.py.
# -*- coding: utf-8 -*-
Created on Thu May 9 16:23:30 2019
D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd ed. (Oxford,2019)
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
sm = x + y + z
xp = x/sm
yp = y/sm
f = np.sqrt(3)/2
y0 = f*xp
x0 = -0.5*xp - yp + 1;
lines = plt.plot(x0,y0)
plt.plot([0, 1],[0, 0],'k',linewidth=1)
plt.plot([0, 0.5],[0, f],'k',linewidth=1)
plt.plot([1, 0.5],[0, f],'k',linewidth=1)
def flow_deriv(y, t0):
#"""Compute the time-derivative ."""
f = np.zeros(shape=(N,))
for iloop in range(N):
ftemp = 0
for jloop in range(N):
ftemp = ftemp + A[iloop,jloop]*y[jloop]
f[iloop] = ftemp
phitemp = phi0 # Can adjust this from 0 to 1 to stabilize (but Nth population is no longer independent)
for loop in range(N):
phitemp = phitemp + f[loop]*y[loop]
phi = phitemp
yd = np.zeros(shape=(N,))
for loop in range(N-1):
yd[loop] = y[loop]*(f[loop] - phi);
if np.abs(phi0) < 0.01: # average fitness maintained at zero
yd[N-1] = y[N-1]*(f[N-1]-phi);
else: # non-zero average fitness
ydtemp = 0
for loop in range(N-1):
ydtemp = ydtemp - yd[loop]
yd[N-1] = ydtemp
# Solve for the trajectories
t = np.linspace(0, tspan, 701)
x_t = integrate.odeint(flow_deriv,y,t)
return t, x_t
# model_case 1 = zero diagonal
# model_case 2 = zero trace
# model_case 3 = asymmetric (zero trace)
print('Case: 1 = antisymm zero diagonal')
print('Case: 2 = antisymm zero trace')
print('Case: 3 = random')
model_case = int(input('Enter the Model Case (1-3)'))
N = 3
asymm = 3 # 1 = zero diag (replicator eqn) 2 = zero trace (autocatylitic model) 3 = random (but zero trace)
phi0 = 0.001 # average fitness (positive number) damps oscillations
T = 100;
if model_case == 1:
Atemp = np.zeros(shape=(N,N))
for yloop in range(N):
for xloop in range(yloop+1,N):
Atemp[yloop,xloop] = 2*(0.5 - np.random.random(1))
Atemp[xloop,yloop] = -Atemp[yloop,xloop]
if model_case == 2:
Atemp = np.zeros(shape=(N,N))
for yloop in range(N):
for xloop in range(yloop+1,N):
Atemp[yloop,xloop] = 2*(0.5 - np.random.random(1))
Atemp[xloop,yloop] = -Atemp[yloop,xloop]
Atemp[yloop,yloop] = 2*(0.5 - np.random.random(1))
tr = np.trace(Atemp)
A = Atemp
for yloop in range(N):
A[yloop,yloop] = Atemp[yloop,yloop] - tr/N
Atemp = np.zeros(shape=(N,N))
for yloop in range(N):
for xloop in range(N):
Atemp[yloop,xloop] = 2*(0.5 - np.random.random(1))
tr = np.trace(Atemp)
A = Atemp
for yloop in range(N):
A[yloop,yloop] = Atemp[yloop,yloop] - tr/N
im = plt.matshow(A,3,cmap=plt.cm.get_cmap('seismic')) # hsv, seismic, bwr
cbar = im.figure.colorbar(im)
M = 20
delt = 1/M
ep = 0.01;
tempx = np.zeros(shape = (3,))
for xloop in range(M):
tempx = delt*(xloop)+ep;
for yloop in range(M-xloop):
tempx = delt*yloop+ep
tempx = 1 - tempx - tempx
x0 = tempx/np.sum(tempx); # initial populations
tspan = 70
t, x_t = solve_flow(x0,tspan)
y1 = x_t[:,0]
y2 = x_t[:,1]
y3 = x_t[:,2]
lines = plt.plot(t,y1,t,y2,t,y3)
Topics in Modern Dynamics
These three examples are just the tip of the iceberg. The topics in modern dynamics are almost numberless. Any system that changes in time is a potential object of study in modern dynamics. Here is a list of a few topics that spring to mind.
In one of my previous blog posts, as I was searching for Schwarzschild’s original papers on Einstein’s field equations and quantum theory, I obtained a copy of the January 1916 – June 1916 volume of the Proceedings of the Royal Prussian Academy of Sciences through interlibrary loan. The extremely thick volume arrived at Purdue about a week after I ordered it online. It arrived from Oberlin College in Ohio that had received it as a gift in 1928 from the library of Professor Friedrich Loofs of the University of Halle in Germany. Loofs had been the Haskell Lecturer at Oberlin for the 1911-1912 semesters.
As I browsed through the volume looking for Schwarzschild’s papers, I was amused to find a cornucopia of turn-of-the-century science topics recorded in its pages. There were papers on the overbite and lips of marsupials. There were papers on forgotten languages. There were papers on ancient Greek texts. On the origins of religion. On the philosophy of abstraction. Histories of Indian dramas. Reflections on cancer. But what I found most amazing was a snapshot of the field of physics and mathematics in 1916, with historic papers by historic scientists who changed how we view the world. Here is a snapshot in time and in space, a period of only six months from a single journal, containing papers from authors that reads like a who’s who of physics.
In 1916 there were three major centers of science in the world with leading science publications: London with the Philosophical Magazine and Proceedings of the Royal Society; Paris with the Comptes Rendus of the Académie des Sciences; and Berlin with the Proceedings of the Royal Prussian Academy of Sciences and Annalen der Physik. In Russia, there were the scientific Journals of St. Petersburg, but the Bolshevik Revolution was brewing that would overwhelm that country for decades. And in 1916 the academic life of the United States was barely worth noticing except for a few points of light at Yale and Johns Hopkins.
Berlin in 1916 was embroiled in war, but science proceeded relatively unmolested. The six-month volume of the Proceedings of the Royal Prussian Academy of Sciences contains a number of gems. Schwarzschild was one of the most prolific contributors, publishing three papers in just this half-year volume, plus his obituary written by Einstein. But joining Schwarzschild in this volume were Einstein, Planck, Born, Warburg, Frobenious, and Rubens among others—a pantheon of German scientists mostly cut off from the rest of the world at that time, but single-mindedly following their individual threads woven deep into the fabric of the physical world.
Karl Schwarzschild (1873 – 1916)
Schwarzschild had the unenviable yet effective motivation of his impending death to spur him to complete several projects that he must have known would make his name immortal. In this six-month volume he published his three most important papers. The first (pg. 189) was on the exact solution to Einstein’s field equations to general relativity. The solution was for the restricted case of a point mass, yet the derivation yielded the Schwarzschild radius that later became known as the event horizon of a non-roatating black hole. The second paper (pg. 424) expanded the general relativity solutions to a spherically symmetric incompressible liquid mass.
The subject, content and success of these two papers was wholly unexpected from this observational astronomer stationed on the Russian Front during WWI calculating trajectories for German bombardments. He would not have been considered a theoretical physicist but for the importance of his results and the sophistication of his methods. Within only a year after Einstein published his general theory, based as it was on the complicated tensor calculus of Levi-Civita, Christoffel and Ricci-Curbastro that had taken him years to master, Schwarzschild found a solution that evaded even Einstein.
Schwarzschild’s third and final paper (pg. 548) was on an entirely different topic, still not in his official field of astronomy, that positioned all future theoretical work in quantum physics to be phrased in the language of Hamiltonian dynamics and phase space. He proved that action-angle coordinates were the only acceptable canonical coordinates to be used when quantizing dynamical systems. This paper answered a central question that had been nagging Bohr and Einstein and Ehrenfest for years—how to quantize dynamical coordinates. Despite the simple way that Bohr’s quantized hydrogen atom is taught in modern physics, there was an ambiguity in the quantization conditions even for this simple single-electron atom. The ambiguity arose from the numerous possible canonical coordinate transformations that were admissible, yet which led to different forms of quantized motion.
Schwarzschild’s doctoral thesis had been a theoretical topic in astrophysics that applied the celestial mechanics theories of Henri Poincaré to binary star systems. Within Poincaré’s theory were integral invariants that were conserved quantities of the motion. When a dynamical system had as many constraints as degrees of freedom, then every coordinate had an integral invariant. In this unexpected last paper from Schwarzschild, he showed how canonical transformation to action-angle coordinates produced a unique representation in terms of action variables (whose dimensions are the same as Planck’s constant). These action coordinates, with their associated cyclical angle variables, are the only unambiguous representations that can be quantized. The important points of this paper were amplified a few months later in a publication by Schwarzschild’s friend Paul Epstein (1871 – 1939), solidifying this approach to quantum mechanics. Paul Ehrenfest (1880 – 1933) continued this work later in 1916 by defining adiabatic invariants whose quantum numbers remain unchanged under slowly varying conditions, and the program started by Schwarzschild was definitively completed by Paul Dirac (1902 – 1984) at the dawn of quantum mechanics in Göttingen in 1925.
Albert Einstein (1879 – 1955)
In 1916 Einstein was mopping up after publishing his definitive field equations of general relativity the year before. His interests were still cast wide, not restricted only to this latest project. In the 1916 Jan. to June volume of the Prussian Academy Einstein published two papers. Each is remarkably short relative to the other papers in the volume, yet the importance of the papers may stand in inverse proportion to their length.
The first paper (pg. 184) is placed right before Schwarzschild’s first paper on February 3. The subject of the paper is the expression of Maxwell’s equations in four-dimensional space time. It is notable and ironic that Einstein mentions Hermann Minkowski (1864 – 1909) in the first sentence of the paper. When Minkowski proposed his bold structure of spacetime in 1908, Einstein had been one of his harshest critics, writing letters to the editor about the absurdity of thinking of space and time as a single interchangeable coordinate system. This is ironic, because Einstein today is perhaps best known for the special relativity properties of spacetime, yet he was slow to adopt the spacetime viewpoint. Einstein only came around to spacetime when he realized around 1910 that a general approach to relativity required the mathematical structure of tensor manifolds, and Minkowski had provided just such a manifold—the pseudo-Riemannian manifold of space time. Einstein subsequently adopted spacetime with a passion and became its greatest champion, calling out Minkowski where possible to give him his due, although he had already died tragically of a burst appendix in 1909.
The importance of Einstein’s paper hinges on his derivation of the electromagnetic field energy density using electromagnetic four vectors. The energy density is part of the source term for his general relativity field equations. Any form of energy density can warp spacetime, including electromagnetic field energy. Furthermore, the Einstein field equations of general relativity are nonlinear as gravitational fields modify space and space modifies electromagnetic fields, producing a coupling between gravity and electromagnetism. This coupling is implicit in the case of the bending of light by gravity, but Einstein’s paper from 1916 makes the connection explicit.
Einstein’s second paper (pg. 688) is even shorter and hence one of the most daring publications of his career. Because the field equations of general relativity are nonlinear, they are not easy to solve exactly, and Einstein was exploring approximate solutions under conditions of slow speeds and weak fields. In this “non-relativistic” limit the metric tensor separates into a Minkowski metric as a background on which a small metric perturbation remains. This small perturbation has the properties of a wave equation for a disturbance of the gravitational field that propagates at the speed of light. Hence, in the June 22 issue of the Prussian Academy in 1916, Einstein predicts the existence and the properties of gravitational waves. Exactly one hundred years later in 2016, the LIGO collaboration announced the detection of gravitational waves generated by the merger of two black holes.
Max Planck (1858 – 1947)
Max Planck was active as the secretary of the Prussian Academy in 1916 yet was still fully active in his research. Although he had launched the quantum revolution with his quantum hypothesis of 1900, he was not a major proponent of quantum theory even as late as 1916. His primary interests lay in thermodynamics and the origins of entropy, following the theoretical approaches of Ludwig Boltzmann (1844 – 1906). In 1916 he was interested in how to best partition phase space as a way to count states and calculate entropy from first principles. His paper in the 1916 volume (pg. 653) calculated the entropy for single-atom solids.
Max Born (1882 – 1970)
Max Born was to be one of the leading champions of the quantum mechanical revolution based at the University of Göttingen in the 1920’s. But in 1916 he was on leave from the University of Berlin working on ranging for artillery. Yet he still pursued his academic interests, like Schwarzschild. On pg. 614 in the Proceedings of the Prussian Academy, Born published a paper on anisotropic liquids, such as liquid crystals and the effect of electric fields on them. It is astonishing to think that so many of the flat-panel displays we have today, whether on our watches or smart phones, are technological descendants of work by Born at the beginning of his career.
Ferdinand Frobenius (1849 – 1917)
Like Schwarzschild, Frobenius was at the end of his career in 1916 and would pass away one year later, but unlike Schwarzschild, his career had been a long one, receiving his doctorate under Weierstrass and exploring elliptic functions, differential equations, number theory and group theory. One of the papers that established him in group theory appears in the May 4th issue on page 542 where he explores the series expansion of a group.
Heinrich Rubens (1865 – 1922)
Max Planck owed his quantum breakthrough in part to the exquisitely accurate experimental measurements made by Heinrich Rubens on black body radiation. It was only by the precise shape of what came to be called the Planck spectrum that Planck could say with such confidence that his theory of quantized radiation interactions fit Rubens spectrum so perfectly. In 1916 Rubens was at the University of Berlin, having taken the position vacated by Paul Drude in 1906. He was a specialist in infrared spectroscopy, and on page 167 of the Proceedings he describes the spectrum of steam and its consequences for the quantum theory.
Emil Warburg (1946 – 1931)
Emil Warburg’s fame is primarily as the father of Otto Warburg who won the 1931 Nobel prize in physiology. On page 314 Warburg reports on photochemical processes in BrH gases. In an obscure and very indirect way, I am an academic descendant of Emil Warburg. One of his students was Robert Pohl who was a famous early researcher in solid state physics, sometimes called the “father of solid state physics”. Pohl was at the physics department in Göttingen in the 1920’s along with Born and Franck during the golden age of quantum mechanics. Robert Pohl’s son, Robert Otto Pohl, was my professor when I was a sophomore at Cornell University in 1978 for the course on introductory electromagnetism using a textbook by the Nobel laureate Edward Purcell, a quirky volume of the Berkeley Series of physics textbooks. This makes Emil Warburg my professor’s father’s professor.
Papers in the 1916 Vol. 1 of the Prussian Academy of Sciences
Schulze, Alt– und Neuindisches
Orth, Zur Frage nach den Beziehungen des Alkoholismus zur Tuberkulose
Schulze, Die Erhabunen auf der Lippin- und Wangenschleimhaut der Säugetiere
von Wilamwitz-Moellendorff, Die Samie des Menandros
Engler, Bericht über das >>Pflanzenreich<<
von Harnack, Bericht über die Ausgabe der griechischen Kirchenväter der dri ersten Jahrhunderte
Meinecke, Germanischer und romanischer Geist im Wandel der deutschen Geschichtsauffassung
Rubens und Hettner, Das langwellige Wasserdampfspektrum und seine Deutung durch die Quantentheorie
Einstein, Eine neue formale Deutung der Maxwellschen Feldgleichungen der Electrodynamic
Schwarschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
Helmreich, Handschriftliche Verbesserungen zu dem Hippokratesglossar des Galen
Prager, Über die Periode des veränderlichen Sterns RR Lyrae
Holl, Die Zeitfolge des ersten origenistischen Streits
Lüders, Zu den Upanisads. I. Die Samvargavidya
Warburg, Über den Energieumsatz bei photochemischen Vorgängen in Gasen. VI.
Hellman, Über die ägyptischen Witterungsangaben im Kalender von Claudius Ptolemaeus
Meyer-Lübke, Die Diphthonge im Provenzaslischen
Diels, Über die Schrift Antipocras des Nikolaus von Polen
Müller und Sieg, Maitrisimit und >>Tocharisch<<
Meyer, Ein altirischer Heilsegen
Schwarzschild, Über das Gravitationasfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie
Brauer, Die Verbreitung der Hyracoiden
Correns, Untersuchungen über Geschlechtsbestimmung bei Distelarten
Brahn, Weitere Untersuchungen über Fermente in der Lever von Krebskranken
Erdmann, Methodologische Konsequenzen aus der Theorie der Abstraktion
Bang, Studien zur vergleichenden Grammatik der Türksprachen. I.
Frobenius, Über die Kompositionsreihe einer Gruppe
Schwarzschild, Zur Quantenhypothese
Fischer und Bergmann,Über neue Galloylderivate des Traubenzuckers und ihren Vergleich mit der Chebulinsäure
Schuchhardt, Der starke Wall und die breite, zuweilen erhöhte Berme bei frügeschichtlichen Burgen in Norddeutschland
Born, Über anisotrope Flüssigkeiten
Planck, Über die absolute Entropie einatomiger Körper
Haberlandt, Blattepidermis und Lichtperzeption
Einstein, Näherungsweise Integration der Feldgleichungen der Gravitation
Lüders, Die Saubhikas. Ein Beitrag zur Gecschichte des indischen Dramas
In an ironic twist of the history of physics, Karl Schwarzschild’s fame has eclipsed his own legacy. When asked who was Karl Schwarzschild (1873 – 1916), you would probably say he’s the guy who solved Einstein’s Field Equations of General Relativity and discovered the radius of black holes. You may also know that he accomplished this Herculean feat while dying slowly behind the German lines on the Eastern Front in WWI. But asked what else he did, and you would probably come up blank. Yet Schwarzschild was one of the most wide-ranging physicists at the turn of the 20th century, which is saying something, because it places him into the same pantheon as Planck, Lorentz, Poincaré and Einstein. Let’s take a look at the part of his career that hides in the shadow of his own radius.
Radius of Interest
Karl Schwarzschild was born in Frankfurt, Germany, shortly after the Franco-Prussian war thrust Prussia onto the world stage as a major political force in Europe. His family were Jewish merchants of longstanding reputation in the city, and Schwarzschild’s childhood was spent in the vibrant Jewish community. One of his father’s friends was a professor at a university in Frankfurt, whose son, Paul Epstein (1871 – 1939), became a close friend of Karl’s at the Gymnasium. Schwarzshild and Epstein would partially shadow each other’s careers despite the fact that Schwarzschild became an astronomer while Epstein became a famous mathematician and number theorist. This was in part because Schwarzschild had large radius of interests that spanned the breadth of current mathematics and science, practicing both experiments and theory.
Schwarzschild’s application of the Hamiltonian formalism for quantum systems set the stage for the later adoption of Hamiltonian methods in quantum mechanics. He came dangerously close to stating the uncertainty principle that catapulted Heisenberg to fame.
By the time Schwarzschild was sixteen, he had taught himself the mathematics of celestial mechanics to such depth that he published two papers on the orbits of binary stars. He also became fascinated in astronomy and purchased lenses and other materials to construct his own telescope. His interests were helped along by Epstein, two years older and whose father had his own private observatory. When Epstein went to study at the University of Strasbourg (then part of the German Federation) Schwarzschild followed him. But Schwarzschild’s main interest in astronomy diverged from Epstein’s main interest in mathematics, and Schwarzschild transferred to the University of Munich where he studied under Hugo von Seeliger (1849 – 1924), the premier German astronomer of his day. Epstein remained at Strasbourg where he studied under Bruno Christoffel (1829 – 1900) and eventually became a professor, but he was forced to relinquish the post when Strasbourg was ceded to France after WWI.
Birth of Stellar Interferometry
the Hubble space telescope was launched in 1990 no star had ever been resolved
as a direct image. Within a year of its
launch, using its spectacular resolving power, the Hubble optics resolved—just
barely—the red supergiant Betelgeuse. No
other star (other than the Sun) is close enough or big enough to image the
stellar disk, even for the Hubble far above our atmosphere. The reason is that the diameter of the
optical lenses and mirrors of the Hubble—as big as they are at 2.4 meter
diameter—still produce a diffraction pattern that smears the image so that
stars cannot be resolved. Yet
information on the size of a distant object is encoded as phase in the light
waves that are emitted from the object, and this phase information is
accessible to interferometry.
The first physicist who truly grasped the power of optical interferometry and who understood how to design the first interferometric metrology systems was the French physicist Armand Hippolyte Louis Fizeau (1819 – 1896). Fizeau became interested in the properties of light when he collaborated with his friend Léon Foucault (1819–1868) on early uses of photography. The two then embarked on a measurement of the speed of light but had a falling out before the experiment could be finished, and both continued the pursuit independently. Fizeau achieved the first measurement using a toothed wheel rotating rapidly , while Foucault came in second using a more versatile system with a spinning mirror . Yet Fizeau surpassed Foucault in optical design and became an expert in interference effects. Interference apparatus had been developed earlier by Augustin Fresnel (the Fresnel bi-prism 1819), Humphrey Lloyd (Lloyd’s mirror 1834) and Jules Jamin (Jamin’s interferential refractor 1856). They had found ways of redirecting light using refraction and reflection to cause interference fringes. But Fizeau was one of the first to recognize that each emitting region of a light source was coherent with itself, and he used this insight and the use of lenses to design the first interferometer.
Fizeau’s interferometer used a lens with a with a tight focal spot masked off by an opaque screen with two open slits. When the masked lens device was focused on an intense light source it produced two parallel pencils of light that were mutually coherent but spatially separated. Fizeau used this apparatus to measure the speed of light in moving water in 1859 .
The working principle of the Fizeau refractometer is shown in Fig. 1. The light source is at the bottom, and it is reflected by the partially-silvered beam splitter to pass through the lens and the mask containing two slits. (Only the light paths that pass through the double-slit mask on the lens are shown in the figure.) The slits produce two pencils of mutually coherent light that pass through a system (in the famous Fizeau ether drag experiment it was along two tubes of moving water) and are returned through the same slits, and they intersect at the view port where they produce interference fringes. The fringe spacing is set by the separation of the two slits in the mask. The Rayleigh region of the lens defines a region of spatial coherence even for a so-called “incoherent” source. Therefore, this apparatus, by use of the lens, could convert an incoherent light source into a coherent probe to test the refractive index of test materials, which is why it was called a refractometer.
Fizeau became adept at thinking of alternative optical designs of his refractometer and alternative applications. In an address to the French Physical Society in 1868 he suggested that the double-slit mask could be used on a telescope to determine sizes of distant astronomical objects . There were several subsequent attempts to use Fizeau’s configuration in astronomical observations, but none were conclusive and hence were not widely known.
An optical configuration and astronomical application that was very similar to Fizeau’s idea was proposed by Albert Michelson in 1890 . He built the apparatus and used it to successfully measure the size of several moons of Jupiter . The configuration of the Michelson stellar interferometer is shown in Fig. 2. Light from a distant star passes through two slits in the mask in front of the collecting optics of a telescope. When the two pencils of light intersect at the view port, they produce interference fringes. Because of the finite size of the stellar source, the fringes are partially washed out. By adjusting the slit separation, a certain separation can be found where the fringes completely wash out. The size of the star is then related to the separation of the slits for which the fringe visibility vanishes. This simple principle allows this type of stellar interferometry to measure the size of stars that are large and relatively close to Earth. However, if stars are too far away even this approach cannot be used to measure their sizes because telescopes aren’t big enough. This limitation is currently being bypassed by the use of long-baseline optical interferometers.
One of the open questions in the history of interferometry is whether Michelson was aware of Fizeau’s proposal for the stellar interferometer made in 1868. Michelson was well aware of Fizeau’s published research and acknowledged him as a direct inspiration of his own work in interference effects. But Michelson also was unaware of the undercurrents in the French school of optical interference. When he visited Paris in 1881, he met with many of the leading figures in this school (including Lippmann and Cornu), but there is no mention or any evidence that he met with Fizeau. By this time Fizeau’s wife had passed away, and Fizeau spent most of his time in seclusion at his home outside Paris. Therefore, it is unlikely that he would have been present during Michelson’s visit. Because Michelson viewed Fizeau with such awe and respect, if he had met him, he most certainly would have mentioned it. Therefore, Michelson’s invention of the stellar interferometer can be considered with some confidence to be a case of independent discovery. It is perhaps not surprising that he hit on the same idea that Fizeau had in 1868, because Michelson was one of the few physicists who understood coherence and interference at the same depth as Fizeau.
The physics of the Michelson stellar interferometer is very similar to the physics of Young’s double slit experiment. The two slits in the aperture mask of the telescope objective act to produce a simple sinusoidal interference pattern at the image plane of the optical system. The size of the stellar diameter is determined by using the wash-out effect of the fringes caused by the finite stellar size. However, it is well known to physicists who work with diffraction gratings that a multiple-slit interference pattern has a much greater resolving power than a simple double slit.
This realization must have hit von Seeliger and Schwarzschild, working together at Munich, when they saw the publication of Michelson’s theoretical analysis of his stellar interferometer in 1890, followed by his use of the apparatus to measure the size of Jupiter’s moons. Schwarzschild and von Seeliger realized that by replacing the double-slit mask with a multiple-slit mask, the widths of the interference maxima would be much narrower. Such a diffraction mask on a telescope would cause a star to produce a multiple set of images on the image plane of the telescope associated with the multiple diffraction orders. More interestingly, if the target were a binary star, the diffraction would produce two sets of diffraction maxima—a double image! If the “finesse” of the grating is high enough, the binary star separation could be resolved as a doublet in the diffraction pattern at the image, and the separation could be measured, giving the angular separation of the two stars of the binary system. Such an approach to the binary separation would be a direct measurement, which was a distinct and clever improvement over the indirect Michelson configuration that required finding the extinction of the fringe visibility.
Schwarzschild enlisted the help of a fine German instrument maker to create a multiple slit system that had an adjustable slit separation. The device is shown in Fig. 3 from Schwarzschild’s 1896 publication on the use of the stellar interferometer to measure the separation of binary stars . The device is ingenious. By rotating the chain around the gear on the right-hand side of the apparatus, the two metal plates with four slits could be raised or lowered, cause the projection onto the objective plane to have variable slit spacings. In the operation of the telescope, the changing height of the slits does not matter, because they are near a conjugate optical plane (the entrance pupil) of the optical system. Using this adjustable multiple slit system, Schwarzschild (and two colleagues he enlisted) made multiple observations of well-known binary star systems, and they calculated the star separations. Several of their published results are shown in Fig. 4.
Schwarzschild’s publication demonstrated one of the very first uses of stellar interferometry—well before Michelson himself used his own configuration to measure the diameter of Betelgeuse in 1920. Schwarzschild’s major achievement was performed before he had received his doctorate, on a topic orthogonal to his dissertation topic. Yet this fact is virtually unknown to the broader physics community outside of astronomy. If he had not become so famous later for his solution of Einstein’s field equations, Schwarzschild nonetheless might have been famous for his early contributions to stellar interferometry. But even this was not the end of his unique contributions to physics.
As Schwarzschild worked for his doctorate under von Seeliger, his dissertation topic was on new theories by Henri Poincaré (1854 – 1912) on celestial mechanics. Poincaré had made a big splash on the international stage with the publication of his prize-winning memoire in 1890 on the three-body problem. This is the publication where Poincaré first described what would later become known as chaos theory. The memoire was followed by his volumes on “New Methods in Celestial Mechanics” published between 1892 and 1899. Poincaré’s work on celestial mechanics was based on his earlier work on the theory of dynamical systems where he discovered important invariant theorems, such as Liouville’s theorem on the conservation of phase space volume. Schwarzshild applied Poincaré’s theorems to problems in celestial orbits. He took his doctorate in 1896 and received a post at an astronomical observatory outside Vienna.
While at Vienna, Schwarzschild performed his most important sustained contributions to the science of astronomy. Astronomical observations had been dominated for centuries by the human eye, but photographic techniques had been making steady inroads since the time of Hermann Carl Vogel (1841 – 1907) in the 1880’s at the Potsdam observatory. Photographic plates were used primarily to record star positions but were known to be unreliable for recording stellar intensities. Schwarzschild developed a “out-of-focus” technique that blurred the star’s image, while making it larger and easier to measure the density of the exposed and developed photographic emulsions. In this way, Schwarzschild measured the magnitudes of 367 stars. Two of these stars had variable magnitudes that he was able to record and track. Schwarzschild correctly explained the intensity variation caused by steady oscillations in heating and cooling of the stellar atmosphere. This work established the properties of these Cepheid variables which would become some of the most important “standard candles” for the measurement of cosmological distances. Based on the importance of this work, Schwarzschild returned to Munich as a teacher in 1899 and subsequently was appointed in 1901 as the director of the observatory at Göttingen established by Gauss eighty years earlier.
Schwarzschild’s years at Göttingen brought him into contact with some of the greatest mathematicians and physicists of that era. The mathematicians included Felix Klein, David Hilbert and Hermann Minkowski. The physicists included von Laue, a student of Woldemar Voigt. This period was one of several “golden ages” of Göttingen. The first golden age was the time of Gauss and Riemann in the mid-1800’s. The second golden age, when Schwarzschild was present, began when Felix Klein arrived at Göttingen and attracted the top mathematicians of the time. The third golden age of Göttingen was the time of Born and Jordan and Heisenberg at the birth of quantum mechanics in the mid 1920’s.
In 1906, the Austrian Physicist Paul Ehrenfest, freshly out of his PhD under the supervision of Boltzmann, arrived at Göttingen only weeks before Boltzmann took his own life. Felix Klein at Göttingen had been relying on Boltzmann to provide a comprehensive review of statistical mechanics for the Mathematical Encyclopedia, so he now entrusted this project to the young Ehrenfest. It was a monumental task, which was to take him and his physicist wife Tatyanya nearly five years to complete. Part of the delay was the desire by the Ehrenfests to close some open problems that remained in Boltzmann’s work. One of these was a mechanical theorem of Boltzmann’s that identified properties of statistical mechanical systems that remained unaltered through a very slow change in system parameters. These properties would later be called adiabatic invariants by Einstein.
Ehrenfest recognized that Wien’s displacement law, which had been a guiding light for Planck and his theory of black body radiation, had originally been derived by Wien using classical principles related to slow changes in the volume of a cavity. Ehrenfest was struck by the fact that such slow changes would not induce changes in the quantum numbers of the quantized states, and hence that the quantum numbers must be adiabatic invariants of the black body system. This not only explained why Wien’s displacement law continued to hold under quantum as well as classical considerations, but it also explained why Planck’s quantization of the energy of his simple oscillators was the only possible choice. For a classical harmonic oscillator, the ratio of the energy of oscillation to the frequency of oscillation is an adiabatic invariant, which is immediately recognized as Planck’s quantum condition .
Ehrenfest published his observations in 1913 , the same year that Bohr published his theory of the hydrogen atom, so Ehrenfest immediately applied the theory of adiabatic invariants to Bohr’s model and discovered that the quantum condition for the quantized energy levels was again the adiabatic invariants of the electron orbits, and not merely a consequence of integer multiples of angular momentum, which had seemed somewhat ad hoc.
After eight exciting years at Göttingen, Schwarzschild was offered the position at the Potsdam Observatory in 1909 upon the retirement from that post of the famous German astronomer Carl Vogel who had made the first confirmed measurements of the optical Doppler effect. Schwarzschild accepted and moved to Potsdam with a new family. His son Martin Schwarzschild would follow him into his profession, becoming a famous astronomer at Princeton University and a theorist on stellar structure. At the outbreak of WWI, Schwarzschild joined the German army out of a sense of patriotism. Because of his advanced education he was made an officer of artillery with the job to calculate artillery trajectories, and after a short time on the Western Front in Belgium was transferred to the Eastern Front in Russia. Though he was not in the trenches, he was in the midst of the chaos to the rear of the front. Despite this situation, he found time to pursue his science through the year 1915.
Schwarzschild was intrigued by Ehrenfest’s paper on adiabatic invariants and their similarity to several of the invariant theorems of Poincaré that he had studied for his doctorate. Up until this time, mechanics had been mostly pursued through the Lagrangian formalism which could easily handle generalized forces associated with dissipation. But celestial mechanics are conservative systems for which the Hamiltonian formalism is a more natural approach. In particular, the Hamilton-Jacobi canonical transformations made it particularly easy to find pairs of generalized coordinates that had simple periodic behavior. In his published paper , Schwarzschild called these “Action-Angle” coordinates because one was the action integral that was well-known in the principle of “Least Action”, and the other was like an angle variable that changed steadily in time (see Fig. 5). Action-angle coordinates have come to form the foundation of many of the properties of Hamiltonian chaos, Hamiltonian maps, and Hamiltonian tapestries.
During lulls in bombardments, Schwarzschild translated the Hamilton-Jacobi methods of celestial mechanics to apply them to the new quantum mechanics of the Bohr orbits. The phrase “quantum mechanics” had not yet been coined (that would come ten years later in a paper by Max Born), but it was clear that the Bohr quantization conditions were a new type of mechanics. The periodicities that were inherent in the quantum systems were natural properties that could be mapped onto the periodicities of the angle variables, while Ehrenfest’s adiabatic invariants could be mapped onto the slowly varying action integrals. Schwarzschild showed that action-angle coordinates were the only allowed choice of coordinates, because they enabled the separation of the Hamilton-Jacobi equations and hence provided the correct quantization conditions for the Bohr electron orbits. Later, when Sommerfeld published his quantized elliptical orbits in 1916, the multiplicity of quantum conditions and orbits had caused concern, but Ehrenfest came to the rescue, showing that each of Sommerfeld’s quantum conditions were precisely Schwarzschild’s action-integral invariants of the classical electron dynamics .
The works by Schwarzschild, and a closely-related paper that amplified his ideas published by his friend Paul Epstein several months later , were the first to show the power of the Hamiltonian formulation of dynamics for quantum systems, foreshadowing the future importance of Hamiltonians for quantum theory. An essential part of the Hamiltonian formalism is the concept of phase space. In his paper, Schwarzschild showed that the phase space of quantum systems was divided into small but finite elementary regions whose areas were equal to Planck’s constant h-bar (see Fig. 6). The areas were products of a small change in momentum coordinate Delta-p and a corresponding small change in position coordinate Delta-x. Therefore, the product DxDp = h-bar. This observation, made in 1915 by Schwarzschild, was only one step away from Heisenberg’s uncertainty relation, twelve years before Heisenberg discovered it. However, in 1915 Born’s probabilistic interpretation of quantum mechanics had not yet been made, nor the idea of measurement uncertainty, so Schwarzschild did not have the appropriate context in which to have made the leap to the uncertainty principle. However, by introducing the action-angle coordinates as well as the Hamiltonian formalism applied to quantum systems, with the natural structure of phase space, Schwarzschild laid the foundation for the future developments in quantum theory made by the next generation.
Quiet on the Eastern Front
the end of his second stay in Munich in 1900, prior to joining the Göttingen
faculty, Schwarzschild had presented a paper at a meeting of the German Astronomical Society held in Heidelberg in
August. The topic was unlike anything he
had tackled before. It considered the
highly theoretical question of whether the universe was non-Euclidean, and more
specifically if it had curvature. He
concluded from observation that if the universe were curved, the radius of
curvature must be larger than between 50 light years and 2000 light years,
depending on whether the geometry was hyperbolic or elliptical. Schwarzschild was working out ideas of
differential geometry and applying them to the universe at large at a time when
Einstein was just graduating from the ETH where he skipped his math classes and
had his friend Marcel Grossmann take notes for him.
The topic of Schwarzschild’s talk tells an important story about the warping of historical perspective by the “great man” syndrome. In this case the great man is Einstein who is today given all the credit for discovering the warping of space. His development of General Relativity is often portrayed as by a lone genius in the wilderness performing a blazing act of creation out of the void. In fact, non-Euclidean geometry had been around for some time by 1900—five years before Einstein’s Special Theory and ten years before his first publications on the General Theory. Gauss had developed the idea of intrinsic curvature of a manifold fifty years earlier, amplified by Riemann. By the turn of the century alternative geometries were all the rage, and Schwarzschild considered whether there were sufficient astronomical observations to set limits on the size of curvature of the universe. But revisionist history is just as prevalent in physics as in any field, and when someone like Einstein becomes so big in the mind’s eye, his shadow makes it difficult to see all the people standing behind him.
This is not meant to take away from the feat that Einstein accomplished. The General Theory of Relativity, published by Einstein in its full form in 1915 was spectacular . Einstein had taken vague notions about curved spaces and had made them specific, mathematically rigorous and intimately connected with physics through the mass-energy source term in his field equations. His mathematics had gone beyond even what his mathematician friend and former collaborator Grossmann could achieve. Yet Einstein’s field equations were nonlinear tensor differential equations in which the warping of space depended on the strength of energy fields, but the configuration of those energy fields depended on the warping of space. This type of nonlinear equation is difficult to solve in general terms, and Einstein was not immediately aware of how to find the solutions to his own equations.
Therefore, it was no small surprise to him when he received a letter from the Eastern Front from an astronomer he barely knew who had found a solution—a simple solution (see Fig. 7) —to his field equations. Einstein probably wondered how he could have missed it, but he was generous and forwarded the letter to the Reports of the Prussian Physical Society where it was published in 1916 .
In the same paper, Schwarzschild used his exact solution to find the exact equation that described the precession of the perihelion of Mercury that Einstein had only calculated approximately. The dynamical equations for Mercury are shown in Fig. 8.
Schwarzschild’s solution to Einstein’s Field Equation of General Relativity was not a general solution, even for a point mass. He had constants of integration that could have arbitrary values, such as the characteristic length scale that Schwarzschild called “alpha”. It was David Hilbert who later expanded upon Schwarzschild’s work, giving the general solution and naming the characteristic length scale (where the metric diverges) after Schwarzschild. This is where the phrase “Schwarzschild Radius” got its name, and it stuck. In fact it stuck so well that Schwarzschild’s radius has now eclipsed much of the rest of Schwarzschild’s considerable accomplishments.
Unfortunately, Schwarzschild’s accomplishments were cut short when he contracted an autoimmune disease that may have been hereditary. It is ironic that in the carnage of the Eastern Front, it was a genetic disease that caused his death at the age of 42. He was already suffering from the effects of the disease as he worked on his last publications. He was sent home from the front to his family in Potsdam where he passed away several months later having shepherded his final two papers through the publication process. His last paper, on the action-angle variables in quantum systems , was published on the day that he died.
Schwarzschild’s legacy was assured when he solved Einstein’s field equations and Einstein communicated it to the world. But his hidden legacy is no less important.
Schwarzschild’s application of the Hamiltonian formalism of canonical transformations and phase space for quantum systems set the stage for the later adoption of Hamiltonian methods in quantum mechanics. He came dangerously close to stating the uncertainty principle that catapulted Heisenberg to later fame, although he could not express it in probabilistic terms because he came too early.
Schwarzschild is considered to be the greatest German astronomer of the last hundred years. This is in part based on his work at the birth of stellar interferometry and in part on his development of stellar photometry and the calibration of the Cepheid variable stars that went on to revolutionize our view of our place in the universe. Solving Einsteins field equations was just a sideline for him, a hobby to occupy his active and curious mind.
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In the fall semester of 1947, a brilliant young British mathematician arrived at Cornell University to begin a yearlong fellowship paid by the British Commonwealth. Freeman Dyson (1923 –) had received an undergraduate degree in mathematics from Cambridge University and was considered to be one of their brightest graduates. With strong recommendations, he arrived to work with Hans Bethe on quantum electrodynamics. He made rapid progress on a relativistic model of the Lamb shift, inadvertently intimidating many of his fellow graduate students with his mathematical prowess. On the other hand, someone who intimidated him, was Richard Feynman.
Initially, Dyson considered Feynman to be a bit of a buffoon and slacker, but he started to notice that Feynman could calculate QED problems in a few lines that took him pages.
I think like most science/geek types, my first introduction to the unfettered mind of Freeman Dyson was through the science fiction novel Ringworld by Larry Niven. The Dyson ring, or Dyson sphere, was conceived by Dyson when he was thinking about the ultimate fate of civilizations and their increasing need for energy. The greatest source of energy on a stellar scale is of course a star, and Dyson envisioned an advanced civilization capturing all that emitted stellar energy by building a solar collector with a radius the size of a planetary orbit. He published the paper “Search for Artificial Stellar Sources of Infra-Red Radiation” in the prestigious magazine Science in 1960. The practicality of such a scheme has to be seriously questioned, but it is a classic example of how easily he thinks outside the box, taking simple principles and extrapolating them to extreme consequences until the box looks like a speck of dust. I got a first-hand chance to see his way of thinking when he gave a physics colloquium at Cornell University in 1980 when I was an undergraduate there. Hans Bethe still had his office at that time in the Newman laboratory. I remember walking by and looking into his office getting a glance of him editing a paper at his desk. The topic of Dyson’s talk was the fate of life in the long-term evolution of the universe. His arguments were so simple they could not be refuted, yet the consequences for the way life would need to evolve in extreme time was unimaginable … it was a bazaar and mind blowing experience for me as an undergrad … and and example of the strange worlds that can be imagined through simple physics principles.
Initially, as Dyson settled into his life at Cornell under Bethe, he considered Feynman to be a bit of a buffoon and slacker, but he started to notice that Feynman could calculate QED problems in a few lines that took him pages. Dyson paid closer attention to Feynman, eventually spending more of his time with him than Bethe, and realized that Feynman had invented an entirely new way of calculating quantum effects that used cartoons as a form of book keeping to reduce the complexity of many calculations. Dyson still did not fully understand how Feynman was doing it, but knew that Feynman’s approach was giving all the right answers. Around that time, he also began to read about Schwinger’s field-theory approach to QED, following Schwinger’s approach as far as he could, but always coming away with the feeling that it was too complicated and required too much math—even for him!
Road Trip Across America
That summer, Dyson had time to explore America for the first time because Bethe had gone on an extended trip to Europe. It turned out that Feynman was driving his car to New Mexico to patch things up with an old flame from his Los Alamos days, so Dyson was happy to tag along. For days, as they drove across the US, they talked about life and physics and QED. Dyson had Feynman all to himself and began to see daylight in Feynman’s approach, and to understand that it might be consistent with Schwinger’s and Tomonaga’s field theory approach. After leaving Feynman in New Mexico, he travelled to the University of Michigan where Schwinger gave a short course on QED, and he was able to dig deeper, talking with him frequently between lectures.
At the end of the summer, it had been arranged that he would spend the second year of his fellowship at the Institute for Advanced Study in Princeton where Oppenheimer was the new head. As a final lark before beginning that new phase of his studies he spent a week at Berkeley. The visit there was uneventful, and he did not find the same kind of open camaraderie that he had found with Bethe in the Newman Laboratory at Cornell, but it left him time to think. And the more he thought about Schwinger and Feynman, the more convinced he became that the two were equivalent. On the long bus ride back east from Berkeley, as he half dozed and half looked out the window, he had an epiphany. He saw all at once how to draw the map from one to the other. What was more, he realized that many of Feynman’s techniques were much simpler than Schwinger’s, which would significantly simplify lengthy calculations. By the time he arrived in Chicago, he was ready to write it all down, and by the time he arrived in Princeton, he was ready to publish. It took him only a few weeks to do it, working with an intensity that he had never experienced before. When he was done, he sent the paper off to the Physical Review.
Dyson knew that he had achieved something significant even though he was essentially just a second-year graduate student, at least from the point of view of the American post-graduate system. Cambridge was a little different, and Dyson’s degree there was more than the standard bachelor’s degree here. Nonetheless, he was now under the auspices of the Institute for Advanced Study, where Einstein had his office, and he had sent off an unsupervised manuscript for publication without any imprimatur from the powers at be. The specific power that mattered most was Oppenheimer, who arrived a few days after Dyson had submitted his manuscript. When he greeted Oppenheimer, he was excited and pleased to hand him a copy. Oppenheimer, on the other hand, was neither excited nor pleased to receive it. Oppenheimer had formed a particularly bad opinion of Feynman’s form of QED at the conference held in the Poconos (to read about Feynman’s disaster at the Poconos conference, see my blog) half-a-year earlier and did not think that this brash young grad student could save it. Dyson, on his part, was taken aback. No one who has ever met Dyson would ever call him brash, but in this case he fought for a higher cause, writing a bold memo to Oppenheimer—that terrifying giant of a personality—outlining the importance of the Feynman theory.
Battle for the Heart of Quantum Field Theory
Oppenheimer decided to give Dyson a chance, and arranged for a series of seminars where Dyson could present the story to the assembled theory group at the Institute, but Dyson could make little headway. Every time he began to make progress, Oppenheimer would bring it crashing to a halt with scathing questions and criticisms. This went on for weeks, until Bethe visited from Cornell. Bethe by then was working with the Feynman formalism himself. As Bethe lectured in front of Oppenheimer, he seeded his talk with statements such as “surely they had all seen this from Dyson”, and Dyson took the opportunity to pipe up that he had not been allowed to get that far. After Bethe left, Oppenheimer relented, arranging for Dyson to give three seminars in one week. The seminars each went on for hours, but finally Dyson got to the end of it. The audience shuffled out of the seminar room with no energy left for discussions or arguments. Later that day, Dyson found a note in his box from Oppenheimer saying “Nolo Contendre”—Dyson had won!
With that victory under his belt, Dyson was in a position to communicate the new methods to a small army of postdocs at the Institute, supervising their progress on many outstanding problems in quantum electrodynamics that had resisted calculations using the complicated Schwinger-Tomonaga theory. Feynman, by this time, had finally published two substantial papers on his approach, which added to the foundation that Dyson was building at Princeton. Although Feynman continued to work for a year or two on QED problems, the center of gravity for these problems shifted solidly to the Institute for Advanced Study and to Dyson. The army of postdocs that Dyson supervised helped establish the use of Feynman diagrams in QED, calculating ever higher-order corrections to electromagnetic interactions. These same postdocs were among the first batch of wartime-trained theorists to move into faculty positions across the US, bringing the method of Feynman diagrams with them, adding to the rapid dissemination of Feynman diagrams into many aspects of theoretical physics that extend far beyond QED .
As a graduate student at Berkeley in the 1980’s I ran across a very simple-looking equation called “the Dyson equation” in our graduate textbook on relativistic quantum mechanics by Sakurai. The Dyson equation is the extraordinarily simple expression of an infinite series of Feynman diagrams that describes how an electron interacts with itself through the emission of virtual photons that link to virtual electron-positron pairs. This process leads to the propagator Green’s function for the electron and is the starting point for including the simple electron in more complex particle interactions.
I had no feel for the use of the Dyson equation, barely limping through relativistic quantum mechanics, until a few years later when I was working at Lawrence Berkeley Lab with Mirek Hamera, a visiting scientist from Warwaw Poland who introduced me to the Haldane-Anderson model that applied to a project I was working on for my PhD. Using the theory, with Dyson’s equation at its heart, we were able to show that tightly bound electrons on transition-metal impurities in semiconductors acted as internal reference levels that allowed us to measure internal properties of semiconductors that had never been accessible before. A few years later, I used Dyson’s equation again when I was working on small precipitates of arsenic in the semiconductor GaAs, using the theory to describe an accordion-like ladder of electron states that can occur within the semiconductor bandgap when a nano-sphere takes on multiple charges .
I last saw Dyson when he gave the Hubert James Memorial Lecture at Purdue University in 1996. The title of his talk was “How the Dinosaurs Might Have Been Saved: Detection and Deflection of Earth-Impacting Bodies”. As always, his talk was wild and wide ranging, using the simplest possible physics to derive the most dire consequences of our continued existence on this planet.
Dyson, F. J. (1949). “THE RADIATION THEORIES OF TOMONAGA,
SCHWINGER, AND FEYNMAN.” Physical Review75(3): 486-502.
Feynman, R. P. (1949). “THE THEORY OF POSITRONS.” Physical
Feynman, R. P. (1949). “SPACE-TIME APPROACH TO QUANTUM
ELECTRODYNAMICS.” Physical Review76(6): 769-789.
 Kaiser, D., K. Ito and K. Hall (2004). “Spreading the tools of theory: Feynman diagrams in the USA, Japan, and the Soviet Union.” Social Studies of Science34(6): 879-922.
 Nolte, D. D. (1998). “Mesoscopic Point-like Defects in Semiconductors.” Phys. Rev. B58(12): pg. 7994