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