Like all people, Albert Einstein made mistakes, and like many physicists he sometimes published them. For most of us, the times when we go astray are happily forgettable. In Einstein's case, even the mistakes are noteworthy. They offer insight into the evolution of his thinking and with it the surrounding shifts in scientific conceptions of the universe. Einstein's errors also lay bare the challenges of discovery at the leading edge. When pushing the limits of understanding, it is difficult to know whether ideas written down on paper correspond to real phenomena and whether a radically new idea will lead to profound insights or will fizzle out.

Over the years Einstein—the man who brazenly redefined the meaning of space and time—underestimated his discoveries and second-guessed himself surprisingly often. Today three whole flourishing areas of cosmology are built on ideas he misjudged: gravitational lensing, gravitational waves and the accelerating expansion of our universe.

## Einstein's distorted lens

In the case of gravitational lensing, Einstein's crucial error was to downplay one of his most famous results: his prediction that light bends in a gravitational field. In December 1936 he published a short paper in the journal *Science*, with the title “Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field.” It began with a kind of innocence that would be impossible to find in modern academic literature: *“Some time ago, R. W. Mandl [a Czech engineer] paid me a visit and asked me to publish the results of a little calculation which I had made at his request. This note complies with his wish.”*

The “little calculation” examined the possibility of extreme deflections of light caused by gravity. It was a simple matter for Einstein to show that given a massive enough intervening object and a sufficiently close approach, light rays originating from well behind the object would be bent so strongly by gravity that they could converge, producing a magnified image or multiple images of the distant source—akin to the bending of light through a lens, hence the name gravitational lensing. Lensing has developed into one of the most important observational tools in modern cosmology because it offers a way to deduce the distribution of mass in the universe even in places where the matter is invisible.

Einstein did not recognize either the magnitude or the importance of the lensing effect, however. Rather he concluded in his 1936 paper that the splitting of images caused by light passing a nearby star would be so small as to be essentially immeasurable, which undoubtedly explains the self-deprecating nature of the introduction to his paper. He was technically correct, but apparently it did not occur to him that stars are not the only objects that could produce such bending.

Einstein's obliviousness is all the more surprising given the huge impact of gravitational lensing on his scientific reputation. Deflection of light by a massive object was a key observational prediction of general relativity. In 1919 an expedition led by physicist Arthur Eddington observed a solar eclipse and determined that starlight passing by the sun bent just as Einstein expected. News of the confirmation appeared on the front pages of newspapers around the world, with the drama of a British expedition confirming the work of a German scientist right at the end of World War I no doubt contributing to the public's fascination. Einstein rapidly attained a level of scientific fame unequaled ever since.

There is a further twist to the story. Einstein had done the same light-bending calculation years earlier, in 1911.* He had not recognized the cosmological importance of his result then, either. Even worse, he had made a near-disastrous mathematical error: he performed his calculation using an early version of general relativity that predicted a light deflection by gravity half as big as the true value. An expedition had been planned to search for the bending of starlight by the sun during a 1914 solar eclipse, but it was preempted by the outbreak of World War I. Einstein was lucky that the observation never happened. If it had, the first prediction of Einstein's emerging theory of gravity would have disagreed with the data. How that would have affected his life, and the subsequent history of science, is anyone's guess.

After the 1936 article was published, Einstein wrote to the editor with a charmingly incorrect assessment of his research: *“Let me also thank you for your cooperation with the little publication, which Mister Mandl squeezed out of me. It is of little value, but it makes the poor guy happy.”*

What Einstein missed—as the irascible but brilliant California Institute of Technology astronomer Fritz Zwicky pointedly argued in a paper he submitted to the *Physical Review* within months of Einstein's publication—was that stars combine to form galaxies. Individual stars might produce unobservably small lensing effects, Zwicky noted, but lensing by massive galaxies, containing perhaps 100 billion stars, might be observable.

Zwicky's one-page paper, published in 1937, was remarkable. In it he proposed three uses for gravitational lensing that presage almost all the applications that astronomers have managed to achieve in the intervening decades: testing general relativity, using lensing by galaxies to magnify more distant objects that would otherwise be unobservable, and using lensing to measure the masses of the largest structures in the universe. Zwicky missed a fourth application that has turned out to be equally important, using the lensing by galaxies to probe the geometry and evolution of the universe on its largest scales.

It is hard to imagine a larger underestimation of the significance of any calculation in physics.

## Stymied by imaginary singularities

In the case of gravitational waves—ripples in spacetime—Einstein understood early on that they were implied by his theory but for a time backtracked from his original, correct claims for their existence. Today the detection of gravitational waves from colliding black holes and exploding stars or from the inflationary era (an epoch of hyperfast expansion immediately after the big bang) promises to open a vast new window on the universe.

Einstein first predicted gravitational waves shortly after he finalized his general theory of relativity in 1916. Although the mathematics behind the waves is complex, the line of reasoning he employed is not. According to the laws of electromagnetism, if we move an electrical charge back and forth, we generate an oscillating disturbance that manifests itself as an electromagnetic wave such as light. Likewise, if we move a pebble back and forth across the surface of a pond, we generate a pattern of water waves. Einstein had demonstrated that matter curves space, so matter in motion should produce an analogous, oscillating disturbance of space. But then he started to doubt whether such disturbances were physically real.

Einstein announced this change of heart in a 1936 paper submitted to *Physical Review* (the same prestigious American journal that published Zwicky's lensing paper). The tale of how he made the error and later discovered his mistake is almost comically twisted. He had moved to the U.S. from Germany three years earlier, and clearly he was still not used to the way things were done in the new world. Around the time he submitted his paper, entitled “Do Gravitational Waves Exist?” Einstein wrote a letter to his colleague Max Born, stating, *“Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist, though they had been assumed a certainty to the first approximation. This shows us that the non-linear general relativistic field equations can tell us more or, rather, limit us more than we have believed up to now.”*

The paper that Einstein sent to the *Physical Review* no longer exists because it was never published there. Following normal procedure, the editor of the journal had sent his paper (co-authored with Nathan Rosen, then Einstein's research assistant at the Institute for Advanced Study in Princeton, N.J.) out for peer review. A critical report came back from an anonymous referee and was forwarded to Einstein for a response. He was stunned to have had his work subject to review, given that this policy was not the norm in the German publications he previously had submitted to.

In response, Einstein wrote a haughty letter to the editor: *“We (Mr. Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed. I see no reason to address the—in any case erroneous—comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.”* He never again submitted a paper to the *Physical Review*. Apparently he also never read the referee's report, written by the distinguished U.S. cosmologist Howard Percy Robertson, which correctly explained the crucial error in his thinking.

Einstein and Rosen had tried to write a formula for gravitational plane waves (flat, evenly spaced waves, analogous to pond ripples from a rock that was dropped extremely far away), but in doing so they encountered a singularity—a place where quantities become infinitely large. That nonsensical result led them to infer that such waves could not exist. In reality, Einstein misunderstood the mathematics of his own theory. General relativity tells us that nature is independent of the particular way that scientists choose to define coordinates in space; many seemingly bizarre results that come out of solving relativity's equations are now understood as mere artifacts of using the wrong coordinate system. For example, around a black hole there is a radius, called the event horizon, inside of which one can never escape the pull of the black hole. When writing down the geometry around a black hole, many quantities—including distance and time—seem to blow up at the event horizon. These infinities are unphysical, however. In another set of coordinates, defined by the way that light moves through space, they disappear. The same is true for gravitational waves. There is no single coordinate system in which planar gravitational waves can be described without apparent singularities, but these are not real. By using two different, overlapping coordinates, the singularities disappear.

Still convinced of his argument, Einstein resubmitted his paper to the *Journal of the Franklin Institute*, but before it could be published, he, too, realized his mistake and informed the editors he had discovered errors. The final published form, retitled “On Gravitational Waves,” presents a solution to the general relativity equations that use a different coordinate system—one appropriate for cylindrical rather than planar gravitational waves—in which no singularities appear, just as Robertson had suggested.

How did Einstein come to the correct conclusion in the end? According to his later assistant, Leopold Infeld, Robertson sought out Infeld and kindly explained to him both the error in the original paper and the possible resolution, which Infeld related to Einstein. Robertson apparently never revealed that he was the paper's referee, nor did Einstein ever mention the original referee's report. The upshot is that Einstein never published his erroneous claim disputing the existence of gravitational waves, but only thanks to the intervention of a particularly diligent peer reviewer.

Einstein did not fare as well with regard to black holes. He remained confused by the unphysical singularity at the event horizon and assumed that nature must prohibit it somehow. He argued that conservation of angular momentum would cause particles in a collapsing object to stabilize in orbits of finite radius, making it impossible for an event horizon to form. He never accepted black holes as physically real objects.

## A brilliant blunder?

The most famous of Einstein's errors is his modification of general relativity to allow a universe that is not expanding. It became widely known because he reportedly denounced it himself as a “blunder.” When he completed general relativity in 1915, the prevailing wisdom held that our galaxy, the Milky Way, was surrounded by an infinite void that was both static and eternal. But Einstein recognized that the gravitational force caused by matter in general relativity (as in Newton's theory) is universally attractive, making a static solution impossible. Gravity should cause the matter to collapse inward.

In a 1917 paper, “Cosmological Considerations in the General Theory of Relativity,” Einstein therefore introduced an additional, constant term in his equations for general relativity to ensure a static universe. The cosmological term would provide a counteracting gravitational repulsion throughout all of space, “holding back gravity” as Einstein hoped. There was no physical justification for this term, other than staving off collapse.

Within a decade after the introduction of the cosmological constant, evidence began to mount that the universe wasn't static after all. At first, Einstein was resistant. Belgian physicist and Catholic priest Georges Lemaître developed a model of an expanding universe, complete with a kind of big bang, in 1927, which was two years before Edwin Hubble published his landmark paper documenting the recession of galaxies. Lemaître later recalled being admonished by Einstein, “Your calculations are correct, but your physics is abominable!”

Eventually Einstein came around. He went to visit Hubble and looked through his telescope at Mount Wilson Observatory near Pasadena, Calif., and in 1933 Einstein reportedly praised Lemaître's cosmological theory: “This is the most beautiful and satisfactory explanation of creation to which I have ever listened.”

It was not lost on Einstein that in an expanding universe there was no longer any need for a cosmological constant to keep things static. Even in 1919 he wrote that the constant was “gravely detrimental to the formal beauty of the theory.” And in an oft-quoted reference in George Gamow's book, *My World Line: An Informal Autobiography*, Gamow related the following anecdote: “Much later, when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder he ever made in his life.”

In retrospect, Einstein was completely mistaken in thinking that the cosmological constant was worthless, but his introduction of it was a blunder, for two reasons. Had he had the courage of his convictions, he would have recognized that general relativity's inconsistency with a static universe was a *prediction*. At a time when no one expected that the universe was dynamic on large scales, Einstein could have predicted cosmic expansion instead of having to grudgingly accept it later.

The introduction of the cosmological constant was also a blunder in a more fundamental way. Simply put, the constant could not work the way he intended: it would not allow the kind of static universe that he was trying to match. That error arose in part because once again Einstein used the wrong coordinate frame for his calculations. But his conception was wrong from a physical perspective as well. Although it is possible to briefly balance the gravitational attraction of matter with the repulsion from a cosmological constant, the smallest perturbation will produce runaway expansion or collapse. With or without the cosmological constant, the universe must be dynamic.

The cosmological constant ultimately proved far more durable than the limited astronomical knowledge that inspired it. Although the constant was an ad hoc addition to his equations, physicists now understand that when viewed through the lens of quantum theory, it corresponds to a possible energy residing in empty space. In fact, quantum physics requires the presence of such a cosmological term. Moreover, the energy content of empty space is not just a theoretical concept. In one of the most astonishing measurements in recent history, two groups in 1998 observed that the expansion of the universe is accelerating, driven outward by something that seems to act just like a cosmological constant. In this instance, one might say that Einstein actually blundered twice: by introducing the cosmological constant for the wrong reason and again by throwing it out instead of exploring its implications.

## The error he never admitted

Einstein's errors were intellectually fertile because they were all rooted in grand, provocative ideas about how physics works. That is true even of what is generally regarded as his greatest error of all: his refusal to accept quantum mechanics as a fundamental theory of nature.

Although Einstein had created the basis for quantum mechanics with his theory of the photoelectric effect (for which he later won the Nobel Prize), he never completely shed the mind-set of classical physics. The idea that the location of a particle is a matter of probability or that one particle can instantaneously influence another one from a great distance struck him as absurd, although his views on the quandaries of quantum theory were more nuanced than he is usually given credit for [see “Is the Cosmos Random?” by George Musser]. He spent most of his later years attempting to merge the equations of gravity and electromagnetism within a classical framework, into a so-called unified field theory.

As part of that effort, Einstein became fascinated by a speculation introduced by German mathematician Theodor Kaluza in 1921 and later elaborated on by Swedish physicist Oskar Klein. They suggested that if the universe contains five dimensions—three of familiar space, one of time and a fifth dimension curled up so as to be invisible—it would be possible to create a single, combined description of electromagnetism and gravity. For Einstein, one of the attractive facets of the theory was that it was purely classical. Klein had shown that, in the model, the apparent quantization of electrical charge could be a consequence of electromagnetism reflecting the geometry of the closed, circular shape of the fifth dimension.

Einstein's effort to construct a unified field theory ultimately went nowhere, but his flawed ideas once again led to important breakthroughs. In calling attention to the extra dimensions of Kaluza and Klein, Einstein may have helped inspire the higher-dimensional mathematics of modern string theory, a currently popular proposal for incorporating general relativity into quantum mechanics. Einstein probably would have been repelled by the idea of having general relativity arise out of a quantum landscape rather than the other way around. But as we have seen, he was anything but infallible.

**Editor's Note (11/4/15): This sentence from the print article was edited after posting to correct the year Einstein made his light-bending calculation.*