Einstein’s Field Equations: A Long Road of Trial and Error

From the lecture series: What Einstein Got Wrong

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

Einstein’s field equations would lead him down the path to perfecting his general theory of relativity— a path, fraught with frustration and demoralizing setbacks.

Blackboard containing Einstein field equations, Hamiltonian and Gauss' Law
(Image: Asmus Koefoed/Shutterstock)

Gravitational Field Equations: The Missing Piece

By 1912 or so, Einstein had most of the major conceptual pieces in place for what became his general theory of relativity. But this doesn’t mean he had a working theory yet.

There was still a long way for him to go. To complete his theory, Einstein needed to produce an equation—or perhaps a set of equations—that could be used to relate the distribution of matter and energy with the geometry of space and time. These equations are known as the gravitational field equations or just the field equations.

With the correct gravitational field equations, one could calculate things like how objects should move through space under the influence of gravity. Without them, Einstein’s theory isn’t functional. The missing equations were essential, and Einstein knew it.

This is a transcript from the video series What Einstein Got Wrong. Watch it now, Wondrium.

Einstein spent much of 1912 working with his friend and colleague Marcel Grossmann on precisely this problem. In doing so, they found themselves taking two very different approaches. At times, Einstein adopted a mostly “physical strategy”.

To do this, he relied primarily on his intuition for physics—something Einstein had in spades. He thought it was important that he come up with a set of field equations that mimicked the Newtonian equations of gravity under certain circumstances, and that respected some basic and long-standing physical principles, like the laws of conservation of energy, and the conservation of momentum.

Einstein also insisted that the equivalence principle must somehow be manifest in these equations.

Inventing a New System of Math

At other times, however, Einstein took a very different and much less physical approach. In these instances, Einstein focused instead on the formal mathematics of the problem, as this was no ordinary math problem.

Einstein, photo from 1920
From the more mathematical approach, Einstein came up with some very elegant, and entirely covariant, field equations. But at this point, he became convinced that these equations didn’t align well enough with the predictions of Newtonian gravity. (Image: Unknown photographer – “The Solar Eclipse of May 29, 1919, and the Einstein Effect,” The Scientific Monthly 10:4 (1920), 418-422, on p. 418 and ETH-Archiv/Public domain)

To incorporate gravity into a system of non-Euclidean geometry is an incredibly difficult task that involves tensor analysis. Graduate students in physics departments sometimes take a whole course on this topic, often seen as one of the most difficult.

Keep in mind that in a course like that, the students are trying to learn the math; Einstein was trying to invent it.

Neither of these two strategies worked out well for him. His physical strategy led to equations that had some features he liked but also had serious mathematical problems. In particular, these equations were not covariant, which means that they couldn’t be consistently self-applied in all frames of reference.

Einstein knew any equations that were not covariant couldn’t be the right equations. From the more mathematical approach, Einstein came up with some very elegant, and entirely covariant, field equations.

These equations were quite similar, yet different, from those that ultimately appeared in the final version of Einstein’s theory.

Einstein was convinced these equations didn’t align well enough with the predictions of Newtonian gravity. If this had been true, these new field equations would lead to erroneous predictions for some well-measured things—like the orbits of planets, for example.

We now know, however, that Einstein was wrong about this. This early set of field equations does mimic the Newtonian predictions in the correct limit—but Einstein didn’t know that at the time.

Learn more about how Einstein was able to break out of the classical mode of thinking

Einstein also objected because these equations don’t respect the conservation of energy or momentum. For these and other reasons, Einstein jettisoned this set of field equations.

Considering how close they were to the right answer, this was almost certainly a mistake. Instead, Einstein embraced the equations that came from his physical strategy—which were much more problematic than the ones he had decided to throw out.

Einstein’s Findings on Mercury’s Orbit

In 1913, Einstein and Grossmann published a paper entitled an “Outline of a Generalized Theory of Relativity and of a Theory of Gravitation.” Conceptually, this paper contained all of the major elements that later made up the general theory of relativity.

The perihelion precession of Mercury
According to Newtonian mechanics in the absence of any other forces, a particle orbiting another under the influence of Newtonian gravity follows the same perfect ellipse eternally. But Mercury’s orbit doesn’t match with this orbit predicted by Newtonian gravity. Its orbit is seen to rotate gradually. (Image: Benutzer: Rainer Zenz/Public domain)

But in this version, many of the details were far from correct. Importantly, this version of the theory was not covariant and thus was not mathematically self-consistent.

By calling this paper an “outline”, Einstein appears to have acknowledged that this couldn’t be the final answer. But it still represented an important landmark on the way to Einstein’s ultimate theory.

For decades, scientists had noticed that the orbit of the planet Mercury, while close, doesn’t precisely agree with the behavior predicted by Newtonian gravity.

More specifically, the orientation of the ellipse that makes up Mercury’s orbit rotates a small amount each year. This is called the precession of the perihelion of Mercury’s orbit.

By Einstein’s day, the rate of this precession had been measured to be off—or in disagreement with the Newtonian prediction—by about 43 arcseconds per century, approximately 0.01 degrees per century.

Some scientists had imagined that there might be another planet nearby that was gently tugging on Mercury and slightly altering its orbit—a planet they called Vulcan. But Vulcan doesn’t exist.

We now know that Mercury’s orbit doesn’t agree with the Newtonian prediction because the Newtonian prediction is slightly wrong. To make a more accurate prediction, we need Einstein’s theory of general relativity.

Learn more about Einstein’s “blunders” concerning space

But the version of this theory that Einstein published in 1913 doesn’t lead to the right answer to this question either. Instead of the correct rate of 43 arcseconds per century, this version of Einstein’s theory predicted only 18. Einstein worked out this calculation himself, and he knew that it was a problem for this theory.

The Search for the Perfect Theory Continues

As time went on, Einstein also became increasingly concerned that his theory wasn’t covariant—and therefore wasn’t internally self-consistent. In early 1914, Einstein wrote something in a letter that does a good job of capturing his feelings at the time:

Nature shows us only the tail of the lion. But I have no doubt that the lion belongs with it, even if he cannot reveal himself all at once.

Einstein was confident that there was a great theory out there to be discovered—a theory that would connect the geometry of space and time with the force of gravity.

But he also knew that he hadn’t found that theory yet. He’d seen the lion’s tail, but not yet the lion.

Learn more about why Einstein rejected the idea of black holes

A Blessing in Disguise

By this time, Einstein had spent three years searching for the correct field equations that would complete his theory of gravity, geometry, and acceleration. In 1914, a solar eclipse was predicted to occur, allowing Einstein and other astronomers a chance to test the theory.

With this event, it was thought that the deflection of starlight around the Sun could be measured with enough accuracy to test Einstein’s notion of the equivalence principle, thus testing the basis of the general theory of relativity.

With this goal in mind, a group of astronomers set out on an expedition to Crimea, where the eclipse of the Sun would be total. But only a few weeks before the eclipse, the First World War broke out, and the astronomers were captured by the Russian army.

Though they were held as prisoners for a matter of weeks, it was long enough to make it impossible for them to make any measurements of that year’s solar eclipse.

Einstein was disappointed by this missed opportunity, but in reality, he had dodged a bullet. Einstein didn’t know it at the time, but the equations he was using were incorrect, leading him to predict the wrong amount of deflection by the Sun.

Learn more about the phenomenon of gravitational waves

The correct amount of deflection was twice as large as the value that Einstein had calculated and published. If the team of astronomers had been able to carry out their measurement, they likely would have shown that Einstein was wrong, discrediting him and all the work he had done up to that point.

Thankfully, though, Einstein was getting closer. It would be November of 1915 when he would finally arrive at a solution.

Common Questions About Einstein’s Field Equations

Q: What exactly are Einstein’s Field Equations? 

E=MC2 is Einstein’s equation within his theory of general relativity that states that mass and energy are essentially the same things in different forms. The actual terms mean “energy is equivalent to mass times the speed of light squared.”

Q: What does E=MC2 mean?

E=MC2 is Einstein’s equation within his theory of general relativity that states that mass and energy are essentially the same things in different forms. The actual terms mean “energy is equivalent to mass times the speed of light squared.”

Q: Has Einstein’s equation E=MC2 been proven?

Einstein’s field equation E=MC2 was reported to have been proven in 2005 by a team of researchers from the Institute Laue Langevin, Genoble, France (ILL), the National Institute of Standards and Technology (NIST), and Massachusetts Institute of Technology. There is much conjecture, and researchers in recent years have shown that it is limited in that it only describes effects in very isolated parameters.

Q: What is Einstein’s equation E=MC2 used for?

Einstein’s equation E=MC2 provided a fundamental understanding of nuclear fission, which resulted in the creation of nuclear power and nuclear weapons.

This article was updated on December 11, 2019

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