Karl Schwarzschild’s Radius: How Fame Eclipsed a Physicist’s own Legacy

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.

A 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. 

The Birth of Stellar Interferometry

Until 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 [1], while Foucault came in second using a more versatile system with a spinning mirror [2].  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 [3]

Fig. 1  Optical configuration of the source element of the Fizeau refractometer.

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 [4].  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 [5].  He built the apparatus and used it to successfully measure the size of several moons of Jupiter [6].  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.

Fig. 2  Optical configuration of the Michelson stellar interferometer.  Fringes at the view port are partially washed out by the finite size of the star.  By adjusting the slit separation, the fringes can be made to vanish entirely, yielding an equation that can be solved for the size of the star.

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.

Schwarzschild’s Stellar Interferometer

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 [7].  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.

Fig. 3  Illustration from Schwarzschild’s 1896 paper describing an improvement of the Michelson interferometer for measuring the separation of binary star systems Ref. [7].
Fig. 4  Data page from Schwarzschild’s 1896 paper measuring the angular separation of two well-known binary star systems: gamma Leonis and chsi Ursa Major. Ref. [7]

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.

Adiabatic 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 [8], 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 [9], 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.

Fig. 5  Description of the canonical transformation to action-angle coordinates (Ref. [9] pg. 549). Schwarzschild names the new coordinates “Wirkungsvariable” and “Winkelvariable”.

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 [10].

The works by Schwarzschild, and a closely-related paper that amplified his ideas published by his friend Paul Epstein several months later [11], 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.

Fig. 6  Expression of the division of phase space into elemental areas of action equal to h-bar (Ref. [9] pg. 550).

All Quiet on the Eastern Front

Towards 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 [12].  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 [13].

Fig. 7  Schwarzschild’s solution of the Einstein Field Equations (Ref. [13] pg. 194).

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.

Fig. 8  Explanation for the precession of the perihelion of Mercury ( Ref. [13]  pg. 195)

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

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.


[1] Fizeau, H. L. (1849). “Sur une expérience relative à la vitesse de propagation de la lumière.” Comptes rendus de l’Académie des sciences 29: 90–92, 132.

[2] Foucault, J. L. (1862). “Détermination expérimentale de la vitesse de la lumière: parallaxe du Soleil.” Comptes rendus de l’Académie des sciences 55: 501–503, 792–596.

[3] Fizeau, H. (1859). “Sur les hypothèses relatives à l’éther lumineux.” Ann. Chim. Phys.  Ser. 4 57: 385–404.

[4] Fizeau, H. (1868). “Prix Bordin: Rapport sur le concours de l’annee 1867.” C. R. Acad. Sci. 66: 932.

[5] Michelson, A. A. (1890). “I. On the application of interference methods to astronomical measurements.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 30(182): 1-21.

[6] Michelson, A. A. (1891). “Measurement of Jupiter’s Satellites by Interference.” Nature 45(1155): 160-161.

[7] Schwarzschild, K. (1896). “Über messung von doppelsternen durch interferenzen.” Astron. Nachr. 3335: 139.

[8] P. Ehrenfest, “Een mechanische theorema van Boltzmann en zijne betrekking tot de quanta theorie (A mechanical theorem of Boltzmann and its relation to the theory of energy quanta),” Verslag van de Gewoge Vergaderingen der Wis-en Natuurkungige Afdeeling, vol. 22, pp. 586-593, 1913.

[9] Schwarzschild, K. (1916). “Quantum hypothesis.” Sitzungsberichte Der Koniglich Preussischen Akademie Der Wissenschaften: 548-568.

[10] P. Ehrenfest, “Adiabatic invariables and quantum theory,” Annalen Der Physik, vol. 51, pp. 327-352, Oct 1916.

[11] Epstein, P. S. (1916). “The quantum theory.” Annalen Der Physik 51(18): 168-188.

[12] Einstein, A. (1915). “On the general theory of relativity.” Sitzungsberichte Der Koniglich Preussischen Akademie Der Wissenschaften: 778-786.

[13] Schwarzschild, K. (1916). “Über das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie.” Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften: 189.

Orbiting Photons around a Black Hole

The physics of a path of light passing a gravitating body is one of the hardest concepts to understand in General Relativity, but it is also one of the easiest.  It is hard because there can be no force of gravity on light even though the path of a photon bends as it passes a gravitating body.  It is easy, because the photon is following the simplest possible path—a geodesic equation for force-free motion.

         This blog picks up where my last blog left off, having there defined the geodesic equation and presenting the Schwarzschild metric.  With those two equations in hand, we could simply solve for the null geodesics (a null geodesic is the path of a light beam through a manifold).  But there turns out to be a simpler approach that Einstein came up with himself (he never did like doing things the hard way).  He just had to sacrifice the fundamental postulate that he used to explain everything about Special Relativity.

Throwing Special Relativity Under the Bus

The fundamental postulate of Special Relativity states that the speed of light is the same for all observers.  Einstein posed this postulate, then used it to derive some of the most astonishing consequences of Special Relativity—like E = mc2.  This postulate is at the rock core of his theory of relativity and can be viewed as one of the simplest “truths” of our reality—or at least of our spacetime. 

            Yet as soon as Einstein began thinking how to extend SR to a more general situation, he realized almost immediately that he would have to throw this postulate out.   While the speed of light measured locally is always equal to c, the apparent speed of light observed by a distant observer (far from the gravitating body) is modified by gravitational time dilation and length contraction.  This means that the apparent speed of light, as observed at a distance, varies as a function of position.  From this simple conclusion Einstein derived a first estimate of the deflection of light by the Sun, though he initially was off by a factor of 2.  (The full story of Einstein’s derivation of the deflection of light by the Sun and the confirmation by Eddington is in Chapter 7 of Galileo Unbound (Oxford University Press, 2018).)

The “Optics” of Gravity

The invariant element for a light path moving radially in the Schwarzschild geometry is

The apparent speed of light is then

where c(r) is  always less than c, when observing it from flat space.  The “refractive index” of space is defined, as for any optical material, as the ratio of the constant speed divided by the observed speed

Because the Schwarzschild metric has the property

the effective refractive index of warped space-time is

with a divergence at the Schwarzschild radius.

            The refractive index of warped space-time in the limit of weak gravity can be used in the ray equation (also known as the Eikonal equation described in an earlier blog)

where the gradient of the refractive index of space is

The ray equation is then a four-variable flow

These equations represent a 4-dimensional flow for a light ray confined to a plane.  The trajectory of any light path is found by using an ODE solver subject to the initial conditions for the direction of the light ray.  This is simple for us to do today with Python or Matlab, but it was also that could be done long before the advent of computers by early theorists of relativity like Max von Laue  (1879 – 1960).

The Relativity of Max von Laue

In the Fall of 1905 in Berlin, a young German physicist by the name of Max Laue was sitting in the physics colloquium at the University listening to another Max, his doctoral supervisor Max Planck, deliver a seminar on Einstein’s new theory of relativity.  Laue was struck by the simplicity of the theory, in this sense “simplistic” and hence hard to believe, but the beauty of the theory stuck with him, and he began to think through the consequences for experiments like the Fizeau experiment on partial ether drag.

         Armand Hippolyte Louis Fizeau (1819 – 1896) in 1851 built one of the world’s first optical interferometers and used it to measure the speed of light inside moving fluids.  At that time the speed of light was believed to be a property of the luminiferous ether, and there were several opposing theories on how light would travel inside moving matter.  One theory would have the ether fully stationary, unaffected by moving matter, and hence the speed of light would be unaffected by motion.  An opposite theory would have the ether fully entrained by matter and hence the speed of light in moving matter would be a simple sum of speeds.  A middle theory considered that only part of the ether was dragged along with the moving matter.  This was Fresnel’s partial ether drag hypothesis that he had arrived at to explain why his friend Francois Arago had not observed any contribution to stellar aberration from the motion of the Earth through the ether.  When Fizeau performed his experiment, the results agreed closely with Fresnel’s drag coefficient, which seemed to settle the matter.  Yet when Michelson and Morley performed their experiments of 1887, there was no evidence for partial drag.

         Even after the exposition by Einstein on relativity in 1905, the disagreement of the Michelson-Morley results with Fizeau’s results was not fully reconciled until Laue showed in 1907 that the velocity addition theorem of relativity gave complete agreement with the Fizeau experiment.  The velocity observed in the lab frame is found using the velocity addition theorem of special relativity. For the Fizeau experiment, water with a refractive index of n is moving with a speed v and hence the speed in the lab frame is

The difference in the speed of light between the stationary and the moving water is the difference

where the last term is precisely the Fresnel drag coefficient.  This was one of the first definitive “proofs” of the validity of Einstein’s theory of relativity, and it made Laue one of relativity’s staunchest proponents.  Spurred on by his success with the Fresnel drag coefficient explanation, Laue wrote the first monograph on relativity theory, publishing it in 1910. 

Fig. 1 Front page of von Laue’s textbook, first published in 1910, on Special Relativity (this is a 4-th edition published in 1921).

A Nobel Prize for Crystal X-ray Diffraction

In 1909 Laue became a Privatdozent under Arnold Sommerfeld (1868 – 1951) at the university in Munich.  In the Spring of 1912 he was walking in the Englischer Garten on the northern edge of the city talking with Paul Ewald (1888 – 1985) who was finishing his doctorate with Sommerfed studying the structure of crystals.  Ewald was considering the interaction of optical wavelength with the periodic lattice when it struck Laue that x-rays would have the kind of short wavelengths that would allow the crystal to act as a diffraction grating to produce multiple diffraction orders.  Within a few weeks of that discussion, two of Sommerfeld’s students (Friedrich and Knipping) used an x-ray source and photographic film to look for the predicted diffraction spots from a copper sulfate crystal.  When the film was developed, it showed a constellation of dark spots for each of the diffraction orders of the x-rays scattered from the multiple periodicities of the crystal lattice.  Two years later, in 1914, Laue was awarded the Nobel prize in physics for the discovery.  That same year his father was elevated to the hereditary nobility in the Prussian empire and Max Laue became Max von Laue.

            Von Laue was not one to take risks, and he remained conservative in many of his interests.  He was immensely respected and played important roles in the administration of German science, but his scientific contributions after receiving the Nobel Prize were only modest.  Yet as the Nazis came to power in the early 1930’s, he was one of the few physicists to stand up and resist the Nazi take-over of German physics.  He was especially disturbed by the plight of the Jewish physicists.  In 1933 he was invited to give the keynote address at the conference of the German Physical Society in Wurzburg where he spoke out against the Nazi rejection of relativity as they branded it “Jewish science”.  In his speech he likened Einstein, the target of much of the propaganda, to Galileo.  He said, “No matter how great the repression, the representative of science can stand erect in the triumphant certainty that is expressed in the simple phrase: And yet it moves.”  Von Laue believed that truth would hold out in the face of the proscription against relativity theory by the Nazi regime.  The quote “And yet it moves” is supposed to have been muttered by Galileo just after his abjuration before the Inquisition, referring to the Earth moving around the Sun.  Although the quote is famous, it is believed to be a myth.

            In an odd side-note of history, von Laue sent his gold Nobel prize medal to Denmark for its safe keeping with Niels Bohr so that it would not be paraded about by the Nazi regime.  Yet when the Nazis invaded Denmark, to avoid having the medals fall into the hands of the Nazis, the medal was dissolved in aqua regia by a member of Bohr’s team, George de Hevesy.  The gold completely dissolved into an orange liquid that was stored in a beaker high on a shelf through the war.  When Denmark was finally freed, the dissolved gold was precipitated out and a new medal was struck by the Nobel committee and re-presented to von Laue in a ceremony in 1951. 

The Orbits of Light Rays

Von Laue’s interests always stayed close to the properties of light and electromagnetic radiation ever since he was introduced to the field when he studied with Woldemor Voigt at Göttingen in 1899.  This interest included the theory of relativity, and only a few years after Einstein published his theory of General Relativity and Gravitation, von Laue added to his earlier textbook on relativity by writing a second volume on the general theory.  The new volume was published in 1920 and included the theory of the deflection of light by gravity. 

         One of the very few illustrations in his second volume is of light coming into interaction with a super massive gravitational field characterized by a Schwarzschild radius.  (No one at the time called it a “black hole”, nor even mentioned Schwarzschild.  That terminology came much later.)  He shows in the drawing, how light, if incident at just the right impact parameter, would actually loop around the object.  This is the first time such a diagram appeared in print, showing the trajectory of light so strongly affected by gravity.

Fig. 2 A page from von Laue’s second volume on relativity (first published in 1920) showing the orbit of a photon around a compact mass with “gravitational cutoff” (later known as a “black hole:”). The figure is drawn semi-quantitatively, but the phenomenon was clearly understood by von Laue.

Python Code

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Tue May 28 11:50:24 2019

@author: nolte
"""

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
import time
import os

plt.close('all')

def create_circle():
	circle = plt.Circle((0,0), radius= 10, color = 'black')
	return circle

def show_shape(patch):
	ax=plt.gca()
	ax.add_patch(patch)
	plt.axis('scaled')
	plt.show()
    
def refindex(x,y):
    
    A = 10
    eps = 1e-6
    
    rp0 = np.sqrt(x**2 + y**2);
        
    n = 1/(1 - A/(rp0+eps))
    fac = np.abs((1-9*(A/rp0)**2/8))   # approx correction to Eikonal
    nx = -fac*n**2*A*x/(rp0+eps)**3
    ny = -fac*n**2*A*y/(rp0+eps)**3
     
    return [n,nx,ny]

def flow_deriv(x_y_z,tspan):
    x, y, z, w = x_y_z
    
    [n,nx,ny] = refindex(x,y)
        
    yp = np.zeros(shape=(4,))
    yp[0] = z/n
    yp[1] = w/n
    yp[2] = nx
    yp[3] = ny
    
    return yp
                
for loop in range(-5,30):
    
    xstart = -100
    ystart = -2.245 + 4*loop
    print(ystart)
    
    [n,nx,ny] = refindex(xstart,ystart)


    y0 = [xstart, ystart, n, 0]

    tspan = np.linspace(1,400,2000)

    y = integrate.odeint(flow_deriv, y0, tspan)

    xx = y[1:2000,0]
    yy = y[1:2000,1]


    plt.figure(1)
    lines = plt.plot(xx,yy)
    plt.setp(lines, linewidth=1)
    plt.show()
    plt.title('Photon Orbits')
    
c = create_circle()
show_shape(c)
axes = plt.gca()
axes.set_xlim([-100,100])
axes.set_ylim([-100,100])

# Now set up a circular photon orbit
xstart = 0
ystart = 15

[n,nx,ny] = refindex(xstart,ystart)

y0 = [xstart, ystart, n, 0]

tspan = np.linspace(1,94,1000)

y = integrate.odeint(flow_deriv, y0, tspan)

xx = y[1:1000,0]
yy = y[1:1000,1]

plt.figure(1)
lines = plt.plot(xx,yy)
plt.setp(lines, linewidth=2, color = 'black')
plt.show()

One of the most striking effects of gravity on photon trajectories is the possibility for a photon to orbit a black hole in a circular orbit. This is shown in Fig. 3 as the black circular ring for a photon at a radius equal to 1.5 times the Schwarzschild radius. This radius defines what is known as the photon sphere. However, the orbit is not stable. Slight deviations will send the photon spiraling outward or inward.

The Eikonal approximation does not strictly hold under strong gravity, but the Eikonal equations with the effective refractive index of space still yield semi-quantitative behavior. In the Python code, a correction factor is used to match the theory to the circular photon orbits, while still agreeing with trajectories far from the black hole. The results of the calculation are shown in Fig. 3. For large impact parameters, the rays are deflected through a finite angle. At a critical impact parameter, near 3 times the Schwarzschild radius, the ray loops around the black hole. For smaller impact parameters, the rays are captured by the black hole.

Fig. 3 Photon orbits near a black hole calculated using the Eikonal equation and the effective refractive index of warped space. One ray, near the critical impact parameter, loops around the black hole as predicted by von Laue. The central black circle is the black hole with a Schwarzschild radius of 10 units. The black ring is the circular photon orbit at a radius 1.5 times the Schwarzschild radius.

Photons pile up around the black hole at the photon sphere. The first image ever of the photon sphere of a black hole was made earlier this year (announced April 10, 2019). The image shows the shadow of the supermassive black hole in the center of Messier 87 (M87), an elliptical galaxy 55 million light-years from Earth. This black hole is 6.5 billion times the mass of the Sun. Imaging the photosphere required eight ground-based radio telescopes placed around the globe, operating together to form a single telescope with an optical aperture the size of our planet.  The resolution of such a large telescope would allow one to image a half-dollar coin on the surface of the Moon, although this telescope operates in the radio frequency range rather than the optical.

Fig. 4 Scientists have obtained the first image of a black hole, using Event Horizon Telescope observations of the center of the galaxy M87. The image shows a bright ring formed as light bends in the intense gravity around a black hole that is 6.5 billion times more massive than the Sun.

Further Reading

Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd Ed. (Oxford University Press, 2019)

B. Lavenda, The Optical Properties of Gravity, J. Mod. Phys, 8 8-3-838 (2017)

How to Teach General Relativity to Undergraduate Physics Majors

As a graduate student in physics at Berkeley in the 1980’s, I took General Relativity (aka GR), from Bruno Zumino, who was a world-famous physicist known as one of the originators of super-symmetry in quantum gravity (not to be confused with super-asymmetry of Cooper-Fowler Big Bang Theory fame).  The class textbook was Gravitation and cosmology: principles and applications of the general theory of relativity, by Steven Weinberg, another world-famous physicist, in this case known for grand unification of the electro-weak force with electromagnetism.  With so much expertise at hand, how could I fail but to absorb the simple essence of general relativity? 

The answer is that I failed miserably.  Somehow, I managed to pass the course, but I walked away with nothing!  And it bugged me for years.  What was so hard about GR?  It took me almost a decade teaching undergraduate physics classes at Purdue in the 90’s before I realized that it my biggest obstacle had been language:  I kept mistaking the words and terms of GR as if they were English.  Words like “general covariance” and “contravariant” and “contraction” and “covariant derivative”.  They sounded like English, with lots of “co” prefixes that were hard to keep straight, but they actually are part of a very different language that I call Physics-ese

Physics-ese is a language that has lots of words that sound like English, and so you think you know what the words mean, but the words have sometimes opposite meanings than what you would guess.  And the meanings of Physics-ese are precisely defined, and not something that can be left to interpretation.  I learned this while teaching the intro courses to non-majors, because so many times when the students were confused, it turned out that it was because they had mistaken a textbook jargon term to be English.  If you told them that the word wasn’t English, but just a token standing for a well-defined object or process, it would unshackle them from their misconceptions.

Then, in the early 00’s when I started to explore the physics of generalized trajectories related to some of my own research interests, I realized that the primary obstacle to my learning anything in the Gravitation course was Physics-ese.   So this raised the question in my mind: what would it take to teach GR to undergraduate physics majors in a relatively painless manner?  This is my answer. 

More on this topic can be found in Chapter 11 of the textbook IMD2: Introduction to Modern Dynamics, 2nd Edition, Oxford University Press, 2019

Trajectories as Flows

One of the culprits for my mind block learning GR was Newton himself.  His ubiquitous second law, taught as F = ma, is surprisingly misleading if one wants to have a more general understanding of what a trajectory is.  This is particularly the case for light paths, which can be bent by gravity, yet clearly cannot have any forces acting on them. 

The way to fix this is subtle yet simple.  First, express Newton’s second law as

which is actually closer to the way that Newton expressed the law in his Principia.  In three dimensions for a single particle, these equations represent a 6-dimensional dynamical space called phase space: three coordinate dimensions and three momentum dimensions.  Then generalize the vector quantities, like the position vector, to be expressed as xa for the six dynamics variables: x, y, z, px, py, and pz

Now, as part of Physics-ese, putting the index as a superscript instead as a subscript turns out to be a useful notation when working in higher-dimensional spaces.  This superscript is called a “contravariant index” which sounds like English but is uninterpretable without a Physics-ese-to-English dictionary.  All “contravariant index” means is “column vector component”.  In other words, xa is just the position vector expressed as a column vector

This superscripted index is called a “contravariant” index, but seriously dude, just forget that “contravariant” word from Physics-ese and just think “index”.  You already know it’s a column vector.

Then Newton’s second law becomes

where the index a runs from 1 to 6, and the function Fa is a vector function of the dynamic variables.  To spell it out, this is

so it’s a lot easier to write it in the one-line form with the index notation. 

The simple index notation equation is in the standard form for what is called, in Physics-ese, a “mathematical flow”.  It is an ODE that can be solved for any set of initial conditions for a given trajectory.  Or a whole field of solutions can be considered in a phase-space portrait that looks like the flow lines of hydrodynamics.  The phase-space portrait captures the essential physics of the system, whether it is a rock thrown off a cliff, or a photon orbiting a black hole.  But to get to that second problem, it is necessary to look deeper into the way that space is described by any set of coordinates, especially if those coordinates are changing from location to location.

What’s so Fictitious about Fictitious Forces?

Freshmen physics students are routinely admonished for talking about “centrifugal” forces (rather than centripetal) when describing circular motion, usually with the statement that centrifugal forces are fictitious—only appearing to be forces when the observer is in the rotating frame.  The same is said for the Coriolis force.  Yet for being such a “fictitious” force, the Coriolis effect is what drives hurricanes and the colossal devastation they cause.  Try telling a hurricane victim that they were wiped out by a fictitious force!  Looking closer at the Coriolis force is a good way of understanding how taking derivatives of vectors leads to effects often called “fictitious”, yet it opens the door on some of the simpler techniques in the topic of differential geometry.

To start, consider a vector in a uniformly rotating frame.  Such a frame is called “non-inertial” because of the angular acceleration associated with the uniform rotation.  For an observer in the rotating frame, vectors are attached to the frame, like pinning them down to the coordinate axes, but the axes themselves are changing in time (when viewed by an external observer in a fixed frame).  If the primed frame is the external fixed frame, then a position in the rotating frame is

where R is the position vector of the origin of the rotating frame and r is the position in the rotating frame relative to the origin.  The funny notation on the last term is called in Physics-ese a “contraction”, but it is just a simple inner product, or dot product, between the components of the position vector and the basis vectors.  A basis vector is like the old-fashioned i, j, k of vector calculus indicating unit basis vectors pointing along the x, y and z axes.  The format with one index up and one down in the product means to do a summation.  This is known as the Einstein summation convention, so it’s just

Taking the time derivative of the position vector gives

and by the chain rule this must be

where the last term has a time derivative of a basis vector.  This is non-zero because in the rotating frame the basis vector is changing orientation in time.  This term is non-inertial and can be shown fairly easily (see IMD2 Chapter 1) to be

which is where the centrifugal force comes from.  This shows how a so-called fictitious force arises from a derivative of a basis vector.  The fascinating point of this is that in GR, the force of gravity arises in almost the same way, making it tempting to call gravity a fictitious force, despite the fact that it can kill you if you fall out a window.  The question is, how does gravity arise from simple derivatives of basis vectors?

The Geodesic Equation

To teach GR to undergraduates, you cannot expect them to have taken a course in differential geometry, because most of them just don’t have the time in their schedule to take such an advanced mathematics course.  In addition, there is far more taught in differential geometry than is needed to make progress in GR.  So the simple approach is to teach what they need to understand GR with as little differential geometry as possible, expressed with clear English-to-Physics-ese translations. 

For example, consider the partial derivative of a vector expressed in index notation as

Taking the partial derivative, using the always-necessary chain rule, is

where the second term is just like the extra time-derivative term that showed up in the derivation of the Coriolis force.  The basis vector of a general coordinate system may change size and orientation as a function of position, so this derivative is not in general zero.  Because the derivative of a basis vector is so central to the ideas of GR, they are given their own symbol.  It is

where the new “Gamma” symbol is called a Christoffel symbol.  It has lots of indexes, both up and down, which looks daunting, but it can be interpreted as the beta-th derivative of the alpha-th component of the mu-th basis vector.  The partial derivative is now

For those of you who noticed that some of the indexes flipped from alpha to mu and vice versa, you’re right!  Swapping repeated indexes in these “contractions” is allowed and helps make derivations a lot easier, which is probably why Einstein invented this notation in the first place.

The last step in taking a partial derivative of a vector is to isolate a single vector component Va as

where a new symbol, the del-operator has been introduced.  This del-operator is known as the “covariant derivative” of the vector component.  Again, forget the “covariant” part and just think “gradient”.  Namely, taking the gradient of a vector in general includes changes in the vector component as well as changes in the basis vector.

Now that you know how to take the partial derivative of a vector using Christoffel symbols, you are ready to generate the central equation of General Relativity:  The geodesic equation. 

Everyone knows that a geodesic is the shortest path between two points, like a great circle route on the globe.  But it also turns out to be the straightest path, which can be derived using an idea known as “parallel transport”.  To start, consider transporting a vector along a curve in a flat metric.  The equation describing this process is

Because the Christoffel symbols are zero in a flat space, the covariant derivative and the partial derivative are equal, giving

If the vector is transported parallel to itself, then there is no change in V along the curve, so that

Finally, recognizing

and substituting this in gives

This is the geodesic equation! 

Fig. 1 The geodesic equation of motion is for force-free motion through a metric space. The curvature of the trajectory is analogous to acceleration, and the generalized gradient is analogous to a force. The geodesic equation is the “F = ma” of GR.

Putting this in the standard form of a flow gives the geodesic flow equations

The flow defines an ordinary differential equation that defines a curve that carries its own tangent vector onto itself.  The curve is parameterized by a parameter s that can be identified with path length.  It is the central equation of GR, because it describes how an object follows a force-free trajectory, like free fall, in any general coordinate system.  It can be applied to simple problems like the Coriolis effect, or it can be applied to seemingly difficult problems, like the trajectory of a light path past a black hole.

The Metric Connection

Arriving at the geodesic equation is a major accomplishment, and you have done it in just a few pages of this blog.  But there is still an important missing piece before we are doing General Relativity of gravitation.  We need to connect the Christoffel symbol in the geodesic equation to the warping of space-time around a gravitating object. 

The warping of space-time by matter and energy is another central piece of GR and is often the central focus of a graduate-level course on the subject.  This part of GR does have its challenges leading up to Einstein’s Field Equations that explain how matter makes space bend.  But at an undergraduate level, it is sufficient to just describe the bent coordinates as a starting point, then use the geodesic equation to solve for so many of the cool effects of black holes.

So, stating the way that matter bends space-time is as simple as writing down the length element for the Schwarzschild metric of a spherical gravitating mass as

where RS = GM/c2 is the Schwarzschild radius.  (The connection between the metric tensor gab and the Christoffel symbol can be found in Chapter 11 of IMD2.)  It takes only a little work to find that

This means that if we have the Schwarzschild metric, all we have to do is take first partial derivatives and we will arrive at the Christoffel symbols that go into the geodesic equation.  Solving for any type of force-free trajectory is then just a matter of solving ODEs with initial conditions (performed routinely with numerical ODE solvers in Python, Matlab, Mathematica, etc.).

The first problem we will tackle using the geodesic equation is the deflection of light by gravity.  This is the quintessential problem of GR because there cannot be any gravitational force on a photon, yet the path of the photon surely must bend in the presence of gravity.  This is possible through the geodesic motion of the photon through warped space time.  I’ll take up this problem in my next Blog.