einstein

Beginning in that summer of 1912, Einstein struggled to develop gravitational field equations using tensors along the lines developed by
Riemann, Ricci, and others. His first round of fitful efforts are preserved in a scratchpad notebook. Over the years, this revealing “Zurich Notebook”
has been dissected and analyzed by a team of scholars including Jürgen Renn, John D. Norton, Tilman Sauer, Michel Janssen, and John
Stachel. 17
In it Einstein pursued a two-fisted approach. On the one hand, he engaged in what was called a “physical strategy,” in which he tried to build the
correct equations from a set of requirements dictated by his feel for the physics. At the same time, he pursued a “mathematical strategy,” in which
he tried to deduce the correct equations from the more formal math requirements using the tensor analysis that Gross-mann and others
recommended.
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. 18
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. 19
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. 20
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. 21
The Entwurf and Newton’s Bucket, 1913
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.” 22
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.” 23
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. 24
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