Einstein's Washington Manuscript on Unified Field Theory
aa r X i v : . [ phy s i c s . h i s t - ph ] A ug Einstein’s Washington Manuscript on UnifiedField Theory
Tilman Sauer ∗ and Tobias Schütz † Institute of MathematicsJohannes Gutenberg University MainzD-55099 Mainz, Germany
Version of August 25, 2020
Abstract
In this note, we point attention to and briefly discuss a curious manu-script of Einstein, composed in 1938 and entitled “Unified Field Theory,”the only such writing, published or unpublished, carrying this title withoutany further specification. Apparently never intended for publication, themanuscript sheds light both on Einstein’s modus operandi as well as on thepublic role of Einstein’s later work on a unified field theory of gravitationand electromagnetism. ∗ [email protected] † [email protected] The “Washington manuscript”
In July 1938, the Princeton based journal
Annals of Mathematics published apaper
On a Generalization of Kaluza’s Theory of Electricity in its Vol. 39, issueNo. 3 (Einstein and Bergmann, 1938). The paper was co-authored by AlbertEinstein (1879–1955) and his then assistant Peter Gabriel Bergmann (1915–2002). It presented a new discussion of an approach toward a unified theoryof the gravitational and electromagnetic fields based on an extension of thenumber of physical dimensions characterizing space-time. Such five-dimensionaltheories had been discussed already a number of times, notably by TheodorKaluza in 1921, and then again in the late twenties by Oskar Klein and others(Goenner, 2004). Einstein had contributed to the discussion already in 1923and in 1927, but had given up the approach in favor of another one basedon distant parallelism (Sauer, 2014). After giving up on distant parallelism,he had pursued another version of a five-dimensional approach, together withWalter Mayer in 1931 and 1932. The paper of 1938 indicated a return to theidea of five-dimensional theories, that he would pursue until the early forties.But he published only one other paper on the approach, this time together withBergmann and Valentine Bargmann (1908–1989) before giving up this approachaltogether for the rest of his life (Einstein, Bargmann, and Bergmann, 1941).The manuscript for the published paper was received by the
Annals onApril 8, 1938, but as correspondence between Einstein and Bergmann shows,corrections to the proofs were still made in late July during a phase of intensecorrespondence between Einstein and Bergmann during that time. The corre-spondence concerned, among other things, final corrections to the proofs of thispaper. In fact, in a letter to his friend Michele Besso (1873–1955) from August8, 1938 (AEA 70-368), Einstein indicated that the paper had not been printedas late as then.It turns out that during that same time Einstein composed another paper,which he referred to as “my Washington manuscript” in several letters to Berg-mann, e.g. in AEA 6-271 on July 15, 1938. Indeed, the Library of Congress inWashington owns a 12 pp. holographic manuscript in Einstein’s hand, signedand dated 6 July 1938. It is written in German and entitled simply “UnifiedField Theory” [“Einheitliche Feldtheorie”]. The Albert Einstein Archives at theHebrew University of Jerusalem has two identical Xerox copies of a 15pp. type-script, also entitled “Einheitliche Feldtheorie.” The typescripts are typed ver- An English phrase of the proofs was still discussed in a letter Einstein sent to Bergmannon July 22 or 23, see Albert Einstein Archives (AEA) Call No. 6-266. In the following, ourdating of archival items in the Einstein Archives is based on a reconstruction of the Einstein-Bergmann correspondence (to be published) and at times differs from the dating given in theoriginal archival catalogue. Einstein was at Nassau Point, New York from June 15, 1938, see AEA 54-240; Bergmannwas at Robinhood, Maine. The document carries the Library of Congress Identification Number MSS19596. The documents carry the archival signatures AEA 2-121 and 5-008, respectively. A copyof the holograph of the Library of Congress was recently accessioned by the AEA as welland was given archival signature AEA 97-487. In addition to these complete versions of the The handwritten equations of the typescript were addedin Einstein’s hand.Our analysis of the extant documentary evidence suggests the following ori-gin of the Washington manuscript. According to Einstein’s letter of submission, the manuscript was deposited with the Library of Congress at the suggestion ofEinstein’s friend Elias Avery Lowe (1879–1969). Einstein and Lowe had metin 1936 when Lowe became professor for palaeography at the Institute for Ad-vanced Study (John, 1994). Lowe, however, was based in Oxford. At the sametime, he held a position as reader for palaeography at Oxford where he hadbeen teaching since 1913. When he retired as professor emeritus in Princetonin 1945, he still kept his appointment in Oxford (John, 1970). Furthermore, heworked as consultant in palaeography to the Library of Congress (AEA 97-491),(John, 1994). Not only was Lowe a good friend of Einstein, but their familieswere friends as well, as Lowe’s daughter described in her memoir of her parents(Lowe, 2006). In particular, Lowe’s wife Helen Tracy Lowe-Porter (1877–1963), whom he had married in 1911, worked as a translator for Einstein, and Einsteinheld her work in high esteem, as he expressed it several times in letters. Inone of these letters from 1939, Einstein tried to advise Lowe regarding sometroubles he had at the Institute. Einstein also spoke out for Lowe in 1944when Lowe was about to retire, and he endorsed the latter’s wish for betterpension arrangements. Considering the good relation to “the Lowes” (AEA30-815), it seems natural that Lowe, who as a palaeograph was interested inhandwritten manuscripts, suggested that Einstein donate a holograph to theLibrary of Congress.In any case, Einstein complied with his wish and composed a 12pp. manuscript,which he signed on July 6. Before sending it off to Washington, he apparentlyhad his secretary Helen Dukas (1896–1982) prepare a typed copy of it which hethen completed by filling in the equations. The original manuscript was then document, AEA 2-119 is a single page with a handwritten draft version of its last paragraph,and AEA 62-789 contains a draft version of a paragraph from that document as well aspertinent calculations related to the document. AEA 62-789 is part of a batch of manuscriptpages in Einstein’s hand containing mainly unidentified and undated calculations, see Sauer(2019). As an example of how these unpublished working sheets can reveal Einstein’s thinkingand theorizing, see Sauer and Schütz (2019). The transcripts contain all corrections made in the manuscript. In cases where thetyped script differs from the manuscript, it was later corrected by hand to conform withthe manuscript. Einstein to Herbert Putnam (1861–1955), July 13, 1938 (AEA 97-494). Lowe’s literary estate is located at Morgan Library in New York (Mayo and Sharma,1990). She mainly became known as the translator of Thomas Mann (Romero, 1980). See, for instance, Einstein to Lowe, summer 1940, AEA 55-635. (AEA 53-892), this draft of a letter was not dated by Einstein. However, he mentioned atranslation regarding Gandhi’s birthday, which was made in 1939, see AEA 28-459.1. We know from several letters (AEA 38-093, 38-094, 52-503 and 53-783) about conflicts inthe Institute at the time. The Institute’s faculty demanded a say regarding the succession ofFlexner, who was director at the Institute from 1930 to 1939 (Bonner, 1998). However, it isnot clear whether these are the troubles Einstein hinted at. See Einstein’s letters to Leidesdorf and to Fulton, AEA 55-632 and 55-633. and received there on July 19. Einstein apparently then gave the typed version to Bergmann, in whose posses-sion it remained. Apparently, no copy was retained with Einstein or Dukas, asindicated by her in later years in letters to Bergmann from 1964 and 1965 (AEA6-321 and 6-322). Bergmann, then at Syracuse University, had copies made oforiginal Einstein documents in his possession and sent them to Helen Dukasfor inclusion into the Albert Einstein Archives. After receiving the copies,Dukas first incorporated the correspondence into the archives, a process duringwhich she learned about the existence of the so-called “Washington manuscript.”When she turned her attention to the unpublished scientific manuscripts, shecontacted Bergmann for further information about the typescript. Upon herinstigation, Bergmann, who did not remember himself any details, wrote to theLibrary of Congress and was informed that Einstein had indeed donated themanuscript as a gift to the Library in July 1938 (AEA 6-324 and 6-325).
We have found no evidence whatsoever that Einstein intended to publish themanuscript, even though Einstein considered it an improved presentation of thetheory. The lack of any such contextual information already puzzled Dukas andBergmann, who had no recollection of the circumstances of its composition. Given its substantial character, as we will see, the question therefore arises asto the reasons why Einstein would have written the Washington manuscript?We can think of three different, not mutually exclusive motivations for Ein-stein to compose the manuscript.1) The submission letter as well as the letters of acknowledgment mentionthat Einstein wrote the manuscript and donated it to the Library of Congressat the suggestion of Lowe. Lowe was indeed acting as a consultant for pa-leography for the Library, and may have been involved in the creation of acollection of autographs. We know from other examples that Einstein was verywilling to comply with similar requests. Already in 1924, the Lautabteilung See his submission letter to Herbert Putnam, director of the Library of Congress (AEA97-494). See the memorandum of receipt in AEA 97-493. This is indicated by Einstein in a letter to Bergmann (AEA 6-269), probably was writtenbetween July 29 and August 03, 1938. The AEA only has Xerox copies of the typescript version (AEA 2-121, 5-008), we havenot been able to locate the original typescript. In fact, Dukas even conjectured in 1964 that it might not even have been typed by herself(AEA 6-321). She changed her mind when she observed later in 1965 that the types lookedlike those of her old “Remington portable” (AEA 6-322). In her own reconstruction she alsorecollected that she might have been away for some time in July and August 1938. Indeed,Einstein mentioned in a letter from July 1938 that Dukas would be back on August 1 (AEA54-583). It seems more likely to us that she did do the typing and only forgot. Since she wasaway in July, she could have made the transcription in a hurry which is why she may haveforgotten to make a copy for her own records. (AEA 97-494) and letters from Chief Assistant Librarian Martin A. Roberts (1875-1940)to Einstein (AEA 97-491) and to Lowe (AEA 97-489). n der Preußischen Staatsbibliothek , for instance, recorded Einstein’s voice forhistorical interest as part of a project that collected voices of famous people. Furthermore, Einstein was interested in the study of handwriting, as a visit toa graphologist in 1930 shows (Anonymous, 1930).Although we have found no evidence for it in the case of this particulardonation, such autograph donations were also serving, both before and after1938, specific purposes of fund raising. For instance, Einstein donated, aftersome dispute with the astronomer Erwin Freundlich (1885–1964), a manuscriptof his publication “The Foundation of the General Theory of Relativity” (“DieGrundlage der allgemeinen Relativitätstheorie”) from 1916 (Einstein, 1916) tothe
Jewish National and University Library in 1925 in order to support the uni-versity library as well as other charities (Gutfreund and Renn, 2015). A similardonation by Einstein happened in 1943. In order to promote the sale of warbonds, Einstein prepared a handwritten copy of his famous special relativitypaper “On the electrodynamics of moving bodies” (“Zur Elektrodynamik be-wegter Körper”) (Einstein, 1905). This manuscript was then auctioned andpresented by the buyer to the Library of Congress. Again, a later manuscripton unified field theory entitled “the bi-vector field” (“Das Bi-Vektor Feld”) wasauctioned for the same purpose and given to the Library in late 1944 or early1945 (Library of Congress, 2010; Brasch, 1945).2) Abraham J.Karp (1921–2003) suggested that the donation was given byEinstein as a sign of gratitude to the United States (Karp, 1991). Given Ein-stein’s situation this appears quite possible. He was himself in the process ofnaturalization, having applied in May 1935 for US citizenship, which was thenobtained in 1940 (Calaprice, Kennefick, and Schulmann, 2015). He was alsovery active in helping friends and colleagues who were suffering persecution inNazi Germany to emigrate to the United States. Karp does not provide anyfurther evidence to support his conjecture. On receiving the manuscript, theLibrary of Congress issued a press release which, however, mentioned or evenonly suggested no such connection.
3) Einstein may also have welcomed the writing of a holograph for the Li-brary as occasion for a moment of reflection. This interpretation is suggested byEinstein’s comments about the manuscript to Bergmann. There he emphasizedthat he had “reworked the whole theory anew” and “presented the new theorysystematically” and found that “the whole thing now takes on a really beautifulform, and I really have joy with it.” This interpretation is also suggested bya closer look at the manuscript itself. See AEA 5-025 for a copy of that manuscript. See AEA 97-492, for a draft of the press release. We have not found evidence for receptionof the press announcement. Einstein to Bergmann in AEA 6-256 on July 12, 1938. See also AEA 6-271. Characteristics of the manuscript
The substance of the Washington manuscript repeats what is contained alreadyin the published Einstein-Bergmann paper. But there are significant differences.The essence of the Einstein-Bergmann paper is to extend Kaluza’s originalfive-dimensional theory (Kaluza, 1921) by giving the fifth dimension reality,and at the same time replacing its so-called cylinder condition by a periodicitycondition. On the basis of this “generalization,” Einstein and Bergmann thenderived the following field equations α (cid:18) R kl + 12 R lk − g kl R (cid:19) + α (cid:18) A km A ml − g kl A mn A mn (cid:19) + α (cid:18) − ∂ ∂ g kl + 2 g rs ∂ g kr ∂ g ls − g rs ∂ g rs ∂ g kl − ∂ g rs ∂ g rs g kl (cid:19) + α g kl (cid:18)
12 ( g mn ∂ g mn ) + 2 g mn ∂ ∂ g mn + 2 ∂ g mn ∂ g mn (cid:19) = 0 , (1)and Z (cid:16) α ( g mn ∂ Γ smn − g ms ∂ Γ nmn ) − α ∇ t A st (cid:17) √− g dx = 0 (2)for a five-dimensional theory, derived from a variational principle (Einstein and Bergmann,1938). In these equations, the g ab are components of a five-dimensional metrictensor, which effectively reduces to a four dimensional one since its componentswith index zero vanish. The difference between Kaluza’s theory and that ofEinstein and Bergmann lies in the fact that the components of this metric arenow periodic functions of x , the fifth coordinate. This also gives rise to theremaining integral in the second equation. The quantities A mn are the antisym-metrized derivatives of the vector field A m , which was interpreted as representingthe electric potential. R kl and R are the (5-d) Ricci tensor and Ricci scalar,while Γ smn (appearing only in the second equation) denotes the (5-d) Christoffelsymbol. The quantities α i are constants. A discussion of these constants wasgiven by Einstein in a later unpublished manuscript. As regards a unified field theory based on these field equations, the Washing-ton manuscript contains nothing new, but the differences between the manuscriptand the published paper are nevertheless interesting and revealing.A most obvious difference between the published paper and the Washingtonmanuscript concerns the title and authorship. Given that the manuscript did notcontain anything new of substance and Einstein’s and Bergmann’s cooperationwas ongoing, one might have expected that Bergmann would have been co-author here as well. Instead, we learn from the correspondence that Einsteindid not even bother to consult with Bergmann about this manuscript. Equally See “Ein Gesichtspunkt für eine spezielle Wahl der in der verallgemeinerten Kaluza-Theorieauftretenden Konstanten” (AEA 1-136), dated to 1941 by the Albert Einstein Archives cata-logue. See also van Dongen (2010). Both the change inauthorship and the change of title are compatible with the donation requestas the primary motivation for its composition. If Einstein never intended topublish this manuscript, he may instead have wanted to comply with Lowe’swish of having an autograph manuscript and the Library’s interest in obtainingsomething that could be put on display to a larger public.Notwithstanding the fact that Bergmann no longer figured as a co-author,Einstein made the latter’s co-responsibility clear in the introductory paragraphof the manuscript which also makes explicit what he intended to be the newfeature of the manuscript. The introductory paragraph reads:In the last months, I have developed, together with my assistantP. Bergmann, a unified field theory, which emerged as a generaliza-tion of Kaluza’s theory of the electric field. In the following thistheory shall be presented independently from its historical roots, inorder that its logical structure may come to the fore as clearly aspossible. That Einstein presented their theory independently from Kaluza’s theory isone of the main differences to the publication. There, Einstein and Bergmannfirst gave a recapitulation of Kaluza’s theory. They then illustrated how to alterKaluza’s theory in order to ascribe a “physical reality to the fifth dimension”(Einstein and Bergmann, 1938, p. 683) and, therefore, they extended Kaluza’stheory. In Einstein’s Washington manuscript, on the other hand, the theorywas developed from scratch on the basis of independent axioms.The theory was now exclusively based on three axioms. The first one pos-tulated a five-dimensional space, equipped with a regular Riemannian metric.A five-dimensional space with a regular Riemann-metric dσ = γ µν dx µ dx ν Similar titles containing the phrase “Unified Field Theory” include a 1925 paper onthe metric-affine approach, entitled “Einheitliche Feldtheorie von Gravitation und Elektriz-ität” (Einstein, 1925a), or a paper with his assistant Mayer on another variant of theKaluza-Klein approach, entitled “Einheitliche Theorie von Gravitation und Elektrizität"(Einstein and Mayer, 1931, 1932). Perhaps closest to the manuscript is a paper entitled “Zureinheitlichen Feldtheorie” (Einstein, 1929) or a French paper “Théorie unitaire du champphysique” (Einstein, 1930). The latter two items are from his teleparallel approach. Inter-estingly, the 1929 paper was the one that created out-of-proportion public attention when itcame out in 1929, as a result of international media coverage, see (Sauer, 2006, p. 414) and(Sauer, 2014). “In den letzten Monaten habe ich zusammen mit meinem Assistenten P. Bergmann eineeinheitliche Feldtheorie entwickelt, welche durch Verallgemeinerung von Kaluza’s Theorie deselektrischen Feldes entstanden ist. Im Folgenden soll diese Theorie unabhängig von ihrenhistorischen Wurzeln dargestellt werden, damit ihre logische Struktur möglichst deutlich her-vortrete.” (AEA 2-121, 97-487).
7s assumed to be the basis for the theory. The axiom further stipulated that the metric could locally be transformed to adiagonal metric of the form dσ = dx + dx + dx − dx + dx . The second axiom introduced spatial compactification of the additional di-mension by requiring periodicity along the fifth coordinate.With respect to the dimension characterized by the coordinate x ,the space be closed. This requirement was then further concretized by stipulating the possibility offinding coordinates in which the γ µν are periodic functions of x .The third axiom introduced a congruence of geodesics by requiring the exis-tence of a unique, singularity-free, closed space-like geodesic through each point.Through each point of our space there shall exist one and only onegeodesic line that is closed without singularities and “space-like”. Again this point was concretized by stipulating that in the periodic representa-tion the geodesic should be unique and pass through all periodically repeatedpoints.By these three axioms, Einstein wrote, “the space structure, which underliesthe theory, is characterized completely” (“Damit ist die Raumstruktur, welcheder Theorie zugrundeliegt, vollständig charakterisiert.”)Einstein here emphasizes the fact that the substance of the theory is cap-tured in just three axioms. This explicitness is remarkable in view of his earliermethodological reflections on the axiomatic method. The axiomatic formulationof the Washington manuscript is clearly an example of his understanding that[t]he goal of theoretical physics is to create a logical system of con-cepts based on the fewest possible mutually independent hypotheses,allowing a causal understanding of the entire complex of physicalprocesses,as he expressed it in 1922 in reflections “on the present crisis of theoreticalphysics” (Einstein, 1922, p. 1). Nevertheless, Einstein’s attitude towards the roleof axioms has been somewhat ambivalent ever since his experience of competitiveefforts in completing the general theory of relativity with David Hilbert (1862–1943) in 1915. While Hilbert made the use of the axiomatic method the hallmarkof his heuristics, Einstein acknowledged its use only somewhat hesitantly. In hisown exposition of the new theory of general relativity in 1916, he emphasizedthat it was not his purpose “Es wird ein fünf-dimensionaler Raum mit einer regulären Riemann-Metrik dσ = γ µν dx µ dx ν zugrunde gelegt.” “Bezüglich der durch die Koordinate x charakterisierten Dimension sei der Raum in sichgeschlossen.” “Durch jeden Punkt unseres Raumes soll es eine und nur eine in sich singularitätsfreigeschlossene “raumartige” geodätische Linie geben.”
8o represent the general theory of relativity as a system that is assimple and logical as possible, and with a minimum number of ax-ioms;and that instead his aim was todevelop this theory in such a way that the reader will feel that thepath we have entered upon is psychologically the natural one, andthat the underlying assumptions will seem to have the highest pos-sible degree of security. (Einstein, 1916, p. 777)And in contemporary correspondence, Einstein expressed himself rather criticalabout the axiomatic method , which he felt could not help anything in findingthe suitable hypotheses. Nevertheless, an axiomatic formulation of a physicaltheory seems to have been his ultimate goal, and, as an example, in his discussionof Eddington’s unified theory based on a general affine connection he was readyto refer to the variational principle underlying the theory as an axiom, justas Hilbert had done (Einstein, 1925b, p. 367). The transition from a geneticexposition of explaining the difficulties of the earlier Kaluza-Klein theory as ajustification for the new ansatz toward a purely axiomatic presentation againreflects this tension of the role of axiomatics in Einstein’s thinking.The second main difference from the publication was pointed out by Einsteinin a letter to Bergmann (AEA 6-256), when he noted that he incorporatedthe mathematical appendix of their publication into the main text. Instead oftreating tensor densities and the mathematical part of the derivation of the fieldequations separately, Einstein now incorporated this part into the main text. In another letter to Bergmann (AEA 6-271), Einstein noted that, regarding thestructure, he preferred his manuscript over their publication.When Einstein and Bergmann first considered Kaluza’s theory and then gen-eralized it, they also considered first the classical theory of general relativity intheir appendix by introducing tensor densities and by deriving the field equa-tions for the four-dimensional case. They then transferred the procedure to thefive-dimensional generalized Kaluza theory. But just as Einstein developed thetheory independently from Kaluza’s theory in his manuscript, he also did notconsider the four-dimensional theory first. Instead he started axiomatically withthe new five-dimensional theory as an independent theory.In addition to the above mentioned differences, the two works also differ withrespect to minor aspects, as regards e.g. the notation. For instance, instead ofdenoting the equivalent to the electric potential by A m , he used the notation ϕ m . This change in notation apparently created some confusion for Einsteinhimself, when he miswrote A mn instead of ϕ mn in his manuscript. In summary, the Washington manuscript, as we have seen, essentially gaveanother derivation of the field equations of the Einstein-Bergmann paper but See, e.g., his letter to Hermann Weyl, 23 November 1916 (Schulmann et al., 2012,Doc. 278). In fact, Einstein’s manuscript does not have an appendix. See page 9 of his holographic version AEA 97-487. The confusion about the notationsalso appears in his correspondence with Bergmann, see his letter AEA 6-266.
In contrast to the published Einstein-Bergmann paper which does not indicatein which direction Einstein and Bergmann wanted to proceed further, the Wash-ington manuscript ends with the following paragraph:The theory developed here provides a unified conception of the struc-ture of physical space which is completely satisfactory from a formalpoint of view. Further investigations will have to show whether itcontains a theory (free of statistical elements) of elementary particlesand of the quantum phenomena. Three years later, in 1941, Einstein and Bergmann, now together with Valen-tine Bargmann, published a follow-up to the first Einstein-Bergmann paper(Einstein, Bargmann, and Bergmann, 1941). In it, they began with a brief de-scription of the earlier theory, going back in spirit, if not in exact phrasing,to Einstein’s Washington manuscript, characterizing the core of the theory interms of three “axioms.” But this paper by Einstein, Bargmann, and Bergmannwas Einstein’s last published attempt along higher-dimensional approaches tounified field theory. Their paper ends, much more sceptically, by listing theirreasons for the non-viability of the five-dimensional approach. Two years later,Einstein and Pauli (1943) closed the lid on this approach by publishing a proofof “the non-existence of regular stationary solutions of relativistic field equa-tions” in both four and five dimensions. As van Dongen (2002, 2010) has al-ready emphasized, foremost among the reasons for Einstein’s giving up on thefive-dimensional approach figured their failure to find particle-like solutions, atask which had been formulated at the end of the Washington manuscript, butnot in the published paper. The failed attempt to find particle solutions aredocumented only in working sheets and unpublished calculational notes (Sauer,2019). This fact and the characteristics of the Washington manuscript showthe necessity of including unpublished notes and correspondence for a properhistorical understanding of the unified field theory program.Despite the highly technical character of Einstein’s manuscript which wouldhave made it understandable only for a handful of contemporaries, the holograph “Die im Vorstehenden entwickelte Theorie gibt eine formal völlig befriedigende einheitlicheAuffassung von der Struktur des physikalischen Raumes. Weitere Untersuchungen müssenzeigen, ob sie eine (von statistischen Elementen freie) Theorie der Elementar-Teilchen sowieder Quanten-Phänomene enthält.” (AEA 2-121, 97-487). References
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