Einstein (47 page)

Einstein’s “physical strategy” began with his mission to generalize the principle of relativity so that it applied to observers who were accelerating or moving in an arbitrary manner. Any gravitational field equation he devised would have to meet the following physical requirements:

• It must revert to Newtonian theory in the special case of weak and static gravitational fields. In other words, under certain normal conditions, his theory would describe Newton’s familiar laws of gravitation and motion.

• It should preserve the laws of classical physics, most notably the conservation of energy and momentum.

• It should satisfy the principle of equivalence, which holds that observations made by an observer who is uniformly accelerating would be equivalent to those made by an observer standing in a comparable gravitational field.

Einstein’s “mathematical strategy,” on the other hand, focused on using generic mathematical knowledge about the metric tensor to find a gravitational field equation that was generally (or at least broadly) covariant.

The process worked both ways: Einstein would examine equations that were abstracted from his physical requirements to check their covariance properties, and he would examine equations that sprang from elegant mathematical formulations to see if they met the requirements of his physics. “On page after page of the notebook, he approached the problem from either side, here writing expressions suggested by the physical requirements of the Newtonian limit and energy-momentum conservation, there writing expressions naturally suggested by the generally covariant quantities supplied by the mathematics of Ricci and Levi-Civita,” says John Norton.
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But something disappointing happened. The two groups of requirements did not mesh. Or at least Einstein thought not. He could not get the results produced by one strategy to meet the requirements of the other strategy.

Using his mathematical strategy, he derived some very elegant equations. At Grossmann’s suggestion, he had begun using a tensor developed by Riemann and then a more suitable one developed by Ricci. Finally, by the end of 1912, he had devised a field equation using a tensor that was, it turned out, pretty close to the one that he would eventually use in his triumphant formulation of late November 1915. In
other words, in his Zurich Notebook he had come up with what was quite close to the right solution.
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But then he rejected it, and it would stagnate in his discard pile for more than two years. Why? Among other considerations, he thought (somewhat mistakenly) that this solution did not reduce, in a weak and static field, to Newton’s laws. When he tried it a different way, it did not meet the requirement of the conservation of energy and momentum. And if he introduced a coordinate condition that allowed the equations to satisfy one of these requirements, it proved incompatible with the conditions needed to satisfy the other requirement.
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As a result, Einstein reduced his reliance on the mathematical strategy. It was a decision that he would later regret. Indeed, after he finally returned to the mathematical strategy and it proved spectacularly successful, he would from then on proclaim the virtues—both scientific and philosophical—of mathematical formalism.
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In May 1913, having discarded the equations derived from the mathematical strategy, Einstein and Grossmann produced a sketchy alternative theory based more on the physical strategy. Its equations were constructed to conform to the requirements of energy-momentum conservation and of being compatible with Newton’s laws in a weak static field.

Even though it did not seem that these equations satisfied the goal of being suitably covariant, Einstein and Grossmann felt it was the best they could do for the time being. Their title reflected their tentativeness: “Outline of a Generalized Theory of Relativity and of a Theory of Gravitation.” The paper thus became known as the
Entwurf,
which was the German word they had used for “outline.”
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For a few months after producing the
Entwurf,
Einstein was both pleased and depleted. “I finally solved the problem a few weeks ago,” he wrote Elsa. “It is a bold extension of the theory of relativity, together with a theory of gravitation. Now I must give myself some rest, otherwise I will go kaput.”
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However, he was soon questioning what he had wrought. And the
more he reflected on the
Entwurf,
the more he realized that its equations did not satisfy the goal of being generally or even broadly covariant. In other words, the way the equations applied to people in arbitrary accelerated motion might not always be the same.

His confidence in the theory was not strengthened when he sat down with his old friend Michele Besso, who had come to visit him in June 1913, to study the implications of the
Entwurf
theory. They produced more than fifty pages of notes on their deliberations, each writing about half, which analyzed how the
Entwurf
accorded with some curious facts that were known about the orbit of Mercury.
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Since the 1840s, scientists had been worrying about a small but unexplained shift in the orbit of Mercury. The perihelion is the spot in a planet’s elliptical orbit when it is closest to the sun, and over the years this spot in Mercury’s orbit had slipped a tiny amount more—about 43 seconds of an arc each century—than what was explained by Newton’s laws. At first it was assumed that some undiscovered planet was tugging at it, similar to the reasoning that had earlier led to the discovery of Neptune. The Frenchman who discovered Mercury’s anomaly even calculated where such a planet would be and named it Vulcan. But it was not there.

Einstein hoped that his new theory of relativity, when its gravitational field equations were applied to the sun, would explain Mercury’s orbit. Unfortunately, after a lot of calculations and corrected mistakes, he and Besso came up with a value of 18 seconds of an arc per century for how far Mercury’s perihelion should stray, which was not even halfway correct. The poor result convinced Einstein not to publish the Mercury calculations. But it did not convince him to discard his
Entwurf
theory, at least not yet.

Einstein and Besso also looked at whether rotation could be considered a form of relative motion under the equations of the
Entwurf
theory. In other words, imagine that an observer is rotating and thus experiencing inertia. Is it possible that this is yet another case of relative motion and is indistinguishable from a case where the observer is at rest and the rest of the universe is rotating around him?

The most famous thought experiment along these lines was that described by Newton in the third book of his
Principia.
Imagine a
bucket that begins to rotate as it hangs from a rope. At first the water in the bucket stays rather still and flat. But soon the friction from the bucket causes the water to spin around with it, and it assumes a concave shape. Why? Because inertia causes the spinning water to push outward, and therefore it pushes up the side of the bucket.

Yes, but if we suspect that all motion is relative, we ask: What is the water spinning relative to? Not the bucket, because the water is concave when it is spinning along with the bucket, and also when the bucket stops and the water keeps spinning inside for a while. Perhaps the water is spinning relative to nearby bodies such as the earth that exert gravitational force.

But imagine the bucket spinning in deep space with no gravity and no reference points. Or imagine it spinning alone in an otherwise empty universe. Would there still be inertia? Newton believed so, and said it was because the bucket was spinning relative to absolute space.

When Einstein’s early hero Ernst Mach came along in the mid-nineteenth century, he debunked this notion of absolute space and argued that the inertia existed because the water was spinning relative to the rest of the matter in the universe. Indeed, the same effects would be observed if the bucket was still and the rest of the universe was rotating around it, he said.
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The general theory of relativity, Einstein hoped, would have what he dubbed “Mach’s Principle” as one of its touchstones. Happily, when he analyzed the equations in his
Entwurf
theory, he concluded that they
did
seem to predict that the effects would be the same whether a bucket was spinning or was motionless while the rest of the universe spun around it.

Or so Einstein thought. He and Besso made a series of very clever calculations designed to see if indeed this was the case. In their notebook, Einstein wrote a joyous little exclamation at what appeared to be the successful conclusion of these calculations: “Is correct.”

Unfortunately, he and Besso had made some mistakes in this work. Einstein would eventually discover those errors two years later and realize, unhappily, that the
Entwurf
did not in fact satisfy Mach’s principle. In all likelihood, Besso had already warned him that this might be the case. In a memo that he apparently wrote in August 1913, Besso
suggested that a “rotation metric” was not in fact a solution permitted by the field equations in the
Entwurf.

But Einstein dismissed these doubts, in letters to Besso as well as to Mach and others, at least for the time being.
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If experiments upheld the theory, “your brilliant investigations on the foundations of mechanics will have received a splendid confirmation,” Einstein wrote to Mach days after the
Entwurf
was published. “For it shows that inertia has its origin in some kind of interaction of the bodies, exactly in accordance with your argument about Newton’s bucket experiment.”
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What worried Einstein most about the
Entwurf,
justifiably, was that its mathematical equations did not prove to be generally covariant, thus deflating his goal of assuring that the laws of nature were the same for an observer in accelerated or arbitrary motion as they were for an observer moving at a constant velocity. “Regrettably, the whole business is still so very tricky that my confidence in the theory is still rather hesitant,” he wrote in reply to a warm letter of congratulations from Lorentz.“The gravitational equations themselves unfortunately do not have the property of general covariance.”
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He was soon able to convince himself, at least for a while, that this was inevitable. In part he did so through a thought experiment, which became known as the “hole argument,”
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that seemed to suggest that the holy grail of making the gravitational field equations generally covariant was impossible to reach, or at least physically uninteresting. “The fact that the gravitational equations are not generally covariant, something that quite disturbed me for a while, is unavoidable,” he wrote a friend. “It can easily be shown that a theory with generally covariant equations cannot exist if the demand is made that the field is mathematically completely determined by matter.”
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For the time being, very few physicists embraced Einstein’s new theory, and many came forth to denounce it.
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Einstein professed pleasure that the issue of relativity “has at least been taken up with the requisite vigor,” as he put it to his friend Zangger. “I enjoy controversies. In the manner of Figaro: ‘Would my noble Lord venture a little dance? He should tell me! I will strike up the tune for him.’ ”
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Through it all, Einstein continued to try to salvage his
Entwurf
approach. He was able to find ways, or so he thought, to achieve enough
covariance to satisfy most aspects of his principle about the equivalence of gravity and acceleration. “I succeeded in proving that the gravitational equations hold for arbitrarily moving reference systems, and thus that the hypothesis of the equivalence of acceleration and gravitational field is absolutely correct,” he wrote Zangger in early 1914. “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. We see him only the way a louse that sits upon him would.”
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There was, Einstein knew, one way to quell doubts. He often concluded his papers with suggestions for how future experiments could confirm whatever he had just propounded. In the case of general relativity, this process had begun in 1911, when he specified with some precision how much he thought light from a star would be deflected by the gravity of the sun.

This was something that could, he hoped, be measured by photographing stars whose light passed close to the sun and determining whether there appeared to be a tiny shift in their position compared to when their light did not have to pass right by the sun. But this was an experiment that had to be done during an eclipse, when the starlight would be visible.

So it was not surprising that, with his theory arousing noisy attacks from colleagues and quiet doubts in his own mind, Einstein became keenly interested in what could be discovered during the next suitable total eclipse of the sun, which was due to occur on August 21, 1914. That would require an expedition to the Crimea, in Russia, where the path of the eclipse would fall.

Einstein was so eager to have his theory tested during the eclipse that, when it seemed there might be no money for such an expedition, he offered to pay part of the costs himself. Erwin Freundlich, the young Berlin astronomer who had read the light-bending predictions in Einstein’s 1911 paper and become eager to prove him correct, was ready to take the lead. “I am extremely pleased that you have taken up the question of the bending of light with so much zeal,” Einstein wrote
him in early 1912. In August 1913, he was still bombarding the astronomer with encouragement.“Nothing more can be done by the theorists,” he wrote. “In this matter it is only you, the astronomers, who can next year perform a simply invaluable service to theoretical physics.”
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