How Einstein Solved the General Theory of Relativity

From the lecture series: What Einstein Got Wrong

By Dan Hooper, Ph.D., University of Chicago

The general theory of relativity is recognized as Einstein’s greatest achievement, but arriving at the final version was not an easy task. It involved years of missteps and disappointments. Discover how Einstein finally connected all the dots and how the theory holds up in light of today’s scientific advancements.

Einstein (gravitational) field equations including the cosmological constant by Albert Einstein from his theory of general relativity (Stephen Weinberg, Gravitation and cosmology, New York, 1972 p. 155)
Einstein’s field equation on a wall in Leiden, Netherlands (Image: painting by Jan-Willem Bruins (TegenBeeld); photograph by Vysotsky/Public domain)

Einstein Presents His Theory Publicly

In 1914, Einstein had already spent three years searching for the correct field equations that would complete his theory of gravity, geometry, and acceleration, known as general relativity. This theory explains how the force of gravity and acceleration are the same.

He was getting closer to reaching a solution; however, the current version of his field equations had problems: They were still noncovariant.

But despite this, Einstein gradually became more—instead of less—confident in the validity of his incorrect result.

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

A full decade had passed since he first published his special theory of relativity. It was around this time that Einstein began to present publicly the incorrect version of his theory. In a week-long series of lectures in June of 1915, Einstein presented the incorrect version of his theory to a group of physicists and mathematicians at a university in Germany, going into considerable detail.

David Hilbert (1912)
Using Einstein’s incorrect version of the theory, David Hilbert began to work toward producing a correct and complete form of general relativity. (Image: Von Unbekannt – Possibly Reid, Constance (1970) Hilbert, Berlin, Heidelberg/Public domain)

Among those in attendance were David Hilbert—one of the world’s most brilliant mathematicians and perhaps, one of the greatest and most influential mathematicians of all time. Hilbert immediately took a great interest in Einstein’s new theory.

In hindsight, we can see that Einstein already had all of the most important physical pieces of his theory correctly in place—elements like the equivalence principle, among others. But Einstein’s math was inconsistent and at times incorrect.

Hilbert seems to have recognized this, and he began to work toward producing a correct and complete form of general relativity, perhaps before Einstein would be able to do so. With Einstein and Hilbert both working toward this same goal, the remaining months 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.

Learn more about the mistakes of three other great thinkers

Starting Fresh on the Theory of Relativity

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

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

Albert Einstein.
After spending weeks studying his old notes carefully, Einstein began to notice some of the mistakes he had made at the time. (Image: Ernestoectorado/Public domain)

After spending weeks looking over the notebooks that he had produced in those earlier years, Einstein began to recognize some of the conceptual mistakes he had made at the time.

Looking at it with fresh eyes, 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.

Furthermore, he could see both 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 Einstein’s problems with time travel

The Race to Find a Solution

Over the entire month of November 1915, Einstein worked feverishly toward the goal of producing the final, correct, and completely covariant form of his field equations. By the middle of the month, he had gotten close but was not yet to the final answer.

However, he saw how it would be possible to correctly predict the details of Mercury’s orbit—something he had failed to do before. Einstein also recognized by this time that his former prediction for the deflection of light had been incorrect.

He now knew it was a lucky break that the eclipse of 1914, which would have allowed him to publicly test his theory, had never been measured.

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 work. He had made many of the same realizations Einstein had recently made.

Learn more about Einstein’s deterministic version of quantum theory

But neither Hilbert nor Einstein had the correct version of the equations that they were both looking for, at least not yet.

Einstein’s Final Equations

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 correctly.

They suffer from no physical or mathematical problems, and they describe how our universe truly is and how it truly behaves.

Einstein's Field Equation
Einstein’s Field Equation

Einstein’s final equations are also mathematically elegant. Despite being unusually hard to put to use in practice, they are 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.

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

These equations are particularly difficult to solve because they are what mathematicians call non-linear. This means that when one input is changed, it invariably ends up changing several 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.

Learn more about Einstein’s search for a unified field theory

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

Overturning Conventional Views on Gravity

Before Einstein, physicists thought of gravity simply as a force that attracts massive objects toward one another. In a sense, this is correct.

Gravity pulls us downward and toward the Earth, and it 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 called 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.

Conceptual illustration of both 3D space and time distortion near a mass.
The presence of mass changes the geometry of the surrounding space & tie by curving or warping it.(Image: Lucas Vieira Barbosa authored the original OGV. Selected frames (10 percent of the original) extracted and re-timed by Stigmatella aurantiaca/Public domain)

According to Einstein, the presence of mass and other energy changes the geometry of the surrounding space and time, curving or warping it. 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, without being pulled or pushed by any forces, it simply moves forward in a straight line. According to Einstein, when the Earth moves in its orbit around the Sun, it too is moving in a straight line.

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, but 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 is not only profoundly creative and mathematically elegant, but it is also right. The predictions of this theory agree extremely well with many observations that have been made.

Learn more about Einstein’s major contributions to physics

How Einstein’s Theory Holds Up Today

To date, no experiment or other test has been found to conflict with the predictions of general relativity. Maybe one day there will be some circumstances under which Einstein’s theory fails, but nothing has been found so far.

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

For example, for the satellites that make up the global positioning system to determine locations on the surface of the Earth with the five-to-ten-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, the global positioning system would only be accurate within a kilometer or so.

Navigation technology concept
GPS satellites can determine your location with an accuracy of few meters because they take into account the effects of general relativity. (Image: AleksOrel/Shutterstock)

The fact that the GPS satellites can determine a location within a distance of a few meters is only possible because they take into account the effects of general relativity.

Learn more about the strange phenomenon of entanglement

To summarize, the general theory of relativity is Einstein’s greatest achievement. Although this is certainly an example of something that Einstein got right, he made many mistakes along the way.

Common Questions About Einstein’s General Theory of Relativity

Q: What is Einstein’s General Theory of Relativity?

Einstein’s general theory of relativity is a foundational theory in physics which states that gravity is the warping of spacetime by a large mass.

Q: What is Einstein’s general theory of relativity used for?

Einstein’s general theory of relativity is the bedrock of astrophysics and has helped us understand everything from black holes to exoplanets and more by understanding the curvature of spacetime.

Q: Has Einstein’s general theory of relativity been proven?

Einstein’s general theory of relativity was first proven by Arthur Stanley Eddington in 1919. During a solar eclipse, Eddington photographed the sun from multiple places on the globe and proved Einstein’s prediction that starlight would be deflected. This has been proven time and time again in the years since.

Q: How does one test Einstein’s general theory of relativity?

Einstein’s general theory of relativity is tested in three ways: deflection of starlight by the Sun, Mercury’s orbit in perihelion procession, and gravitational redshift of light.

This article was updated on December 11, 2018

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About Kate Findley 100 Articles
Kate is a writer, novelist, and blogger living in Los Angeles. She has been writing for The Great Courses since 2017. It incorporates her two favorite things: writing and learning.