The General Theory of Relativity

The Equivalence Principle

Recall that the special theory of relativity provided a simple answer to a question raised by the Michelson-Morley experiment. Why does light behave as though it travels the same speed in all reference frames? Because it actually does travel the same speed in all reference frames. Simple as that. Yet, as we have seen, this seemingly straightforward answer revolutionized our understanding of space, time, causation, and even mass. Einstein’s second theory, likewise, provides a simple answer to a difficult question. And as before, the answer has mind-blowing implications.

Shortly after formulating the special theory of relativity, Einstein realized that his theory was incompatible with Newton’s theory of gravity. According to Newton’s law of universal gravitation, the gravitational force between two objects depends on the distance between them. So, whenever that distance changes, the gravitational force changes too. If a distant star moves closer to the sun, for instance, the force it exerts on the sun should increase immediately, even if the star is still many light-years away. In other words, the distant star’s motion and the resulting increase of force on the sun occur simultaneously, according to Newton. This is problematic, because special relativity implies that there is no fact of the matter whether those two things happen simultaneously. The same problem could have been raised with Coulomb’s law, which has a mathematical form similar to Newton’s law of universal gravitation. However, Coulomb’s law had already been superseded by Maxwell’s equations, which don’t imply that changes in the electromagnetic force are instantaneous. Maxwell’s equations are fully compatible with both of Einstein’s theories.

To solve this problem, Einstein needed a new theory of gravity. In 1915, he finally succeeded in formulating a theory that is now—more than a century later—still the best-confirmed theory of gravity ever devised: the general theory of relativity, also called general relativity for short. Its predictions have proven extremely accurate, and its practical applications are used by ordinary people every day: GPS navigation systems rely on general relativity, for example.

But I’m getting ahead of myself. Let’s rewind to 1907. That year, just two years after publishing his first theory of relativity, Einstein had an epiphany that he would later describe as “the happiest thought of my life.”This famous quote comes from Einstein’s unpublished manuscript “Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development” (later published in Nature, without the famous quote, in 1921). An English translation of the manuscript appears in The Collected Papers of Albert Einstein, Vol. 7: The Berlin Years: Writings, 1918-1921 (English translation supplement), trans. Alfred Engel (Princeton: Princeton University Press, 1998), 113-150. The full text is available online here; the quote is on this page. In a uniform gravitational field, free-falling objects behave exactly as they would if there were no gravity at all.

You can see this for yourself, if you like. Hold a couple of small objects near your face at eye level, jump into the air, and release the items as you do so. While you are airborne, the objects will appear to float next to each other. (If you try this experiment on a trampoline, you can jump higher so the effect will last longer.) They appear weightless, because you and the other objects are all falling at the same rate.

You experience this zero-gravity phenomenon briefly, as you fall a short distance; but astronauts in orbit around the earth—for example, those aboard the International Space Station (ISS)—get to experience it for weeks or months on end. The ISS orbits the earth at an altitude only 400 km above sea level (on average), and at that distance Earth’s gravity is still quite strong, almost 90% as strong as it is here on the surface. However, the ISS is essentially “falling” toward Earth: it is constantly accelerating in a “downward” direction (toward Earth’s center) as it moves in a circle around our planet. That’s why astronauts aboard the ISS don’t feel Earth’s gravitational pull. From their perspective, it appears as though gravity just doesn’t exist inside the ISS.

There were no astronauts or space stations in 1907, of course, but Einstein had a good imagination. In order to visualize what it would be like to experience free fall for a long period of time, he considered how things would appear to someone trapped inside a falling elevator. For a less terrifying version of his thought experiment, let’s imagine an astronaut inside a free-falling space capsule with no windows. From the astronaut’s perspective, it would seem exactly as if the capsule were floating far outside the solar system somewhere in deep space, where the gravitational pull of planets and stars is too weak to detect. No experiment performed inside the capsule could tell the astronaut whether she is falling toward a planet, or floating out in deep space. There would be no discernable difference between those two possibilities.

This fact was too remarkable to be mere coincidence, Einstein thought. There must be some reason why gravity seems to disappear in the falling astronaut’s frame of reference. Why does gravity have no detectable effects inside the falling space capsule? The answer suddenly seemed all too obvious. Perhaps there just isn’t any gravitational force on the falling astronaut, in her own reference frame! Maybe gravitational force, too, is a relative quantity!

To extend this idea further, Einstein considered what would happen if the windowless space capsule (or elevator, in his original thought experiment) were accelerated at a constant rate somewhere out in deep space. As the capsule accelerates, it pushes the astronaut in a certain direction, which she will regard as “upward.” She’ll be able to stand on the “downward” side of the capsule, which pushes on her feet as the capsule accelerates. To her, the situation feels exactly as though she is standing on the surface of a planet. If she drops something, it will appear to fall toward her feet, since her feet are accelerating upward toward the object. Moreover, all objects will appear to fall at the same rate. If the spacecraft happens to be accelerating at a rate of 9.8m/s² (the same rate at which objects fall near Earth’s surface), the astronaut might think that her windowless space capsule is still at rest on the launchpad, rather than accelerating out in deep space. (Perhaps she dozed off before the countdown, and has just woken from a long nap.)

In fact, no experiment performed inside the capsule will tell the astronaut whether she is at rest in a gravitational field, or accelerating in deep space. So long as the spacecraft’s acceleration is constant (i.e., it doesn’t change magnitude or direction), things inside the spacecraft will behave exactly as though they are in a uniform gravitational field—a field in which the strength and direction of the gravitational force is the same everywhere. A planet’s gravitational field isn’t really uniform. Earth’s gravitational force decreases with altitude; and since it pulls toward Earth’s center, the direction of this force also varies from one place to another. If the astronaut’s spacecraft is very large, she may be able to detect these tiny differences in the gravitational pull at different locations inside her spacecraft. But in a sufficiently small capsule, she wouldn’t be able to detect these differences, and it will seem as though she’s in a uniform gravitational field. There is no physical difference between the two. This is known as the equivalence principle:

Constant acceleration is physically equivalent to a uniform gravitational field.

This simple idea is the foundation of Einstein’s theory of gravity. As we saw with Einstein’s previous theory, however, simple ideas can have complex implications. It took Einstein years to work out the details of his new theory, and he had to recruit his friend Marcel Grossmann to help him with the complicated mathematics needed to describe it precisely. In what follows, I’ll leave the mathematical complexities aside and give a basic overview of the theory’s fascinating ramifications.