EINSTEIN, Albert and Michele BESSO. Autograph manuscript, comprising a series of calculations using the early version ("Entwurf") of the field equations of Einstein's general theory of relativity, the aim of which was to test whether the theory could account for the well-known anomaly in the motion of the perihelion of Mercury. 26 pages in Einstein's hand; 25 pages in Besso's; 3 pages with entries of both collaborators (many pages with contributions of one to entries of the other). No place, [Berlin and Zurich], n.d. [mostly June 1913; additions from early 1914]. Published in Einstein, Collected Papers, 4:344-359 (introduction), 4:360-473 (transcription, with notes), and 4:630-682 (facsimile).
54 (of 56) pages, 4to, written in ink (a few portions by Besso in pencil), on 37 separate sheets of foolscap and squared paper of various types, mostly 273 x 212 mm. (10¾ x 8 3/8 in.), leaf [10-11] an oblong folded sheet bearing unpublished partial diagrams and calculations on its verso (counted in above total), irregular partial pagination, now arranged as published in Collected Papers, one sheet with lower portion torn away (probably by the collaborators), page [3] written on the back of a printed announcement (dated "Ende April 1913"), many pages with extensive corrections to the formulae or with whole sections of calculations crossed out, a few pages with minor corner defects, pp.[16-17] not present (these pages, in Besso's hand, are written on the verso of a letter from C.-E. Guye to Einstein, 31 May 1913, now in a private collection), p.[1] with minor rust markings, otherwise in fine condition for a working scientific manuscript.
THE SEARCH FOR PROOF OF EINSTEIN'S GENERAL THEORY OF RELATIVITY: THE EINSTEIN-BESSO CALCULATIONS OF THE PRECESSION OF THE PERIHELION OF MERCURY. ONE OF ONLY TWO SURVIVING WORKING SCIENTIFIC MANUSCRIPTS FROM THIS KEY PERIOD, AND THE MOST IMPORTANT SCIENTIFIC MANUSCRIPT OF EINSTEIN EVER TO BE OFFERED AT AUCTION
THE EINSTEIN-BESSO MANUSCRIPT: A GLIMPSE BEHIND THE CURTAIN OF THE WIZARD
The Einstein-Besso manuscript is a set of research notes produced by Albert Einstein (1879-1955) and his closest friend and confidant Michele Besso (1873-1955). They are from the period 1913-1914, when Einstein was still developing his general theory of relativity. Einstein first published the theory in the form in which physicists still use it today in the last of a series of four papers in the Proceedings of the Berlin Academy of Science in November 1915. Manuscript material such as the Einstein-Besso manuscript offers historians and philosophers of science a window into how Einstein arrived at his theory.
A BRIEF CHARACTERIZATION OF THE EINSTEIN-BESSO MANUSCRIPT AND ITS IMPORTANCE
There are only two manuscripts still extant with research notes documenting Einstein's work toward the general theory of relativity. These two manuscripts are the Zurich Notebook of late 1912/early 1913 and the Einstein-Besso manuscript, the bulk of which stems from June 1913. Of these two manuscripts, only the latter is in private hands. The Zurich Notebook is part of the Einstein Archives at Hebrew University in Jerusalem.
The manuscript consists of roughly 50 pages, about half of them in Einstein's hand, the other half in Besso's. There is no continuous numbering, which makes it hard to establish the exact order of the pages. The manuscript, which was found in the Besso Nachlass, was brought to the attention of the editors of the Einstein Papers Project in 1988. It was published in 1995, both in transcription and in facsimile, with extensive annotation in Vol. 4 of The Collected Papers of Albert Einstein. In the summer of 1998, fourteen pages closely related to the manuscript were discovered and there might well be more. These pages, still in the possession of Besso's heirs, do not have any entries by Einstein and are all in Besso's hand. It documents Besso's efforts to complete the work he had begun with Einstein.
The aim of most of the calculations in the Einstein-Besso manuscript is to determine whether an early version of the general theory of relativity, which Einstein had published in June 1913 in a paper co-authored with his friend, the mathematician Grossmann, can account for a tiny discrepancy between the observed motion of Mercury and the motion predicted on the basis of Newton's theory of gravity. This discrepancy is known as the anomalous advance of Mercury's perihelion (see Fig. 1 below). The results of the calculations by Einstein and Besso were disappointing. The Einstein-Grossmann theory (or Entwurf theory as it is commonly called after the title of the Einstein-Grossmann paper) could only account for part of the discrepancy between observation and Newtonian theory. However, Einstein and Besso's efforts would not be in vain. The techniques developed in the manuscript for doing these calculations would be taken over virtually unchanged in November 1915 to compute the motion of Mercury predicted by the general theory of relativity in its final form. Einstein found that the final theory can account for the full effect left unexplained by Newtonian theory. As he later told a colleague, he was so excited about this result that it gave him heart palpitations (Pais 'Subtle is the Lord'..., p. 253).
The explanation of the anomalous advance of Mercury's perihelion would become one of the three classical tests of general relativity. Einstein found and published it in a time span of no more than a week. With the discovery of the Einstein-Besso manuscript, this impressive feat becomes more understandable: Einstein had done essentially the same calculation together with Besso two years earlier. The study of these earlier calculations has contributed to a fuller understanding of the 1915 perihelion paper (see lot 85).
The Einstein-Besso manuscript is not only the key document for understanding the celebrated application of the general theory of relativity to the problem of Mercury's perihelion, but is also of great significance for the historical reconstruction of the genesis of the theory. In letters written shortly after his papers of November 1915, Einstein listed three reasons for abandoning the earlier version of his theory. The fact that the perihelion motion did not come out right was one of them. Another problem he mentioned was that the earlier theory was incompatible with the relativity of rotation. The Einstein-Besso manuscript contains an ingenious calculation, the whole purpose of which was to check whether the theory is compatible with this notion. Einstein was able to convince himself that the theory was. However, he made some trivial errors in this calculation. One of the recently discovered pages in the Besso Nachlass shows that Besso had found this error by August 1913. Einstein, it seems, first accepted Besso's result, then changed his mind again, all without checking or redoing the calculation he had done in the Einstein-Besso manuscript. In what was clearly intended as the definitive exposition of the Entwurf theory, a lengthy paper published in November 1914, Einstein gave pride of place to the specious result of this calculation. It was not until September 1915 that he carefully redid the calculation and discovered, to his dismay, that the Entwurf theory was incompatible with the relativity of rotation. This discovery triggered the unraveling of the theory and led Einstein to return to ideas considered and rejected in the Zurich Notebook of some three years earlier. Using the same ideas, he was able to complete, in a little over a month, the general theory of relativity as we know it today.
EINSTEIN AND BESSO: THE EAGLE AND THE SPARROW
Einstein's historic 1905 paper "On the Electrodynamics of Moving Bodies," in which the theory now known as special relativity was announced, was unusual for a scientific paper in that it carried none of the usual references to the literature of theoretical physics. The only individual credited with any contribution to the 1905 paper was Michele Angelo Besso, whom Einstein thanked for "many useful suggestions." Besso, whom Einstein once characterized as a perpetual student, had studied mechanical engineering at the Zurich Polytechnic during the years Einstein was enrolled in the physics section. The two met at a musical evening in Zurich and remained lifelong friends. In 1904, on the recommendation of Einstein, Besso took a position at the Swiss Patent Office. Whenever they could, the two friends engaged in long discussions of physics and mathematics. Besso played a very important role as a "sounding board" for Einstein, and when Einstein moved to Zurich and later Berlin, the two men visited and kept up a lively correspondence.
When, shortly after taking up his position in Berlin in 1914, Einstein sent his wife Mileva and his sons Hans-Albert and Eduard back to Zurich, Besso and his wife took on the role of intermediary between the feuding partners as their marriage was dissolving. They even cared for the couple's two sons during Mileva's illness.
In 1913, when Einstein and Besso collaborated on the calculations in the manuscript under discussion here, Besso was living in Gorizia, near Trieste. The manuscript shows that in this case Besso functioned as considerably more than just a sounding board. Although he left the hardest parts to Einstein, he did take responsibility for some important parts of the calculations. In later years, Besso described his scientific collaboration with Einstein with a charming simile: Einstein was an eagle, and he, Besso, a sparrow. Under the eagle's wing, the sparrow had been able to fly higher than on its own.
FROM SPECIAL RELATIVITY TO GENERAL RELATIVITY
Einstein started on the path that would lead him to the general theory of relativity in late 1907. He was writing a review article about his 1905 special theory of relativity (see lot 77). The last section of this article was devoted to gravity. In it, Einstein argued that a satisfactory theory of gravity cannot be achieved within the framework of special relativity and that a generalization of that theory is needed. This was an extraordinary step to take at the time. Other researchers, such as the great mathematicians Henri Poincaré and Hermann Minkowski, felt that a perfectly adequate theory of gravitation could be constructed simply by modifying Newton's theory of gravitation somewhat to meet the demands of special relativity.
As Einstein was pondering the problem of gravitation, a thought occurred to him which he later described as "the happiest thought of my life." It had been known since Galileo that all bodies fall alike in a given gravitational field. This is the point of the famous, though most likely apocryphal, story of Galileo dropping bodies of different mass from the leaning tower of Pisa and watching them hit the ground simultaneously. Galileo's insight was incorporated but not explained in Newton's theory of gravity. It just happens to be the case that two very different quantities in Newton's theory have the same numerical value: the gravitational mass, a measure of a body's susceptibility to gravity, and the inertial mass, a measure of a body's resistance to acceleration.
Einstein found it unsatisfactory that the equality of inertial and gravitational mass was just a coincidence in Newton's theory. He felt that there had to be some deeper reason for it. He proposed to explain it by assuming that acceleration and gravity are essentially just one and the same thing. If that is true, it is not surprising that inertial mass, having to do with resistance to acceleration, and gravitational mass, having to do with susceptibility to gravity, are equal to one another. This idea is the core of what Einstein later dubbed the equivalence principle. Initially, Einstein formulated the equivalence principle as follows: the situation of an observer uniformly accelerating in the absence of a gravitational field is fully equivalent to the situation of an observer at rest in a homogeneous gravitational field. In particular, both observers will find that all free particles have the same acceleration with respect to them.
Formulated in this way, one can readily understand why Einstein felt that the equivalence principle could be used to extend the principle of relativity of his special theory of relativity of 1905, which only holds for uniform motion (i.e., motion with constant velocity), to arbitrary motion (i.e., accelerated motion, motion with changing velocity). An accelerated observer, Einstein reasoned, can always claim to be at rest in some gravitational field equivalent to his or her acceleration. In this manner, Einstein thought he could eliminate once and for all Newton's concept of absolute acceleration and formulate a theory in which all motion is relative: a general theory of relativity.
Unfortunately, the relation between acceleration and gravity turned out not to be quite as simple as Einstein initially thought. It took him till early 1918--more than two years after he had published the theory in the form in which it is still used today--to give a more accurate formulation of the relation between gravity and acceleration. He changed the definition of the equivalence principle accordingly, although he continued to use the old definition in popular expositions of his theory. The connection between gravity and acceleration or inertia is not that the two are always interchangeable. Rather it is that the effects we ascribe to gravity and the effects we ascribe to inertia are both produced by one and the same structure, a structure we now call the inertio-gravitational field, and which is represented in Einstein's theory by curved space-time. Observers in accelerated motion with respect to one another will disagree over how to split the effects of this inertio-gravitational field into effects of inertia and effects of gravitation.
As Einstein explained in an unpublished article intended for Nature in 1920 (the manuscript of which is now in the Pierpoint Morgan Library in New York) the situation is similar to the one he had arrived at in his work on special relativity. There he had replaced the electric field and the magnetic field by one electro-magnetic field. Observers in relative motion with respect to one another will disagree over how to break down this one field into its electric and magnetic components. In working out the geometry of the new concepts of space and time of special relativity, the Göttingen mathematician Hermann Minkowski had done the same for space and time. He replaced space and time by one space-time. Observers in relative motion with respect to one another will disagree over how to break down a certain spatio-temporal distance into a spatial and a temporal distance component. The great novelty of general relativity-and perhaps its greatest lasting value-is that Einstein did the same thing for gravity and inertia. In 1918, he introduced a new version of the equivalence principle to bring out this central notion: there is only one field-a field we now call the inertio-gravitational field-that encompasses both the effects of gravity and of acceleration. The old equivalence principle is no more than a very special case of the new one. The main function of the old principle had been to guide Einstein in his search for his new theory of gravity.
Once the relation between gravity and acceleration has been identified correctly, we have to face up to the fact that the name "general theory of relativity" is a bit of a misnomer. Contrary to what Einstein had originally hoped, the theory does not fully eliminate absolute motion. Recall what absolute motion means in the context of Newton's theory. There it means that there is an absolute distinction between inertial motion--the motion of a body on which no external forces act--and non-inertial motion-the motion of a body subject to external forces. The former motion will be represented by a straight line, the latter by a crooked line. A very similar distinction holds in Einstein's theory. There it is the distinction between inertio-gravitational motion, the motion of a body subject only to the influence of the curved space-time representing the effects of gravity and inertia, and non-inertio-gravitational motion, the motion of a body subject to influences over and above those of the inertio-gravitational field. The former motion is motion on straightest lines in curved space-time (think, for instance, of great circles on the surface of a sphere), the latter motion is motion on crooked lines in the curved space-time. These motions and these lines are called geodesic and non-geodesic, respectively.
All of these subtleties and many more, Einstein only learned along the way in his long and only partially successful quest for a general theory of relativity and a new theory of gravity from 1907 until about 1920. But, as we know from a letter to his friend Conrad Habicht of December 24, 1907, one of the goals that Einstein had set himself early on was to use his new theory of gravity, whatever it might turn out to be, to explain the discrepancy between the observed motion of the perihelion of the planet Mercury and the motion predicted on the basis of Newtonian gravitational theory. This tiny discrepancy had been known for about half a century. Various explanations had been put forward during that time, none to Einstein's liking.
THE MERCURY ANOMALY
Kepler's first law says that a planet moves on an ellipse with the sun in one of its focal points. Newtonian theory confirms this to a very good approximation. However, Newtonian theory also predicts that, because of the influence of other planets, these ellipses are not fixed in space but undergo a slow precession (see Fig. 1). One way to observe this precession is to follow the motion of the perihelion, the point where the planet is closest to the sun. For the perihelion of Mercury, Newtonian theory predicts (in a coordinate system at rest with respect to the sun) a secular advance of about 570" (i.e., 570 seconds of arc per century). A second of arc is a very tiny unit. A circle has 360 degrees, a degree has 60 minutes, and a minute has 60 seconds.
In 1859, the French astronomer Urbain Jean Joseph Le Verrier, after working on the problem for many years, pointed out that there was a discrepancy of about 38" (i.e., about 1/100 of a degree per century!) between the value that Newtonian theory predicts for the secular motion of Mercury's perihelion and the value that was actually observed. Le Verrier suggested that perturbations coming from an additional planet, located between the sun and Mercury, were responsible for this discrepancy. In 1846, he had likewise predicted the existence of an additional planet to account for discrepancies between theory and observation in the case of Uranus. This planet, Neptune, was actually observed shortly afterwards, almost exactly where Le Verrier had predicted it to be. Vulcan, however, the planet to be made responsible for the discrepancy in the case of Mercury, was never found.
In 1895, the American astronomer Simon Newcomb published a new value for the anomalous secular advance of Mercury's perihelion, based on the latest observations. He arrived at about 41". The modern value is about 43". The best explanation for the anomaly that Newcomb could find was a suggestion made a year earlier by Asaph Hall that the gravitational force would not exactly fall off with the inverse square of the distance as in Newton's theory but slightly faster. Newton himself had already noted that any deviation from an exact inverse square law would produce a perihelion motion.
In the years following Newcomb's publication, several other explanations were put forward for the anomaly. The most popular one was offered by the German astronomer Hugo von Seeliger, who suggested the discrepancy was due, not to a planet between Mercury and the sun, but to bands of diffuse matter in that region.
In the decade following the publication of the special theory of relativity, various new theories of gravitation were suggested. For several of these theories--e.g., for theories by Henri Poincaré, Hermann Minkowski, Max Abraham, and Gunnar Nordström--it was checked explicitly whether they could account for the Mercury anomaly. None of them could. In Nordström's theory, the problem was even worse than in Newtonian theory, because the theory predicted a retrogression of about 7 per century instead of the desired advance of a good 40. All theories could, of course, easily be rendered compatible with observation as long as one was willing to accept Seeliger's hypothesis of intra-Mercurial matter.
It is against this background that the project documented in the Einstein-Besso manuscript should be seen. In June 1913, Michele Besso visited Einstein in Zurich. Together with his former classmate, the mathematician Marcel Grossmann, Einstein had just written a paper, which, as the title modestly announces, gives an outline of a generalized theory of relativity and a theory of gravitation. Einstein and Besso set themselves the task to find out whether this new theory could account for the Mercury anomaly.
THE ENTWURF THEORY
The Einstein-Grossmann or Entwurf theory is, in fact, already very close to the version of general relativity published in November 1915 and constitutes an enormous advance over Einstein's first attempt at a generalized theory of relativity and theory of gravitation published in 1912. The crucial breakthrough had been that Einstein had recognized that the gravitational field--or, as we would now say, the inertio-gravitational field--should not be described by a variable speed of light as he had attempted in 1912, but by the so-called metric tensor field. The metric tensor is a mathematical object of 16 components, 10 of which independent, that characterizes the geometry of space and time.* In this way, gravity is no longer a force in space and time, but part of the fabric of space and time itself: gravity is part of the inertio-gravitational field. Einstein had turned to Grossmann for help with the difficult and unfamiliar mathematics needed to formulate a theory along these lines.
Any theory of the gravitational field can naturally be divided into two parts, a part describing how the gravitational field affects matter and a part describing how matter in turn generates gravitational fields. As far as the first part is concerned, the Entwurf theory of June 1913 is identical to the general theory of relativity in its final form. The difference between the two theories concerns only the second part. The theories of 1913 and of 1915 have different field equations for the metric field, the field representing the gravitational field in these theories. Field equations are the equations that tell us what gravitational field is produced by a certain given matter distribution. During his collaboration with Grossmann, Einstein had actually considered field equations that are very close to the ones he would eventually settle on in November 1915. From a purely mathematical point of view, these equations were the natural candidates, but at the time Einstein convinced himself that from a physical point of view they would be unacceptable. Instead, he chose a set of equations, now known as the Einstein-Grossmann equations or the Entwurf field equations, that, although mathematically less elegant, he thought were more satisfactory from a physical point of view. This early struggle to find suitable field equations is documented in the Zurich Notebook of late 1912/early 1913.
APPLYING THE EINSTEIN-GROSSMANN THEORY TO THE PROBLEM OF MERCURY'S PERIHELION
The first step in calculating the perihelion advance predicted by the Einstein-Grossmann theory was to solve the Entwurf field equations to find the metric field produced by the sun. The second step would be to calculate the perihelion advance of a planet moving in this field.
In order to solve the field equations, Einstein came up with an ingenious iterative approximation procedure. First, he calculated the metric field of a point mass (representing the sun) using the field equations in a first-order approximation. He then substituted the result of this calculation back into the field equations, now in a more accurate second-order approximation, and solved these equations to obtain more accurate expressions for the metric field of the point mass. This two-step procedure nicely illustrates the fundamental physical complication that underlies the complexity of the equations Einstein had to work with: due to the equivalence of mass and energy, the gravitational field, by virtue of carrying a certain amount of energy, acts as its own source. So, in Einstein's second-order approximation, both the energy of the point mass and the energy of the metric field has to be taken into account.
Finding a sufficiently accurate expression for the metric field of the sun takes up the first few pages of the Einstein-Besso manuscript (pp. 1-7 of the manuscript as presented in Vol. 4 of the Einstein edition). These pages are all in Einstein's hand, with just a few corrections of minor slips in Besso's hand (p. 1 is shown in Fig. 2). Besso takes over on the next couple of pages (pp. 8-9), deriving an equation for the perihelion motion of a planet in the metric field of the sun (p. 8 is shown in Fig. 3). Notice that this page is not nearly as "clean" as Einstein's. Einstein is clearly much more comfortable with the calculations than Besso is. Besso's insecurity is reflected in the many deletions he makes and also in the fact that he uses a lot more explanatory prose than Einstein does. On the next two pages (pp. 10-11), Einstein takes over again and finds an expression for the perihelion advance of a planet in the field of the sun. Besso then rewrites this equation, making sure that it only contains quantities for which numerical values would be readily available, some astronomical data pertaining to the sun and Mercury along with some constants of nature (p. 14).
It is only later in the manuscript (on p. 26 and p. 28 shown in Figs. 4 and 5, respectively) that actual numbers are inserted in this expression, and that the result is converted from the units used in the calculation to those used in observations. The end result is given by Einstein on p. 28: "1821" = 30' unabängig geprüft" (independently checked). This result is disastrous. The hope was that the sun, which in Newton's theory produces no perihelion motion at all, would, in the Einstein-Grossmann theory, produce the observed motion that cannot be attributed to other planets. Instead, it looks as if the sun alone produces a perihelion motion of more than three times the size of the total perihelion motion that is observed!
Fortunately, there is a mistake in the numerical calculation. The value for the mass of the sun is off by a factor of 10. Since the perihelion motion is proportional to the square of this quantity, the final result is off by a factor 100. Even though neither Einstein nor Besso corrected the bizarre result on p. 28, it is clear that they discovered their mistake. On p. 35 (see Fig. 6), in the context of another numerical calculation, there is a correction in Besso's hand of the erroneous value for the mass of the sun that Einstein had used. And on p. 30 (see Fig. 7), Einstein himself replaced the value 3.4 10-8 which, when converted, gives the bizarre 1800" of p. 28, by 3.4 10-6, which gives the less bizarre but still not quite accurate 18".
Of course, even the corrected value was disappointing. One would still need Seeliger's hypothesis of intra-Mercurial matter to account for the remaining part of the discrepancy. Einstein and Besso therefore considered other effects in the Einstein-Grossmann theory that might contribute to the perihelion motion. In particular, they considered the effect of the rotation of the sun. In Newtonian theory, the sun's rotation would not produce any perihelion motion at all, but in the Einstein-Grossmann theory (as in general relativity in its final form) it does have a small effect. On p. 35, the final result of Einstein and Besso's calculations for this effect is given as 8.7 10-10, which amounts to about .1" per century, which is negligible. In fact, Einstein once again used the wrong value for the mass of the sun, so the result should really be .001". Einstein probably was pleased to find that the effect is small, because the effect of the rotation of the sun is not an advance but a retrogression of the perihelion.
THE FATE OF THE EINSTEIN-BESSO MANUSCRIPT
Before they could finish their joint project in June 1913, Besso had to leave Zurich and go back to Gorizia where he lived at the time. The additional pages found in 1998 make it clear that Besso visited Einstein again in late August 1913. At that time, they worked on the project some more and added a few more pages. The manuscript stayed behind with Einstein in Zurich. Early in 1914, Einstein sent it to Besso, urging his friend to finish their project.
Besso added more material some of which can be found on pp. 45-53 and on pp. 41-42 of the published portion of the manuscript. The remainder is part of the extra pages discovered in 1998 in the Besso Nachlass. He investigated three other possible contributions to the motion of Mercury's perihelion: the effect of the sun's rotation on the motion of the nodes of the planet (i.e., the points where the orbit intersects the ecliptic); a similar effect due to the motion of Jupiter; and, finally, the effect of solar pressure. There are a number of mathematical and arithmetical errors in these calculations, which in some cases led Besso to seriously overestimate the magnitude of these effects. But even with these errors, none of the effects led to contributions to the perihelion motion of the desired magnitude.
Besso also calculated the perihelion motion predicted by the Nordstrvm theory using some of the same techniques he and Einstein had used in the context of the Entwurf theory. Despite the fact that this calculation also contains some (minor) errors, it shows that Besso had gotten quite proficient at performing this type of calculation. Still, Besso did not pursue the topic any further. For the rest of his life, however, he held on to the notes of this joint work with the friend he so ardently admired. It is thanks to this admiration that the manuscript survives at all. Had it remained in Einstein's possession, it would almost certainly have been discarded. After all, it consists entirely of working calculations. It would not have been the kind of document that Einstein would have saved. But it is extremely valuable to historians of relativity, since it gives us a glimpse of Einstein at work at the height of his creative powers.
THE EINSTEIN-GROSSMANN THEORY AND THE PROBLEM OF ROTATION
Einstein did not give up the Einstein-Grossmann theory once he had established that it could not fully explain the Mercury anomaly. He continued to work on the theory and never even mentioned the disappointing result of his work with Besso in print. So Einstein did not do what the influential philosopher of science Sir Karl Popper claimed all good scientists do: once they have found an empirical refutation of their theory, they abandon that theory and go back to the drawing board. If scientist religiously stuck to that principle, very little progress would be made. A certain tenacity in working on a theory even in the face of empirical problems is needed to give that theory a fighting chance. The thing to watch out for is that such tenacity does not turn into the stubbornness of refusing to let go of a beloved theory after it has become clear that no amount of fixing it up is going to turn it into a viable theory.
In late 1914, the Dutch physicist Johannes Droste independently found and published the basic result of the Einstein-Besso manuscript, viz. that the Entwurf theory could account for only 18" of the Mercury anomaly. This was not seen as particularly damning to the Entwurf theory in the scientific community. Einstein, at this point, firmly believed in the correctness of his theory. In November 1914, he had published what he clearly thought was the definitive exposition of the definitive theory, the Entwurf theory.
What eventually prompted Einstein to give up the Einstein-Grossmann theory had nothing to do with the Mercury anomaly. It did have to do, however, with another calculation preserved in the Einstein-Besso manuscript. One of the virtues of the Entwurf theory emphasized in Einstein's lengthy exposition of the theory of November 1914 is that it does away with the notion of absolute rotation. Newton had illustrated the absolute character of rotation in the famous thought experiment of the rotating bucket (see Fig. 8).
A bucket filled with water, initially at rest (stage I), is set spinning. At first the water remains at rest and the water surface stays flat (stage II). After a while the water starts "catching up" with the rotation of the bucket and the water surface becomes concave. At some point the water will be spinning just as fast as the bucket (stage III). Now grab the bucket. The bucket will instantaneously be brought to rest while the water continues to spin. The water surface maintains its concave shape. Now notice that in stages I and III there is no relative rotation of the water with respect to the bucket, whereas in stages II and IV there is. If the shape of the water were determined by the relative rotation of the water with respect to the bucket, the shapes in stages I and III and the shapes in stages II and IV should be the same. However, they clearly are not. Even though the relative rotation of the water and the bucket is the same in stages I and III (and in stages II and IV), the surface is flat in one case and concave in the other. This proves, as Newton argued, that the shape of the surface cannot be due to the relative rotation of the water and the bucket. The shape of the surface is determined not by the relative but by the absolute rotation of the water, the rotation with respect to Newton's absolute space.
In the late 19th century, Ernst Mach suggested that the effect might still be explained in terms of relative rotation, not of the water with respect to the bucket, but of the water and the bucket with respect to the rest of the universe. In order for this explanation to work, it has to be the case that the theory predicts the same effect for the case where the water-filled bucket is rotating and the distant stars are at rest as it does for the case where the bucket is at rest and the distant stars are rotating (see Fig. 13).
Unfortunately, Newton's theory, while predicting the observed effect in the former case, predicts no effect at all in the latter. The Entwurf theory, however, Einstein suggested, does predict the same effect in the two cases and hence allows one to relativize Newton's absolute rotation along Machian lines.
There are two calculations in the Einstein-Besso manuscript pertaining to these claims about rotation, one on pp. 36-37, the other one on pp. 41-42. On pp. 36-37, Einstein calculated, in a first-order approximation, the metric field that a rotating shell would produce near its center. The shell obviously represented the distant stars. As Einstein reported to Mach in June 1913, the Entwurf theory, unlike Newtonian theory, does predict a small effect of the type needed for a Machian account of Newton's bucket experiment.
On pp. 41-42, Einstein arrived at an even more encouraging result. According to the early version of the equivalence principle, rotation should be equivalent to some gravitational field. This is illustrated in Fig. 10. Standing on a rotating merry-go-round your velocity constantly changes direction, which means that you are experiencing a centripetal acceleration. This feels as if there is a force-known as the centrifugal force-that is trying to throw you off the merry-go-round. According to the equivalence principle, that situation should be fully equivalent to one in which you are standing on a merry-go-round at rest in a peculiar centrifugal gravitational field.
One can easily calculate the metric field describing space and time from the point of view of an observer in uniform rotation. The early version of the equivalence principle requires that this metric field can also be interpreted as a gravitational field. This means that it should be a solution of the field equations of the theory. To check this, Einstein used the same approximation procedure he had used to find the field of the sun for the perihelion calculations. In first-order approximation, it is easily verified that the metric field for the rotating observer is indeed a solution of the field equations of the Einstein-Grossmann theory. Moreover, this first-order metric field has the same form as the first-order metric field for the case of the rotating shell found on pp. 36-37, which fits nicely with the idea that this metric field can be interpreted as the field produced by the distant stars rotating with respect to the observer. Einstein then substituted this first-order field into the field equations in a second-order approximation and checked whether the metric field of the rotating observer is also a solution to the equations at this further level of approximation. He concluded that it is. Next to the final result of his calculation on p. 41, he wrote: "stimmt" ("is correct"). The relevant passage is reproduced in Fig. 11.
Unfortunately, Einstein made some trivial mistake in this calculation. The metric field describing space and time for a rotating observer is not a solution of the field equations of the Einstein-Grossmann theory. Einstein had been so convinced that it was that he had not been careful enough in checking this important aspect of the theory.
HOW ROTATION BROUGHT DOWN THE EINSTEIN-GROSSMANN THEORY AND HOW MERCURY'S PERIHELION CONFIRMED ITS SUCCESSOR
In September 1915, Einstein, probably at the instigation of Besso, finally decided to redo the calculation of pp. 41-42 of the Einstein-Besso manuscript and discovered the error he had made over two years earlier. This must have come as a severe blow. This was a problem far worse than the theory's prediction of the perihelion motion of Mercury being a few seconds of arc off. This problem went straight to the core of Einstein's theory, the idea that gravity and acceleration are essentially one and the same thing. Tenacity would have turned into stubbornness if Einstein had continued to hold on to the theory in the face of this debacle. As we saw earlier, when Einstein was ready to abandon his own 1905 theory in 1907 just when others were warming up to it, Einstein was anything but stubborn. Shortly after discovering the problem with rotation for the field equations of the Entwurf theory, Einstein gave up these field equations.
He then returned to the other mathematically more elegant candidates he had considered but rejected nearly three years earlier in his work with Grossmann preserved in the Zurich notebook. Einstein had learned a good deal about theories based on a metric tensor field in the meantime and as a result he was able to quickly overcome his objections to those equations of 1912/1913. On November 4, 1915, he presented a paper to the Berlin Academy retracting the Einstein-Grossmann equations and replacing them with new ones. On November 11, a short addendum to this paper followed, once again changing his field equations. A week later, on November 18, Einstein presented the paper containing his celebrated explanation of the perihelion motion of Mercury on the basis of this new theory (see lot 85). Another week later he changed the field equations once more. These are the equations still used today. This last change did not affect the result for the perihelion of Mercury.
Besso is not acknowledged in Einstein's paper on the perihelion problem. Apparently, Besso's help with this technical problem had not been as valuable to Einstein as his role as sounding board that had earned Besso the famous acknowledgment in the special relativity paper of 1905. Still, an acknowledgment would have been appropriate. After all, what Einstein had done that week in November, was simply to redo the calculation he had done with Besso in June 1913, using his new field equations instead of the Entwurf equations. It is not hard to imagine Einstein's excitement when he inserted the numbers for Mercury into the new expression he found and the result was 43", in excellent agreement with observation. Fortunately, this result is not affected by a final modification of the field equation, which Einstein introduced in one last communication to the Prussian Academy on November 25.
EPILOGUE
On November 19, the famous Göttingen mathematician David Hilbert sent Einstein a postcard congratulating him on his success in explaining the Mercury anomaly. He expressed his admiration for the speed with which Einstein had done the necessary calculations. Einstein did not let on that this was basically because he had already done the same calculation two years earlier with a less happy result. Little did he know at that point that his friend and admirer Michele Besso would preserve these earlier calculations for posterity.
REFERENCES:
Albert Einstein, "Grundgedanken und Methoden der Relativitätstheorie in ihrer Entwicklung dargestellt" ["Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development"]. Manuscript for an article commissioned by but never published in Nature, 1920. Published as Doc. 31 in Vol. 7 of the The Collected Papers of Albert Einstein.
John Earman and Michel Janssen, "Einstein's Explanation of the Motion of Mercury's Perihelion." In: John Earman, Michel Janssen, and John D. Norton (eds.), Einstein Studies. Vol. 5. The Attraction of Gravitation. Boston: Birkhduser, 1993.
Michel Janssen, "Rotation as the Nemesis of Einstein's Entwurf Theory." In: Hubert Goenner, Jürgen Renn, Jim Ritter, Tilman Sauer (eds.) Einstein Studies. Vol. 7. The Expanding Worlds of General Relativity. Boston: Birkhäuser, 1999, pp. 127-157.
John D. Norton, "How Einstein found his field equations, 1912-1915." Historical Studies in the Physical Sciences 14 (1984): 253-316. Reprinted in: Don Howard and John Stachel (eds.), Einstein Studies. Vol. 1. Einstein and the History of General Relativity. Boston: Birkhäuser, 1989, pp. 101-159.
Abraham Pais, 'Subtle is the Lord': The Science and the Life of Albert Einstein. Oxford: Clarendon Press; New York: Oxford University Press, 1982.
Jürgen Renn, Tilman Sauer, Michel Janssen, John Norton, and John Stachel, The Genesis of General Relativity: Sources and Interpretations. Vol. 1. General Relativity in the Making: Einstein's Zurich Notebook. Dordrecht: Kluwer, forthcoming [2002]
N. T. Roseveare, Mercury's Perihelion from Leverrier to Einstein. Oxford: Clarendon Press, 1982.
Pierre Speziali (ed.), Albert Einstein-Michele Besso. Correspondance. Paris: Hermann, 1972.
John Stachel, "Einstein's Search for General Covariance, 1912-1915." In: Don Howard and John Stachel (eds.), Einstein Studies. Vol. 1. Einstein and the History of General Relativity. Boston: Birkhä user, 1989, pp. 63-100.
John Stachel et al. (eds.), The Collected Papers of Albert Einstein. Vols. 1-8. Princeton: Princeton University Press, 1987-2002.
* The following analogy may be helpful to understand the notion of a metric. Consider a map of the earth (which is essentially a way of coordinatizing the globe). We cannot simply take distances on the map (the coordinate distances) to represent distances on the globe (the actual or proper distances). For instance, a horizontal line segment of two inches on the map near the equator will correspond to a larger distance on the globe than a horizontal line segment of two inches on the map near the poles. For every point on the map, we need to specify a set of numbers with which we have to multiply distances on the map in the vicinity of that point (coordinate distances) to convert them to actual distances (proper distances). (It will be clear that we need more than one number because the conversion for north-south distances will be different from the conversion for east-west distances.) The numbers in such a set are called the components of the metric at that point. The metric field is the collection of all such sets of numbers for all points on the map. The same thing we do here with 2-dim. space (representing the 2-dim. curved surface of the earth on a 2-dim. Euclidean plane together with a specification of the metric field to do all conversions from coordinate distances to proper distances) we can do with 4-dim. space-time as well. There will now also be a number (the temporal component of the metric) by which we have to multiply coordinate time differences to convert them to proper time differences.