From Special to General Relativity

From a lecture series presented by The Great Courses

To understand what Einstein got wrong, it’s helpful to take a look at what he very much got right. His special and general theories of relativity gave us a new conception of the universe. Take a look at what was incomplete about the special theory of relativity, and what conceptual elements Einstein used to formulate his general theory of relativity.

Albert Einstein

The general theory of relativity couldn’t have happened without an earlier theory. The theory of relativity that Einstein put forth in 1905—which is now known as the special theory of relativity—describes how lengths in space and durations of time are different to observers moving at different speeds, in different frames of reference.

And although special relativity predicts many observed phenomena correctly, this theory is also incomplete. And even Einstein himself was aware, from early on, that it was incomplete. For one thing, special relativity can only be applied to objects that are moving at a constant rate of speed.

In this sense, special relativity is like a theory for a car that describes how it moves and behaves with its cruise control on, but that doesn’t address anything about the brake pedal or the accelerator pedal.

This is a transcript from the video series What Einstein Got Wrong. Watch it now, on The Great Courses Plus.

Second, in the theory of gravity that existed at the time—the Newtonian theory of gravity—gravity’s attraction works instantaneously, pulling bodies together across great distances in space without any time delay. But according to special relativity, nothing can move faster than the speed of light. This made it hard to reconcile special relativity with this aspect of Newtonian gravity. So, in at least two ways, Einstein’s theory of special relativity left us with important and unanswered questions.

Learn more about mysteries of modern physics, including time

Gravity and Acceleration

Shortly after publishing his special theory of relativity, Einstein began to work toward creating an even more complete and far-reaching theory of space and time. It took him another decade, but eventually Einstein came up with an expanded and completely general form of his theory. This theory—the general theory of relativity—was not only a theory of space and time, but it also provided us with a deeper and more powerful way of thinking about the force of gravity.

Approximately in 1907, Einstein had his first important conceptual breakthrough that would put him on the road to general relativity. This was a couple of years after special relativity and his other breakthrough papers from 1905. Thinking about how he might be able to incorporate acceleration and gravity into his theory, he came up with something we now call the equivalence principle.

To understand this concept, imagine that you’re in an impenetrable chamber—you can’t hear, see, or otherwise know anything about what’s going on outside of the chamber. Toward one side of the chamber, you feel a force. This force feels just like gravity does. It pulls you toward one side of the chamber, and it allows you to walk normally along what feels like the bottom of the chamber. But is this really the force of gravity? Instead, what feels like gravity to you might be the consequence of the chamber being accelerated. When you’re in an elevator that’s speeding up or accelerating, you feel a downward force that makes you feel slightly heavier than normal. And when the elevator is slowing down, you feel an upward force, making you feel slightly lighter.

The fact is that the force of gravity feels exactly the same as the effects of acceleration. So, to someone sealed in the chamber, there is no way to know whether the force that they are experiencing is in fact gravity, or is instead the consequence of the chamber being accelerated. This is the essence of Einstein’s equivalence principle. And although he didn’t yet know exactly where it would lead him, this insight made Einstein begin to speculate that acceleration and gravity might be very deeply interconnected.

To better appreciate the nature of the equivalence principle, consider what we mean when we use the word “mass.” In Newtonian physics, there are two very different kinds of quantities that we sometimes call “mass.” The first of these is the kind of mass that resists acceleration. We call this inertial mass. Something with a lot of inertial mass—like a boulder, for example—requires a lot more force to move than something with much less inertial mass—like a baseball. The second kind of mass is what gravity acts upon. We call this kind of mass gravitational mass. The weird and surprising thing is that the inertial mass of an object always seems to be exactly equal to its gravitational mass.

As far as we know, there are no objects in our universe with more inertial mass than gravitational mass, or vice versa. For some reason—unknown before Einstein—the inertial mass and gravitational mass of an object were always exactly the same. But Einstein’s equivalence principle provided us with an insight as to why this was the case. After all, Einstein was beginning to think that the force of gravity was really just acceleration in some sense. If this were the case, then it might not be surprising at all that gravitational mass was really just the same thing as inertial mass.

Learn more about gravitational waves

Gravity and Light

Well before Einstein constructed his theory of general relativity, he recognized a particularly important consequence of the equivalence principle—beams of light should be subtly deflected or bent by the force of gravity. A few years later—in 1911—he published an article that pointed this out. He entitled this article “On the Influence of Gravity on the Propagation of Light,” and in it, Einstein presented a calculation showing that a ray of light passing by the Sun should be deflected by about 0.83 arcseconds, or about one four-thousandth of a degree. A very subtle effect indeed, but one that could be tested, at least in principle.

But under normal circumstances, any light that was deflected by the Sun would be lost in the much brighter sea of ordinary sunlight. In order to see or detect the deflected beam of light as it skims past the Sun, the light of the Sun would have to be blocked out. So, in order for such a measurement to succeed, it would have to be made under the conditions of a nearly perfect solar eclipse. The next solar eclipse was predicted to take place three years later, in 1914. At that time, Einstein hoped that his prediction—and the equivalence principle along with it—would be proven correct.

Solar eclipse diagram
Einstein planned to measure the effect of gravity on propagation of light during a total solar eclipse. It was easy to detect small deviations in light path when the sun is completely blocked.

Einstein spent the years leading up to the scheduled eclipse considering some of the conceptual questions that were raised by the possibility of the gravitational deflection of light. In many applications, beams of light had long been used as the very definition of a “straight line.” If the Sun’s gravity could bend the trajectory of a ray of light, then—at least in some sense—gravity could change the geometry of space.

With this insight, Einstein began to recognize the deep connection that exists between what we call gravity, and the geometry of space and time. But even Einstein was not yet in any position to really understand this connection. In order to build the theory he was beginning to imagine, Einstein would have to dig much deeper into the mathematics of geometry. Deeper than any physicist had ever gone before.

A New Kind of Geometry

In high school, you probably took a geometry class. And in that class, you were almost certainly taught a system of geometry that is known as Euclidean geometry. Your teacher might not have told you that they were teaching you Euclidean geometry, but they were. Until Einstein came along, physics was entirely based on Euclidean geometry. To almost everyone at the time, Euclidean geometry was seen as the only reasonable way to think about space.

Learn more about the geometry of space

Euclidean geometry is named after the ancient Greek philosopher and mathematician Euclid. And everything about it can be derived from five basic rules, sometimes called axioms or postulates. When you first hear these postulates, they all seem very self-evident. For example, one of Euclid’s postulates says that “any two points in space can be connected by a straight line.” And another says that “all right angles are equal to each other.” Pretty uncontroversial, right? But one of Euclid’s postulates—his fifth postulate—turns out to be on less solid footing. This fifth postulate says that “for any straight line there is exactly one straight line that is parallel to it that passes through any given point in space.”

Among other things, this last postulate can be used to show that two parallel lines will never meet or cross one another. In your high school geometry class, you were probably taught this postulate as an indisputable fact. After all, it seems so obvious. It’s hard to even imagine that it might not be true. Throughout most of history, Euclid’s postulates were treated as self-evident and indisputable. But in the first half of the 19th century, a few mathematicians started to think about systems of geometry that broke one or more of these postulates.

In particular, a number of mathematicians had managed to develop self-consistent geometrical frameworks that do not adhere to Euclid’s fifth postulate—the one about parallel lines. In these new non-Euclidean geometries, two parallel lines do not necessarily remain parallel.

Instead, two straight lines that are parallel to each other at one point in space can come together or diverge from one another as you follow them along their paths. In these geometrical systems, it can be shown that the three angles of a triangle don’t always have to add up to 180 degrees—they can add up to a larger or a smaller number. And the ratio of a circle’s circumference to its diameter doesn’t have to be equal to the number pi. Within these non-Euclidean systems, much of what you learned in high school geometry turns out not to be true.

In Non-Euclidean geometry systems, such as on a sphere, the three angles of a triangle don’t always add up to 180 degrees. But locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the Earth, the sum of the angles is very nearly 180 degrees.

But just because a mathematician can write down a weird geometrical system, it doesn’t mean that it’s real in any physical sense. Mathematics is certainly useful to physicists, but not all mathematical possibilities are realized in nature. What these 19th-century mathematicians had done was to prove that logic and reason alone don’t force us to accept Euclidean geometry—there are other self-consistent possibilities. Whether or not those possibilities have anything to do with our physical world remained an open question. Intrigued by these strange new systems of geometry, a handful of mathematicians and physicists began to consider whether they might have anything to do with our physical world. But despite a few intermittent shows of interest, most physicists didn’t take these exotic geometries very seriously. That is until Einstein placed them at the very heart of the general theory of relativity.

From the lecture series What Einstein Got Wrong, taught by Professor Dan Hooper

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by Ferdinand Schmutzer [Public domain] via Wikimedia Commons
by Lars H. Rohwedder, Sarregouset [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)] via Wikimedia Commons