• How Einstein Arrived at His Theory of General Relativity

    “He was struck that planets, stars, and other celestial objects all pull on each other.”

    In 1905, physicists understood something of the laws governing two types of forces: those of electricity and magnetism and of gravity. We’ve seen that the laws of electricity and magnetism, encoded in Maxwell’s equations, forced a rethinking of the most basic concepts of space and time. But what about Newton’s law of gravity? How might it be purged of the notions of absolute space and absolute time?

    In 1907, two years after putting forward his special relativity, Einstein was asked to write a review of the subject. In the course of this project, he confronted the question: How does Newton’s theory of gravity fit in with his principles? The simple answer: It doesn’t. This was actually related to a deficiency of Newton’s law of gravitation that was clear from the moment he promulgated it.

    Newton—and perhaps more importantly his critics—were very troubled by a feature of his theory called action at a distance. In Newton’s laws, if, say, the sun were suddenly to “jump” (for the moment, take the far-fetched possibility of some space invader attaching rockets to it), the effect on the planets in the solar system would be instantaneous. This despite the fact that the planets are far away. Neptune, for example, is so far away that it takes light from the sun four hours to reach it, but it would move immediately in response to the sun’s sudden motion.

    Newton was criticized for this—was he suggesting that some higher being was responsible for the force between stars and planets? But his law was extremely successful, and, for almost two centuries, this question was largely put aside. Indeed, it was only in the early 20th century that it was possible to seriously test this disturbing feature.

    But with special relativity, one could no longer look away. It didn’t make sense that this principle should apply to electro-magnetism but not to the force of gravity. It was hard to see how Einstein’s assertion, that events at one place and time can affect events at another only after at least enough time has passed for light to travel from one to the other, would not have to apply to any law of nature. This was not a crisis in the sense of an obvious experimental or observational problem. Newton’s theory works so well because the speed of light is so large. Light moves so fast that in most of the situations that astronomers encountered in the two centuries after Newton put forward his law, the effects of the finite (as opposed to infinite!) speed of the propagation of information and interactions were impossible to see.

    The process of arriving at what Einstein called his general theory of relativity involved a struggle of eight years and a combination of extraordinary scientific insight and sheer hard work.

    Still, Einstein began to consider how one might modify Newton’s theory so as to retain its enormous successes while accommodating the relativity principle. In other words, the new laws, in situations where the objects under study are moving much more slowly than the speed of light, or in which the force of gravity is not too strong, should reduce to Newton’s.

    The process of arriving at what Einstein called his general theory of relativity involved a struggle of eight years and a combination of extraordinary scientific insight and sheer hard work. Along the way, there were many missteps. But the theory even more fully revealed Einstein’s genius than did his accomplishments of 1905. Einstein might have gone at the problem by observing that Newton’s gravitational force law is almost the same as Coulomb’s force law for charged particles. Just replace electric charges by masses, and they look alike. The electric force is described by Maxwell’s equations. So he might have tried to write equations like Maxwell’s, but for the gravitational force.

    This is how I would likely have proceeded, and the result would have been failure. But Einstein thought much more deeply before starting his struggle. He was struck by the fact that planets, stars, and other celestial objects all pull on each other; they never push each other apart. This is different from electric forces, where while a proton attracts an electron—pulls the electron toward itself—two protons repel each other. The force of gravity always seems to be attractive, never repulsive. This is hard to mimic with Coulomb’s law. Einstein instead took his cue from observations that predated Newton.

    Among Galileo’s most famous experiments were his studies of falling objects. Archimedes, the ancient Greek philosopher, had asserted that heavy objects fall faster than lighter ones.

    This was a plausible guess, but not a statement based on careful observations. Galileo was skeptical and studied the question experimentally. Whether he actually dropped objects of different mass from the Leaning Tower of Pisa is a subject of scholarly debate, but he did perform experiments in which he established that objects of different mass fall to earth at the same rate, neglecting the fact that the air tends to slow everything down. (On the surface of the earth, a piece of paper falls much more slowly than a brick, due to the resistance of the air, but you can easily do a version of this experiment dropping two heavy objects, of different weight, from the same height.)

    These observations had been improved over the intervening centuries by various investigators, including Newton. Very sensitive experiments were conducted in the late 19th century by Baron Loránd Eötvös, a Hungarian physicist, who used a different strategy, attaching various objects to a rod. The device was set up so that it would move if objects of different types responded differently to gravity, but not otherwise. Eötvös established that, for a range of substances, these responses were the same to better than a part in a million; present experiments do thousands of times better.

    In Newton’s laws, mass has to do with inertia, the rate at which things accelerate in response to a force. But it also has to do with the strength of the force of gravity between two objects. Newton, presumably under the influence of Galileo, assumed that these two kinds of mass were the same. But as far as Newton was concerned, this was simply a fact; no deep principle forced this relationship. Eötvös (and others) established that the inertial mass is the same as the gravitational mass to a high degree of accuracy.

    Einstein’s theory presented a radical new conception of space and time. No longer were they fixed eternally, but they responded to the presence of matter.

    Einstein started with this observation and assumed that the equivalence is exact. He then performed a very simple but very ingenious thought experiment, in a setting from daily life. In developing special relativity, Einstein had reasoned by analogy with experiences of one important technology of his day—railroads. He now reasoned by invoking another, newer technology—elevators.

    He imagined cutting an elevator cable so that the elevator would fall freely (a rather scary prospect). He noted that due to this assumed equivalence of inertial and gravitational mass, observers in an elevator in free fall would experience what we now call weightlessness. They could, for example, float around in the elevator, or pass a ball back and forth with no sense of gravity. It would be as if no gravitational force acted on the objects in the elevators. For the passengers, unfortunately, this would last only until the elevator hit the bottom of the shaft.

    But nowadays we routinely achieve weightlessness in space travel. The International Space Station, when it orbits the earth, is in free fall. It falls due to the earth’s gravity. It stays in orbit because the downward pull of gravity competes with the energy of motion provided by the initial launch, to keep the spacecraft constantly circling around the earth. The effects of free fall can also be achieved with aircraft by shutting off the engines for a period. This is routinely done as part of astronaut training. Famously, Stephen Hawking, one of the great gravity theorists, was treated to such a flight in 2007.

    Einstein didn’t have the advantage of this experience, and the tallest buildings of his day would have allowed a fall of only 4 or 5 seconds. But he realized the phenomenon of weightlessness would follow from the observations of Galileo and Eötvös. Einstein called his realization “the happiest thought of my life” and elevated this to a principle: No experiment can distinguish free fall in a gravitational field from motion with uniform acceleration (as in the elevator). He put forward the hypothesis, his “Principle of Equivalence,” that this should apply to all laws of nature: gravity, electromagnetism, and laws yet to be discovered.

    From here to mathematical equations was a long struggle. Einstein knew roughly what he was looking for, but when he set out on his journey he did not possess a suitable mathematics for achieving it. David Hilbert, a professor in Gottingen, Germany, and one of the greatest mathematicians of the day, did know the required mathematics and was also in a quest for a theory of gravity; it is likely that had he fully understood the physics issues, he would have gotten to general relativity first, and indeed he almost did.

    In 1915, however, Einstein completed and published his general theory. The theory met his basic requirements. First, it was consistent with the principles of special relativity. For example, the gravitational interaction propagated at the speed of light; there was no action at a distance. Second, it incorporated the principle of equivalence. Finally, it reduced to Newton’s laws except in very exceptional circumstances. Around typical stars and planets, the corrections would be very tiny.

    Einstein’s theory presented a radical new conception of space and time. No longer were they fixed eternally, but they responded to the presence of matter.

    Einstein’s theory presented a radical new conception of space and time. No longer were they fixed eternally, but they responded to the presence of matter. Space might be curved, time might run faster or slower near larger or smaller concentrations of matter. Most physicists and mathematicians familiar with the theory would describe it as beautiful, but while the principles are simple, the mathematics is rather complicated, and calculations can be challenging. Einstein, however, focused not only on the great principles and the beautiful mathematics but on the observational consequences of the theory. Because in most circumstances the corrections to Newton’s laws are extremely tiny, he had to look for situations where these effects, though small, would be sufficiently prominent to be detectable. He made three predictions that one could realistically hope to test with the technologies then available.

    One of these predictions might be more properly described as a “postdiction,” an explanation of an already known puzzle in the motion of the planet Mercury. The sun exerts the dominant force on each of the planets; the planets also pull on each other, but these effects are relatively small. Taking into account, first, only the force due to the sun, Newton had shown that the planets would move in orbits the shape of ellipses, just as Kepler observed. According to Newton, these orbits should retain their shape and orientation for all time, ignoring the pull of the other planets.

    Even in Newton’s day, astronomers studied the motion of the planets with precision. They carefully calculated the orbits on paper, making corrections for all sorts of small effects, such as the pull of the planets on each other. They compared these calculations with equally careful observations. They concluded that small corrections due to the other planets and other effects would lead to a slow deviation from Newton’s results; the ellipse would gradually rotate over time. This is referred to (by those with a better memory than mine of their high school analytic geometry) as the precession of the perihelion. Already in the 1850s, astronomers noted that the precession of Mercury was not quite at the speeds predicted by Newton’s laws; there was a tiny deviation. They proposed a variety of explanations, such as a small, unseen planet or dust, but none was compelling.

    Einstein was aware of the discrepancy in Mercury’s motion. He realized that Mercury, being the planet closest to the sun, experiences the strongest force of gravity and was thus a promising testing ground for his theory. Einstein set out to calculate the correction to Newton’s result. He found it was exactly what was needed to account for the observed precession. I can only begin to imagine how he felt. For a physicist, discovery of a new law of nature is the supreme accomplishment. I have speculated as to several, but the likelihood that anyone is true is, typically, not high. Einstein indeed recalled that he was enormously excited—he said he had palpitations—and with the correct result for Mercury’s perihelion, he became convinced that his theory was correct.

    But inventing theories to explain possible observational discrepancies is still within the realm of more “routine” science. Even better was to come. The second prediction was a real prediction in the sense that he proposed a measurement that had not yet been performed and predicted the outcome. In Newton’s theory, one describes the force of gravity as acting on mass. The path of a satellite passing near the sun would be bent by the pull of the sun’s gravity. But in special relativity, mass is just a form of energy, and in the general theory, gravity acts on all forms of energy. Light has no mass, but it does carry energy. So the paths of light rays should be altered from simple straight lines as they pass near objects with strong gravity. In 1911, before the theory was fully developed, Einstein tried to calculate the effect. He found that one should be able to see a slight alteration in the position of stars lined up with the sun during a solar eclipse.

    Einstein was a genius—and he was also lucky. As I said, the mathematics of general relativity is complicated and was, at that time, also rather unfamiliar. It turned out that in his first calculation of the bending of light by the sun, before he had his theory in its final form, Einstein had made a mistake. He actually obtained the value that Newton would have obtained if the energy of the light was treated as equivalent to mass, through E=mc2.

    Already in 1912, and again in 1914, expeditions to observe the bending of light during eclipses failed to obtain results, the first time due to rain, the second when it was canceled due to the outbreak of the First World War. In 1915, the year in which he published the final version of the general theory, he got the correct result for the bending of light, finding double the Newton value. The war prevented further measurements until 1919.

    In that year, two expeditions, one to Príncipe island led by the English astronomer Arthur Eddington, and one to Brazil by Andrew Crommelin of the Greenwich observatory, succeeded in observing the effect. The results were announced in a joint meeting of the Royal Society and the Royal Astronomical Society: Einstein’s prediction was confirmed. By this time, Einstein was already well known in the scientific community, and occasional articles about him had appeared in the popular press, but now his name became a household word.

    The headline of the London Times of November 17, 1919, was typical: “Revolution in Science. New Theory of the Universe. Newtonian Ideas Overthrown.”

    When I was a student, Einstein’s theory of general relativity was a subject of fascination—something any self-described theoretical physicist should know something about. At the same time, actually saying that you might work on it would lead to rolling of eyes. There was, in those days, only very limited evidence that the theory was correct—beyond the perihelion and the bending of light, only a phenomenon called the redshift—and it seemed that only dreamers imagined there would be new tests.

    Perhaps even worse, the theory, when combined with quantum mechanics, the subject of the next chapter, did not seem to make sense. Attacking that problem put you even more on the fringe. Still, most of the great theorists of the era had taken a stab at these issues, including Richard Feynman and Lev Landau (one of the greatest of 20th century Russian theoretical physicists). In the 1980s, perhaps more famously, Stephen Hawking raised issues that challenged the notion that general relativity and quantum mechanics could be reconciled and argued that a reformulation of quantum mechanics would be necessary.

    Over the course of my career, all that has changed dramatically. Einstein’s theory is now a well-tested theory. Our understanding of general relativity is an important tool in our explorations of the universe. Observation of black holes is almost routine. General relativity is a crucial tool in determining the composition of the present universe, and essential, as we’ll see, to our understanding of the big bang. Recently, the discovery of the gravitational waves predicted by the theory over a century ago has opened up a new window on astrophysical phenomena. General relativity even plays a role in our navigation apps (through the Global Positioning System, or GPS). On the quantum mechanical side, we have learned a great deal as well, although experimental verification of what we do understand (and clues as to what we don’t) is probably not around the corner.


    Excerpted from This Way to the Universe by Michael Dine. Copyright © 2022. Available from Dutton Books.

    Michael Dine
    Michael Dine
    Michael Dine is Distinguished Professor of Physics at the Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, and one of the world's leading physicists. He has been a Sloan Fellow, a Guggenheim Fellow, a Fellow of the American Physical Society, and in 2010 was elected a Fellow of the American Academy of Arts and Sciences. Dine is a recipient of the 2018 J. J. Sakurai Prize honoring outstanding achievement in particle physics theory. In April 2019, he was elected to membership of the National Academy of Sciences. He has served the physics profession in numerous roles, including as Chair of the Committee on the Future of Theoretical Particle Physics of the American Physical Society.

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