## Die formale Grundlage der allgemeinen Relativitätstheorie. Author’s presentation offprint from *Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften*, XLI, 19 November 1914.

Berlin: Königlichen Akademie der Wissenschaften, 1914.

First edition of this extremely rare offprint, a remarkable association copy inscribed by the theoretical physicist Gunnar Nordström, often designated by modern writers as ‘The Einstein of Finland’. Einstein had an extended correspondence with Nordström on the subject of Nordström’s own competing theory of gravitation, which at the time was considered a serious competitor to Einstein’s, and which he completed in the same year as the present paper. A few years later Nordström also assisted Einstein in his work on gravitational waves. The present paper was the crucial step between Einstein’s *Entwurf* theory of 1913 and the final form of general relativity which Einstein completed in November 1915: it develops the mathematical techniques necessary for the final formulation, namely the ‘absolute differential calculus’ of Tullio Levi-Civita, as well as the expression of the field equations in terms of a variational principle, which later proved to be of great importance. This author’s presentation offprint, with “Überreicht vom Verfasser” printed on upper wrapper, must not to be confused with the much more common trade offprint which lacks this printed statement (see below). We have located only one copy of this author’s presentation offprint at auction, in the collection belonging to Einstein’s son Hans Albert sold at Christie’s in 2006 (there was no copy in Einstein’s own collection of his offprints sold by Christie’s in 2008).

*Provenance*: Gunnar Nordström (1881-1923) (‘G. Nordström’ written in pencil on upper wrapper in Nordström’s hand). Mathematical annotations in pencil to margin of p. 1077 (in Nordström’s hand?). Later inscription in Russian on upper wrapper.

“In summer 1914, Einstein felt that the new theory should be presented in a comprehensive review. He also felt that a mathematical derivation of the field equations that would determine them uniquely was still missing. Both tasks are addressed in a long paper, presented in October 1914 to the Prussian Academy for publication in its *Sitzungsberichte*. It is entitled ‘The formal foundation of the general theory of relativity’; here, for the first time, Einstein gave the new theory of relativity the epithet ‘general’ in lieu of the more cautious 'generalized' that he had used for the *Entwurf*” (*Landmark Writings in Western Mathematics 1640-1940*). “In the year that he was called to Berlin, on October 29, 1914, Einstein was able to present his work “Die formale Grundlage der allgemeinen Relativitätstheorie” … The “formal foundation” of the general theory of relativity was the tensor calculus. Without the tensor calculus, the general theory of relativity could not have been formulated … By October 1914, Einstein was finally able to present his results in mathematical form, and indeed in a manner that became the basis of his general theory of relativity of 1916. He introduced general covariants, contravariants, and also—what was new—mixed tensors, in order to represent the individual arithmetic operations, above all, the various types of multiplication. Thus the mathematical calculus necessary for the general theory of relativity was at the ready in 1914” (Reich). “The principal novelty [in the present paper] lies in the mathematical formulation of the theory. Drawing on earlier work with [Marcel] Grossman, Einstein formulated his gravitational field equations using a variation principle” (Calaprice, 47).

The first important stage in the development of Einstein’s theory of gravitation was accomplished, with his friend and classmate the mathematician Marcel Grossmann, in their 1913 work *Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation*. “In this book, Einstein and Grossman investigated curved space and curved time as they relate to a theory of gravity. They presented virtually all the elements of the general theory of relativity with the exception of one striking omission: gravitational field equations that were not generally covariant. Einstein soon reconciled himself to this lack of general covariance through the ‘hole argument,’ which sought to establish that generally covariant gravitational field equations would be physically uninteresting” (Calaprice 40). Einstein’s ‘hole argument’, he believed, implied that general covariance was incompatible with the requirement that the distribution of mass-energy should determine the gravitational field *uniquely*. He believed, therefore, that the field equations should only be valid in certain coordinate systems, which he called ‘adapted’, and that only coordinate transformations from one adapted system to another adapted system should be allowed – he called these ‘justified coordinate transformations’.

“Einstein’s move to Berlin in April 1914 marked the end of his collaboration with Grossmann. Fortunately, by this time Einstein no longer seems to have needed Grossmann’s mathematical guidance. By October 1914, he had completed a lengthy summary article [offered here] on his new theory, whose form and detailed nature suggest that Einstein felt his theory had reached its final form. The article contained a review of the methods of tensor calculus used in the theory and, flexing his newfound mathematical muscles, Einstein could even promise to give new and simpler derivations of the basic laws of the ‘absolute differential calculus’. Of great importance was the fact that Einstein had taken the new mathematical techniques of his last paper with Grossmann, generalized them and found in them a quite new derivation of the field equations” (Norton, p. 293).

This new derivation made use, for the first time in Einstein’s work on the theory of gravity, of an action principle (or variational principle). Einstein worked initially with an action that was an arbitrary function of the metric tensor and its first derivatives, and then showed that with a particular choice of the action he could recover the *Entwurf *field equations. He further believed that he had found a simple general covariance condition which forced the action to take the *Entwurf *form. “Einstein had good reason to be pleased with this result. For it seemed to show that his theory was not just a theory of gravitation, but a generalized theory of relativity, in so far as it was concerned with establishing the widest covariance possible in its equations. His original derivation of the field equations [in *Entwurf*] had been based squarely on considerations in gravitation theory ... The new derivation, however, focused on covariance considerations. He had found a simple way of formulating field equations that would have exactly the maximum covariance allowed by the ‘hole argument’, and they led him almost directly to his original *Entwurf *field equations. As a result he could promise to “recover the equations of the gravitational field in a purely-covariant-theoretical way” and claim to “have arrived at quite definite field equations in a purely formal way, i.e., without directly drawing on our physical knowledge of gravitation” …

“Einstein appears to have remained satisfied with the theory he developed in 1914 through the first half of 1915. In March, April and early May he defended the theory wholeheartedly in an intense correspondence with Levi-Cività, who challenged Einstein’s derivation of the covariance properties of his gravitation tensor. But it seems that by mid-July he was less certain … By mid-October Einstein’s points of dissatisfaction with his theory had grown in number and intensity. They soon culminated in some of the most agitated and strenuous works of his life, in which generally covariant field equations were discovered … Einstein’s work [in the present paper] had brought him both temporally and conceptually closer than ever before to a generally covariant theory … It is hard to imagine that Einstein was unprepared for the ease with which his formalism of 1914 could be applied to his final generally covariant theory” (Norton, pp. 296-303).

After publishing the generally covariant theory in November 1915, Einstein gave a further treatment of the variational formulation (Hamiltonsches Prinzip und allgemeine Relativitätstheorie, *Sitzungsberichte* (1916), pp. 1111-1116). By this time, the great German mathematician David Hilbert had published his own account of general relativity in terms of a variational principle [Die Grundlagen der Physik (Erste Mitteilung), *Nachrichten der**Königliche Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse, *November 1915, 395-407]. This led to some controversy over who had been the first to publish the final version of general relativity (although Hilbert himself never claimed priority). “Hilbert, through his important paper of November 1915, is generally thought of as introducing the comprehensive use of these action principles to the theory. My analysis shows that although Einstein might have drawn some of his work of 1916 in this area from Hilbert’s, his basic mathematical apparatus and even the notation itself had its ancestry in his own work earlier in 1914 and 1915” (Norton, p. 303).

Gunnar Nordström first studied at the University of Helsinki (1903-7), and then spent a year at Göttingen, where he became a convert to the theory of relativity in its Minkowskian formulation. His remaining published work was focused almost exclusively on relativity, the most important being his theory of gravitation, developed between 1912 and 1914. “After an initial enchantment and subsequent disillusionment with [Max] Abraham’s theory of gravitation, Einstein found himself greatly impressed by a Lorentz covariant gravitation theory due to the Finnish physicist Gunnar Nordström. In fact, by late 1913, Einstein had nominated [in a lecture at the 85th Congress of the German Natural Scientists and Physicians] Nordström’s theory as the only viable competitor to his own emerging theory of relativity. This selection came, however, only after a series of exchanges between Einstein and Nordström that led Nordström to significant modifications of his theory … under continued pressure from Einstein, Nordström made his theory compatible with the equality of inertial and gravitational mass by assuming that rods altered their length and clocks their rate upon falling into a gravitational field so that the background Minkowskian space-time had become inaccessible to direct measurement. As Einstein and Fokker showed in 1914, the space-time actually revealed by direct clock and rod measurement had become curved, much like the space-times of Einstein’s own theory. Moreover, Nordström’s gravitational field equation was equivalent to a geometrical equation in which the Riemann-Christoffel curvature tensor played the central role. In it, the curvature scalar is set proportional to the trace of the stress-energy tensor. What is remarkable about this field equation is that it comes almost two years *before* Einstein recognized the importance of the curvature tensor in constructing field equations for his own general theory of relativity! In this regard, the conservative approach actually anticipated Einstein’s more daring approach” (Norton in Earman et al, pp. 4-5). As late as 1917, more than a year after Einstein published his final version of general relativity, Max von Laue published an exposition of Nordström’s theory: it was still considered by some a potential competitor to Einstein’s. This finally changed with the confirmation of the bending of light rays during the solar eclipse of 1919 as predicted by general relativity: Nordström’s theory predicted no such bending.

Nordström is remembered today for two other contributions. In 1914 he introduced an additional space dimension to his theory, which provided coupling to electromagnetism. This was the first of the extra-dimensional theories, which later came to be known as Kaluza-Klein theories, although Kaluza and Klein did not publish their work until the 1920s. Today extra-dimensional theories are widely researched. Then in 1918 Nordström obtained the solution of Einstein’s field equations for a spherically symmetric charged body, used today in the description of charged black holes (this is now known as the ‘Reissner-Nordström solution,’ as Hans Reissner (1874-1902) had in 1916 given the solution for a charged point mass).

Nordström also assisted Einstein in his work on gravitational waves. In Einstein’s first published paper on gravitational waves (1916), “he made use of a somewhat controversial mathematical construct known as a pseudotensor to describe the energy in the gravitational field. He made a mistake in doing so, however, which was only discovered when Nordström attempted to use the pseudotensor from Einstein’s linearized approximation paper to calculate the energy in the gravitational field of an isolated mass. After some to and fro between himself and Nordström, Einstein realized the nature of his mistake, which had given rise to an incorrect formula for the energy transmitted in a gravitational wave. He presented a new paper in 1918” (*Cambridge Companion to Einstein*, pp. 272-3).

This author’s presentation offprint is of extreme rarity, and must be distinguished from other so-called ‘offprints’ of papers from the Berlin *Sitzungsberichte*, many of which are commonly available on the market. The celebrated bookseller Ernst Weil (1919-1981), in the introduction to his Einstein bibliography, wrote: “I have often been asked about the number of those offprints. It seems to be certain that there were few before 1914. They were given only to the author, and mostly ‘Überreicht vom Verfasser’ (Presented by the Author) is printed on the wrapper. Later on, I have no doubt, many more offprints were made, and also sold as such, especially by the Berlin Academy.” If the term ‘offprint’ means, as we believe it should, a separate printing of a journal article given (only) to the author for distribution to colleagues, then ‘offprints’ were not commercially available. Although there is certainly some truth in Weil’s remark, in our view it requires clarification and explanation.

Until about 1916, most of Einstein’s papers were published in *Annalen der Physik*; from 1916 until he left Germany for the United States in 1933, most were published in the Berlin *Sitzungsberichte*. The *Sitzungsberichte* differed from other journals in which Einstein published in that it made separate printings of its papers commercially available. These separate printings have ‘Sonderabdruck’ printed on the front wrapper, the usual German term for offprint, but they are not offprints according to our definition. They were available to anyone; indeed a price list of these ‘trade offprints’ is printed on the rear wrapper. True author’s presentation offprints can be distinguished from these trade offprints by the presence of ‘Überreicht vom Verfasser’ on the front wrapper.

In the period 1916 to 1919 or 1920, the *Sitzungsberichte *trade offprints are themselves rare: for example, ABPC/RBH list only three ‘offprints’ of Einstein’s famous 1917 *Sitzungsberichte *paper ‘Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie’ (the auction records do not distinguish between trade and author’s presentation offprints). After 1919 or 1920, however, the trade offprints become much more common, although the author’s presentation offprints are still very rare. The reason for this change is that it was only in 1919 that Einstein became famous among the general public.

It might seem obvious that Einstein’s fame dates from 1905, his ‘annus mirabilis’, in which he published his epoch-making papers on special relativity and the light quantum. However, these works did not make him immediately well known even in the physics community – many physicists did not understand or accept his work, and it was two or three years before his genius was fully accepted even by his colleagues. He secured his first academic position, at the University of Bern, in 1908. Among the general public, Einstein became well known only in late 1919, following the success of Eddington’s expedition to observe the bending of light by the Sun, which confirmed Einstein’s general theory of relativity. This was front-page news, and made Einstein universally famous. (See Chapter 16, ‘The suddenly famous Doctor Einstein’, in Pais, *Subtle is the Lord*, for an account of these events). Before 1919 the trade offprints of Einstein’s papers would probably only have been purchased by professional physicists; after 1919 everyone wanted a memento of the famous Dr. Einstein, whether or not they understood anything of theoretical physics, and the trade offprints of his papers were printed and sold in far greater numbers than before to meet the demand. It is telling that when these post-1919 trade offprints appear on the market, they are often in mint condition – they were never read simply because their owners were unable to understand them.

In our view, Einstein’s author’s presentation offprints are rare, from any journal and any period, though of course some are rarer than others. Before 1919 or 1920, the *Sitzungsberichte* trade offprints are also quite rare, although not nearly as rare as the author’s presentation offprints; after 1919 or 1920, the trade offprints are much more common.

Boni 65; Weil 68. Calaprice, *The Einstein Almanac; *Norton, ‘How Einstein Found His Field Equations: 1912-1915’, *Historical Studies in the Physical Sciences* 14 (1984), pp. 253-316; Norton, ‘Einstein and Nordström: some lesser-known thought experiments in gravitation,’ pp. 3-30 in *The Attraction of Gravitation*: *New Studies in the History of General Relativity***, **edited by John Earman, Michel Janssen, John D. Norton, 1993; Reich, Einstein’s “Formal Foundations of the General Theory of Relativity” (1914) (http://mathineurope.eu/images/information_pic/hist_phil_pic/calendar_pic/2014einstein/Einstein_English.pdf). For the history of tensor calculus, including Einstein’s application of it to general relativity, see Reich, *Die Entwicklung des Tensorkalküls*, Birkhäuser 2012.

8vo (254 x 176 mm), pp. 1030-1085. Original printed wrappers, light wear to upper and lower part of spine, very light vertical crease from having been folded (for post?), small ink stain to rear wrapper, outer margin of all text leaves and wrapper have been unevenly cut - this might have been done by an early owner as a way of opening all the text leaves at once, instead of having to cut open each of the closed pages one at a time.

Item #4585

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Price:
$5,000.00
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