Although astronomers and physicists had learned much about stars in the fifteen years since the discovery of general relativity—including the likely existence of white dwarfs and neutron stars—Albert Einstein’s opinions on the subject had not changed substantially.

In 1939, he wrote his first and only paper about black holes, entitled “On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses.” In this article, Einstein set out to calculate how a large group of particles would behave as they collapsed under the force of gravity.

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

Einstein argued that the particles’ angular momentum would prevent them from collapsing indefinitely and that this would prevent a black hole from ever forming. In this conclusion, he was completely wrong.

Einstein’s prejudice that black holes could never exist in nature blinded him from all of the arguments to the contrary, leading him to reject one of the most incredible facets of his theory.

Worse still, Einstein’s opinion was so highly regarded that most physicists specializing in relativity tended to dismiss all talk of black holes for many years afterward. For decades, such objects were seldom mentioned in scientific literature.

Furthermore, interest in general relativity declined considerably during this period. It wasn’t so much that physicists doubted the validity of Einstein’s theory; they simply hadn’t found many practical uses for it.

The predictions made by general relativity are, in most cases, similar to the old Newtonian predictions. There wasn’t much that could be done in a laboratory either to test the theory further or explore its implications.

Learn more about what Einstein Got Right: Special Relativity

A few years after Einstein’s death in 1955, interest in general relativity began to see a resurgence. All around the world, small groups of physicists started to actively explore the deeper—and stranger—implications of Einstein’s general theory.

One of the key figures in general relativity’s renaissance was the young British physicist and mathematician, Roger Penrose. Penrose first became interested in relativity while he was an undergraduate at University College London.

During this period, however, few physicists knew very much about general relativity. Penrose had little choice but to teach himself about the subject, managing to learn general relativity from books and papers instead of from his professors.

Penrose then went on to study mathematics at Cambridge, where he earned his Ph.D., and then researched for brief stints at Princeton, London, Syracuse and the University of Texas at Austin.

At the time, Austin was the location of one of the few concentrations of physicists who were actively studying general relativity. Among others, Penrose met the physicist Roy Kerr in Austin.

Kerr was able to find a solution to Einstein’s field equations that is more general and more powerful than those found by Karl Schwarzschild. In particular, while Schwarzschild’s result only describes stationary objects, Kerr’s solution also allows for the possibility that black holes could be rotating.

Learn more about what Einstein got right: general relativity

At the time, few physicists thought that black holes genuinely existed—if they gave any thought to the matter at all. But in 1965, Penrose made a discovery that would upend that viewpoint.

Using a type of mathematics that was very different from anything Einstein had ever used, Penrose was able to rigorously prove that, under certain circumstances, a collapsing star would be guaranteed to form a black hole. In particular, if the collapsing star is massive enough, then the formation of a black hole is entirely inevitable.

In January of 1965, Penrose published a short, three-page paper, entitled “Gravitational Collapse and Space-Time Singularities.” At the time, Penrose’s argument went strongly against the conventional wisdom of the physics community.

Many argued, as Einstein had long done, that the complexities of real collapsing stars would prevent them from forming black holes.

But Penrose’s mathematical argument was compelling. Over the next few years, the opinions of many physicists were swayed. By the end of the 1960s, it had become a mainstream view that black holes were, in fact, likely—if not guaranteed—to exist in nature.

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As more and more physicists became convinced that black holes exist, interest began to grow about the ways that these objects might be detected or observed. One of the first scientists to actively work on this question was the incredibly prolific and versatile Russian physicist Yakov Zel’dovich.

Throughout his career, Zel’dovich made major contributions to almost every field of physics and astronomy, including material science, particle physics, relativity, astrophysics, cosmology, and nuclear physics—including work that he did on the Soviet weapons program.

In the early 1960s, Zel’dovich proposed that the presence of black holes could be indirectly inferred by studying the motion of other nearby stars. The invisible black hole, he argued, would cause another star within its own solar system to wobble back and forth with a regular period.

If scientists could somehow observe such a wobbling star, they could identify the black hole, and even measure its mass.

Alternatively, Zel’dovich argued that under certain circumstances, a black hole could have a dramatic impact on the material surrounding it. All astrophysical bodies attract and accumulate matter through the force of their gravity.

But unlike ordinary stars or planets, the matter that falls toward a black hole will be accelerated to nearly the speed of light as it approaches. Furthermore, this infalling material will spiral around the black hole like a fluid running down a drain.

Since this material moves at nearly the speed of light, it reaches temperatures in the millions of degrees. Zel’dovich argued that such systems would release huge amounts of energy and could be observed by astronomers, even at very great distances.

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The mystery of the strange astronomical object known as Cygnus X-1 left astronomers perplexed. First detected by astronomers in 1964, observations of this object in 1970, however, revealed some of its more bizarre characteristics.

Cygnus X-1 was observed to release very bright flashes of X-rays multiple times each second. The short duration of this X-ray light indicated that whatever was emitting them was not very big by astronomical standards—no more than a fraction of a light second across.

In other words, the object would be no more than 100,000 kilometers or so. X-rays are produced only in very hot environments at millions of degrees.

In the following year, radio observations in the direction of Cygnus X-1 discovered a blue supergiant star. This star, however, is far too big to generate the rapid X-ray flickering that had been observed.

To explain the production of the observed X-rays, astronomers deduced that a portion of this star’s gas was somehow being torn off, then heated to very high temperatures. Later in the same year, other observations began to detect the wobble of the blue supergiant—just as Zel’dovich had suggested a decade earlier.

From the observed wobble, it was clear to astronomers that the nearby object was massive—far too massive to even be a neutron star.

As the quality of the observations continued to improve over the years that followed, it became clearer that Cygnus X-1 was a black hole. By the late 1970s, most astrophysicists had come to accept this conclusion, as well as the conclusion that black holes indeed exist in our universe.

It is now known that Cygnus X-1 is a black hole about 6,000 light-years away from us, and about 15 times as massive as the Sun. At this mass, the Schwarzschild radius of this black hole is about 44 kilometers.

Anything within this radius is forever lost from our view. And, in a sense, is lost from our universe itself.

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In the decades following the determination that Cygnus X-1 is a black hole, astronomers and astrophysicists discovered numerous other black holes in our universe. This includes dozens of black holes that were once thought to be massive stars, similar to Cygnus X-1.

Also, many larger and more massive black holes have been discovered. The center of the Milky Way galaxy, for example, is the host of an enormous black hole, with a mass equal to about four million times the mass of the Sun.

It is now generally thought that most spiral and elliptical galaxies contain a supermassive black hole at their centers.

Although most of these supermassive black holes are similar in mass to the one at the center of the Milky Way, some galaxies harbor even larger black holes, with masses that are measured in the billions, rather than mere millions, of solar masses.

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Black holes are a consequence of Einstein’s Theory of General Relativity. Yet Einstein never came to accept that black holes did—or even could—exist in our universe.

Even though Einstein’s general relativity predicted black holes, Karl Schwarzschild is often credited with discovering them. Even this fact is tricky to state with absolute certainty, though, as Kerr after him better defined what black holes were. It was Roger Penrose who proved their existence as collapsed stars.

Scientists estimate that nearly all large galaxies have super massive black holes in their center, which would result in billions of billions.

Yes. Scientists have confirmed a super massive black hole at the center of the Milky Way.

Scientists believe that when a galaxy forms, its black hole is formed at the same time.

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If so, do they exist in our universe? And if they do, how do they form?

The answers to these questions would gradually be revealed over the decades that followed. As it turns out, to understand the formation of black holes, scientists first needed to understand the inner working and evolution of stars.

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

During the period when Einstein’s Theory of General Relativity was being developed, scientists knew very little about how stars evolved or even how they were powered.

The question of where the Sun gets the energy needed to produce its sunlight had been a stubbornly unanswered question for a long time.

Ordinary ways of storing and releasing energy didn’t come close to accounting for the huge quantity of energy that had been released by the Sun over its lifetime. For example, imagine that the Sun generated energy through chemical processes—like the way that a car extracts energy from fossil fuels, or that our bodies exact energy from food.

Even if the entire mass of the Sun—all two million trillion-trillion kilograms of it—was made up of gasoline, at the current output of sunlight, the Sun would run out of fuel in about 7,000 years.

To account for the quantity of energy that has been released by the Sun over the past 4.6 billion years, there had to be a fuel that was at least hundreds of thousands of times more efficient.

Throughout the 1800s, many physicists thought that the Sun was powered by its gradual gravitational collapse. As gravity compressed the Sun, they argued, the gravitational potential energy could be transformed into heat and sunlight.

The most optimistic estimates, however, suggested that this might be able to generate enough energy to power the Sun for a few tens of millions of years. In 1800, this seemed plausible—geologists didn’t know yet about how old the Earth and Sun were.

But by 1900 or so, it was clear that the Earth was billions—not millions—of years old. Gravity could not be the primary energy source of the Sun.

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With the introduction of the theory of relativity, however, there appeared another possibility for the source of the Sun’s energy. According to Einstein’s most famous equation, E=mc2, mass could—at least in principle—be transformed into energy, and at a very generous exchange rate.

A single gram of matter contains 90 trillion joules of energy—the equivalent of more than 20,000 tons of TNT. If there was some process going on in the Sun that was able to convert even a tiny amount of the Sun’s mass into energy, that process could plausibly provide enough energy to power the Sun for tens of billions of years.

By around 1920 or so, physicists had begun to recognize that the most likely way that stars could convert their mass into energy was through the process known as nuclear fusion.

In particular, the English astronomer and physicist Arthur Eddington proposed that stars like the Sun might generate their energy through the gradual transformation of hydrogen into helium nuclei. This process destroys a small fraction of the star’s mass and steadily releases a great deal of energy in its place.

A few years after making this proposal, Eddington wrote a book entitled *The Internal Constitution of Stars*. In this book, he described stars as being in a constant balance between the contracting force of gravity and the outward pressure of nuclear fusion.

Today, this is how we understand how stars work. In his book, Eddington recognized one particularly interesting consequence of his theory.

Whereas gravity will continue to compress a star forever, a star’s ability to undergo nuclear fusion will eventually run out once it has exhausted all or at least most of the hydrogen in its core.

Once the process of fusion ends in a star, it seems logical that gravity should be expected to compress the star into a much smaller volume. Eddington remained essentially neutral regarding what takes place when a star runs out of its nuclear fuel, remarking only that one could, “make many fanciful suggestions as to what actually will happen.”

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Although no one had realized it yet, astronomers had already found a clue to the mystery of what happens to stars when they run out of nuclear fuel. For years, astronomers had been studying a strange star that they called Sirius B.

By studying the orbits of this star and its binary companion star, astronomers had learned that the mass of Sirius B is similar to that of the Sun. But the light emitted by this star told astronomers that Sirius B was very different—both much hotter and more luminous than ordinary stars are.

Strangest of all, the density of Sirius B is about a billion times the density of water—wildly more dense than ordinary stars. Sirius B is a tiny star, which contains a Sun’s worth of mass within a volume that is about the size of the Earth.

It was unlike any star that had ever been seen before. No one understood what it was made of or how it came to be that way.

It was the British astrophysicist Ralph Fowler who first offered an answer to the question of the nature of Sirius B. In so doing, he also provided an answer to Eddington’s question of what happens to stars when they run out of nuclear fuel.

Fowler argued than when a star could no longer support itself with nuclear fusion, gravity would suddenly compress it into a much smaller volume. The novel element that Fowler introduced came from the new theory of quantum mechanics.

According to quantum theory, there seemed to be a minimum size to which matter can be compressed. This leads to a phenomenon called quantum degeneracy pressure, which forces electrons to keep their distance from one another, and thus prevents the matter that makes up a star from becoming compressed into too small of a volume.

As it turns out, the minimum size that Fowler calculated for a typical star was similar to the observed size of Sirius B. Sirius B and other stars like it are supported against the force of gravity, not by nuclear fusion like ordinary stars, but by the strange effects of quantum mechanics. Stars like this are called white dwarfs.

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Although Fowler’s insights into the nature of white dwarf stars were certainly important, it turns out that his arguments only apply to stars with a relatively modest mass. The scientist who first reached this conclusion was the young Indian physics student, Subrahmanyan Chandrasekhar.

Chandrasekhar grew up in India as part of a wealthy and privileged family. But even given those advantages, his talents were staggering and evident from a young age.

Upon finishing college at the age of 19, Chandrasekhar was invited to pursue graduate studies at Cambridge and study under Ralph Fowler, who had recently proposed the existence of white dwarf stars.

Chandrasekhar was not one to waste time and he spent the long sea voyage to England reading physics papers, including the paper on white dwarfs by his soon-to-be advisor, Ralph Fowler. Not only did the 19-year-old Chandrasekhar understand Fowler’s paper, but he was the first to really recognize its limitations.

Fowler’s arguments—Chandrasekhar deduced—were valid for medium-sized stars, but break down for stars that are too massive. For any star with more mass than about 1.4 times the mass of the Sun, the force of gravity will be strong enough to overcome the quantum degeneracy pressure that supports a white dwarf.

Any star that is more massive than this will continue to collapse when it runs out of nuclear fuel, shrinking to a size even smaller than a white dwarf.

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Over the next decade or so, Chandrasekhar, Fowler, Eddington, and others debated back and forth about what happens to massive stars when their nuclear fuel becomes exhausted. By the end of the 1930s, it had become clear that massive stars collapse well beyond the white dwarf stage, forming something denser, called a neutron star.

A neutron star is an object that consists almost entirely of neutrons—without the protons or electrons that are found in all forms of ordinary matter. Because it contains no electrically charged particles, this all-neutron matter can be compressed into ridiculously small volumes of space, and reach unimaginably high densities.

A typical neutron star contains a couple of Sun’s worth of mass, confined within a volume the size of a small city—about seven miles in radius. To put it another way, the density of a neutron star is roughly equivalent to that of the entire Earth if you were to squeeze it into a sphere the width of a football field.

But even this was the end of the story for the very most massive of stars. For those stars that are more massive than a few times the mass of the Sun, even this all-neutron state isn’t stable.

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The calculations of several physicists, including Robert Oppenheimer—who would later become the head of the Manhattan Project—had shown that even a neutron star will collapse if it’s heavier than a few times the mass of the Sun. Once this threshold is passed, there is nothing that can prevent a star from collapsing indefinitely.

A very massive star, once out of nuclear fuel, will inevitably collapse beyond the size of a white dwarf or even a neutron star. Such a star will become a black hole.

While stars are growing, they fuse atomic matter and emit energy. As they begin to stop fusing matter, they cool off inside, and this creates a pressure imbalance that results in collapse.

There are many types of stars and all act differently when dying, depending on their mass. A core-collapse star, or supernova, takes millions of years to die and around 15 seconds for core-collapse. After this, it takes a few hours as a shockwave reaches the surface and blows precious materials outward; this process continues for about a few months and then after a few years, they fade away in brightness.

While not emitting energy, neutron stars do glow for some time before eventually becoming a black dwarf, which is essentially a ball of iron ash.

The Sun is a very common star, the type of which comprises around 99 percent of known stars in the universe. The Sun is called G-type and is considered a main-sequence star in the category of sequences.

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Dozens of popular books have been written about these strange and fascinating objects by authors ranging from Kip Thorne and Stephen Hawking to Neil deGrasse Tyson. Black holes are, in fact, a direct consequence of Einstein’s Theory of General Relativity.

But Einstein himself never realized or accepted this fact.

This is a transcript from the video series

What Einstein Got Wrong. Watch it now, on The Great Courses Plus.

In the paper that he wrote in 1915 introducing the Theory of General Relativity, Einstein used the field equations of his theory to make a series of different predictions. Most notably, these predictions included Einstein’s calculation of Mercury’s orbit, which agreed well with the observations while the equations of Newtonian gravity did not.

Learn more about Einstein’s general theory of relativity

Einstein’s field equations are notoriously difficult to manipulate, even for physicists who are experts in relativity. Technically speaking, it is because these equations are nonlinear. This means that when you change one input, you end up changing many other things as well.

Einstein himself originally thought these equations couldn’t be solved precisely. Instead, he found mathematical techniques to identify approximate solutions. Today, physicists often employ the help of powerful computers to solve these ruthlessly formidable equations.

It turns out, however, that in some special and simple cases, exact solutions to Einstein’s field equations do exist. The first person to find one of these exact solutions was not Einstein himself, or even another leading scientist from among the German universities, to which general relativity was first widely disseminated.

Instead, the first exact solution came from a German lieutenant, fighting in the trenches of the Russian front of World War I. This lieutenant was an astronomer and mathematician named Karl Schwarzschild.

When World War I broke out in 1914, Schwarzschild was the director of the Astrophysical Observatory in Potsdam. Despite being 41-years-old at the time, Schwarzschild volunteered for military service shortly after hostilities began.

Over the next several months, he put his mathematical abilities to use for the German army, calculating the trajectories of artillery shells, and in the process, seeing action in France, Belgium and Russia.

While in the trenches of the Eastern Front, Schwarzschild somehow got his hands on a copy of the most recent issue of the *Proceedings of the Royal Prussian Academy of Sciences*, which included a brief account by Einstein describing his new Theory of General Relativity.

From the contents of this article, Schwarzschild became one of the first physicists to learn to skillfully manipulate the equations of Einstein’s new theory. In his exploration of general relativity, Schwarzschild focused on an extremely simple—albeit physically important—case.

Learn more about what Einstein got right: special relativity

Schwarzchild imagined a situation with a spherical mass—like a perfectly round star or a planet—that wasn’t rotating or otherwise changing. For this simple case, Schwarzschild calculated the effects of gravity using Einstein’s field equations.

Far away from the spherical mass, Schwarzschild found that gravity acts in the same way that Isaac Newton had predicted more than two centuries before. But as you move in closer to the spherical mass, Schwarzschild’s solution begins to depart from the Newtonian prediction.

Among other things, Schwarzschild’s solution showed that Einstein’s theory could perfectly explain the long-standing discrepancy observed in the orbit of Mercury. By finding the exact solution for Mercury’s orbit—rather than Einstein’s approximate solution—Schwarzschild demonstrated with greater rigor that observations of Mercury favored general relativity over Newtonian gravity.

In December of 1915, Schwarzschild wrote a letter to Einstein, describing the solution he had discovered to the field equations. The next month, Einstein wrote back to Schwarzschild and shortly thereafter presented the new solution at a meeting of the Prussian Academy.

Einstein was very pleased—albeit surprised—that such a simple and exact solution could be found. Einstein had previously thought that the nonlinearity of his field equations would make it impossible to find any exact solutions, but upon examining Schwarzschild’s letter, Einstein happily conceded that he was mistaken—at least in this special case.

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Over the next few months, Schwarzschild wrote two papers on general relativity, which among other things, detailed his important and exact solution to the field equations. Despite the ongoing hardships of war, Schwarzschild was at the peak of his scientific skills and achievements.

Tragically, in March of 1916, Schwarzschild came down with a rare skin disease, that caused his immune system to attack his own skin cells, leading to painful blistering and other symptoms. Two months later, Schwarzschild died, never to see the legacy of his work.

The Schwarzschild solution to the field equations is famous today not because it was the first exact solution of general relativity, or because it predicts the orbit of Mercury correctly. It is most often talked about today because of some of the other interesting and bizarre predictions it makes.

According to Schwarzschild’s solution, if you could compress enough mass into a small enough volume, the geometry of the surrounding space would go haywire, and spacetime itself would become infinitely curved. The radius around an object at which the spacetime becomes infinitely curved is known as the “Schwarzschild radius,” and it is proportional to the mass of the object.

For example, if the mass of the entire Sun were compressed from its current radius of 430,000 miles into a radius of only 1.9 miles, the space around it would become infinitely curved.

For an object with the mass of the Earth, the Schwarzschild radius is about a third of an inch. This infinite curvature would prevent anything, even including light, from ever passing through the Schwarzschild radius.

To a stationary observer viewing such an object from the outside, the infinite curvature of space means that it would take an infinitely long time for anything to pass through the Schwarzschild radius.

Therefore, nothing can ever escape from or reach such an object. Decades later, physicists would begin to call these objects by the name we use today—black holes.

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Einstein seemed to give little thought to the possibility that such exotic objects might exist. He instead focused on more pragmatic issues, such as comparing the predictions of the Schwarzschild solution to the observations of Mercury’s orbit.

At the time, there were good reasons for scientists to be skeptical that black holes exist. Even if a black hole could hypothetically form if there were enough mass compressed into a small enough volume of space, this doesn’t mean that it has ever happened.

Even if there were a law of physics that said a unicorn is born every time the Queen of England does a jig on the North Pole, this wouldn’t mean that there are any unicorns in our universe today. Maybe a black hole could in principle be formed, but in practice maybe it has never formed.

Furthermore, most physicists at the time thought that there would likely be things that would prevent a black hole from forming, even in principle. After all, there was a lot of uncharted territory between the kinds of stars that had been observed, and the kinds of conditions that could potentially lead to the formation of a black hole.

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There could very well have been new laws of physics yet to be discovered that would somehow prevent black holes from forming in the real world. It was far from obvious at the time that black holes did, or even could, exist.

In 1916, Einstein was not alone in doubting that black holes were real.

Einstein denied that black holes could exist and published a paper arguing his point.

It is currently thought that black holes do evaporate and disappear, but any current black holes would take longer than the life of the universe to do so.

Karl Schwarzschild had developed math while solving Einstein’s general relativity equations that hinted at a Schwarzschild radius where gravity went into a singularity in the center of large masses. This was extrapolated over the years until observations were made that appear to back up the predictions about black holes, and one was observed in 1971.

Einstein did not discover black holes, but his unproven equations describing general relativity predicted their existence.

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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 seriesWhat 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.

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.

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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.

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.

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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.

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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 correctly.

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

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.

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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.

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.

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

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.

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.

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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.

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.

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.

**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.

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.

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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 seriesWhat Einstein Got Wrong. Watch it now, on The Great Courses Plus.

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.

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.

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.

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.

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.

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

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.

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.”

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.”

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.

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

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Einstein’s theory of relativity—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.

Although special relativity predicts many observed phenomena correctly, this theory is also incomplete. Even Einstein himself was aware that it was incomplete from early on.

Firstly, 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 that describes how a car moves and behaves with its cruise control on but knows nothing about the brake pedal or the accelerator.

This is a transcript from the video series

What Einstein Got Wrong. Watch it now, on The Great Courses Plus.

Secondly, in the Newtonian theory of gravity, the accepted theory of the time, gravity’s attraction works instantaneously, pulling bodies together across great distances in space without any time delay. According to special relativity, nothing can move faster than the speed of light.

This made it difficult to reconcile special relativity with this aspect of Newtonian gravity. In at least two ways, Einstein’s theory of special relativity left us with important and unanswered questions.

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. Although it took him another decade, Einstein eventually came up with an expanded and completely general form of his theory.

The general theory of relativity was not only a theory of space and time, but also provided us with a deeper, more powerful way of thinking about the force of gravity.

In 1907, Einstein had his first important conceptual breakthrough that placed him on the road to general relativity. This occurred a few 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 that 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 genuinely 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. When the elevator is slowing down, you feel an upward force, making you slightly lighter.

The fact is that the force of gravity feels the same as the effects of acceleration. 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. Although he didn’t know yet where it would lead him, this insight made Einstein begin to speculate that acceleration and gravity might be very deeply interconnected.

Learn more about Einstein’s special theory of relativity

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 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 the same.

But Einstein’s equivalence principle provided us with an insight as to why this was the case. He began to think that the force of gravity was just acceleration in some sense. If this was the case, then it might not be surprising at all that gravitational mass was just the same thing as inertial mass.

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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. 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, 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. 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.

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.

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 truly understand this connection. 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.

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In high school, you probably took a geometry class, where you were almost certainly taught a system known as Euclidean geometry.

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.

Euclidean geometry is named after the ancient Greek philosopher and mathematician Euclid. Everything about it can be derived from five basic rules, sometimes called axioms or postulates. These postulates seem very self-evident.

For example, one of Euclid’s postulates states 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”.

But Euclid’s fifth postulate turns out to be on less solid footing. The fifth postulate states 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. It seems obvious.

Throughout most of history, Euclid’s postulates were treated as self-evident and indisputable. But in the first half of the nineteenth century, a few mathematicians started to think about systems of geometry that broke one or more of these postulates.

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Many mathematicians had managed to develop self-consistent geometrical frameworks that do not adhere to the fifth postulate’s position on 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.

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.

Just because a mathematician can write down a weird geometrical system, 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.

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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 seriously.

That was until Einstein placed them at the very heart of the general theory of relativity.

The Weak Principle of Equivalence concerns the laws of motion of bodies in free fall and states they are the same as in an unaccelerated reference frame.

Einstein’s Equivalence Principle is crucial to Einstein’s theory of general relativity in that it states that mass is the same whether inertial or gravitational, and so these types of movement are not altered by mass.

Einstein’s statement that the speed of light is the speed limit in the universe hinges on his discovery that particles gain mass as they accelerate and thus would require infinite amounts of energy to accelerate to the speed of photons which have no mass.

Einstein’s Principle of General Relativity essentially states that spacetime is curved and the laws of physics remain the same in any inertial frame of reference but can change in non-inertial frames.

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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.

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 seriesWhat 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.

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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.

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.

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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.

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.

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.

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.

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.

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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.

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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

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 sometimes 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 these equations, you can’t do much at all with Einstein’s theory. The missing equations were essential, and Einstein knew it.

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.” In doing so, 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 longstanding 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.

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

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. It was no ordinary math problem. To incorporate gravity into a system of non-Euclidean geometry is an incredibly difficult task, and involves what is known as tensor analysis. Graduate students in physics departments sometimes take a whole course on this topic, and that course is often seen as one of the most difficult. The students are just trying to learn the math. Einstein was trying to invent it.

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

Any equations that were not covariant couldn’t be the right equations, and Einstein knew it. From the more mathematical approach, Einstein came up with some very elegant, and entirely covariant, field equations. In fact, these equations were quite similar—but yet different—from those that would ultimately appear in the final version of Einstein’s theory.

But at this point in time, Einstein became convinced that 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, in fact, mimic the Newtonian predictions in the correct limit—but Einstein didn’t know that at the time. Einstein also objected on the grounds that these equations don’t respect the conservation of energy or momentum. For these and other reasons, Einstein jettisoned this set of field equations.

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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 in fact much more problematic that the ones he had decided to throw out.

So, in 1913, Einstein and Grossmann published a paper titled, “Outline of a Generalized Theory of Relativity and of a Theory of Gravitation.” Conceptually, this paper contained all of the major elements that would later make up the general theory of relativity. But in this version, many of the details were far from correct.

And most important, this version of the theory was not covariant and, thus, was not mathematically self-consistent. By calling this paper an “outline,” Einstein seems to have been acknowledging 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 doesn’t precisely agree with the behavior that is predicted by Newtonian gravity. It’s close, but it’s not in perfect agreement. 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. And 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, or about 0.01 degrees per century.

Some scientists had even imagined that there might be another planet somewhere nearby that was gently tugging on Mercury and slightly altering its orbit—a planet they called Vulcan. But Vulcan, it turns out, 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. 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.

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. He said: “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, in fact, 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.

At this point in the story, Einstein had spent three years searching for the correct field equations that would complete his theory of gravity, geometry, and acceleration. And now it was 1914, the year of that the solar eclipse that was predicted to take place. With this event, Einstein thought he could measure the deflection of starlight around the Sun with enough accuracy to test his notion of the equivalence principle, and to test the basis of the general theory of relativity.

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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. They must have seemed like likely spies to the Russians. But in any case, they were held as prisoners for a matter of weeks, which was long enough to make it impossible for them to make any measurements of that year’s solar eclipse.

Einstein, of course, was very disappointed by this missed opportunity; but in reality, he had just dodged a bullet. Einstein didn’t know it at the time, the equations he was using were incorrect, leading him to predict the wrong amount of deflection by the Sun. The correct amount of deflection was actually 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 very likely would have shown that Einstein was wrong, discrediting him and all the work that he had done up to that point.

But of course, no one could have known this at the time. Einstein did know, however, that the current version of his field equations had problems. They were still not covariant. And on top of this, Einstein knew that they predicted the wrong behavior for Mercury’s orbit. But despite these problems, Einstein gradually became more—instead of less—confident in the validity of his incorrect result. A full decade had passed since he first published his special theory of relativity, and he must have been very frustrated and exhausted after so many years of effort.

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 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. Among those in attendance was 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.

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by Unknown photographer [Public domain] via Wikimedia Commons

By adfadfdf [Public domain] via Wikimedia Public Domain

By Harris & Ewing, photographer. (ca. 1940) Albert Einstein speaking. , ca. 1940. [Photograph] Retrieved from the Library of Congress

By Benutzer:Rainer Zenz via Wikimedia Public Domain

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

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

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.

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.

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.

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.

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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

The mathematization of physics was a crucial step in the advancement of science. It was realized that the mathematical tools we had at the time weren’t strong enough.

However, in the following generations, Rene Descartes developed analytic geometry and Isaac Newton created calculus, giving scientists increasingly powerful methods to develop mathematical theories to describe the workings of the world.

This is a transcript from the video seriesRedefining Reality: The Intellectual Implications of Modern Science. Watch it now, on The Great Courses Plus.

Johannes Kepler, a contemporary of Galileo, took the data of the Danish master astronomer Tycho Brahe and worked to find an account of planetary motion that would rid humanity once and for all of the epicycles—the circles in circles.

By trial and error, Kepler worked and worked until finally, he hit upon the shape that worked—elliptical orbits with the Sun at one focus. It turned out to perfectly fit the known observations.

Kepler derived three numerical laws setting out not only the shape of the orbits, but also the relations between the distance from the Sun and the period of revolution. They were stunning results, but no one knew why they would be true. Aristotle’s circular orbits had a philosophical basis—the perfection of the aether from which everything out there was made.

The Church’s adherence to the circles of their modified Aristotelianism had a theological foundation. But why ellipses, an egg shape? Why? It was an open problem of the highest order. It was an anomaly in the classical paradigm. None of this was revolutionary. The basic concepts which ordered the universe and the picture of reality they gave rise to had become wobbly, but had not fallen.

Isaac Newton developed a simple theory—four basic laws: three laws of motion and the law of universal gravitation.

Newton’s first law of motion concerns any object that has no force applied to it. An object not subject to an external force will continue in its state of motion at a constant speed in a straight line. Now, suppose someone is on ice skates, just standing in the middle of an ice rink. What’s going to happen? The person just stays in the middle of the rink. But if they are on ice skates and moving forward at two miles an hour, they will continue to move straight ahead at two miles an hour until something pushes them or stops them.

So, the first law describes the behavior of an object subjected to no external force. The second law then describes the behavior of an object that is subjected to an external force.

So again, if a person is on ice skates moving forward at two miles an hour and they are pushed from behind, they now go faster in the same direction. If they are pulled from behind, they slow down.

If pushed from the side, they change direction. The bigger the push, the more the change; the heavier the object, the less the change. An object is either subject to a force or it isn’t, so the first two laws are sufficient to describe the behavior of the object.

But what about the object or thing that applied the force? What happens to it? The force felt from a push is felt in the opposite direction, but in the same amount. Again, if a person is on ice skates and someone pushes them, they accelerate forward because of the force and the other person goes backwards because of it. To every action there is always an equal, but opposite reaction.

These three simple laws explained a lot, but they become incredibly powerful when combined with the fourth law—the law of universal gravitation, which says that gravitation is an attractive force, a very attractive force.

Take any two objects with mass and there will be an attraction between them, along the lines connecting their centers of mass. This pull will be proportional to the product of their masses—make one twice as heavy, twice the attraction. And it will be inversely proportional to the square of the distance between them—move them twice as far away, feel only one-fourth the pull.

When these three laws of mechanics and the law of universal gravitation are used together, we suddenly have an explanation for Kepler’s elliptical orbits. Not only that, we can explain the tides, the motion of cannonballs, virtually everything we see in the world around us.

This theory was gigantic in terms of scientific thought. When Edmond Halley, a friend of Newton’s, used it to predict the coming of a comet, it was hailed—rightly so—as one of the greatest achievements of the human mind in all of history.

Learn more about Newton, who inspired the Age of Enlightenment

When these three laws of mechanics and the law of universal gravitation are used together, it was not only successful in terms of explaining and predicting, but, theoretically, it also undermined the old foundation—Aristotle.

Aristotle said that an object’s natural state of motion is at rest in is natural place. Newton has no natural places and says that its natural state of motion is in a straight line at a constant speed. Aristotle says that objects move themselves, seeking their own natural place.

Newton says that an object can’t move itself. Aristotle gives completely different accounts for the motions of objects close to the Earth and heavenly bodies.

Newton’s law of universal gravitation is universal. It applies to everything equally. Aristotle’s worldview was enforced by the centralized power of the Catholic Church. Newton’s worldview came not from authority, but from observing, something anyone could do.

And so Newton’s success supercharged an intellectual movement developing around him, the Enlightenment. The picture of reality that emerged from the Enlightenment is one in which the universe is well-ordered according to principles that are accessible to the human mind.

We live in a world that we can understand. Humans are perfectly rational beings, made to understand the world we inhabit. Since rationality is the hallmark of humanity and all people have it, then none of us is better than any other. Human equality is a basic axiom.

This view stands deeply opposed to the hierarchical structures found in religion and monarchical governments that were the holders of power at the time. The Enlightenment gave to all people the ability to understand the world.

No longer were we subservient to superior authorities, if justice be done. We didn’t need to be told the truth from above. There is no one above, and we could discover the truth ourselves. Let all of us hear the arguments, and we’ll select the best one ourselves.

When it comes to distribution of political power, let us vote. Since humans are rational, we will select the best person to oversee the implementation of laws. If humans are rational, then our choices will reflect that.

We all feel pleasure and pain, preferring the pleasurable to the painful, and so we will act as perfectly rational maximizers of pleasure over pain. Put us all in a marketplace, and we will all act to bring about the best consequences for ourselves.

Since our best interests are often at cross-purposes—I want to spend as little as possible on this sandwich I am buying from you, while you want to charge as much as possible when selling it to me—there will be rational, predictable prices governed by forces in the marketplace that look exactly like Newton’s forces on billiard balls. Like Newton, we just need a combination of observation and reason to derive what must happen.

This is the picture of reality we get from the Enlightenment in the 17th and 18th century. We live in a well-ordered, predictable universe full of things we can observe. We exist in it as perfectly rational agents capable of observing everything there is, capable of using our reason to find the laws that govern their behavior. As such rational agents, we, too, become predictable, allowing new human sciences to explain how we behave.

Learn more about chaos theory

Needless to say, the views derived from the Enlightenment did not please everyone. There were those who thought that it took the mystery away from the world, mystery that gives meaning. Humans were reduced to robots with no passion or love. The Romantic backlash of the 19th century strove to put the irrational back front and center as the basis for true humanity.

What makes us human is not that which makes us glorified billiard balls, but that which sets us apart from the rest of the world. We make and appreciate beauty. We have free will, which we exercise in ways that are often capricious and bizarre. We are not just numbers; we’re romantic beings full of life in a universe that hides the unknowable deep at its core.

This was the battle over the shape of reality that was raging at the start of the 20th century. The Enlightenment ideals, with the scientific advances that they generated, had given rise to human progress as was seen in the incredible advances in every field of scientific endeavor and the emergence of democratic states with market economies.

The romantics objected, but the scientists and their supporters marched on. And then as the 20th century dawned, the cracks started to appear. Strangely, they arose in the last place anyone would have suspected—mathematics.

Who Invented Calculus: Newton or Leibniz?

A Search for the Theory of Everything

Enlightenment Britain

William Blake’s painting “Newton” [Public domain], via Wikimedia Commons

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