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 (10¾ x 8 3/8 in.), leaf [10-11] an oblong folded sheet bearing unpublished partial diagrams and calculations on its verso (counted in above total), irregular partial pagination, now arranged as published in Collected Papers, one sheet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
A scientific manuscript on relativity, gravitation and planetary motion which is of fundamental importance in the development of the General Theory of Relativity. As one of only two working manuscripts from the period of the genesis of the general theory, it provides remarkable insight into Einstein's work and the complex mathematics in which the theory of relativity was conceived, expressed, and, in the November 1915 paper which ultimately resulted from this work, finally proven. As one biographer has written, the successful 1915 proof was "by far the strongest emotional experience in Einstein's scientific life, perhaps in all his life. Nature had spoken to him. He had to be right. Later he told Fokker that 'For a few days I was beside myself with excitement.' What he told de Haas is even more profoundly significant: when he saw that his calculations agreed with the unexplained astronomical observations, he had the feeling something actually snapped in him..." (Abraham Pais, 'Subtle is the Lord...' The Science and the Life of Albert Einstein, Oxford, 1982, 253.)
The Einstein-Besso Manuscript, the Perihelion Motion of Mercury, and the Genesis of the General Theory of Relativity
by Dr. Michel Janssen,
Associate Editor of the Einstein Papers Project
There are only two manuscripts still extant with research notes documenting Einstein's work toward the general theory of relativity, completed in November 1915: the so-called Zurich notebook of late 1912/early 1913 (now in the Einstein Archives at Hebrew University) and the Einstein-Besso manuscript, mostly dating from June 1913, which came to light in 1988.
The aim of most of the calculations in the Einstein-Besso manuscript is to test whether the early version of the general theory of relativity (published by Einstein in June 1913) can account for the anomalous motion of the perihelion of Mercury. The results of these calculations were disappointing. The theory, as it stood, could account for only part of the discrepancy between observation and Newtonian theory. However, Einstein's and Besso's efforts would not be in vain. The techniques developed in this manuscript for performing these calculations were taken over virtually unchanged in November 1915 to compute the perihelion advance predicted by the general theory of relativity in its final form. Einstein found that the final theory did 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.
The explanation of the anomalous advance of Mercury's perihelion would become one of the three classic tests of general relativity. Einstein produced 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 essentially performed the same calculation with Besso two years earlier. The study of these earlier calculations has contributed to a fuller understanding of the 1915 perihelion paper.
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 historic papers of November 1915, Einstein listed three reasons for abandoning the earlier version of the theory. One was the fact that it yielded the wrong result for the perihelion motion of Mercury, another 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 test whether the theory is compatible with this principle. Einstein was able to convince himself that the theory passed this test, but he had made some trivial errors in his calculation. More than two years later, in September 1915, he redid the calculation, this time without errors, and discovered to his dismay that the theory failed. In all likelihood, this discovery triggered the unraveling of the early 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.
Eagle and Sparrow: The Einstein-Besso Collaboration
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, a pattern which would repeat itself in many of Einstein's key papers. The only individual credited with any contribution to the 1905 paper was Michele Angelo Besso, whom Einstein thanked for "many useful suggestions." Besso (1873-1955), 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 Einstein's wife and children moved back to Zurich in 1914, Besso often acted as intermediary between Albert and Mileva and even cared for Einstein's two sons during Mileva Einstein-Maric's illness. In 1913, when the two collaborated on the calculations in the present manuscript, Besso was living in Gorizia, near Trieste. The Einstein-Besso manuscript demonstrates that, in the testing of the field equations for the general theory of relativity, Besso functioned as considerably more than just a sounding board, and took responsibility for some significant parts of the calculations. In later years, Besso himself described their scientific collaboration with a charming simile: Einstein was an eagle, and he, Besso, a sparrow. Under the eagle's wing, the sparrow was able to fly higher than on its own.
Generalizing Special 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 the 1905 special theory of relativity. In the last section of this review, devoted to gravitation, Einstein argued that a satisfactory theory of gravitation cannot be achieved within the framework of special relativity and that a generalization of the theory was needed. This was an extraordinary step to take at the time. Others 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 the guiding principle for the generalization of special relativity, Einstein put forward what a few years later would be called the equivalence principle. According to this principle, the situation of an observer uniformly accelerating in the absence of a gravitational field is physically 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 him or her. This principle allowed Einstein to broaden the principle of relativity of the special theory --that all frames of reference in uniform motion with respect to one another are physically equivalent--to include accelerated frames of reference. Given the proposed equivalence of acceleration and gravitational field, it is clear that this generalized theory of relativity would also constitute a new theory of gravitation. In a letter to his friend Conrad Habicht on 24 December 1907, Einstein wrote that he hoped this new theory of gravitation would explain a discrepancy between the observed motion of the perihelion of Mercury and the motion predicted by Newton's gravitational theory. This tiny discrepancy had been known for about half a century. Various explanations of it had been put forward during that time, none to Einstein's liking.
The Problem of 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 close 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. This precession may be observed by following the motion of the perihelion, the point where the planet is closest to the sun. For the perihelion of Mercury, Newtonian theory predicted (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). 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^{"} between the Newtonian prediction and the value actually observed. Le Verrier suggested that perturbations 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. The unknown planet, Neptune, had actually observed shortly afterwards, almost exactly where Le Verrier had predicted it would be. However, Vulcan, the planet thought to be 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^{"}.
In the years following Newcomb's publication, several explanations were put forward, the most popular 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 after the publication of the special theory of relativity, various new theories of gravitation were proposed. Such theories, typically, could only explain a portion of the missing 43 seconds, and had to rely on Seelinger's hypothesis of inter-Mercurial matter to explain the remainder of the discrepancy.
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 ("Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation", Zeitschrift für Mathematik und Physik, vol.62, 1913, pp.225-261). As the title modestly announces, the paper 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 that new theory could account for the Mercury anomaly.
The Einstein-Grossmann Theory
The Einstein-Grossmann theory--also known as the "Entwurf" ("outline") theory, from the title of their joint paper--is, in fact, already very close to the final version of the general theory of relativity as 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 was Einstein's recognition that the gravitational field should not be described by a variable speed of light (as attempted in his 1912 theory), but by the so-called metric tensor, a mathematical object of 16 components, 10 of which independent, that characterizes the geometry of space and time. In this way, gravitation is no longer a force in space and time, but part of the fabric of space and time itself. Einstein had turned to Grossmann for help with the difficult and unfamiliar mathematics.
Any theory of the gravitational field can be divided into two parts, one describing how the gravitational field affects matter, the other describing how matter in turn generates gravitational fields. As far as the first is concerned, the Einstein-Grossmann 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 use different field equations for the metric field, which represents the gravitational field in these theories. (Field equations determine the gravitational field produced by a given distribution of matter.) During his collaboration with Grossmann, Einstein had considered field equations very similar to those he eventually arrived at in November 1915. From a purely mathematical point of view, these equations were the natural candidates, but at the time, Einstein convinced himself that they were unacceptable from a physical point of view. Instead, he chose a set of equations--now known as the Einstein-Grossmann equations or the "Entwurf" field equations. This early struggle to find suitable field equations is documented in the Zurich notebook of late 1912/early 1913, the only other extant research manuscript of this period.
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 and find the metric field produced by the sun. The second step was 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. Then, he substituted the result of this calculation back into the field equations, now in a more accurate second-order approximation, and solved those equations, in order to obtain more accurate expressions for the metric field of the point mass. This two-step procedure illustrates the fundamental physical complication underlying 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 had 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 published in Collected Papers). These pages are entirely in Einstein's hand, with just a few minor corrections in Besso's hand. Besso takes over on the next two pages (pp. 8-9), and derives an equation for the perihelion motion of a planet in the metric field of the sun. On the next two (pp. 10-11), Einstein solves this equation 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 are known, i.e., astronomical data pertaining to the sun and the planet Mercury, and certain constants of nature (p. 14).
It is only later in the manuscript (on pp. 26 and 28) that actual numerical values are inserted into the equations, and the result converted from the units used in the calculation to those used in observations. The end result is notated by Einstein on p. 28: "1821^{"} = 30^{'} unabhängig geprüft" (independently checked). This result is disastrous. Einstein hoped that the sun, which in Newton's theory produces no perihelion motion at all, would, in the Einstein-Grossmann theory, produce the 43^{"} of the observed 570^{"} that cannot be attributed to other planets. Instead, the result suggested that the sun alone generated a perihelion motion more than three times the size of the total perihelion motion that is observed! Evidently, there was a key mistake in the numerical calculation: the value for the mass of the sun was 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 of 100. Even though neither Einstein nor Besso corrected the bizarre result on p. 28, it is evident that they discovered the mistake. On p. 35, in the context of another numerical calculation, Besso corrects the value for the mass of the sun used by Einstein. And Einstein himself, on p. 30, replaces the value 3.4 10^{-6} (which, when converted, gives the bizarre 1800" of p. 28), with 3.4 10^{-8}, which yields the correct 18^{"}.
However, even this corrected value was disappointing, since it only accounted for part of the anomaly. Einstein and Besso therefore attempted to calculate other effects in the Einstein-Grossmann theory which might affect perihelion motion. In particular, they calculated 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 0.1" per century, which is negligible.
The Fate of the Einstein-Besso Manuscript
Before they were able to finish their joint project, Besso had to leave Zurich and go back to Gorizia, where he lived at the time. It is thanks to this circumstance that the Einstein-Besso manuscript survives at all. It appears that Besso left the manuscript with Einstein in Zurich. Early in 1914, Einstein sent it to Besso, urging his friend to finish the project. The material added by Besso comprises pp. 45-53 and parts of pp. 41-42. Besso 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 elliptic); a similar effect due to the motion of Jupiter; and, finally, the effect of solar pressure. But none of his findings led to contributions to the perihelion motion of the desired magnitude.
Finally, Besso calculated the perihelion motion predicted by another theory (Nordström's), using some of the same techniques he and Einstein had used in the context of the Einstein-Grossmann theory (p.53). He then abandoned the project. For the rest of his life, however, he retained the working manuscript from this pioneering collaboration with the friend he so ardently admired. Had the manuscript remained in Einstein's possession, it would almost certainly have been discarded.
The Einstein-Grossmann Theory and the Problem of Rotation
Contrary to what one might expect on the basis of a naive falsificationist conception of the workings of science, 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. In late 1914, the Dutch physicist Johannes Droste independently found and published the same basic result as in the Einstein-Besso manuscript, that the Einstein-Grossmann theory could account for only 18^{"} of the Mercury anomaly. This was not seen as particularly damning to the Einstein-Grossmann theory in the scientific community. Einstein, at this point, firmly believed in the correctness of his theory. In October 1914, he had presented what looks like a definitive presentation of the theory. What eventually prompted Einstein to give up the Einstein-Grossmann theory had nothing to do with the Mercury anomaly. It did, however, relate to another calculation preserved in the Einstein-Besso manuscript.
One of the virtues claimed for the Einstein-Grossmann theory in Einstein's lengthy exposition of the theory of October 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. The fact that the water surface becomes concave when a bucket filled with water is set spinning, Newton argued, can only be attributed to absolute rotation, i.e., rotation with respect to absolute space, not to relative rotation such as the rotation of the water with respect to the bucket. In the late 19th century, Ernst Mach had 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, the theory must predict the same effect when the water-filled bucket is rotating and the distant stars are at rest as it does when the bucket is at rest and the distant stars are rotating. Unfortunately, Newton's theory, while predicting the observed effect in the former case, predicts no effect at all in the latter. The Einstein-Grossmann 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 on pp. 41-42. Besso does not seem to have been involved in these calculations. Einstein probably did them shortly after Besso had left. On pp. 36-37, Einstein calculated, in a first-order approximation, the metric field that a rotating shell would produce near its center. This shell obviously represented the distant stars. As Einstein reported to Mach in June 1913, the Einstein-Grossmann 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. In the spirit of the equivalence principle, rotation should be equivalent to some 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 equivalence principle requires that this metric field can also be interpreted as a gravitational field. This means that this metric field 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 in 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").
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 would be 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 for some reason redid 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. It demonstrated that the field equations of the Einstein-Grossmann theory were incompatible with the relativity of rotation and therefore unacceptable. Shortly afterwards, Einstein gave up those field equations and 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 quickly overcame his earlier reservations and on 4 November, he presented a paper to the Berlin Academy officially retracting the Einstein-Grossmann equations and replacing them with new ones. On 11 November, a short addendum to this paper followed, and, a week later, on 18 November, Einstein presented the paper containing his celebrated explanation of the perihelion motion of Mercury on the basis of this new theory ("Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie," Preussische Akademie der Wissenschaften, Sitzungsberichte, 1917, Part 1, pp.142-152). Besso is not acknowledged in this historic paper. Apparently, Besso's help with this technical problem had not been, in Einstein's view, as valuable to him as his role as sounding board, which had earned him 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 Einstein-Grossmann 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 equations, which Einstein introduced in one last communication to the Prussian Academy on 25 November.
Epilogue
On 19 November, the famous Göttingen mathematician David Hilbert (1862-1943, who actually published the field equations of general relativity in their final form a few days before Einstein) wrote to Einstein 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 explain that he had essentially already done the same calculation two years earlier, with a less happy result. The manuscript of Einstein's updated calculations does not survive. Fortunately for the history of science, his oldest friend and admirer Michele Besso preserved the earlier calculations for posterity.