June 12, 2025September 5, 2025 by David D. Nolte

The Light in Einstein’s Elevator

At Light Speed, History of Dynamics
Acceleration, albert-einstein, Einstein, Equivalence Principle, Free Fall, general relativity, gravitation, History of Physics, photon, physics, Relativity, science, Spacetime

Gravity bends light!

Of all the audacious proposals made by Einstein, and there were many, this one takes the cake because it should be impossible.

There can be no force of gravity on light because light has no mass. Without mass, there is no gravitational “interaction”. We all know Newton’s Law of gravity … it was one of the first equations of physics we ever learned

which shows the interaction between the masses M and m through their product. For light, this is strictly zero.

How, then did Einstein conclude, in 1907, only two years after he proposed his theory of special relativity, that gravity bends light? If it were us, we might take Newton’s other famous equation and equate the two

and guess that somehow the little mass m (though it equals zero) cancels out to give

so that light would fall in gravity with the same acceleration as anything else, massive or not.

But this is not how Einstein arrived at his proposal, because this derivation is wrong! To do it right, you have to think like an Einstein.

“My Happiest Thought”

Towards the end of 1907, Einstein was asked by Johannes Stark to contribute a review article on the state of the relativity theory to the Jahrbuch of Radioactivity and Electronics. There had been a flurry of activity in the field in the two years since Einstein had published his groundbreaking paper in Annalen der Physik in September of 1905 [1]. Einstein himself had written several additional papers on the topic, along with others, so Stark felt it was time to put things into perspective.

Photo of Einstein around 1905 during his Annis Mirabalis. — Fig. 1 Einstein around 1905.

Einstein was still working at the Patent Office in Bern, Switzerland, which must not have been too taxing, because it gave him plenty of time think. It was while he was sitting in his armchair in his office in 1907 that he had what he later described as the happiest thought of his life. He had been struggling with the details of how to apply relativity theory to accelerating reference frames, a topic that is fraught with conceptual traps, when he had a flash of simplifying idea:

“Then there occurred to me the ‘glucklichste Gedanke meines Lebens,’ the happiest thought of my life, in the following form. The gravitational field has only a relative existence in a way similar to the electric field generated by magnetoelectric induction. Because for an observer falling freely from the roof of a house there exists —at least in his immediate surroundings— no gravitational field. Indeed, if the observer drops some bodies then these remain relative to him in a state of rest or of uniform motion… The observer therefore has the right to interpret his state as ‘at rest.'”[2]

In other words, the freely falling observer believes he is in an inertial frame rather than an accelerating one, and by the principle of relativity, this means that all the laws of physics in the accelerating frame must be the same as for an inertial frame. Hence, his great insight was that there must be complete equivalence between a mechanically accelerating frame and a gravitational field. This is the very first conception of his Equivalence Principle.

Cover of the Jahrbuch for Radioactivity and Electronics from 1907. — Fig. 2 Front page of the 1907 volume of the Jahrbuch. The editor list reads like a “Whos-Who” of early modern physics.

Title page to Einstein's 1907 Jahrbuch review article — Fig. 3 Title page to Einstein’s 1907 Jahrbuch review article “On the Relativity Principle and its Consequences” [3]

After completing his review of the consequences of special relativity in his Jahrbuch article, Einstein took the opportunity to launch into his speculations on the role of the relativity principle in gravitation. He is almost appologetic at the start, saying that:

“This is not the place for a detailed discussion of this question. But as it will occur to anybody who has been following the applications of the principle of relativity, I will not refrain from taking a stand on this question here.”

But he then launches into his first foray into general relativity with keen insights.

Fig. 4 The beginning of the section where Einstein first discusses the effects of accelerating frames and effects of gravity.

He states early in his exposition:

“… in the discussion that follows, we shall therefore assume the complete physical equivalence of a gravitational field and a corresponding accelerated reference system.”

Here is his equivalence principle. And using it, in 1907, he derives the effect of acceleration (and gravity) on ticking clocks, on the energy density of electromagnetic radiation (photons) in a gravitational potential, and on the deflection of light by gravity.

Over the next several years, Einstein was distracted by other things, such as obtaining his first university position, and his continuing work on the early quantum theory. But by 1910 he was ready to tackle the general theory of relativity once again, when he discovered that his equivalence principle was missing a key element: the effects of spatial curvature, which launched him on a 5-year program into the world of tensors and metric spaces that culminated with his completed general theory of relativity that he published in November of 1915 [4].

The Observer in the Chest: There is no Elevator

Einstein was never a stone to gather moss. Shortly after delivering his triumphal exposition on the General Theory of Relativity, he wrote up a popular account of his Special and now General Theories to be published as a book in 1916, first in German [5] and then in English [6]. What passed for a “popular exposition” in 1916 is far from what is considered popular today. Einstein’s little book is full of equations that would be somewhat challenging even for specialists. But the book also showcases Einstein’s special talent to create simple analogies, like the falling observer, that can make difficult concepts of physics appear crystal clear.

In 1916, Einstein was not yet thinking in terms of an elevator. His mental image at this time, for a sequestered observer, was someone inside a spacious chest filled with measurement apparatus that the observer could use at will. This observer in his chest was either floating off in space far from any gravitating bodies, or the chest was being pulled by a rope hooked to the ceiling such that the chest accelerates constantly. Based on the measurement he makes, he cannot distinguish between gravitational fields and acceleration, and hence they are equivalent. A bit later in the book, Einstein describes what a ray of light would do in an accelerating frame, but he does not have his observer attempt any such measurement, even in principle, because the deflection of the ray of light from a linear path would be far too small to measure.

But Einstein does go on to say that any curvature of the path of the light ray requires that the speed of light changes with position. This is a shocking admission, because his fundamental postulate of relativity from 1905 was the invariance of the speed of light in all inertial frames. It was from this simple assertion that he was eventually able to derive E = mc². Where, on the one hand, he was ready to posit the invariance of the speed of light, on the other hand, as soon as he understood the effects of gravity on light, Einstein did not hesitate to cast this postulate adrift.

Position-dependent speed of light in relativity.

Fig. 5 Einstein’s argument for the speed of light depending on position in a gravitational field.

(Einstein can be forgiven for taking so long to speak in terms of an elevator that could accelerate at a rate of one g, because it was not until 1946 that the rocket plane Bell X-1 achieved linear acceleration exceeding 1 g, and jet planes did not achieve 1 g linear acceleration until the F-15 Eagle in 1972.)

Fig. 6 Aircraft with greater than 1:1 thrust to weight ratios.

The Evolution of Physics: Enter Einstein’s Elevator

Years passed, and Einstein fled an increasingly autocratic and belligerent Germany for a position at Princeton’s Institute for Advanced Study. In 1938, at the instigation of his friend Leopold Infeld, they decided to write a general interest book on the new physics of relativity and quanta that had evolved so rapidly over the past 30 years.

Title page of "Evolution of Physics" 1938 written with his friend Leopold Infeld at Princeton's Institute for Advanced Study. — Fig. 7 Title page of “Evolution of Physics” 1938 written with his friend Leopold Infeld at Princeton’s Institute for Advanced Study.

Here, in this obscure book that no-one remembers today, we find Einstein’s elevator for the first time, and the exposition talks very explicitly about a small window that lets in a light ray, and what the observer sees (in principle) for the path of the ray.

One of the only figures in the Einstein and Infeld book: The origin of "Einstein's Elevator"! — Fig. 8 One of the only figures in the Einstein and Infeld book: The origin of “Einstein’s Elevator”!

By the equivalence principle, the observer cannot tell whether they are far out in space, being accelerated at the rate g, or whether they are statinary on the surface of the Earth subject to a gravitational field. In the first instance of the accelerating elevator, a photon moving in a straight line through space would appear to deflect downward in the elevator, as shown in Fig. 9, because the elevator is accelerating upwards as the photon transits the elevator. However, by the equivalence principle, the same physics should occur in the gravitational field. Hence, gravity must bend light. Furthermore, light falls inside the elevator with an acceleration g, just as any other object would.

The accelerating elevator and what an observer inside sees (From "Galileo Unbound" (Oxford, 2018). — Fig. 9 The accelerating elevator and what an observer inside sees (From “Galileo Unbound” (Oxford, 2018). [7])

Light Deflection in the Equivalence Principle

A photon enters an elevator at right angles to its acceleration vector g. Use the geodesic equation and the elevator (Equivalence Principle) metric [8]

to show that the trajectory is parabolic. (This is a classic HW problem from Introduction to Modern Dynamics.)

Solution:

The geodesic equation with time as the dependent variable

This gives two coordinate equations

Note that x⁰ = ct and x¹ = ct are both large relative to the y-motion of the photon. The metric component that is relevant here is

and the others are unity. The geodesic becomes (assuming dy/dt = 0)

The Christoffel symbols are

which give

Therefore

where the photon falls with acceleration g, as anticipated.

Light Deflection in the Schwarzschild Metric

Do the same problem of the light ray in Einstein’s Elevator, but now using the full Schwarzschild solution to the Einstein Field equations.

Solution:

Einstein’s elevator is the classic test of virtually all heuristic questions related to the deflection of light by gravity. In the previous Example, the deflection was attributed to the Equivalence Principal in which the observer in the elevator cannot discern whether they are in an acceleration rocket ship or standing stationary on Earth. In that case, the time-like metric component is the sole cause of the free-fall of light in gravity. In the Schwarzschild metric, on the other hand, the curvature of the field near a spherical gravitating body also contributes. In this case, the geodesic equation, assuming that dr/dt = 0 for the incoming photon, is

where, as before, the Christoffel symbol for the radial displacements are

Evaluating one of these

The other Christoffel symbol that contributes to the radial motion is

and the geodesic equation becomes

with

The radial acceleration of the light ray in the elevator is thus

The first term on right is free-fall in gravity, just as was obtained from the Equivalence Principal. The second term is a higher-order correction caused by curvature of spacetime. The third term is the motion of the light ray relative to the curved ceiling of the elevator in this spherical geometry and hence is a kinematic (or geometric) artefact. (It is interesting that the GR correction on the curved-ceiling correction is of the same order as the free-fall term, so one would need to be very careful doing such an experiment … if it were at all measurable.) Therefore, the second and third terms are curved-geometry effects while the first term is the free fall of the light ray.

Post-Script: The Importance of Library Collections

I was amused to see the library card of the scanned Internet Archive version of Einstein’s Jahrbuch article, shown below. The volume was checked out in August of 1981 from the UC Berkeley Physics Library. It was checked out again 7 years later in September of 1988. These dates coincide with when I arrived at Berkeley to start grad school in physics, and when I graduated from Berkeley to start my post-doc position at Bell Labs. Hence this library card serves as the book ends to my time in Berkeley, a truly exhilarating place that was the top-ranked physics department at that time, with 7 active Nobel Prize winners on its faculty.

During my years at Berkeley, I scoured the stacks of the Physics Library looking for books and journals of historical importance, and was amazed to find the original volumes of Annalen der Physik from 1905 where Einstein published his famous works. This was the same library where, ten years before me, John Clauser was browsing the stacks and found the obscure paper by John Bell on his inequalities that led to Clauser’s experiment on entanglement that won him the Nobel Prize of 2022.

That library at UC Berkeley was recently closed, as was the Physics Library in my department at Purdue University (see my recent Blog), where I also scoured the stacks for rare gems. Some ancient books that I used to be able to check out on a whim, just to soak up their vintage ambience and to get a tactile feel for the real thing held in my hands, are now not even available through Interlibrary Loan. I may be able to get scans from Internet Archive online, but the palpable magic of the moment of discovery is lost.

References:

[1] Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Annalen der Physik, 17(10), 891–921.

[2] Pais, A (2005). Subtle is the Lord: The Science and Life of Albert Einstein (Oxford University Press). pg. 178

[3] Einstein, A. (1907). Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen. Jahrbuch der Radioaktivität und Elektronik, 4, 411–462.

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

[5] Einstein, A. (1916). Über die spezielle und die allgemeine Relativitätstheorie (Gemeinverständlich). Braunschweig: Friedr. Vieweg & Sohn.

[6] Einstein, A. (1920). Relativity: The Special and the General Theory (A Popular Exposition) (R. W. Lawson, Trans.). London: Methuen & Co. Ltd.

[7] Nolte, D. D. (2018). Galileo Unbound. A Path Across Life, the Universe and Everything. (Oxford University Press)

[8] Nolte, D. D. (2019). Introduction to Modern Dynamics: Chaos, Networks, Space and Time (Oxford University Press).

Read more in Books by David Nolte at Oxford University Press.

December 27, 2023September 5, 2025 by David D. Nolte

A Brief History of Nothing: The Physics of the Vacuum from Atomism to Higgs

It may be hard to get excited about nothing … unless nothing is the whole ball game.

The only way we can really know what is, is by knowing what isn’t. Nothing is the backdrop against which we measure something. Experimentalists spend almost as much time doing control experiments, where nothing happens (or nothing is supposed to happen) as they spend measuring a phenomenon itself, the something.

Even the universe, full of so much something, came out of nothing during the Big Bang. And today the energy density of nothing, so-called Dark Energy, is blowing our universe apart, propelling it ever faster to a bitter cold end.

So here is a brief history of nothing, tracing how we have understood what it is, where it came from, and where is it today.

With sturdy shoulders, space stands opposing all its weight to nothingness. Where space is, there is being.
Friedrich Nietzsche

40,000 BCE – Cosmic Origins

This is a human history, about how we homo sapiens try to understand the natural world around us, so the first step on a history of nothing is the Big Bang of human consciousness that occurred sometime between 100,000 – 40,000 years ago. Some sort of collective phase transition happened in our thought process when we seem to have become aware of our own existence within the natural world. This time frame coincides with the beginning of representational art and ritual burial. This is also likely the time when human language skills reached their modern form, and when logical arguments–stories–first were told to explain our existence and origins.

Roughly two origin stories emerged from this time. One of these assumes that what is has always been, either continuously or cyclically. Buddhism and Hinduism are part of this tradition as are many of the origin philosophies of Indigenous North Americans. Another assumes that there was a beginning when everything came out of nothing. Abrahamic faiths (Let there be light!) subscribe to this creatio ex nihilo. What came before creation? Nothing!

500 BCE – Leucippus and Democritus Atomism

The Greek philosopher Leucippus and his student Democritus, living around 500 BCE, were the first to lay out the atomic theory in which the elements of substance were indivisible atoms of matter, and between the atoms of matter was void. The different materials around us were created by the different ways that these atoms collide and cluster together. Plato later adhered to this theory, developing ideas along these lines in his Timeaus.

300 BCE – Aristotle Vacuum

Aristotle is famous for arguing, in his Physics Book IV, Section 8, that nature abhors a vacuum (horror vacui) because any void would be immediately filled by the imposing matter surrounding it. He also argued more philosophically that nothing, by definition, cannot exist.

1644 – Rene Descartes Vortex Theory

Fast forward a millennia and a half, and theories of existence were finally achieving a level of sophistication that can be called “scientific”. Rene Descartes followed Aristotle’s views of the vacuum, but he extended it to the vacuum of space, filling it with an incompressible fluid in his Principles of Philosophy (1644). Just like water, laminar motion can only occur by shear, leading to vortices. Descartes was a better philosopher than mathematician, so it took Christian Huygens to apply mathematics to vortex motion to “explain” the gravitational effects of the solar system.

Rene Descartes, Vortex Theory, 1644. Image Credit

1654 – Otto von Guericke Vacuum Pump

Otto von Guericke is one of those hidden gems of the history of science, a person who almost no-one remembers today, but who was far in advance of his own day. He was a powerful politician, holding the position of Burgomeister of the city of Magdeburg for more than 30 years, helping to rebuild it after it was sacked during the Thirty Years War. He was also a diplomat, playing a key role in the reorientation of power within the Holy Roman Empire. How he had free time is anyone’s guess, but he used it to pursue scientific interests that spanned from electrostatics to his invention of the vacuum pump.

With a succession of vacuum pumps, each better than the last, von Geuricke was like a kid in a toy factory, pumping the air out of anything he could find. In the process, he showed that a vacuum would extinguish a flame and could raise water in a tube.

His most famous demonstration was, of course, the Magdeburg sphere demonstration. In 1657 he fabricated two 20-inch hemispheres that he attached together with a vacuum seal and used his vacuum pump to evacuate the air from inside. He then attached chains from the hemispheres to a team of eight horses on each side, for a total of 16 horses, who were unable to separate the spheres. This dramatically demonstrated that air exerts a force on surfaces, and that Aristotle and Descartes were wrong—nature did allow a vacuum!

1667 – Isaac Newton Action at a Distance

When it came to the vacuum, Newton was agnostic. His universal theory of gravitation posited action at a distance, but the intervening medium played no direct role.

Nothing comes from nothing, Nothing ever could.
Rogers and Hammerstein, The Sound of Music

This would seem to say that Newton had nothing to say about the vacuum, but his other major work, his Optiks, established particles as the elements of light rays. Such light particles travelled easily through vacuum, so the particle theory of light came down on the empty side of space.

Statue of Isaac Newton by Sir Eduardo Paolozzi based on a painting by William Blake. Image Credit

1821 – Augustin Fresnel Luminiferous Aether

Today, we tend to think of Thomas Young as the chief proponent for the wave nature of light, going against the towering reputation of his own countryman Newton, and his courage and insights are admirable. But it was Augustin Fresnel who put mathematics to the theory. It was also Fresnel, working with his friend Francois Arago, who established that light waves are purely transverse.

For these contributions, Fresnel stands as one of the greatest physicists of the 1800’s. But his transverse light waves gave birth to one of the greatest red herrings of that century—the luminiferous aether. The argument went something like this, “if light is waves, then just as sound is oscillations of air, light must be oscillations of some medium that supports it – the luminiferous aether.” Arago searched for effects of this aether in his astronomical observations, but he didn’t see it, and Fresnel developed a theory of “partial aether drag” to account for Arago’s null measurement. Hippolyte Fizeau later confirmed the Fresnel “drag coefficient” in his famous measurement of the speed of light in moving water. (For the full story of Arago, Fresnel and Fizeau, see Chapter 2 of “Interference”. [1])

But the transverse character of light also required that this unknown medium must have some stiffness to it, like solids that support transverse elastic waves. This launched almost a century of alternative ideas of the aether that drew in such stellar actors as George Green, George Stokes and Augustin Cauchy with theories spanning from complete aether drag to zero aether drag with Fresnel’s partial aether drag somewhere in the middle.

1849 – Michael Faraday Field Theory

Micheal Faraday was one of the most intuitive physicists of the 1800’s. He worked by feel and mental images rather than by equations and proofs. He took nothing for granted, able to see what his experiments were telling him instead of looking only for what he expected.

This talent allowed him to see lines of force when he mapped out the magnetic field around a current-carrying wire. Physicists before him, including Ampere who developed a mathematical theory for the magnetic effects of a wire, thought only in terms of Newton’s action at a distance. All forces were central forces that acted in straight lines. Faraday’s experiments told him something different. The magnetic lines of force were circular, not straight. And they filled space. This realization led him to formulate his theory for the magnetic field.

Others at the time rejected this view, until William Thomson (the future Lord Kelvin) wrote a letter to Faraday in 1845 telling him that he had developed a mathematical theory for the field. He suggested that Faraday look for effects of fields on light, which Faraday found just one month later when he observed the rotation of the polarization of light when it propagated in a high-index material subject to a high magnetic field. This effect is now called Faraday Rotation and was one of the first experimental verifications of the direct effects of fields.

Nothing is more real than nothing.
Samuel Beckett

In 1949, Faraday stated his theory of fields in their strongest form, suggesting that fields in empty space were the repository of magnetic phenomena rather than magnets themselves [2]. He also proposed a theory of light in which the electric and magnetic fields induced each other in repeated succession without the need for a luminiferous aether.

1861 – James Clerk Maxwell Equations of Electromagnetism

James Clerk Maxwell pulled the various electric and magnetic phenomena together into a single grand theory, although the four succinct “Maxwell Equations” was condensed by Oliver Heaviside from Maxwell’s original 15 equations (written using Hamilton’s awkward quaternions) down to the 4 vector equations that we know and love today.

One of the most significant and most surprising thing to come out of Maxwell’s equations was the speed of electromagnetic waves that matched closely with the known speed of light, providing near certain proof that light was electromagnetic waves.

However, the propagation of electromagnetic waves in Maxwell’s theory did not rule out the existence of a supporting medium—the luminiferous aether. It was still not clear that fields could exist in a pure vacuum but might still be like the stress fields in solids.

Late in his life, just before he died, Maxwell pointed out that no measurement of relative speed through the aether performed on a moving Earth could see deviations that were linear in the speed of the Earth but instead would be second order. He considered that such second-order effects would be far to small ever to detect, but Albert Michelson had different ideas.

1887 – Albert Michelson Null Experiment

Albert Michelson was convinced of the existence of the luminiferous aether, and he was equally convinced that he could detect it. In 1880, working in the basement of the Potsdam Observatory outside Berlin, he operated his first interferometer in a search for evidence of the motion of the Earth through the aether. He had built the interferometer, what has come to be called a Michelson Interferometer, months earlier in the laboratory of Hermann von Helmholtz in the center of Berlin, but the footfalls of the horse carriages outside the building disturbed the measurements too much—Postdam was quieter.

But he could find no difference in his interference fringes as he oriented the arms of his interferometer parallel and orthogonal to the Earth’s motion. A simple calculation told him that his interferometer design should have been able to detect it—just barely—so the null experiment was a puzzle.

Seven years later, again in a basement (this time in a student dormitory at Western Reserve College in Cleveland, Ohio), Michelson repeated the experiment with an interferometer that was ten times more sensitive. He did this in collaboration with Edward Morley. But again, the results were null. There was no difference in the interference fringes regardless of which way he oriented his interferometer. Motion through the aether was undetectable.

(Michelson has a fascinating backstory, complete with firestorms (literally) and the Wild West and a moment when he was almost committed to an insane asylum against his will by a vengeful wife. To read all about this, see Chapter 4: After the Gold Rush in my recent book Interference (Oxford, 2023)).

Michelson-Morley Experiment. Reprinted from Interference: The History of Optical Interferometry and the Scientists Who Tamed Light (Oxford, 2023)

The Michelson Morley experiment did not create the crisis in physics that it is sometimes credited with. They published their results, and the physics world took it in stride. Voigt and Fitzgerald and Lorentz and Poincaré toyed with various ideas to explain it away, but there had already been so many different models, from complete drag to no drag, that a few more theories just added to the bunch.

But they all had their heads in a haze. It took an unknown patent clerk in Switzerland to blow away the wisps and bring the problem into the crystal clear.

1905 – Albert Einstein Relativity

So much has been written about Albert Einstein’s “miracle year” of 1905 that it has lapsed into a form of physics mythology. Looking back, it seems like his own personal Big Bang, springing forth out of the vacuum. He published 5 papers that year, each one launching a new approach to physics on a bewildering breadth of problems from statistical mechanics to quantum physics, from electromagnetism to light … and of course, Special Relativity [3].

Whereas the others, Voigt and Fitzgerald and Lorentz and Poincaré, were trying to reconcile measurements of the speed of light in relative motion, Einstein just replaced all that musing with a simple postulate, his second postulate of relativity theory:

2. Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence …
Albert Einstein, Annalen der Physik, 1905

And the rest was just simple algebra—in complete agreement with Michelson’s null experiment, and with Fizeau’s measurement of the so-called Fresnel drag coefficient, while also leading to the famous E = mc² and beyond.

There is no aether. Electromagnetic waves are self-supporting in vacuum—changing electric fields induce changing magnetic fields that induce, in turn, changing electric fields—and so it goes.

The vacuum is vacuum—nothing! Except that it isn’t. It is still full of things.

1931 – P. A. M Dirac Antimatter

The Dirac equation is the famous end-product of P. A. M. Dirac’s search for a relativistic form of the Schrödinger equation. It replaces the asymmetric use in Schrödinger’s form of a second spatial derivative and a first time derivative with Dirac’s form using only first derivatives that are compatible with relativistic transformations [4].

One of the immediate consequences of this equation is a solution that has negative energy. At first puzzling and hard to interpret [5], Dirac eventually hit on the amazing proposal that these negative energy states are real particles paired with ordinary particles. For instance, the negative energy state associated with the electron was an anti-electron, a particle with the same mass as the electron, but with positive charge. Furthermore, because the anti-electron has negative energy and the electron has positive energy, these two particles can annihilate and convert their mass energy into the energy of gamma rays. This audacious proposal was confirmed by the American physicist Carl Anderson who discovered the positron in 1932.

The existence of particles and anti-particles, combined with Heisenberg’s uncertainty principle, suggests that vacuum fluctuations can spontaneously produce electron-positron pairs that would then annihilate within a time related to the mass energy

Although this is an exceedingly short time (about 10^-21 seconds), it means that the vacuum is not empty, but contains a frothing sea of particle-antiparticle pairs popping into and out of existence.

1938 – M. C. Escher Negative Space

Scientists are not the only ones who think about empty space. Artists, too, are deeply committed to a visual understanding of our world around us, and the uses of negative space in art dates back virtually to the first cave paintings. However, artists and art historians only talked explicitly in such terms since the 1930’s and 1940’s [6]. One of the best early examples of the interplay between positive and negative space was a print made by M. C. Escher in 1938 titled “Day and Night”.

M. C. Escher. Day and Night. Image Credit

1946 – Edward Purcell Modified Spontaneous Emission

In 1916 Einstein laid out the laws of photon emission and absorption using very simple arguments (his modus operandi) based on the principles of detailed balance. He discovered that light can be emitted either spontaneously or through stimulated emission (the basis of the laser) [7]. Once the nature of vacuum fluctuations was realized through the work of Dirac, spontaneous emission was understood more deeply as a form of stimulated emission caused by vacuum fluctuations. In the absence of vacuum fluctuations, spontaneous emission would be inhibited. Conversely, if vacuum fluctuations are enhanced, then spontaneous emission would be enhanced.

This effect was observed by Edward Purcell in 1946 through the observation of emission times of an atom in a RF cavity [8]. When the atomic transition was resonant with the cavity, spontaneous emission times were much faster. The Purcell enhancement factor is

where Q is the “Q” of the cavity, and V is the cavity volume. The physical basis of this effect is the modification of vacuum fluctuations by the cavity modes caused by interference effects. When cavity modes have constructive interference, then vacuum fluctuations are larger, and spontaneous emission is stimulated more quickly.

1948 – Hendrik Casimir Vacuum Force

Interference effects in a cavity affect the total energy of the system by excluding some modes which become inaccessible to vacuum fluctuations. This lowers the internal energy internal to a cavity relative to free space outside the cavity, resulting in a net “pressure” acting on the cavity. If two parallel plates are placed in close proximity, this would cause a force of attraction between them. The effect was predicted in 1948 by Hendrik Casimir [9], but it was not verified experimentally until 1997 by S. Lamoreaux at Yale University [10].

Two plates brought very close feel a pressure exerted by the higher vacuum energy density external to the cavity.

1949 – Shinichiro Tomonaga, Richard Feynman and Julian Schwinger QED

The physics of the vacuum in the years up to 1948 had been a hodge-podge of ad hoc theories that captured the qualitative aspects, and even some of the quantitative aspects of vacuum fluctuations, but a consistent theory was lacking until the work of Tomonaga in Japan, Feynman at Cornell and Schwinger at Harvard. Feynman and Schwinger both published their theory of quantum electrodynamics (QED) in 1949. They were actually scooped by Tomonaga, who had developed his theory earlier during WWII, but physics research in Japan had been cut off from the outside world. It was when Oppenheimer received a letter from Tomonaga in 1949 that the West became aware of his work. All three received the Nobel Prize for their work on QED in 1965. Precision tests of QED now make it one of the most accurately confirmed theories in physics.

Richard Feynman’s first “Feynman diagram”.

1964 – Peter Higgs and The Higgs

The Higgs particle, known as “The Higgs”, was the brain-child of Peter Higgs, Francois Englert and Gerald Guralnik in 1964. Higgs’ name became associated with the theory because of a response letter he wrote to an objection made about the theory. The Higg’s mechanism is spontaneous symmetry breaking in which a high-symmetry potential can lower its energy by distorting the field, arriving at a new minimum in the potential. This mechanism can allow the bosons that carry force to acquire mass (something the earlier Yang-Mills theory could not do).

Spontaneous symmetry breaking is a ubiquitous phenomenon in physics. It occurs in the solid state when crystals can lower their total energy by slightly distorting from a high symmetry to a low symmetry. It occurs in superconductors in the formation of Cooper pairs that carry supercurrents. And here it occurs in the Higgs field as the mechanism to imbues particles with mass .

Conceptual graph of a potential surface where the high symmetry potential is higher than when space is distorted to lower symmetry. Image Credit

The theory was mostly ignored for its first decade, but later became the core of theories of electroweak unification. The Large Hadron Collider (LHC) at Geneva was built to detect the boson, announced in 2012. Peter Higgs and Francois Englert were awarded the Nobel Prize in Physics in 2013, just one year after the discovery.

The Higgs field permeates all space, and distortions in this field around idealized massless point particles are observed as mass. In this way empty space becomes anything but.

1981 – Alan Guth Inflationary Big Bang

Problems arose in observational cosmology in the 1970’s when it was understood that parts of the observable universe that should have been causally disconnected were in thermal equilibrium. This could only be possible if the universe were much smaller near the very beginning. In January of 1981, Alan Guth, then at Cornell University, realized that a rapid expansion from an initial quantum fluctuation could be achieved if an initial “false vacuum” existed in a positive energy density state (negative vacuum pressure). Such a false vacuum could relax to the ordinary vacuum, causing a period of very rapid growth that Guth called “inflation”. Equilibrium would have been achieved prior to inflation, solving the observational problem.Therefore, the inflationary model posits a multiplicities of different types of “vacuum”, and once again, simple vacuum is not so simple.

Energy density as a function of a scalar variable. Quantum fluctuations create a “false vacuum” that can relax to “normal vacuum: by expanding rapidly. Image Credit

1998 – Saul Pearlmutter Dark Energy

Einstein didn’t make many mistakes, but in the early days of General Relativity he constructed a theoretical model of a “static” universe. A central parameter in Einstein’s model was something called the Cosmological Constant. By tuning it to balance gravitational collapse, he tuned the universe into a static Ithough unstable) state. But when Edwin Hubble showed that the universe was expanding, Einstein was proven incorrect. His Cosmological Constant was set to zero and was considered to be a rare blunder.

Fast forward to 1999, and the Supernova Cosmology Project, directed by Saul Pearlmutter, discovered that the expansion of the universe was accelerating. The simplest explanation was that Einstein had been right all along, or at least partially right, in that there was a non-zero Cosmological Constant. Not only is the universe not static, but it is literally blowing up. The physical origin of the Cosmological Constant is believed to be a form of energy density associated with the space of the universe. This “extra” energy density has been called “Dark Energy”, filling empty space.

The expanding size of the Universe. Image Credit

Bottom Line

The bottom line is that nothing, i.e., the vacuum, is far from nothing. It is filled with a froth of particles, and energy, and fields, and potentials, and broken symmetries, and negative pressures, and who knows what else as modern physics has been much ado about this so-called nothing, almost more than it has been about everything else.

References:

[1] David D. Nolte, Interference: The History of Optical Interferometry and the Scientists Who Tamed Light (Oxford University Press, 2023)

[2] L. Peirce Williams in “Faraday, Michael.” Complete Dictionary of Scientific Biography, vol. 4, Charles Scribner’s Sons, 2008, pp. 527-540.

[3] A. Einstein, “On the electrodynamics of moving bodies,” Annalen Der Physik 17, 891-921 (1905).

[4] Dirac, P. A. M. (1928). “The Quantum Theory of the Electron”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624.

[5] Dirac, P. A. M. (1930). “A Theory of Electrons and Protons”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 126 (801): 360–365.

[6] Nikolai M Kasak, Physical Art: Action of positive and negative space, (Rome, 1947/48) [2d part rev. in 1955 and 1956].

[7] A. Einstein, “Strahlungs-Emission un -Absorption nach der Quantentheorie,” Verh. Deutsch. Phys. Ges. 18, 318 (1916).

[8] Purcell, E. M. (1946-06-01). “Proceedings of the American Physical Society: Spontaneous Emission Probabilities at Ratio Frequencies”. Physical Review. American Physical Society (APS). 69 (11–12): 681.

[9] Casimir, H. B. G. (1948). “On the attraction between two perfectly conducting plates”. Proc. Kon. Ned. Akad. Wet. 51: 793.

[10] Lamoreaux, S. K. (1997). “Demonstration of the Casimir Force in the 0.6 to 6 μm Range”. Physical Review Letters. 78 (1): 5–8.

March 2, 2020October 24, 2023 by David D. Nolte

A Commotion in the Stars: The History of the Doppler Effect

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 18^th 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 [1] 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 30^th 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).

Salzburg Austria

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 [2]. 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 [3]. 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” [4].

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 [5] 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).

Christian Doppler

Voigt’s Transformation

It is difficult today to appreciate just how deeply engrained the reality of the luminiferous ether was in the psyche of the 19^th 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 20^th 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 [6]. 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 [7].

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) [8], 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 [9]. 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 [10]. 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 [11].

Notes

[1] pg. 15, Eden, A. (1992). The search for Christian Doppler. Wien, Springer-Verlag.

[2] pg. 30, Eden

[3] Bradley, J (1729). “Account of a new discoved Motion of the Fix’d Stars”. Phil Trans. 35: 637–660.

[4] 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).

[5] 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.

[6] 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.

[7] A. Einstein, “On the electrodynamics of moving bodies,” Annalen Der Physik, vol. 17, pp. 891-921, 1905.

[8] 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.

[9] A. Einstein, “”Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips”,” vol. 328, pp. 197–198, 1907.

[10] 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.

[11] B. Rossi and D. B. Hall, “Variation of the Rate of Decay of Mesotrons with Momentum,” Physical Review, vol. 59, pp. 223–228, 1941.

September 30, 2019July 24, 2023 by David D. Nolte

Science 1916: Schwarzschild, Einstein, Planck, Born, Frobenius et al.

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.

Schwarzschild’s solution to Einstein’s field equations for a point mass.

Schwarzschild’s extension of the field equation solutions to a finite incompressible fluid.

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 proposal of action-angle variables for quantization of dynamical systems.

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.

Relativistic energy density of electromagnetic fields.

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.

Einstein’s weak-field low-velocity approximation solutions of his field equations.

Einstein’s prediction of gravitational waves.

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 4^th 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.

Rubens and the infrared spectrum of steam.

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

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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

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Erdmann, Methodologische Konsequenzen aus der Theorie der Abstraktion

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Dirac relativistic quantum wave equation

February 23, 2019November 29, 2025 by David D. Nolte

Dirac: From Quantum Field Theory to Antimatter

Paul Adrian Maurice Dirac (1902 – 1984) was given the moniker of “the strangest man” by Niels Bohr while he was reminiscing about the many great scientists with whom he had worked over the years [1]. It is a moniker that resonates with the innumerable “Dirac stories” that abound in the mythology of the hallways of physics departments around the world. Dirac was awkward, shy, a loner, rarely said anything, was completely literal, had not the slightest comprehension of art or poetry, nor any clear understanding of human interpersonal interaction. Dirac was also brilliant, providing the theoretical foundation for the central paradigm of modern physics—quantum field theory. The discovery of the Higgs boson in 2012, a human achievement that capped nearly a century of scientific endeavor, rests solidly on the theory of quantum fields that permeate space. The Higgs particle, when it pops into existence at the Large Hadron Collider in Geneva, is a singular quantum excitation of the Higgs field, a field that usually resides in a vacuum state, frothing with quantum fluctuations that imbue all particles—and you and me—with mass. The Higgs field is Dirac’s legacy.

… all of a sudden he had a new equation with four-dimensional space-time symmetry.

Copenhagen and Bohr

Although Dirac as a young scientist was initially enthralled with relativity theory, he was working under Ralph Fowler (1889 – 1944) in the physics department at Cambridge in 1925 when he had the chance to read advanced proofs of Heisenberg’s matrix mechanics paper. This chance event launched him on his own trajectory in quantum theory. After Dirac was awarded his doctorate from Cambridge in 1926, he received a stipend that sent him to work with Niels Bohr (1885 – 1962) in Copenhagen—ground zero of the new physics. During his time there, Dirac became famous for taking long walks across Copenhagen as he played about with things in his mind, performing mental juggling of abstract symbols, envisioning how they would permute and act. His attention was focused on the electromagnetic field and how it interacted with the quantized states of atoms. Although the electromagnetic field was the classical field of light, it was also the quantum field of Einstein’s photon, and he wondered how the quantized harmonic oscillators of the electromagnetic field could be generated by quantum wavefunctions acting as operators. But acting on what? He decided that, to generate a photon, the wavefunction must operate on a state that had no photons—the ground state of the electromagnetic field known as the vacuum state.

In late 1926, nearing the end of his stay in Copenhagen with Bohr, Dirac put these thoughts into their appropriate mathematical form and began work on two successive manuscripts. The first manuscript contained the theoretical details of the non-commuting electromagnetic field operators. He called the process of generating photons out of the vacuum “second quantization”. This phrase is a bit of a misnomer, because there is no specific “first quantization” per se, although he was probably thinking of the quantized energy levels of Schrödinger and Heisenberg. In second quantization, the classical field of electromagnetism is converted to an operator that generates quanta of the associated quantum field out of the vacuum (and also annihilates photons back into the vacuum). The creation operators can be applied again and again to build up an N-photon state containing N photons that obey Bose-Einstein statistics, as they must, as required by their integer spin, agreeing with Planck’s blackbody radiation.

Dirac then went further to show how an interaction of the quantized electromagnetic field with quantized energy levels involved the annihilation and creation of photons as they promoted electrons to higher atomic energy levels, or demoted them through stimulated emission. Very significantly, Dirac’s new theory explained the spontaneous emission of light from an excited electron level as a direct physical process that creates a photon carrying away the energy as the electron falls to a lower energy level. Spontaneous emission had been explained first by Einstein more than ten years earlier when he derived the famous A and B coefficients, but Einstein’s arguments were based on the principle of detailed balance, which is a thermodynamic argument. It is impressive that Einstein’s deep understanding of thermodynamics and statistical mechanics could allow him to derive the necessity of both spontaneous and stimulated emission, but the physical mechanism for these processes was inferred rather than derived. Dirac, in late 1926, had produced the first direct theory of photon exchange with matter. This was the birth of quantum electrodynamics, known as QED, and the birth of quantum field theory [2].

Göttingen and Born

Dirac’s next stop on his postodctoral fellowship was in Göttingen to work with Max Born (1882 – 1970) and the large group of theoreticians and mathematicians who were like electrons in a cloud orbiting around the nucleus represented by the new quantum theory. Göttingen was second only to Copenhagen as the Mecca for quantum theorists. Hilbert was there and von Neumann too, as well as the brash American J. Robert Oppenheimer (1904 – 1967) who was finishing his PhD with Born. Dirac and Oppenheimer struck up an awkward friendship. Oppenheimer was considered arrogant by many others in the group, but he was in awe of Dirac who arrived with his manuscript on quantum electrodynamics ready for submission. Oppenheimer struggled at first to understand Dirac’s new approach to quantizing fields, but he quickly grasped the importance, as did Pascual Jordan (1902 – 1980), who was also in Göttingen.

Jordan had already worked on ideas very close to Dirac’s on the quantization of fields. He and Dirac seemed to be going down the same path, independently arriving at very similar conclusions around the same time. In fact, Jordan was often a step ahead of Dirac, tending to publish just before Dirac, as with non-commuting matrices, transformation theory and the relationship of canonical transformations to second quantization. However, Dirac’s paper on quantum electrodynamics was a masterpiece in clarity and comprehensiveness, launching a new field in a way that Jordan had not yet achieved with his own work. But because of the closeness of Jordan’s thinking to Dirac’s, he was able to see immediately how to extend Dirac’s approach. Within the year, he published a series of papers that established the formalism of quantum electrodynamics as well as quantum field theory. With Pauli, he systematized the operators for creation and annihilation of photons [3]. With Wigner, he developed second quantization for de Broglie matter waves, defining creation and annihilation operators that obeyed the Pauli exclusion principle of electrons[4]. Jordan was on a roll, forging ahead of Dirac on extensions of quantum electrodynamics and field theory, but Dirac was about to eclipse Jordan once and for all.

St. John’s at Cambridge

At the end of the Spring semester in 1927, Dirac was offered a position as a fellow of St. John’s College at Cambridge, which he accepted, returning to England to begin his life as a college professor. During the summer and into the Fall, Dirac returned to his first passion in physics, relativity, which had yet to be successfully incorporated into quantum physics. Oskar Klein and Walter Gordon had made initial attempts at formulating relativistic quantum theory, but they could not correctly incorporate the spin properties of the electron, and their wave equation had the bad habit of producing negative probabilities. Probabilities went negative because the Klein-Gordon equation had two time derivatives instead of one. The reason it had two (while the non-relativistic Schrödinger equation has only one) is because space-time symmetry required the double space derivative of the Schrödinger equation to be paired with a double time derivative. Dirac, with creative insight, realized that the problem could be flipped by requiring the single time derivative to be paired with a single space derivative. The problem was that a single space derivative did not seem to make any sense [5].

As Dirac puzzled how to get an equation with only single derivatives, he was playing around with Pauli spin matrices and hit on a simple identity that related the spin matrices to the electron momentum. At first he could not get the identity to apply to four-dimensional relativistic momenta using the usual 2×2 spin matrices. Then he realized that four-dimensional space-time could be captured if he expanded Pauli’s 2×2 spin matrices to 4×4 spin matrices, and all of a sudden he had a new equation with four-dimensional space-time symmetry with single derivatives on space and time. As a test of his new equation, he calculated fine details of the experimentally-measured hydrogen spectrum, known as the fine structure, which had resisted theoretical explanation, and he derived answers in close agreement with experiment. He also showed that the electron had spin-1/2, and he calculated its magnetic moment. He finished his manuscript at the end of the Fall semester in 1927, and the paper was published in early 1928[6]. His relativistic quantum wave equation was an instant sensation, becoming known for all time as “the Dirac Equation”. He had succeeded at finding a correct and long-sought relativistic quantum theory where many others had failed, such as Oskar Klein and Paul Gordon. It was a crowning achievement, placing Dirac firmly in the firmament of the quantum theorists.

Fig. 1 The relativistic Dirac equation. The wavefunction is a four-component spinor. The gamma-del product is a 4×4 matrix operator. The time and space derivatives are both first-order operators.

Antimatter

In the process of ridding the Klein-Gordon equation of negative probability, which Dirac found abhorent, his new equation created an infinite number of negative energy states, which he did not find abhorent. It is perhaps a matter of taste what one theoriest is willing to accept over another, and for Dirac, negative energies were better than negative probabilities. Even so, one needed to deal with an infinite number of negative energy states in quantum theory, because they are available to quantum transitions. In 1929 and 1930, as Dirac was writing his famous textbook on quantum theory, he became intrigued by the similarity between the positive and negative electron states of the vacuum and the energy levels of valence electrons on atoms. An electron in a state outside a filled electron shell behaves very much like a single-electron atom, like sodium and lithium with their single valence electrons. Conversely, an atomic shell that has one electron less than a full complement can be described as having a “hole” that behaves “as if” it were a positive particle. It is like a bubble in water. As water sinks, the bubble rises to the top of the water level. For electrons, if all the electrons go one way in an electric field, then the hole goes the opposite direction, like a positive charge.

Dirac took this analogy of nearly-filled atomic shells and applied it to the vacuum states of the electron, viewing the filled negative energy states like the filled electron shells of atoms. If there is a missing electron, a hole in this infinite sea, then it would behave as if it had positive charge. Initially, Dirac speculated that the “hole” was the proton, and he even wrote a paper on that possibility. But Oppenheimer pointed out that the idea was inconsistent with observations, especially the inability of the electron and proton to annihilate, and that the ground state of the infinite electron sea must be completely filled. Hermann Weyl further pointed out that the electron-proton theory did not have the correct symmetry, and Dirac had to rethink. In early 1931 he hit on an audacious solution to the puzzle. What if the hole in the infinite negative energy sea did not just behave like a positive particle, but actually was a positive particle, a new particle that Dirac dubbed the “anti-electron”? The anti-electron would have the same mass as the electron, but would have positive charge. He suggested that such particles might be generated in high-energy collisions in vacuum, and he finished his paper with the suggestion that there also could be an anti-proton with the mass of the proton but with negative charge. In this singular paper, titled “Quantized Singularities of the Electromagnetic Field” published in 1931, Dirac predicted the existence of antimatter. A year later the positron was discovered by Carl David Anderson at Cal Tech. Anderson had originally called the particle the positive electron, but a journal editor of the Physical Review changed it to positron, and the new name stuck.

Fig. 3 An electron-positron pair is created by the absorption of a photon (gamma ray). Positrons have negative energy and can be viewed as a hole in a sea of filled electron states. (Momentum conservation is satisfied if a near-by heavy particle takes up the recoil momentum.)

The prediction and subsequent experimental validation of antmatter stands out in the history of physics in the 20^th Century. In previous centuries, theory was performed mainly in the service of experiment, explaining interesting new observed phenomena either as consequences of known physics, or creating new physics to explain the observations. Quantum theory, revolutionary as a way of understanding nature, was developed to explain spectroscopic observations of atoms and molecules and gases. Similarly, the precession of the perihelion of Mercury was a well-known phenomenon when Einstein used his newly developed general relativity to explain it. As a counter example, Einstein’s prediction of the deflection of light by the Sun was something new that emerged from theory. This is one reason why Einstein became so famous after Eddington’s expedition to observe the deflection of apparent star locations during the total eclipse. Einstein had predicted something that had never been seen before. Dirac’s prediction of the existence of antimatter similarly is a triumph of rational thought, following the mathematical representation of reality to an inevitable conclusion that cannot be ignored, no matter how wild and initially unimaginable it is. Dirac went on to receive the Nobel prize in Physics in 1933, sharing the prize that year with Schrödinger (Heisenberg won it the previous year in 1932).

Read the stories behind the history of quantum field theory, in Galileo Unbound from Oxford University Press

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References

[1] Framelo, “The Strangest Man: The Hidden Life of Paul Dirac” (Basic Books, 2011)

[2] Dirac, P. A. M. (1927). “The quantum theory of the emission and absorption of radiation.” Proceedings of the Royal Society of London Series A114(767): 243-265.; Dirac, P. A. M. (1927). “The quantum theory of dispersion.” Proceedings of the Royal Society of London Series A114(769): 710-728.

[3] Jordan, P. and W. Pauli, Jr. (1928). “To quantum electrodynamics of free charge fields.” Zeitschrift Fur Physik 47(3-4): 151-173.

[4] Jordan, P. and E. Wigner (1928). “About the Pauli’s equivalence prohibited.” Zeitschrift Fur Physik 47(9-10): 631-651.

[5] This is because two space derivatives measure the curvative of the wavefunction which is related to the kinetic energy of the electron.

[6] Dirac, P. A. M. (1928). “The quantum theory of the electron.” Proceedings of the Royal Society of London Series A 117(778): 610-624.; Dirac, P. A. M. (1928). “The quantum theory of the electron – Part II.” Proceedings of the Royal Society of London Series A118(779): 351-361.

“My Happiest Thought”

The Observer in the Chest: There is no Elevator

The Evolution of Physics: Enter Einstein’s Elevator

Light Deflection in the Equivalence Principle

Light Deflection in the Schwarzschild Metric

Post-Script: The Importance of Library Collections

References:

40,000 BCE – Cosmic Origins

500 BCE – Leucippus and Democritus Atomism

300 BCE – Aristotle Vacuum

1644 – Rene Descartes Vortex Theory

1654 – Otto von Guericke Vacuum Pump

1667 – Isaac Newton Action at a Distance

1821 – Augustin Fresnel Luminiferous Aether

1849 – Michael Faraday Field Theory

1861 – James Clerk Maxwell Equations of Electromagnetism

1887 – Albert Michelson Null Experiment

1905 – Albert Einstein Relativity

1931 – P. A. M Dirac Antimatter

1938 – M. C. Escher Negative Space

1946 – Edward Purcell Modified Spontaneous Emission

1948 – Hendrik Casimir Vacuum Force

1949 – Shinichiro Tomonaga, Richard Feynman and Julian Schwinger QED

1964 – Peter Higgs and The Higgs

1981 – Alan Guth Inflationary Big Bang

1998 – Saul Pearlmutter Dark Energy

Bottom Line

References:

Doppler in Prague

Voigt’s Transformation

Further Reading

Notes

Karl Schwarzschild (1873 – 1916)

Albert Einstein (1879 – 1955)

Max Planck (1858 – 1947)

Max Born (1882 – 1970)

Ferdinand Frobenius (1849 – 1917)

Heinrich Rubens (1865 – 1922)

Emil Warburg (1946 – 1931)

Papers in the 1916 Vol. 1 of the Prussian Academy of Sciences

Copenhagen and Bohr

Göttingen and Born

St. John’s at Cambridge

Antimatter

Read the stories behind the history of quantum field theory, in Galileo Unbound from Oxford University Press

References