Einstein’s Grand Achievement: A General Theory of Relativity

From a lecture series presented by The Great Courses

In his quest to develop a general theory of relativity, Einstein spent many years working to resolve inconsistent and, at times, incorrect math. Another mathematician, David Hilbert, was also working toward solving the equations. So, the year of 1915 became a race to see who would be the first to complete the greatest and most celebrated theory in the history of physics.

Illustration of gravitational field lines around Earth and moon
According to the general theory of relativity, the gravity of the Earth and Moon bends space around them.

But before Einstein would be able to make any further progress toward his goal of completing a general theory of relativity, he would first have to recognize the fumbles and missteps that he had already made. This process happened gradually, and by October he had finally realized how serious the problems were with the current version of this theory.

At long last, Einstein abandoned it completely. In its place, Einstein returned to his earlier work, focusing on the results that he had produced years earlier while pursuing the more mathematical version of his strategy.

Racing toward a Solution

After spending weeks looking over the notebooks that he had produced in those earlier years, Einstein began to recognize some of the conceptual mistakes that he had made at the time. Apparently, there is something to be said for looking at something with fresh eyes.

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

Einstein gradually became convinced that, using this approach, it was possible to construct an entirely covariant form of the field equations—exactly the thing his theory was missing at the time. Furthermore, he could see how those equations would incorporate the equivalence principle, and how they would match the predictions of Newtonian gravity for things like planetary orbits. There was still an excruciating math problem ahead of him, but for the first time, Einstein saw the way forward to the final version of his theory.

Learn more about how Einstein unifies space, time, and light

Over the entire month of November 1915, Einstein worked feverishly toward the goal of producing the final, correct ,and totally covariant form of his field equations. By the middle of the month, he had gotten pretty close, but not yet to the final answer. At this point, however, he saw how it would be possible to correctly predict the details of Mercury’s orbit—a key marker for the theory and something that he had failed to do up to that point. Einstein also recognized by this time that his old prediction for the deflection of light had been incorrect.

A Little Competition

David Hilbert (1912)
David Hilbert was a German mathematician who worked in parallel with Einstein to develop field equations of general relativity. By only the thinnest of margins, Einstein beats Hilbert to the correct answer.

During this time, however, Einstein was extremely anxious that Hilbert was going to beat him to the final answer. After spending an entire decade on this problem, the thought of Hilbert getting the credit must have been consuming for Einstein. But in mid-November, Einstein received a copy of Hilbert’s paper, presenting his version of the field equations.

In many ways, Hilbert’s results were similar to Einstein’s own work. Hilbert had, in fact, made many of the same realizations Einstein had recently made. But neither Hilbert nor Einstein had the correct version of the equations that they were both looking for.

At least, not yet. Finally, on November 25, 1915, Einstein presented the equations that are today found in every textbook on relativity. By only the thinnest of margins, Einstein had beaten Hilbert to the correct answer.

This final version of the gravitation field equations is entirely covariant and completely mathematically self-consistent. They predict the orbit of Mercury and the deflection of light entirely correctly. And they suffer from no physical or mathematical problems. They describe how our universe truly is and how it truly behaves.

Learn more about cosmology in Einstein’s universe

Einstein’s final equations are also mathematically elegant. And despite being unusually hard to put to use in practice, they are actually quite simple from a conceptual point of view. These equations relate a set of mathematical quantities known as tensors. Some of these tensors describe the geometry of space, while another describes how matter and other forms of energy are distributed throughout space.

The Einstein field equations
Einstein’s final field equations relate a set of mathematical quantities known as tensors.

Technically speaking, Einstein’s field equations are a set of 10 different equations. Each of these the equations are related and interconnected to the others, and to find a useful solution you generally have to solve all ten of these equations at the same time.

These equations are particularly difficult to solve because they are what mathematicians call non-linear, which means that when you change one input, you invariably end up changing a bunch of other things at the same time. Einstein himself had to use various simplifying approximations to make the initial predictions of his theory. These days, relativists often use supercomputers to find approximate—but potentially very accurate—solutions to these equations.

A Theory at Long Last

Einstein’s general theory of relativity is his greatest contribution to science.

So, in 1915, Einstein completed and published his general theory of relativity. This theory is widely considered Einstein’s greatest contribution to science, and perhaps the greatest scientific accomplishment of the 20th century, if not, of all time.

Before Einstein, physicists thought of gravity simply as a force that attracts massive objects toward one another. And in a sense, this is correct. Gravity does pull us downward and toward the Earth. And gravity keeps the Earth in its orbit by pulling it toward the Sun. But this view of gravity—the Newtonian view—fails to recognize the greater significance of the phenomena we call gravity. What Einstein had discovered is that gravity is not merely a force, but is instead the very manifestation of the shape or geometry of space and time.

According to Einstein, the presence of mass and other energy changes the geometry of the surrounding space and time, curving or warping it. And this curving or warping causes objects to move through space differently than they would have otherwise.

When an object moves through space far from any massive bodies, and without being pulled or pushed by any forces, it simply moves forward in a straight line. Well, according to Einstein, when the Earth moves in its orbit around the Sun, it, too, is moving in a straight line.

Illustration of spacetime
The presence of the Sun has reshaped the geometry of the solar system, bending space, and transforming the Earth’s trajectory.

The presence of the Sun has reshaped the geometry of the solar system, bending space, and transforming the Earth’s trajectory. Gravity isn’t a force at all, according to Einstein. It is geometry, which is a consequence of mass and energy. By explaining gravity in terms of geometry, Einstein overturned hundreds of years of established physics. Furthermore, his theory was not only profoundly creative and mathematically elegant, but also right. The predictions of this theory agree extremely well with any number of observations that have been made.

Testing the Theory

To date, no experiment or other test has been found to conflict with the predictions of general relativity. Maybe one day we’ll find some circumstance under which Einstein’s theory fails, but nothing like that has been found so far.

Learn more about the issues involved in developing a final theory of everything

In modern times, scientists and engineers have found ways to measure and test the effects of general relativity with incredibly high precision.

For example, in order for the satellites that make up the global positioning system (GPS) to determine locations on the surface of the Earth with the 5-meter to 10-meter precision that is currently possible, they have to keep time with an accuracy of about 20 nanoseconds or so. But according to general relativity, time passes differently for the satellites than it does on the surface of the Earth, because of the differences in the Earth’s gravity and the corresponding curvature of space and time.

Without taking general relativity into account, GPS would only be accurate within a kilometer or so. The fact that the GPS satellites can determine a location to within a distance of a few meters is only possible because they take into account the effects of general relativity.

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

Images courtesy of:
by Possibly Reid, Constance (1970)
Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 230 via Wikimedia Commons
by Harris & Ewing, photographer. Albert Einstein, Washington, D.C. [Between 1921 and 1923] Photograph. Retrieved from the Library of Congress