From Newton to Einstein

From:   An Introduction to Classical Electrodynamics by Keohane and Foy

Sir Isaac Newton produced two great works that revolutionized natural philosophy. One was theoretical, the other experimental. One was written in Latin, the other English. One is now considered his greatest work; the other is remembered for advancing a failed theory. One seemed to answer all the questions of the universe, the other asked more questions than it answered. These works were Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) and Opticks or A Treatise of the Reflections, Refractions, Inflections and Colours of Light.

The Principia presented Newton’s three laws of motion, and his Universal Law of Gravitation. Written in Latin, the universal language of scholarship, it was embraced by scientists across Europe and sparked a mathematical revolution. Unifying physics and astronomy, and providing the link between many disparate disciplines, many in Europe regarded the Principia as presenting the ultimate solution to the laws of nature.

Despite its success, Newton was still worried about several lingering philosophical questions, especially the issue of action at a distance. Newton’s primary theme concerns cause and effect. When one object applies a force on another, according to his second law, the other object accelerates by a proportionate amount. This principle was easy to appreciate in the case of contact forces, such as collisions, where one body directly interacts with another. However, gravity works differently. The moon orbits the earth, and water in the earth’s oceans moves in response. How does this happen? What is it that links the seas to the moon, despite their vast distance? Presumably some mechanism conveys the gravitational force, but Newton did not know what it was, and so he did not speculate about it formally.

Like gravity, light travels through vast distances of free space, but unlike gravity we can see light. Moreover, the intensity of each diminishes in proportion to the inverse square of the distance. So, while Newton may have liked to investigate the invisible gravitational mechanism, he could study light. Not only could he study it, but he could investigate it experimentally, and write about it in a way that any educated person could understand.

While most of Opticks describes the geometrical optics of lenses, mirrors, and prisms, now called Newtonian optics, Newton’s most startling experiments investigated color. Prior to Newton, scientists reasoned that colored light contains some sort of pigment that gives it color, much like dye-soaked cloth or stained glass. Under this hypothesis, overlapping projected colors should combine in the same way as paints mix. Newton, however, shocked the scientific community when he used two prisms to separate light into a rainbow and then “compound Whiteness by mixing their Colours.”

To explain his experimental results, Newton proposed that light is composed of particles, or corpuscles, which travel in straight lines over great distances through free space. He believed the color of light varies with the mass of a light particle. By passing white light through prisms, Newton found that red light bends less than blue light, which he attributed to red light corpuscles being more massive than blue ones. Newton’s particle model sharply contrasted with another theory, proposed by his Dutch contemporary Christiaan Huygens, which modeled light as a wave though a medium, much like sound traveling through air. Newton reasoned, however, that if such a medium exists, it could not be so dense as to degrade planetary orbits, but must be elastic enough to convey forces over long distances, which was highly unlikely. However, Newton also reasoned, that if future experiments disproved his particle model, then this medium, called the aether, must exist. The alternative, after all, would be action at a distance across vast distances of truly empty space, something he could not abide.

Along with gravity, two other forces also appeared magically to act from a distance: electricity and magnetism. To study the mechanism that conveys these forces, Newton’s intellectual descendants had to separate cause from effect. They did this by defining a new concept called a force field.

Consider, for example, the force with which the earth pulls down on you, commonly known as your weight. If you traveled, say, to the moon your weight would be different, but you would still be the same. There must be some property intrinsic to the earth that causes some gravitational field, which, in turn, interacts with some property of you to pull you downward. We call this intrinsic property mass, and we define the gravitational field at your location to be the ratio of your weight to your mass, or in introductory physics notation:  \vec{g}\equiv \frac{{{{\vec{F}}}_{g}}}{m}.

By defining the gravitational field in this way, natural philosophers could circumvent the pernicious problem of action at a distance. Someone, some day, would figure out what this elusive gravitational field really is, scientists reasoned. For the time being, however, the gravitational field is defined operationally based on how it is measured.

The force field appropriate for electric charges can be defined in a similar way to gravity. If you rub two identical balloons with, say, your hair, they will repel. One balloon causes something, call it the electric field, which in turn interacts with the other balloon to push it away.   By defining the electric field as the ratio of the force on an object to some governing quantity, call it charge, we can say the electric field is pushing the balloon. Or, in introductory physics notation: \vec{E}\equiv \frac{{{{\vec{F}}}_{E}}}{q}.

And, finally, if you hold a compass, the compass needle twists until it points approximately in the north-south direction. The torque on the compass needle must be caused by the interaction between some magnetic field and some intrinsic property of the magnet, called the magnetic moment, or \vec{\tau }=\vec{m}\times \vec{B}, is the torque on the compass needle, and \vec{m} and\vec{B} are the magnetic moment of the compass needle and the local magnetic field respectively.

Not everyone was spooked by action at a distance. After all, Newton’s law of gravity was so successful that it made sense to search for similar laws relating the forces between charged objects and magnets. Charles Agustin Coulomb, for example, found inverse square force laws relating charged objects and the poles of compass needles. While his famous law involving charged objects is still considered a great achievement, his analogous law involving magnetic poles was later shown to be an overgeneralization of reasoning by analogy, as magnetic poles fundamentally do not exist.

Electricity and magnetism merged when, in 1820, Hans Christian Ørsted discovered that a current carrying wire deflects a compass needle. This led to a flurry of experimental research, where scientists raced each other to understand this new phenomenon. Within only a short time, scientists had carefully measured the forces between magnets and wires to develop more mathematical laws in the same style as Newton’s law of gravity. By the middle of the nineteenth century, Continental physicists had worked out a comprehensive action at a distance theory of electromagnetism.

In England, Michael Faraday was not so sanguine about action at a distance, so he took a different tack. Using iron filings, Faraday mapped out the direction of the magnetic field around magnets and through wire loops. Like the flow of water, and electric current, these magnetic field lines circulated, always making closed loops. Even more startling was Faraday’s discovery that moving a magnet through loops of wire caused electricity to flow. Faraday’s fluid approach made a great deal of sense conceptually, but Faraday was no mathematician.

After successfully modeling Saturn’s rings, James Clerk Maxwell moved to London and began expressing the known laws of electricity and magnetism in terms of the electric and magnetic fields, using the mathematics of fluid mechanics. In free space, Maxwell discovered, the electric and magnetic fields oscillate between each other, and propagate at a speed numerically equivalent to the speed of light.   Thus, light was now an electromagnetic phenomenon. Moreover, the problem of action at a distance was largely put to rest. A changing current causes an oscillating electric field. This, in turn, causes an oscillating magnetic field, which causes an electric field, which, in turn, causes a magnetic field yet again. Thus, all electromagnetic information travels through space at the characteristic speed of the medium.

Maxwell had now successfully combined electricity, magnetism, and optics into a single branch of physics called electrodynamics. The last frontier for physics would be to determine the nature of the medium that links cause to effect: from battery to motor, from magnet to steel, and from star to telescope, as Maxwell’s final sentence of his Treatise on Electricity and Magnetism so eloquently states:

Hence all these theories lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavor to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.

Maxwell’s theory quickly passed the rigorous experimental tests of Heinrich Hertz, leaving little doubt about its correctness and the existence of the aether. A natural next step was to measure the velocity of the earth with respect to the aether.

Consider taking a round-trip airplane trip on a windy day, with a headwind going toward your destination, and a tailwind on the way back. As it turns out, the additional speed of the return flight cannot make up for the slower flight out. Thus, round-trip flights take longer on windy days. The wind velocity, therefore, could, in principle, be calculated from the differences in round-trip flight times of planes traveling in different directions. This is the approach to finding the speed of the aether wind that a young American physicist took while spending a semester in Berlin.

This is a replica of Michelson’s 1881 experiment on display in Potsdam Germany, where Michelson carried out his experiment due to city vibrations in his Berlin optics lab.

With his interferometer, Albert Michelson attempted to measure differences in the round-trip travel time of light with direction. Michelson found no measurable time differences, no matter the time of year. Therefore, he published the following bold conclusion:

The result of the hypothesis of stationary ether is thus shown to be incorrect, and the necessary conclusion follows that the hypothesis is erroneous.[1]

At the time, however, this result was seen as support for the idea, proposed by George Gabriel Stokes about thirty years before, that the earth drags the aether along as it moves about the sun. While the significance of his null result remained unappreciated, it did launch Michelson’s career. He moved to a larger university, where he and his colleague Edward Morley built the most accurate optical interferometer to that date. Alas, he failed again to measure any time differences, and at this he concluded:

the relative velocity of the earth and the ether is probably less than one sixth the earth’s orbital velocity, and certainly less than one-fourth.[2]

By the turn of the century even true believers in the aether’s existence were having doubts. For example, the elderly Lord Kelvin wrote:

The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds. I. The first came into existence with the undulatory theory of light, and was dealt with by Fresnel and Dr. Thomas Young; it involved the question, How could the earth move through an elastic solid, such as essentially is the luminiferous ether? II. The second is the Maxwell-Boltzmann doctrine regarding the partition of energy. [3]

What could possibly explain the propagation of electromagnetic waves without a medium? It just made no sense for a cause one place to produce an effect somewhere else, without either a particle traveling or a medium carrying the signal.

Albert Einstein cleared Kelvin’s first cloud in 1905 by crafting a new aetherless theory that unified kinematics and electrodynamics. Einstein’s theory of relativity did not modify the equations of classical electrodynamics, but rather our fundamental concepts of time and space.[4]

Einstein’s theory of relativity, while bold, was not considered reckless to his colleagues because it so beautifully unified all of classical physics under a single paradigm. Despite its classical beauty, and immediate acceptance, special relativity was not the end of the story for the nature of light.

By dismissing the aether, Einstein’s theory of relativity opened the door for action at a distance to, once again, cast a pall over the scientific world.   This was not lost on Einstein, who, also in 1905, reincarnated the long-dead particle model of light to address whether light from a hot source could be consistent with “the Maxwell-Boltzmann doctrine regarding the partition of energy.”

Even after the acceptance of relativity theory, few people, least of all Robert Millikan, were persuaded that light was a particle; the evidence for the wave model was too overwhelming. This all changed in 1916, when Millikan finished a research program designed to disprove Einstein’s particle model of light once and for all. He wrote:

It was in 1905 that Einstein made the first coupling of photo effects with any form of quantum theory by bringing forward the bold, not to say the reckless, hypothesis of an electro-magnetic light corpuscle of energy, which energy was transferred upon absorption to an electron.[5]

However, rather than sending Einstein’s particle model of light back to the land of the dead, Millikan’s experimental results directly verified Einstein’s ideas, which opened the door to the quantum electrodynamic theories of the twentieth century.

What, after all, are these electric and magnetic fields that will dominate this 900-page book? Are they really just measures of the density of these electromagnetic particles, called photons? And, if they are, how do they communicate to act collectively? Electric and magnetic fields are simply different manifestations of an overall electromagnetic field, but what about the electromagnetic field? Is it really some sort of quantum field, and if so, what is that?

As far as classical electrodynamics is concerned, the electric field is simply a force per charge; similarly, the magnetic field is simply a torque per magnetic moment—no more and no less. Through 17 chapters of this book, this is exactly what they are. In the last chapter, however, we will come back to Einstein’s reckless hypothesis that light might behave as a collection of particles, or even that all fields are really collections of particles. Or, perhaps the opposite is true, that all particles are actually manifestations of fields.   And if they are, what is a field anyway other than something we construct to explain observed phenomena?

[1] Albert A. Michelson, “The Relative Motion of the Earth and the Luminiferous Ether”, American Journal of Science, 22 (1881), 120-129.

[2] Albert A. Michelson & Edward W. Morley, “On the Relative Motion of the Earth and the Luminiferous Ether”, American Journal of Science 34 (1887), 333–345.

[3] W. Thomson, “Nineteenth century clouds over the dynamical theory of heat and light”, Philosophical Magazine Series 6, 2:7, (1901), 1-40.

[4] A. Einstein, “On the Electrodynamics of Moving Bodies,” Annalen der Physik 17 (1905): 891-921, Translated by Anna Beck, in The Collected Papers of Albert Einstein, Volume 2, The Swiss Years, 1900-1909. John Stachel, ed. (Princeton: Princeton University Press). English Translation, ©1989 by the Hebrew University of Jerusalem.

[5] Millikan, R.A., “A Direct Photoelectric Determination of Planck’s ‘h’,” Physical Review, 7 (1916), 355-388.