On the History of Unified Field Theories, Hubert F. M. Goenner, published in LivingReviews in Relativity on 2004-02-13

1.1 Preface

This historical review of classical unified field theories consists of two parts. In the first, the development of unified field theory between 1914 and 1933, i.e., during the years Einstein^$1$ lived and worked in Berlin, will be covered. In the second, the very active period after 1933 until the 1960s to 1970s will be reviewed. In the first version of Part I presented here, in view of the immense amount of material, neither all shades of unified field theory nor all the contributions from the various scientific schools will be discussed with the same intensity ; I apologise for the shortcoming and promise to improve on it with the next version. At least, even if I do not discuss them all in detail, as many references as are necessary for a first acquaintance with the field are listed here; completeness may be reached only (if at all) by later updates. Although I also tried to take into account the published correspondence between the main figures, my presentation, again, is far from exhaustive in this context. Eventually, unpublished correspondence will have to be worked in, and this may change some of the conclusions. Purposely I included mathematicians and also theoretical physicists of lesser rank than those who are known to be responsible for big advances. My aim is to describe the field in its full variety as it presented itself to the reader at the time.

The review is written such that physicists should be able to follow the technical aspects of the papers (cf. Section 2 ), while historians of science without prior knowledge of the mathematics of general relativity at least might gain an insight into the development of concepts, methods, and scientific communities involved. I should hope that readers find more than one opportunity for further in-depth studies concerning the many questions left open.

I profited from earlier reviews of the field, or of parts of it, by Pauli^$2$ ([245] , Section V); Ludwig [211] ; Whittaker ([415] , pp. 188–196); Lichnerowicz [208] ; Tonnelat ([355] , pp. 1–14); Jordan ([175] , Section III); Schmutzer ([289] , Section X); Treder ([182] , pp. 30–43); Bergmann ([10] , pp. 62–73); Straumann [333, 334] ; Vizgin [384, 385] ^$3$ ; Bergia [9] ; Goldstein and Ritter [145] ; Straumann and O'Raifeartaigh [239] ; Scholz [291] , and Stachel [329] . The section on Einstein's unified field theories in Pais' otherwise superb book presents the matter neither with the needed historical correctness nor with enough technical precision [240] . A recent contribution of van Dongen, focussing on Einstein's methodology, was also helpful [371] . As will be seen, with regard to interpretations and conclusions, my views are different in some instances. In Einstein biographies, the subject of “unified field theories” – although keeping Einstein busy for the second half of his life – has been dealt with only in passing, e.g., in the book of Jordan [176] , and in an unsatisfying way in excellent books by Fölsing [135] and by Hermann [158] . This situation is understandable; for to describe a genius stubbornly clinging to a set of ideas, sterile for physics in comparison with quantum mechanics, over a period of more than 30 years, is not very rewarding.

For the short biographical notes, various editions of J. C. Poggendorff 's Biographisch-Literarischem Handwörterbuch and internet sources have been used (in particular [357] ).

If not indicated otherwise, all non-English quotations have been translated by the author; the original text of quotations is given in footnotes.

^$1$ Albert Einstein (1879-1955). Born in Ulm, Württemberg (Germany). Studied physics and mathematics at the Swiss Federal Polytechnic School (ETH) Zurich and received his doctor's degree in 1905.

Lecturer at the University of Bern (Switzerland), Professor in Zurich, Prague (then belonging to Austria), Berlin (Germany) and Princeton (U.S.A.). Nobel Prize 1921 for his work on the light-electric effect (photon concept). Best known for his special and general relativity theories.

Important results in Brownian motion and the statistical foundations of radiation as a quantum phenomenon. Worked for more than 30 years on Unified Field Theory.

^$2$ Wolfgang Ernst Pauli (1900–1958). Born in Vienna, Austria. Studied at the University of Munich with A. Sommerfeld who recognised his great gifts. Received his doctorate in 1921 for a thesis on the quantum theory of ionised molecular hydrogen. From October 1921 assistant of Max Born in Göttingen. After a year with Bohr, Pauli, became a lecturer at the University of Hamburg in 1923. In 1928 he was appointed professor of theoretical physics at the Federal Institute of Technology in Zürich. From 1945–1950 guest professor at the Institute for Advanced Study, Princeton. He then returned to Zürich. Did important work in quantum mechanics, quantum field theory and elementary particle theory (fourth quantum number (spin), Pauli exclusion principle, prediction of neutrino). Fellow of the Royal Society. Nobel Prize winner in 1954.

^$3$ Vizgin's book is the only one that covers the gamut of approaches during the period considered. Fortunately, he has made accessible contributions in the Russian language by scientists in the Soviet Union. Vizgin also presents and discusses attempts at unification prior to 1914.

1.2 Introduction to part I

Past experience has shown that formerly unrelated parts of physics could be fused into one single conceptual formalism by a new theoretical perspective: electricity and magnetism, optics and electromagnetism, thermodynamics and statistical mechanics, inertial and gravitational forces.

In the second half of the 20th century, the electromagnetic and weak nuclear forces have been bound together as an electroweak force; a powerful scheme was devised to also include the strong interaction (chromodynamics), and led to the standard model of elementary particle physics.

Unification with the fourth fundamental interaction, gravitation, is in the focus of much present research in classical general relativity, supergravity, superstring, and supermembrane theory but has not yet met with success. These types of “unifications” have increased the explanatory power of present day physical theories and must be considered as highlights of physical research.

In the historical development of the idea of unification, i.e., the joining of previously separated areas of physical investigation within one conceptual and formal framework, two closely linked yet conceptually somewhat different approaches may be recognised. In the first, the focus is on unification of representations of physical fields. An example is given by special relativity which, as a framework, must surround all phenomena dealing with velocities close to the velocity of light in vacuum. The theory thus is said to provide “a synthesis of the laws of mechanics and of electromagnetism” ([14] , p. 132). Einstein's attempts at the inclusion of the quantum area into his classical field theories belongs to this path. Nowadays, quantum field theory is such a unifying representation^$4$ . In the second approach, predominantly the unification of the dynamics of physical fields is aimed at, i.e., a unification of the fundamental interactions. Maxwell's theory might be taken as an example, unifying the electrical and the magnetic field once believed to be dynamically different. Most of the unified theories described in this review belong here:

Gravitational and electromagnetic fields are to be joined into a new field. Obviously, this second line of thought cannot do without the first: A new representation of fields is always necessary.

In all the attempts at unification we encounter two distinct methodological approaches: a deductive-hypothetical and an empirical-inductive method. As Dirac pointed out, however,

“The successful development of science requires a proper balance between the method of building up from observations and the method of deducing by pure reasoning from speculative assumptions, [...].” ([232] , p. 1001)

In an unsuccessful hunt for progress with the deductive-hypothetical method alone, Einstein spent decades of his life on the unification of the gravitational with the electromagnetic and, possibly, other fields. Others joined him in such an endeavour, or even preceded him, including Mie, Hilbert, Ishiwara, Nordström, and others^$5$ . At the time, another road was impossible because of the lack of empirical basis due to the weakness of the gravitational interaction. A similar situation obtains even today within the attempts for reaching a common representation of all four fundamental interactions.

Nevertheless, in terms of mathematical and physical concepts, a lot has been learned even from failed attempts at unification, vid. the gauge idea, or dimensional reduction (Kaluza–Klein), and much still might be learned in the future.

In the following I shall sketch, more or less chronologically, and by trailing Einstein's path, the history of attempts at unifying what are now called the fundamental interactions during the period from about 1914 to 1933. Until the end of the thirties, the only accepted fundamental interactions were the electromagnetic and the gravitational, plus, tentatively, something like the “mesonic” or “nuclear” interaction. The physical fields considered in the framework of “unified field theory” including, after the advent of quantum (wave-) mechanics, the wave function satisfying either Schrödinger's or Dirac's equation, were all assumed to be classical fields. The quantum mechanical wave function was taken to represent the field of the electron, i.e., a matter field. In spite of this, the construction of quantum field theory had begun already around 1927 [50, 173, 177, 174, 178] .

For the early history and the conceptual development of quantum field theory, cf. Section 1 of Schweber [321] , or Section 7.2 of Cao [26] ; for Dirac's contributions, cf. [189] . Nowadays, it seems mandatory to approach unification in the framework of quantum field theory.

General relativity's doing away with forces in exchange for a richer (and more complicated) geometry of space and time than the Euclidean remained the guiding principle throughout most of the attempts at unification discussed here. In view of this geometrization, Einstein considered the role of the stress-energy tensor

T^{i k}

(the source-term of his field equations

G^{i k} = - κ T^{i k}

) a weak spot of the theory because it is a field devoid of any geometrical significance.

Therefore, the various proposals for a unified field theory, in the period considered here, included two different aspects:

∙ An inclusion of matter in the sense of a desired replacement, in Einstein's equations and their generalisation, of the energy-momentum tensor of matter by intrinsic geometrical structures, and, likewise, the removal of the electric current density vector as a non-geometrical source term in Maxwell's equations.
∙ The development of a unified field theory more geometrico for electromagnetism and gravitation, and in addition, later, of the “field of the electron” as a classical field of “de Broglie-waves” without explicitly taking into account further matter sources^$6$ .

In a very Cartesian spirit, Tonnelat (Tonnelat 1955 [355] , p. 5) gives a definition of a unified field theory as

“a theory joining the gravitational and the electromagnetic field into one single hyperfield whose equations represent the conditions imposed on the geometrical structure of the universe.”

No material source terms are taken into account^$7$ . If however, in this context, matter terms appear in the field equations of unified field theory, they are treated in the same way as the stress-energy tensor is in Einstein's theory of gravitation: They remain alien elements.

For the theories discussed, the representation of matter oscillated between the point-particle concept in which particles are considered as singularities of a field, to particles as everywhere regular field configurations of a solitonic character. In a theory for continuous fields as in general relativity, the concept of point-particle is somewhat amiss. Nevertheless, geodesics of the Riemannian geometry underlying Einstein's theory of gravitation are identified with the worldlines of freely moving point-particles. The field at the location of a point-particle becomes unbounded, or “singular”, such that the derivation of equations of motion from the field equations is a non-trivial affair. The competing paradigm of a particle as a particular field configuration of the electromagnetic and gravitational fields later has been pursued by J. A. Wheeler under the names “geon” and “geometrodynamics” in both the classical and the quantum realm [412] . In our time, gravitational solitonic solutions also have been found [234, 24] .

Even before the advent of quantum mechanics proper, in 1925–26, Einstein raised his expectations with regard to unified field theory considerably; he wanted to bridge the gap between classical field theory and quantum theory, preferably by deriving quantum theory as a consequence of unified field theory. He even seemed to have believed that the quantum mechanical properties of particles would follow as a fringe benefit from his unified field theory; in connection with his classical teleparallel theory it is reported that Einstein, in an address at the University of Nottingham, said that he

“is in no way taking notice of the results of quantum calculation because he believes that by dealing with microscopic phenomena these will come out by themselves. Otherwise he would not support the theory.” ([90] , p. 610)

However, in connection with one of his moves, i.e., the 5-vector version of Kaluza^$8$ 's theory (cf. Sections 4.2 , 6.3 ), which for him provided “a logical unity of the gravitational and the electromagnetic fields”, he regretfully acknowledged:

“But one hope did not get fulfilled. I thought that upon succeeding to find this law, it would form a useful theory of quanta and of matter. But, this is not the case. It seems that the problem of matter and quanta makes the construction fall apart.”^$9$ “Eine Hoffnung ist aber nicht in Erfüllung gegangen. Ich dachte, wenn es gelingt, dieses Gesetz aufzustellen, dass es eine brauchbare Theorie der Quanten und Materie bilden würde. Aber das ist nicht der Fall. Die Konstruktion scheint am Problem der Materie und der Quanten zu scheitern.” ([95] , p. 442)

Thus, unfortunately, also the hopes of the eminent mathematician Schouten^$10$ , who knew some physics, were unfulfilled:

“[...] collections of positive and negative electricity which we are finding in the positive nuclei of hydrogen and in the negative electrons. The older Maxwell theory does not explain these collections, but also by the newer endeavours it has not been possible to recognise these collections as immediate consequences of the fundamental differential equations studied. However, if such an explanation should be found, we may perhaps also hope that new light is shed on the [...] mysterious quantum orbits.”^$11$ “[...] Anhäufungen von positiver und negativer Elektrizität, die wir in den positiven Wasserstoffkernen und in den negativen Elektronen antreffen. Die ältere Maxwellsche Theorie erklärt diese Anhäufungen nicht, aber auch den neueren Bestrebungen ist es bisher nicht gelungen, diese Anhäufungen als selbstverständliche Folgen der zugrundeliegenden Differentialgleichungen zu erkennen. Sollte aber eine solche Erklärung gefunden werden, so darf man vielleicht auch hoffen, dass die [...] mysteriösen Quantenbahnen in ein neues Licht gerückt werden.” ([300] , p. 39)

In this context, through all the years, Einstein vainly tried to derive, from the field equations of his successive unified field theories, the existence of elementary particles with opposite though otherwise equal electric charge but unequal mass. In correspondence with the state of empirical knowledge at the time (i.e., before the positron was found in 1932/33), but despite theoretical hints pointing into a different direction to be found in Dirac's papers, he always paired electron and proton^$12$ . Of course, by quantum field theory the dichotomy between matter and fields in the sense of a dualism is minimised as every field carries its particle-like quanta. Today's unified field theories appear in the form of gauge theories; matter is represented by operator valued spin-half quantum fields (fermions) while the “forces” mediated by “exchange particles” are embodied in gauge fields, i.e., quantum fields of integer spin (bosons). The space-time geometry used is rigidly fixed, and usually taken to be Minkowski space or, within string and membrane theory, some higher-dimensional manifold also loosely called “space-time”, although its signature might not be Lorentzian and its dimension might be 10, 11, 26, or some other number larger than four. A satisfactory inclusion of gravitation into the scheme of quantum field theory still remains to be achieved.

In the period considered, mutual reservations may have existed between the followers of the new quantum mechanics and those joining Einstein in the extension of his general relativity. The latter might have been puzzled by the seeming relapse of quantum mechanics from general covariance to a mere Galileior Lorentz-invariance, and by the statistical interpretation of the Schrödinger wave function. Lanczos^$13$ , in 1929, was well aware of his being out of tune with those adherent to quantum mechanics:

“I therefore believe that between the `reactionary point of view' represented here, aiming at a complete field-theoretic description based on the usual space-time structure and the probabilistic (statistical) point of view, a compromise [...] no longer is possible.”^$14$ “Ich glaube darum, dass zwischen dem hier vertretenen `reactionären Standpunkt', der eine vollständige feldtheoretische Beschreibung auf Grund der normalen Raum-Zeit-Struktur erstrebt, und dem wahrscheinlichkeitstheoretischen (statistischen) Standpunkt ein Kompromiss [...] nicht mehr möglich ist.” ([197] , p. 486, footnote)

On the other hand, those working in quantum theory may have frowned upon the wealth of objects within unified field theories uncorrelated to a convincing physical interpretation and thus, in principle, unrelated to observation. In fact, until the 1930s, attempts still were made to “geometrize” wave mechanics while, roughly at the same time, quantisation of the gravitational field had also been tried [283] . Einstein belonged to those who regarded the idea of unification as more fundamental than the idea of field quantisation [94] . His thinking is reflected very well in a remark made by Lanczos at the end of a paper in which he tried to combine Maxwell's and Dirac's equations:

“If the possibilities anticipated here prove to be viable, quantum mechanics would cease to be an independent discipline. It would melt into a deepened `theory of matter' which would have to be built up from regular solutions of non-linear differential equations, – in an ultimate relationship it would dissolve in the `world equations' of the Universe. Then, the dualism `matter-field' would have been overcome as well as the dualism `corpuscle-wave'.”^$15$ “Sollten sich die hier vorausgeahnten Möglichkeiten als wirklich lebensfähig erweisen, so würde die Quantenmechanik aufhören, eine selbständige Disziplin zu sein. Sie würde verschmelzen mit einer vertieften `Theorie der Materie', die auf reguläre Lösungen von nicht-linearen Differentialgleichungen aufzubauen hätte, – in letztem Zusammenhang also aufgehen in den `Weltgleichungen' des Universums. Der Dualismus `Materie-Feld' würde dann ebenso überwunden sein, wie der Dualismus `Korpuskel-Welle'.” ([197] , p. 493)

Lanczos' work shows that there has been also a smaller subprogram of unification as described before, i.e., the view that somehow the electron and the photon might have to be treated together.

Therefore, a common representation of Maxwell's equations and the Dirac equation was looked for (cf. Section 7.1 ).

During the time span considered here, there also were those whose work did not help the idea of unification, e.g., van Dantzig^$16$ wrote a series of papers in the first of which he stated:

“It is remarkable that not only no fundamental tensor [first fundamental form] or tensor-density, but also no connection, neither Riemannian nor projective, nor conformal, is needed for writing down the [Maxwell] equations. Matter is characterised by a bivector-density [...].” ([367] , p. 422, and also [363, 364, 365, 366] )

If one of the fields to be united asks for less “geometry”, why to mount all the effort needed for generalising Riemannian geometry?

A methodological weak point in the process of the establishment of field equations for unified field theory was the constructive weakness of alternate physical limits to be taken:

∙ no electromagnetic field $\to$ Einstein's equations in empty space;
∙ no gravitational field $\to$ Maxwell's equations;
∙ “weak” gravitational and electromagnetic fields $\to$ Einstein–Maxwell equations;
∙ no gravitational field but a “strong” electromagnetic field $\to$ some sort of non-linear electrodynamics.

A similar weakness occurred for the equations of motion; about the only limiting equation to be reproduced was Newton's equation augmented by the Lorentz force. Later, attempts were made to replace the relationship “geodesics

\to

freely falling point particles” by more general assumptions for charged or electrically neutral point particles – depending on the more general (non-Riemannian) connections introduced^$17$ . A main hindrance for an eventual empirical check of unified field theory was the persistent lack of a worked out example leading to a new gravito-electromagnetic effect.

In the following Section 2 , a multitude of geometrical concepts (affine, conformal, projective spaces, etc.) available for unified field theories, on the one side, and their use as tools for a description of the dynamics of the electromagnetic and gravitational field on the other will be sketched. Then, we look at the very first steps towards a unified field theory taken by Reichenbächer^$18$ , Förster (alias Bach), Weyl^$19$ , Eddington^$20$ , and Einstein (see Section 3.1 ). In Section 4 , the main ideas are developed. They include Weyl's generalization of Riemannian geometry by the addition of a linear form (see Section 4.1 ) and the reaction to this approach. To this, Kaluza's idea concerning a geometrization of the electromagnetic and gravitational fields within a five-dimensional space will be added (see Section 4.2 ) as well as the subsequent extensions of Riemannian to affine geometry by Schouten, Eddington, Einstein, and others (see Section 4.3 ). After a short excursion to the world of mathematicians working on differential geometry (see Section 5 ), the research of Einstein and his assistants is studied (see Section 6 ). Kaluza's theory received a great deal of attention after O. Klein^$21$ intervention and extension of Kaluza's paper (see Section 6.3.2 ). Einstein's treatment of a special case of a metric-affine geometry, i.e., “distant parallelism”, set off an avalanche of research papers (see Section 6.4.4 ), the more so as, at the same time, the covariant formulation of Dirac's equation was a hot topic. The appearance of spinors in a geometrical setting, and endeavours to link quantum physics and geometry (in particular, the attempt to geometrize wave mechanics) are also discussed (see Section 7 ). We have included this topic although, strictly speaking, it only touches the fringes of unified field theory.

In Section 9 , particular attention is given to the mutual influence exerted on each other by the Princeton (Eisenhart^$22$ , Veblen^$23$ ), French (Cartan^$24$ ), and the Dutch (Schouten, Struik^$25$ ) schools of mathematicians, and the work of physicists such as Eddington, Einstein, their collaborators, and others. In section 10 , the reception of unified field theory at the time is briefly discussed.

^$4$ The inclusion of the quantum corresponds to Vizgin's maximal unification problem [385] , p. 169.

^$5$ See Section 1 of Vizgin's book [385] for a treatment of the history of pre-relativistic unified field theories and an exposition of Mie's, Ishiwara's, and Nordström's approaches.

^$7$ This definition corresponds, in a geometrical framework, to Vizgin's minimal unification problem ([385] , p. 187).

^$8$ Theodor Franz Eduard Kaluza (1885–1954). Born in Ratibor, Germany (now Raciborz, Poland). Studied mathematics at the University of Königsberg (now Kaliningrad, Russia) and became a lecturer there in 1910. In 1929 he received a professorship at the University of Kiel, and in 1935 was made full professor at the University of Göttingen. He wrote only a handful of mathematical papers and a textbook on “Higher mathematics for the practician” (cf. [423] ).

^$10$ Jan Arnoldus Schouten (1883–1971). Born near Amsterdam in the Netherlands. Studied electrical engineering at the Technical University (Hogeschool) of Delft and then mathematics at the University of Leiden. His doctoral thesis of 1914 was on tensor analysis, a topic he worked on during his entire academic career. From 1914 until 1943 he held a professorship in mathematics at the University of Delft, and from 1948 to 1953 he was director of the Mathematical Research Centre at the University of Amsterdam. He was a prolific writer, applying tensor analysis to Lie groups, general relativity, unified field theory, and differential equations.

^$12$ It is true that Dirac, in his first paper, in contrast to what his “hole”-theory implied, had identified the positively charged particle corresponding to the electron also with the proton [53] . However, after Weyl had pointed out that Dirac's hole theory led to equal masses [409] , he changed his mind and gave the new particle the same mass as the electron [54] .

^$13$ Cornelius Lanczos (Kornél Löwy) (1893–1974). Born in Székesfehérvár (Hungary). Studied physics and mathematics at the University of Budapest with Eötvös, Fejér, and Lax. Received his doctorate in 1921, became scientific assistant at the University of Freiburg (Germany) and lecturer at the University of Frankfurt am Main (Germany). Worked with Einstein in Berlin 1928–1929, then returned to Frankurt. Became a visiting professor at Purdue University in 1931 and came back on a professorship in 1932. Worked mainly in mathematical physics and numerical analysis. After 1944 he held various posts in industry and in the National Bureau of Standards. Left the U.S.A. during the McCarthy era and in 1952 followed an invitation by Schrödinger to become head of the Theoretical Physics Department of the Dublin Institute for Advanced Study.

^$16$ David van Dantzig (1900–1959). Born in Rotterdam, Netherlands. Studied mathematics at the University of Amsterdam. Worked first on differential geometry, electrodynamics and unified field theory. Known as co-founder, in 1946, of the Mathematical Centre in Amsterdam and by his role in establishing mathematical statistics as a subdiscipline in the Netherlands.

^$17$ Thus, in a paper of 1934 really belonging to the 2nd part of this review, Schouten and Haantjes exchanged the previously assigned “induced geodesic lines” for “auto-geodesical lines” [310] .

^$18$ Ernst Reichenbächer (1881–1944). Studied mathematics and received his doctorate from the University of Halle in 1903 under the guidance of Albert Wangerin (a student of Franz Neumann in Königsberg). At first, Reichenbächer did not enter an academic career, but started teaching in a Gymnasium in Wilhelmshaven in North Germany, then in Königsberg on the Baltic Sea. In 1929 he became a Privatdozent (lecturer) at the University of Königsberg (now Kaliningrad, Russia). His courses covered special and general relativity, the physics of fixed stars and galaxies with a touch on cosmology, and quantum mechanics. In the fifth year of World War II he finally received the title of professor at the University Königsberg, but in the same year was killed during a bombing raid on the city.

^$19$ Hermann Klaus Hugo Weyl (1885–1955). Born in Elmshorn, Germany. Studied at the Universities of Munich and Göttingen where he received his doctorate in 1908 (Hilbert was his supervisor). From 1913 he held the Chair of Mathematics at the Federal Institute of Technology in Zürich, and from 1930 to 1933 a corresponding Chair at the University of Göttingen. Then until retirement he worked at the Institute for Advanced Study in Princeton. Weyl made important contributions in mathematics (integral equations, Riemannian surfaces, continuous groups, analytic number theory) and theoretical physics (differential geometry, unified field theory, gauge theory). For his papers, cf. also the Collected Works [411]

^$20$ Arthur Stanley Eddington (1882–1944). Born in Kendal, England. Studied mathematics at Owens College, Manchester and Trinity College, Cambridge. After some work in physics, moved into astronomy in 1905 and was appointed to the Royal Observatory at Greenwich. From 1914 director of the Cambridge Observatory. Fellow of the Royal Society. As a Quaker he became a conscientious objector to military service during the First World War. Eddington made important contributions to general relativity and astrophysics (internal structure of stars). In 1918, he led an eclipse expedition from which the first indications resulted that Einstein's general relativity theory was correct. Wrote also on epistemology and the philosophy of science.

^$21$ Oskar Klein (1894–1977). Born in Mörby, Sweden. After work with Arrhenius in physical chemistry, he met Kramers, then a student of Bohr, in 1917. Klein worked with Bohr in the field of molecular physics and received his doctorate in 1921 at Stockholm Högskola. His first research position was at the University of Michigan in Ann Arbor, where he worked on the Zeeman effect. Back in Europe from 1925, he taught at Lund University and tried to connect Kaluza's work with quantum theory. In 1930 he became professor for mathematical physics at Stockholm Högskola until retirement. His later work included quantum theory (Klein–Nishina formula), superconductivity, and cosmology.

^$22$ Luther Pfahler Eisenhart (1876–1965). Born in York, Pennsylvania, U.S.A. Studied mathematics at John Hopkins University, Baltimore and received his doctorate in 1900. Eisenhart taught at the University of Princeton from 1900, was promoted to professor in 1909 and remained there (as Dean of the mathematical Faculty and Dean of the Graduate School) until his retirement in 1945. All his work is in differential geometry, including Riemannian and non-Riemannian geometry and in group theory.

^$23$ Oswald Veblen (1880–1960). Born in Decorah, Iowa, U.S.A. Entered the University of Iowa in 1894, receiving his B.A. in 1898. He obtained his doctorate from the University of Chicago on “a system of axioms in geometry” in 1903. He taught mathematics at Princeton (1905–1932), at Oxford in 1928–1929, and became a professor at the Institute for Advanced Study in Princeton in 1932. Veblen made important contribution to projective and differential geometry, and to topology. He gave a new treatment of spin.

^$24$ Elie Joseph Cartan (1869–1951). Born in Dolomien near Chambéry, France. Student at l`École Normale since 1888, he received his Ph.D. in 1894 with a thesis in which he completed Killing's classification of semisimple algebras. He lectured at Montpellier (1894–1903), Lyon (1896–1903), Nancy (1903–1909), and Paris (1909–1940). His following work on the representation of semisimple Lie groups combines group theory, classical geometry, differential geometry, and topology. From 1904 he worked on differential equations and differential geometry, and developed a theory of moving frames (calculus of differential forms). He also contributed to the geometry of symmetric spaces and published on general relativity and its geometric extensions as well as on the theory of spinors. For his Collected Works, cf. [36] .

^$25$ Dirk J. Struik (1894–2000). Born in Rotterdam in the Netherlands. Studied mathematics and physics at the University of Leiden with Lorentz and de Sitter. Received his doctorate in 1922. Then worked with Schouten at the University of Delft and, with a Rockefeller International Education Fellowship, moved to Rome and Göttingen. After a collaboration with Wiener, in 1926 he received a lectureship at the Massachusetts Institute of Technology (MIT) in Cambridge, Ma. where he became full professor in 1940. He stayed on the MIT mathematics faculty until 1960. As a professed Marxist he was suspended from teaching duties during the McCarthy period but was reinstated in 1956. In 1972, he became an honorary research associate in the History of Science Department of Harvard University.

2 The Possibilities of Generalizing General Relativity: A Brief Overview

As a rule, the point of departure for unified field theory was general relativity. The additional task then was to “geometrize” the electromagnetic field. In this review, we will encounter essentially five different ways to include the electromagnetic field into a geometric setting:

∙ by connecting an additional linear form to the metric through the concept of “gauging” (Weyl);
∙ by introducing an additional space dimension (Kaluza);
∙ by choosing an asymmetric Ricci tensor (Eddington);
∙ by adding an antisymmetric tensor to the metric (Bach, Einstein);
∙ by replacing the metric by a 4-bein field (Einstein).

In order to bring some order into the wealth of these attempts towards “unified field theory,” I shall distinguish four main avenues extending general relativity, according to their mathematical direction: generalisation of

∙ geometry,
∙ dynamics (Lagrangians, field equations),
∙ number field, and
∙ dimension of space,

as well as their possible combinations. In the period considered, all four directions were followed as well as combinations between them like e.g., five-dimensional theories with quadratic curvature terms in the Lagrangian. Nevertheless, we will almost exclusively be dealing with the extension of geometry and of the number of space dimensions.

2.1 Geometry

It is very easy to get lost in the many constructive possibilities underlying the geometry of unified field theories. We briefly describe the mathematical objects occurring in an order that goes from the less structured to the more structured cases. In the following, only local differential geometry is taken into account^$26$ .

The space of physical events will be described by a real, smooth manifold

M_{D}

of dimension

D

coordinatised by local coordinates

x^{i}

, and provided with smooth vector fields

X, Y, \dots

with components

X^{i}, Y^{i}, \dots

and linear forms

ω, ν, \dots

, (

ω_{i}, ν_{i}

) in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connections^$27$ . At each point,

D

linearly independent vectors (linear forms) form a linear space, the tangent space (cotangent space) of

M_{D}

. We will assume that the manifold

M_{D}

is spaceand time-orientable. On it, two independent fundamental structural objects will now be introduced.

^$26$ For the following, it is assumed that readers have some prior knowledge of the mathematics underlying General Relativity.

^$27$ For the precise definition of “geometrical object”, cf. Yano's book [425] .

2.1.1 Metrical structure

The first is a prescription for the definition of the distance

d s

between two infinitesimally close points on

M_{D}

, eventually corresponding to temporal and spatial distances in the external world.

For

d s

, we need positivity, symmetry in the two points, and the validity of the triangle equation.

We know that

d s

must be homogeneous of degree one in the coordinate differentials

d x^{i}

connecting the points. This condition is not very restrictive; it still includes Finsler geometry [280, 125, 223] to be briefly touched, below.

In the following,

d s

is linked to a non-degenerate bilinear form

g (X, Y)

, called the first fundamental form ; the corresponding quadratic form defines a tensor field, the metrical tensor, with

D^{2}

components

g_{i j}

such that

\begin{matrix} d s = \sqrt{g_{i j} d x^{i} d x^{j}}, \end{matrix}

(1)

where the neighbouring points are labeled by

x^{i}

and

x^{i} + d x^{i}

, respectively^$28$ . Besides the norm of a vector

| X | : = \sqrt{g_{i j} X^{i} X^{j}}

, the “angle” between directions

X

Y

can be defined by help of the metric:

cos ((̸ X, Y)) : = \frac{g_{i j} X^{i} Y^{j}}{| X | | Y |} .

From this we note that an antisymmetric part of the metrical tensor does not influence distances and norms but angles.

With the metric tensor having full rank, its inverse

g^{i k}

is defined through^$29$

\begin{matrix} g_{m i} g^{m j} = δ_{i}^{j} \end{matrix}

(2)

We are used to

g

being a symmetric tensor field, i.e., with

g_{i k} = g_{(i k)}

and with only

D (D + 1) / 2

components; in this case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is

\pm (D - 2)

^$30$ . In the following this need not hold, so that the decomposition obtains^$31$ :

\begin{matrix} g_{i k} = γ_{(i k)} + φ_{[i k]} . \end{matrix}

(3)

An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.

For an asymmetric metric, the inverse

\begin{matrix} g^{i k} = h^{(i k)} + f^{[i k]} = h^{i k} + f^{i k} \end{matrix}

(4)

is determined by the relations

\begin{matrix} γ_{i j} γ^{i k} = δ_{j}^{k}, φ_{i j} φ^{i k} = δ_{j}^{k}, h_{i j} h^{i k} = δ_{j}^{k}, f_{i j} f^{i k} = δ_{j}^{k}, \end{matrix}

(5)

and turns out to be [355]

\begin{matrix} h^{(i k)} & = & \frac{γ}{g} γ^{i k} + \frac{φ}{g} φ^{i m} φ^{k n} γ_{m n}, \end{matrix}

(6)

\begin{matrix} f^{(i k)} & = & \frac{φ}{g} φ^{i k} + \frac{γ}{g} γ^{i m} γ^{k n} φ_{m n}, \end{matrix}

(7)

where

g

φ

, and

γ

are the determinants of the corresponding tensors

g_{i k}

φ_{i k}

, and

γ_{i k}

. We also note that

\begin{matrix} g = γ + φ + \frac{γ}{2} γ^{k l} γ^{m n} φ_{k m} φ_{l n}, \end{matrix}

(8)

where

g : = det g_{i k}

φ : = det φ_{i k},

γ : = det γ_{i k}

. The results ( 6 , 7 , 8 ) were obtained already by Reichenbächer ([272] , pp. 223–224)^$32$ and also by Schrödinger [319] . Eddington also calculated Equation ( 8 ); in his expression the term

\sim {φ_{i k}}^{◆} φ^{i k}

is missing (cf. [57] , p. 233).

The manifold is called space-time if

D = 4

and the metric is symmetric and Lorentzian, i.e., symmetric and with signature

s i g g = \pm 2

. Nevertheless, sloppy contemporaneaous usage of the term “space-time” includes arbitrary dimension, and sometimes is applied even to metrics with arbitrary signature.

In a manifold with Lorentzian metric, a non-trivial real conformal structure always exists; from the equation

\begin{matrix} g (X, X) = 0 \end{matrix}

(9)

results an equivalence class of metrics

{λ g}

with

λ

being an arbitrary smooth function. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [65] . For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part

γ_{i k} = γ_{(i k)}

of the metric alone taken to be Lorentzian. A special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by

\begin{matrix} η_{i k} = δ_{i}^{0} δ_{i}^{0} - δ_{i}^{1} δ_{i}^{1} - δ_{i}^{2} δ_{i}^{2} - δ_{i}^{3} δ_{i}^{3} . \end{matrix}

(10)

A geometrical characterization of Minkowski space as an uncurved, flat space is given below.

Let

ℒ_{X}

be the Lie derivative with respect to the tangent vector X^$33$ ; then

ℒ_{X_{p}} η_{i k} = 0

holds for the Lorentz group of generators

X_{p}

The metric tensor

g

may also be defined indirectly through

D

vector fields forming an orthonormal

D

-leg (-bein)

h_{\hat{ı}}^{k}

with

\begin{matrix} g_{l m} = h_{l \hat{j}} h_{m \hat{k}} η^{\hat{j} \hat{k}}, \end{matrix}

(11)

where the hatted indices (“bein-indices”) count the number of legs spanning the tangent space at each point

(\hat{j} = 1, 2, \dots, D)

and are moved with the Minkowski metric^$34$ . From the geometrical point of view, this can always be done (cf. theories with distant parallelism ). By introducing 1-forms

θ^{\hat{k}} : = h_{l}^{\hat{k}} d x^{l}

, Equation ( 11 ) may be brought into the form

d s^{2} = θ^{\hat{ı}} θ^{\hat{k}} η_{\hat{ı} \hat{k}}

A new physical aspect will come in if the

h_{\hat{ı}}^{k}

are considered to be the basic geometric variables satisfying field equations, not the metric. Such tetrad-theories (for the case

D = 4

) are described well by the concept of fibre bundle. The fibre at each point of the manifold contains, in the case of an orthonormal

D

-bein (tetrad), all

D

-beins (tetrads) related to each other by transformations of the group

O (D)

, or the Lorentz group, and so on.

In Finsler geometry, the line element depends not only on the coordinates

x^{i}

of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points

d x^{i}

\begin{matrix} d s^{2} = g_{i j} (x^{n}, d x^{m}) d x^{i} d x^{j} . \end{matrix}

(12)

Again,

g_{i j}

is required to be homogeneous of rank 1.

^$28$ The second fundamental form comes into play when local isometric embedding is considered, i.e., when $M_{D}$ is taken as a submanifold of a larger space such that the metrical relationships are conserved. In the following, all geometrical objects are supposed to be differentiable as often as is needed.

^$29$ Here, the Kronecker-symbol $δ_{k}^{i}$ with value +1 for $i = k$ , and value 0 for $i \neq k$ is used. $δ_{k}^{i}$ keeps its components unchanged under arbitrary coordinate transformations.

^$30$ Latin indices $i, j, k, \dots$ run from 1 to $D$ , or from 0 to $D - 1$ to point to the single timelike direction. We have used the symmetrisation bracket defined by $A_{(i j)} : = 1 / 2 (A_{i j} + A_{j i})$ .

^$31$ In physical applications, special conditions for $γ$ and $φ$ might be needed in order to guarantee that $g$ is a Lorentz metric.

^$32$ Reichenbächer's results agree with Schrödinger's, but, as they are in a different form than that given by Tonnelat [355] , I have not checked whether they agree with Equations ( 6 , 7 ). Neither Schrödinger nor Tonnelat give credit to Reichenbächer.

^$33$ We have $(L_{X} Y)^{k} : = ([X, Y])^{k} = X^{i} \partial_{i} Y^{k} - Y^{i} \partial_{i} X^{k}$ .

^$34$ Eisenhart called his object “ennuple” instead of $n$ -bein [118] . In French, the expressions “ $n$ -pode”, “ $n$ -èdre”, and “polyaxe à $n$ dimensions” were also used.

2.1.2 Affine structure

The second structure to be introduced is a linear connection

L

with

D^{3}

components

L_{i j}^{k}

; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations^$35$ . The connection is a device introduced for establishing a comparison of vectors in different points of the manifold. By its help, a tensorial derivative

\nabla

, called covariant derivative is constructed. For each vector field and each tangent vector it provides another unique vector field. On the components of vector fields

X

and linear forms

ω

it is defined by

\begin{matrix} {\overset{+}{\nabla}}_{k} X^{i} = \frac{\partial X^{i}}{\partial x^{k}} + L_{k j}^{i} X^{j}, {\overset{+}{\nabla}}_{k} ω_{i} = \frac{\partial ω_{i}}{\partial x^{k}} - {L_{k i}}^{j} ω_{j} . \end{matrix}

(13)

The expressions

{\overset{+}{\nabla}}_{k} X^{i}

and

\frac{\partial X^{i}}{\partial x^{k}}

are abbreviated by

X_{∥ k}^{i}

and

X_{, k}^{i}

, respectively, while for a scalar

f

covariant and partial derivative coincide:

\nabla_{i} f = \frac{\partial f}{\partial x_{i}} \equiv \partial_{i} f \equiv f_{, i}

We have adopted the notational convention used by Schouten [299, 309, 389] . Eisenhart and others [120, 233] change the order of indices of the components of the connection:

\begin{matrix} {\bar{\nabla}}_{k} X^{i} = \frac{\partial X^{i}}{\partial x^{k}} + {L_{j k}}^{i} X^{j}, {\bar{\nabla}}_{k} ω_{i} = \frac{\partial ω_{i}}{\partial x^{k}} - L_{i k}^{j} ω_{j} . \end{matrix}

(14)

As long as the connection is symmetric, this does not make any difference as

{\overset{+}{\nabla}}_{k} X^{i} - {\bar{\nabla}}_{k} X^{i} = 2 {L_{[k j]}}^{i} X^{j} .

For both kinds of derivatives we have:

\begin{matrix} {\overset{+}{\nabla}}_{k} (v^{l} w_{l}) = \frac{\partial (v^{l} w_{l})}{\partial x^{k}}, {\bar{\nabla}}_{k} (v^{l} w_{l}) = \frac{\partial (v^{l} w_{l})}{\partial x^{k}} \end{matrix}

(15)

Both derivatives are used in versions of unified field theory by Einstein and others^$36$ .

A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special role: With regard to it the connection transforms as a tensor (cf. Section 2.1.5 ).

For a vector density (cf. Section 2.1.5 ), the covariant derivative of

\hat{X}

contains one more term:

\begin{matrix} {\overset{+}{\nabla}}_{k} {\hat{X}}^{i} = \frac{\partial {\hat{X}}^{i}}{\partial x^{k}} + {L_{k j}}^{i} {\hat{X}}^{j} - {L_{k r}}^{r} {\hat{X}}^{i}, {\bar{\nabla}}_{k} {\hat{X}}^{i} = \frac{\partial X^{i}}{\partial x^{k}} + {L_{j k}}^{i} {\hat{X}}^{j} - {L_{r k}}^{r} {\hat{X}}^{i} . \end{matrix}

(16)

A smooth vector field

Y

is said to be parallely transported along a parametrised curve

λ (u)

with tangent vector

X

if for its components

Y_{∥ k}^{i} X^{k} (u) = 0

holds along the curve. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point^$37$ :

\begin{matrix} X_{∥ k}^{i} X^{k} (u) = σ (u) X^{i} . \end{matrix}

(17)

By a particular choice of the curve's parameter,

σ = 0

may be imposed.

A transformation mapping autoparallels to autoparallels is given by:

\begin{matrix} L_{i k}^{j} \to {L_{i k}}^{j} + δ_{(i}^{j} ω_{k)} . \end{matrix}

(18)

The equivalence class of autoparallels defined by Equation ( 18 ) defines a projective structure on

M_{D}

[404, 403] . The particular set of connections

\begin{matrix} _{(p)} {L_{i j}}^{k} : = {L_{i j}}^{k} - \frac{2}{D + 1} δ_{(i}^{k} L_{j)} \end{matrix}

(19)

with

L_{j} : = L_{i m}^{m}

is mapped into itself by the transformation ( 18 ) [347] .

In Part II of this article, we shall find the set of transformations

{L_{i k}}^{j} \to L_{i k}^{j} + δ_{i}^{j} \frac{\partial ω}{\partial x^{k}}

playing a role in versions of Einstein's unified field theory.

From the connection

L_{i j}^{k}

further connections may be constructed by adding an arbitrary tensor field

T

to its symmetrised part^$38$ :

\begin{matrix} {\bar{L}}_{i j}^{k} = {L_{(i j)}}^{k} + {T_{i j}}^{k} = {Γ_{i j}}^{k} + {T_{i j}}^{k} . \end{matrix}

(20)

By special choice of

T

we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i.e.,

\begin{matrix} {S_{i j}}^{k} = {L_{[i j]}}^{k} = {T_{[i j]}}^{k} \end{matrix}

(21)

is called torsion ; it is a tensor field. The trace of the torsion tensor

S_{i} : = S_{i l}^{l}

is called torsion vector ; it connects to the two traces of the affine connection

L_{i} : = {L_{i l}}^{l}

;

{\tilde{L}}_{j} : = {L_{l j}}^{l}

, as

S_{i} = \frac{1}{2} (L_{i} - {\tilde{L}}_{i}) .

^$35$ In an arbitrary basis for the differential forms (cotangent space), the connection may be represented by a 1-form.

^$36$ In the literature, different notations and conventions are used. Tonnelat [355] writes $A_{k +; j} : = A_{k, j} - {L_{k j}}^{l} A_{l},$ and $A_{k -; j} : = A_{k, j} - {L_{j k}}^{l} A_{l}$ .

^$37$ Many authors replace “autoparallel” by “geodesic”. We will reserve the name geodesic for curves of extreme length; cf. Riemannian geometry.

^$38$ In the following we will note a symmetrical connection by ${Γ_{i j}}^{k} = {L_{(i j)}}^{k}$ .

2.1.3 Different types of geometry

Affine geometry

Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric, i.e., with

{L_{i j}}^{k} = {Γ_{i j}}^{k}

. In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it. This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by^$39$

\begin{matrix} {\overset{+}{K}}_{j k l}^{i} & = & \partial_{k} L_{l j}^{i} - \partial_{l} L_{k j}^{i} + L_{k m}^{i} L_{l j}^{m} - L_{l m}^{i} L_{k j}^{m}, \end{matrix}

(22)

\begin{matrix} {\bar{K}}_{j k l}^{i} & = & \partial_{k} L_{j l}^{i} - \partial_{l} L_{j k}^{i} + L_{m k}^{i} L_{j l}^{m} - L_{m l}^{i} L_{j k}^{m}, \end{matrix}

(23)

respectively. In a geometry with symmetric affine connection both tensors coincide because of

\begin{matrix} \frac{1}{2} ({\overset{+}{K}}_{j k l}^{i} - {\bar{K}}_{j k l}^{i}) = \partial_{[k} S_{] l j}^{i} + 2 S_{j [k}^{m} S_{l] m}^{i} + L_{m [k}^{i} S_{l] j}^{m} - L_{j [k}^{m} S_{l] m}^{i} . \end{matrix}

(24)

In particular, in Riemannian geometry, both affine curvature tensors reduce to the one and only Riemann curvature tensor.

The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity :

\begin{matrix} {\overset{+}{\nabla}}_{[j} {\overset{+}{\nabla}}_{k]} A^{i} & = & \frac{1}{2} {\overset{+}{K}}_{r j k}^{i} A^{r} - S_{j k}^{r} {\overset{+}{\nabla}}_{r} A^{i} \end{matrix}

(25)

\begin{matrix} {\bar{\nabla}}_{[j} {\bar{\nabla}}_{k]} A^{i} & = & \frac{1}{2} {\bar{K}}_{r j k}^{i} A^{r} + S_{j k}^{r} {\bar{\nabla}}_{r} A^{i} \end{matrix}

(26)

For a vector density, the identity is given by

\begin{matrix} {\overset{+}{\nabla}}_{[j} {\overset{+}{\nabla}}_{k]} {\hat{A}}^{i} = \frac{1}{2} {\overset{+}{K}}_{r j k}^{i} {\hat{A}}^{r} - S_{j k}^{r} {\overset{+}{\nabla}}_{r} {\hat{A}}^{i} + \frac{1}{2} V_{j k} {\hat{A}}^{i}, \end{matrix}

(27)

with the homothetic curvature

V_{j k}

to be defined below in Equation ( 31 ).

The curvature tensor ( 22 ) satisfies two algebraic identities:

\begin{matrix} {\overset{+}{K}}_{j [k l]}^{i} & = & 0, \end{matrix}

(28)

\begin{matrix} {\overset{+}{K}}_{{j k l}}^{i} & = & 2 \nabla_{{j} S_{k l}}^{i} + 4 S_{m {j}^{i} S_{k l}}^{m}, \end{matrix}

(29)

where the curly bracket denotes cyclic permutation:

K_{{j k l}}^{i} : = K_{j k l}^{i} + K_{l j k}^{i} + K_{k l j}^{i} .

These identities can be found in Schouten's book of 1924 ([299] , p. 88, 91) as well as the additional single integrability condition, called Bianchi identity :

\begin{matrix} {\overset{+}{K}}_{j {k l ∥ m}}^{i} = 2 K_{r {k l}^{i} S_{m} j}^{r} . \end{matrix}

(30)

A corresponding condition obtains for the curvature tensor

\bar{K}

from Equation ( 23 ).

From both affine curvature tensors we may form two different tensorial traces each. In the first case

V_{k l} : = K_{i k l}^{i} = V_{[k l]}

, and

K_{j k} : = K_{j k i}^{i} . V_{k l}

is called homothetic curvature, while

K_{j k}

is the first of the two affine generalisations from

\overset{+}{K}

and

\bar{K}

of the Ricci tensor in Riemannian geometry. We get^$40$

\begin{matrix} V_{k l} = \partial_{k} L_{l} - \partial_{l} L_{k}, \end{matrix}

(31)

and the following identities hold:

\begin{matrix} V_{k l} + 2 K_{[k l]} & = & 4 \nabla_{[k} S_{l]} + 8 S_{k l}^{m} S_{m} + 2 \nabla_{m} S_{k l}^{m}, \end{matrix}

(32)

\begin{matrix} {\bar{V}}_{k l} + 2 {\bar{K}}_{[k l]} & = & - 4 {\bar{\nabla}}_{[k} S_{l]} + 8 S_{k l}^{m} S_{m} + 2 \nabla_{m} S_{k l}^{m}, \end{matrix}

(33)

where

S_{k} : = S_{k l}^{l}

. While

V_{k l}

is antisymmetric,

K_{j k}

has both tensorial symmetric and antisymmetric parts:

\begin{matrix} K_{[k l]} & = & - \partial_{[k} {\tilde{L}}_{l]} + \nabla_{m} S_{k l}^{m} + L_{m} S_{k l}^{m} + 2 L_{[l | r}^{m} S_{m | k]}^{r}, \end{matrix}

(34)

\begin{matrix} K_{(k l)} & = & \partial_{(k} {\tilde{L}}_{l)} - \partial_{m} L_{(k l)}^{m} - {\tilde{L}}_{m} L_{(k l)}^{m} + L_{(k | m |}^{n} L_{l) n}^{m} . \end{matrix}

(35)

We use the notation

A_{(i | k | l)}

in order to exclude the index

k

from the symmetrisation bracket^$41$ .

In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor (cf., however, [355] ).

For a symmetric affine connection, the preceding results reduce considerably due to

S_{k l}^{m} = 0

From Equations ( 29 , 30 , 32 ) we obtain the identities:

\begin{matrix} K_{{j k l}}^{i} & = & 0, \end{matrix}

(36)

\begin{matrix} K_{j {k l ∥ m}}^{i} & = & 0, \end{matrix}

(37)

\begin{matrix} V_{k l} + 2 K_{[k l]} & = & 0, \end{matrix}

(38)

i.e., only one independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor

K_{[k l]} = - \partial_{[k} {\tilde{L}}_{l]}

holds. This equation will be important for the physical interpretation of affine geometry.

In affine geometry, the simplest way to define a fundamental tensor is to set

g_{i j} : = α K_{(i j)}

, or

g_{i j} : = α {\bar{K}}_{(i j)}

. It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by

det (K_{i j})

^$42$ .

As a final result in this section, we give the curvature tensor calculated from the connection

{\bar{L}}_{i j}^{k} = Γ_{i j}^{k} + T_{i j}^{k}

(cf. Equation ( 20 )), expressed by the curvature tensor of

Γ_{i j}^{k}

and by the tensor

T_{i j}^{k}

\begin{matrix} K_{j k l}^{i} (\bar{L}) = K_{j k l}^{i} (Γ) + 2^{(Γ)} \nabla_{[k} T_{l] j}^{i} - 2 T_{[k | j |}^{m} T_{l] m}^{i} + 2 S_{k l}^{m} T_{m j}^{i}, \end{matrix}

(39)

where

^{(Γ)} \nabla

is the covariant derivative formed with the connection

Γ_{i j}^{k}

(cf. also [309] , p. 141).

^$39$ Schouten denotes the curvature tensor by ${K_{j k l}}^{i}$ .

^$40$ Again, we have two kinds of homothetic curvatures deriving from $\overset{+}{K}$ and $\bar{K}$ . In the following, we will mostly use the $\overset{+}{X}$ -quantities and drop the $+$ sign.

^$41$ For a three-index-tensor the symmetrisation bracket is defined by $A_{(i k l)} : = \frac{1}{3!} (A_{i k l} + A_{l i k} + A_{k l i} + A_{k i l} + A_{l k i} + A_{i l k}) .$

^$42$ This scalar density was used by Einstein [75] (cf. Section 6.4.4 ).

Mixed geometry

A manifold carrying both structural elements, i.e., metric and connection, is called a metric-affine space. If the first fundamental form is taken to be asymmetric, i.e., to contain an antisymmetric part

g_{[i k]} : = \frac{1}{2} (g_{i j} - g_{j i})

, we speak of a mixed geometry. In principle, both metric-affine space and mixed geometry may always be re-interpreted as Riemannian geometry with additional geometric objects: the 2-form field

φ (f)

(symplectic form), the torsion

S

, and the non-metricity

Q

(cf.

Equation 41 ). It depends on the physical interpretation, i.e., the assumed relation between mathematical objects and physical observables, which geometry is the most suitable.

From the symmetric part of the first fundamental form

h_{i j} = g_{(i j)}

, a connection may be constructed, often called after Levi-Civita^$43$ [203] ,

\begin{matrix} {_{i j}^{k}} : = \frac{1}{2} γ^{k l} (γ_{l i, j} + γ_{l j, i} - h_{i j,}), \end{matrix}

(40)

and from it the Riemannian curvature tensor defined as in Equation ( 22 ) with

L_{i j}^{k} = {_{i j}^{k}}

(cf.

Section 2.1.3 );

{_{i j}^{k}}

is called the Christoffel symbol. Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. In the remaining part of this section we will deal with a symmetric fundamental form

γ_{i j}

only, and denote it by

g_{i j}

With the help of the symmetric affine connection, we may define the tensor of non-metricity

Q_{i j}^{k}

by^$44$

\begin{matrix} Q_{i j}^{k} : = g^{k l} \nabla_{l} g_{i j} . \end{matrix}

(41)

Then the following identity holds:

\begin{matrix} Γ_{i j}^{k} = {_{i j}^{k}} + K_{i j}^{k} + \frac{1}{2} (Q_{i j}^{k} + Q_{j i}^{k} - Q_{j i}^{k}), \end{matrix}

(42)

where the contorsion tensor

K_{i j}^{k}

, a linear combination of torsion

S_{i j}^{k}

, is defined by^$45$

\begin{matrix} K_{i j}^{k} : = S_{j i}^{k} + S_{i j}^{k} - S_{i j}^{k} = - K_{i j}^{k} . \end{matrix}

(43)

The inner product of two tangent vectors

A^{i}, B^{k}

is not conserved under parallel transport of the vectors along

X^{l}

if the non-metricity tensor does not vanish:

\begin{matrix} X^{k} {\overset{+}{\nabla}}_{k} (A^{n} B^{m} g_{n m}) = Q_{n m l} A^{n} B^{m} X^{l} \neq 0 . \end{matrix}

(44)

A connection for which the non-metricity tensor vanishes, i.e.,

\begin{matrix} {\overset{+}{\nabla}}_{k} g_{i j} = 0 \end{matrix}

(45)

holds, is called metric-compatible^$46$ .

J. M. Thomas^$47$ introduced a combination of the terms appearing in

\overset{+}{\nabla}

and

\bar{\nabla}

to define a covariant derivative for the metric ([345] , p. 188),

\begin{matrix} g_{i k / l} : = g_{i k, l} - g_{r k} Γ_{i l}^{r} - g_{i r} Γ_{l k}^{r}, \end{matrix}

(46)

and extended it for tensors of arbitrary rank

\geq 3

Einstein later used as a constraint on the metrical tensor

\begin{matrix} 0 = g_{\overset{i k ∥ l}{+ -}} : = g_{i k, l} - g_{r k} Γ_{i l}^{r} - g_{i r} Γ_{l k}^{r}, \end{matrix}

(47)

a condition that cannot easily be interpreted geometrically [96] . We will have to deal with Equation ( 47 ) in Section 6.1 and, more intensively, in Part II of this review.

Connections that are not metric-compatible have been used in unified field theory right from the beginning. Thus, in Weyl's theory [397, 395] we have

\begin{matrix} Q_{i j k} = Q_{k} g_{i j} . \end{matrix}

(48)

In case of such a relationship, the geometry is called semi-metrical [299, 309] . According to Equation ( 44 ), in Weyl's theory the inner product multiplies by a scalar factor under parallel transport:

\begin{matrix} X^{k} {\overset{+}{\nabla}}_{k} (A^{n} B^{m} g_{n m}) = (Q_{l} X^{l}) A^{n} B^{m} g_{n m} . \end{matrix}

(49)

This means that the light cone is preserved by parallel transport.

We may also abbreviate the last term in the identity ( 42 ) by introducing

\begin{matrix} X_{i j}^{k} : = Q_{i j}^{k} + Q_{j i}^{k} - Q_{j i}^{k} . \end{matrix}

(50)

Then, from Equation ( 39 ), the curvature tensor of a torsionless affine space is given by

\begin{matrix} K_{j k l}^{i} (\bar{Γ}) = K_{j k l}^{i} ({_{n m}^{r}}) + 2^{({_{j k}^{i}})} \nabla_{[k} X_{l] j}^{i} - 2 X_{[k | j |}^{m} X_{l] m}^{i}, \end{matrix}

(51)

where

^{({_{j k}^{i}})} \nabla

is the covariant derivative formed with the Christoffel symbol.

Riemann–Cartan geometry is the subcase of a metric-affine geometry in which the metric-compatible connection contains torsion, i.e., an antisymmetric part

L_{[i j]}^{k}

; torsion is a tensor field to be linked to physical observables. A linear connection whose antisymmetric part

S_{i j}^{k}

has the form

\begin{matrix} S_{i j}^{k} = S_{[i} δ_{j]}^{k} \end{matrix}

(52)

is called semi-symmetric [299] .

Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection. In this case, the connection is derived from the metric:

Γ_{i j}^{k} = {_{i j}^{k}}

, where

{_{i j}^{k}}

is the usual Christoffel symbol ( 40 ). The covariant derivative of

A

with respect to the Levi-Civita connection

\overset{{_{i j}^{k}}}{\nabla}

is abbreviated by

A_{; k}

. The Riemann curvature tensor is denoted by

\begin{matrix} R_{j k l}^{i} = \partial_{k} {_{l j}^{i}} - \partial_{l} {_{k j}^{i}} + {_{k m}^{i}} {_{l j}^{m}} - {_{l m}^{i}} {_{k j}^{m}} . \end{matrix}

(53)

An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:

\begin{matrix} R_{j k l}^{i} (η) = 0 . \end{matrix}

(54)

This is an invariant characterisation irrespective of whether the Minkowski metric

η

is given in Cartesian coordinates as in Equation ( 10 ), or in an arbitrary coordinate system. We also have

ℒ_{X} R_{i j k l} = 0

where

ℒ

is the Lie-derivative (see below under “symmetries”), and

X

stands for the generators of the Lorentz group.

In Riemanian geometry, the so-called geodesic equation,

\begin{matrix} X_{; k}^{i} X^{k} (u) = σ (u) X^{i}, \end{matrix}

(55)

determines the shortest and the straightest curve between two infinitesimally close points.

However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished.

A conformal transformation of the metric,

\begin{matrix} g_{i k} \to g_{i k}^{'} = λ g_{i k}, \end{matrix}

(56)

with a smooth function

λ

changes the components of the non-metricity tensor,

\begin{matrix} Q_{i j}^{k} \to Q_{i j}^{k} + g_{i j} g^{k l} \partial_{l} σ, \end{matrix}

(57)

as well as the Levi-Civita connection,

\begin{matrix} {_{i j}^{k}} \to {_{i j}^{k}} + \frac{1}{2} (σ_{i} δ_{j}^{k} - σ_{j} δ_{i}^{k} + g_{i j} g^{k l} σ_{l}), \end{matrix}

(58)

with

σ_{i} : = λ^{- 1} \partial_{i} λ .

As a consequence, the Riemann curvature tensor

R_{j k l}^{i}

is also changed; if, however,

{R^{'}}_{j k l}^{i} = 0

can be reached by a conformal transformation, then the corresponding space-time is called conformally flat. In

M_{D}

, for

D > 3

, the vanishing of the Weyl curvature tensor

\begin{matrix} C_{j k l}^{i} : = R_{j k l}^{i} + \frac{2}{D - 2} (δ_{[k}^{i} R_{l] j} + g_{j [l} R_{k]}^{i}) + \frac{2 R}{(D - 1) (D - 2)} δ_{[l}^{i} g_{k] j} \end{matrix}

(59)

is a necessary and sufficient condition for

M_{D}

to be conformally flat ([397] , p. 404, [299] , p. 170).

Even before Weyl, the question had been asked (and answered) as to what extent the conformal and the projective structures were determining the geometry: According to Kretschmann (and then to Weyl) they fix the metric up to a constant factor ([195] ; see also [401] , Appendix 1; for a modern approach, cf. [65] ).

The geometry needed for the preand non-relativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form (Newton–Cartan geometry, cf. [63, 64] ). In the following we shall deal only with relativistic unified field theories.

^$43$ Tullio Levi-Civita (1873–1941). Born in Padua, Italy. Studied mathematics and received his doctorate at the University of Padua. Was given the Chair of Mechanics there and, in 1918, went to the University of Rome in the same position.

Together with Ricci, he developed tensor calculus and introduced covariant differentiation. He worked also in the mechanical many-body problem, in hydrodynamics, general relativity theory, and unified field theory. Strongly opposed to Fascism in Italy and dismissed from his professorship in 1938.

^$44$ This definition differs from Schouten's by an overall minus sign.

^$45$ See [156] ; his sign conventions are different, though.

^$46$ In the case of an asymmetric metric, this relation must be satisfied separately by the symmetric and antisymmetric parts of g: $\nabla_{k} h_{i j} = 0, \nabla_{k} φ_{i j} = 0 .$

^$47$ Joseph Miller Thomas (1898–1979). Studied mathematics in Philadelphia at the University of Pennsylvania. Received his doctorate in 1923. From 1927 assistant professor at the University of Pennsylvania, from 1930 assistant and in 1935 full professor of mathematics at the Duke University in Durham, North Carolina. His fields were differential geometry and partial differential equations. He was the principle founder of Duke Mathematical Journal.

Projective geometry

Projective geometry is a generalisation of Riemannian geometry in the following sense: Instead of tangent spaces with the light cone

η_{i k} d x^{i} d x^{k} = 0

, where

η

is the Minkowski metric, in each event now a tangent space with a general, non-degenerate surface of second order

γ

will be introduced. This leads to a tangential cone

g_{i k} d x^{i} d x^{k} = 0

in the origin (cf. Equation ( 9 )), and to a hyperplane, the polar plane, formed by the contact points of the tangential cone and the surface

γ

. In place of the

D

inhomogeneous coordinates

x^{i}

M_{D}

D + 1

homogeneous coordinates

X^{α}

(

α = 0, 1, 2, \dots, D

) are defined^$48$ such that they transform as homogeneous functions of first degree:

\begin{matrix} X^{α} \frac{\partial {X^{'}}^{ν}}{\partial X^{μ}} = {X^{'}}^{ν} . \end{matrix}

(60)

The connection to the inhomogeneous coordinates

x^{i}

is given by homogeneous functions of degree zero, e.g., by

x^{i} = \frac{X^{i}}{φ_{α} X^{α}}

^$49$ . Thus, the

X^{α}

themselves form the components of a tangent vector. Furthermore, the quadratic form

g_{α β} X^{α} X^{β} = ε = \pm 1

is adopted with

g_{α β}

being a homogeneous function of degree

- 2

. A tensor field

{T_{n_{1}}^{m_{1}}}_{n_{2}}^{m_{2}}_{n_{3}}^{m_{3}}_{\dots}^{\dots}

(cf. Section 2.1.5 ) depending on the homogeneous coordinates

X^{μ}

with

u

contravariant (upper) and

l

covariant (lower) indices is required to be a homogeneous function of degree

r : = u - l

If we define

γ_{μ}^{i} : = \frac{\partial x^{i}}{\partial X^{μ}}

, with

γ_{μ}^{i} X^{μ} = 0

, then

γ_{μ}^{i}

transforms like a tangent vector under point transformations of the

x^{i}

, and as a covariant vector under homogeneous transformations of the

X^{α}

The

γ_{μ}^{i}

may be used to relate covariant vectors

a_{i}

and

A_{μ}

A_{μ} = γ_{μ}^{i} a_{i} .

Thus, the metric tensor in the space of homogeneous coordinates

g_{α β}

and the metric tensor

g_{i k}

M_{D}

are related by

g_{i k} = γ_{i}^{α} γ_{k}^{β} g_{α β}

with

γ_{μ}^{i} γ_{k}^{μ} = δ_{k}^{i} .

The inverse relationship is given by

g_{α β} = γ_{α}^{i} γ_{β}^{k} g_{i k} + ε X_{α} X_{β}

with

X_{α} = g_{α β} X^{β} .

The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf. Section 2.1.2 ):

\begin{matrix} \nabla_{α} A^{β} (X) = \frac{\partial A^{β} (X)}{\partial X^{α}} + Γ_{α ν}^{β} (X) A^{ν} (X) . \end{matrix}

(61)

The covariant derivative of the quantity

γ_{k}^{μ}

interconnecting both spaces is given by

\begin{matrix} \nabla_{ρ} γ_{k}^{μ} = \frac{\partial γ_{k}^{μ}}{\partial x^{ρ}} + Γ_{ρ σ}^{μ} γ_{k}^{σ} - {_{k l}^{m}} γ_{ρ}^{l} γ_{m}^{μ} . \end{matrix}

(62)

^$48$ The index 0 does not refer to a time-coordinate.

^$49$ Note that $X^{α}$ and $λ X^{α}$ correspond to the same point of $M_{D}$ ; Veblen uses $λ = exp (N x^{0})$ (see [379] , Section II).

2.1.4 Cartan's method

In this section, we briefly present Cartan's one-form formalism in order to make understandable part of the literature. Cartan introduces one-forms

θ^{\hat{a}}

(

\hat{a} = 1, \dots, 4

) by

θ^{\hat{a}} : = h_{l}^{\hat{a}} d x^{l} .

The reciprocal basis in tangent space is given by

e_{\hat{j}} = h_{\hat{j}}^{l} \frac{\partial}{\partial x^{l}}

. Thus,

θ^{\hat{a}} (e_{\hat{j}}) = δ_{\hat{j}}^{\hat{a}}

. The metric is then given by

η_{\hat{ı} \hat{k}} θ^{\hat{ı}} \otimes θ^{\hat{k}}

. The covariant derivative of a tangent vector with bein-components

X^{\hat{k}}

is defined via Cartan's first structure equations,

\begin{matrix} Θ^{i} : = D θ^{\hat{ı}} = d θ^{\hat{ı}} + ω_{\hat{l}}^{\hat{ı}} \land θ^{\hat{l}}, \end{matrix}

(63)

where

ω_{\hat{k}}^{\hat{ı}}

is the connection-1-form, and

Θ^{\hat{ı}}

is the torsion-2-form,

Θ^{\hat{ı}} = - S_{\hat{l} \hat{m}}^{\hat{ı}} θ^{\hat{l}} \land θ^{\hat{m}}

. We have

ω_{\hat{ı} \hat{k}} = - ω_{\hat{k} \hat{ı}} .

The link to the components

L_{[i j]}^{k}

of the affine connection is given by

ω_{\hat{k}}^{\hat{ı}} = h_{l}^{\hat{ı}} h_{\hat{k}}^{m} L_{\hat{r} m}^{l} θ^{\hat{r}}

^$50$ . The covariant derivative of a tangent vector with bein-components

X^{\hat{k}}

then is

\begin{matrix} D X^{\hat{k}} : = d X^{\hat{k}} + ω_{\hat{l}}^{\hat{k}} X^{\hat{l}} . \end{matrix}

(64)

By further external derivation^$51$ on

Θ

we arrive at the second structure relation of Cartan,

\begin{matrix} D Θ^{\hat{k}} = Ω_{\hat{l}}^{\hat{k}} \land θ^{\hat{l}} . \end{matrix}

(65)

In Equation ( 65 ) the curvature-2-form

Ω_{\hat{l}}^{\hat{k}} = \frac{1}{2} R_{\hat{l} \hat{m} \hat{n}}^{\hat{k}} θ^{\hat{m}} \land θ^{\hat{n}}

appears, which is given by

\begin{matrix} Ω_{\hat{l}}^{\hat{k}} = d ω_{\hat{l}}^{\hat{k}} + ω_{\hat{l}}^{\hat{k}} \land ω_{\hat{l}}^{\hat{k}} . \end{matrix}

(66)

Ω_{\hat{k}}^{\hat{k}}

is the homothetic curvature.

^$50$ For an asymmetric connection, this corresponds to the + derivative.

^$51$ The external derivative $d$ of linear forms $ω, μ$ satisfies the following rules:

(1) $d (a ω + b μ) = a d ω + b d μ$ ,
(2) $d (ω \land μ) = d ω \land μ - ω \land d μ$ ,
(3) $d d ω = 0$ .

2.1.5 Tensors, spinors, symmetries

Tensors

Up to here, no definitions of a tensor and a tensor field were given: A tensor

T_{p} (M_{D})

of type

(r, s)

at a point

p

on the manifold

M_{D}

is a multi-linear function on the Cartesian product of

r

cotangentand

s

tangent spaces in

p

. A tensor field is the assignment of a tensor to each point of

M_{D}

. Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:

\begin{matrix} T_{l_{1}^{'} l_{2}^{'} l_{3}^{'} \dots}^{k_{1}^{'} k_{2}^{'}, k_{3}^{'} \dots} = T_{n_{1} n_{2} n_{3} \dots}^{m_{1} m_{2} m_{3} \dots} \frac{\partial x^{n_{1}}}{\partial x^{l_{1}^{'}}} \frac{\partial x^{n_{2}}}{\partial x^{l_{2}^{'}}} \frac{\partial x^{n_{3}}}{\partial x^{l_{3}^{'}}} \frac{\partial x^{k_{1}^{'}}}{\partial x^{m_{1}}} \frac{\partial x^{k_{2}^{'}}}{\partial x^{m_{2}}} \frac{\partial x^{k_{3}^{'}}}{\partial x^{m_{3}}} \dots \end{matrix}

(67)

where

x^{k^{'}} = x^{k^{'}} (x^{i})

with smooth functions on the r.h.s. are taken from the set (“group”) of coordinate transformations (diffeomorphisms). Strictly speaking, tensors are representations of the abstract group at a point on the manifold^$52$ .

A relative tensor

T_{p} (M_{D})

of type

(r, s)

and of weight

ω

at a point

p

on the manifold

M_{D}

transforms like

\begin{matrix} T_{l_{1}^{'} l_{2}^{'} l_{3}^{'} . .}^{k_{1}^{'} k_{2}^{'}, k_{3}^{'} . .} = {[det (\frac{\partial x^{s}}{\partial {x^{'}}^{r}})]}^{ω} T_{n_{1} n_{2} n_{3} . . .}^{m_{1} m_{2} m_{3} . . .} \frac{\partial x^{n_{1}}}{\partial x^{l_{1}^{'}}} \frac{\partial x^{n_{2}}}{\partial x^{l_{2}^{'}}} \frac{\partial x^{n_{3}}}{\partial x^{l_{3}^{'}}} \frac{\partial x^{k_{1}^{'}}}{\partial x^{m_{1}}} \frac{\partial x^{k_{2}^{'}}}{\partial x^{m_{2}}} \frac{\partial x^{k_{3}^{'}}}{\partial x^{m_{3}}} \dots \end{matrix}

(68)

An example is given by the totally antisymmetric object

ε_{i j k l}

with

ε_{i j k l} = \pm 1

, or

ε_{i j k l} = 0

depending on whether

(i j k l)

is an even or odd permutation of (0123), or whether two indices are alike.

ω = - 1

for

ε_{i j k l}

; in this case, the relative tensor is called tensor density. We can form a tensor from

ε_{i j k l}

by introducing

η_{i j k l} : = \sqrt{- g} ε_{i j k l}

, where

g_{i k}

is a Lorentz-metric. Note that

η^{i j k l} : = \frac{1}{\sqrt{- g}} ε^{i j k l} .

The dual to a 2-form (skew-symmetric tensor) then is defined by

^{*} F^{i j} = \frac{1}{2} η^{i j k l} F_{k l} .

In connection with conformal transformations

g \to λ g

, the concept of the gauge-weight of a tensor is introduced. A tensor

T_{\dots}^{\dots}

is said to be of gauge weight

q

if it transforms by Equation ( 56 ) as

\begin{matrix} {T^{'}}_{\dots}^{\dots} = λ^{q} T_{\dots}^{\dots} . \end{matrix}

(69)

Objects that transform as in Equation ( 67 ) but with respect to a subgroup, e.g., the linear group, affine group

G (D)

, orthonormal group

O (D)

, or the Lorentz group

ℒ

, are tensors in a restricted sense; sometimes they are named affine or Cartesian tensors. All the subgroups mentioned are Lie-groups, i.e., continuous groups with a finite number of parameters. In general relativity, both the “group” of general coordinate transformations and the Lorentz group are present. The concept of tensors used in Special Relativity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinate-independent (“generally covariant”) form, then the full tensor concept comes into play.

^$52$ A representation of a group is defined as a map to the vectors of a linear space that is homomorphic in the group operation.

Spinors

Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold. To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group

S L (2, C)

and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group^$53$ . Let

A \in S L (2, C)

; then

A

is a complex (2-by-2)-matrix with

det A = 1

. By picking the special Hermitian matrix

\begin{matrix} S = x^{0} 1 + \sum_{p} σ_{p} x^{p}, \end{matrix}

(70)

where

1

is the (2 by 2)-unit matrix and

σ_{p}

are the Pauli matrices satisfying

\begin{matrix} σ_{i} σ_{k} + σ_{k} σ_{i} = 2 δ_{i k} . \end{matrix}

(71)

Then, by a transformation

A

from

S L (2, C)

\begin{matrix} S^{'} = A S A^{+}, \end{matrix}

(72)

where

A^{+}

is the Hermitian conjugate matrix^$54$ . Moreover,

det S = det S^{'}

which, according to Equation ( 70 ), expresses the invariance of the space-time distance to the origin:

\begin{matrix} (x^{0^{'}})^{2} - (x^{1^{'}})^{2} - (x^{2^{'}})^{2} - (x^{3^{'}})^{2} = (x^{0})^{2} - (x^{1})^{2} - (x^{2})^{2} - (x^{3})^{2} . \end{matrix}

(73)

The link between the representation of a Lorentz transformation

L_{i k}

in space-time and the unimodular matrix

A

mapping spin space (cf. below) is given by

\begin{matrix} L (A)_{i k} = \frac{1}{2} t r (σ_{i} A σ_{k} A^{+}) . \end{matrix}

(74)

Thus, the map is two to one:

+ A

and

- A

give the same

L_{i k}

Now, contravariant 2-spinors

ξ^{A}

(

A = 1, 2

) are the elements of a complex linear space, spinor space, on which the matrices

A

are acting^$55$ .

The spinor is called elementary if it transforms under a Lorentz-transformation as

\begin{matrix} ξ^{A^{'}} = \pm A_{C}^{A} ξ^{C} . \end{matrix}

(75)

Likewise, contravariant dotted spinors

ζ^{\dot{A}}

are those transforming with the complex-conjugate matrix

\bar{A}

\begin{matrix} ζ^{{\dot{B}}^{'}} = \pm {\bar{A}}_{\dot{D}}^{\dot{B}} ζ^{\dot{C}} . \end{matrix}

(76)

Covariant and covariant dotted 2-spinors correspondingly transform with the inverse matrices,

\begin{matrix} ξ_{B^{'}} = \pm (A^{- 1})_{B}^{C} ξ_{C}, \end{matrix}

(77)

and

\begin{matrix} ξ_{{\dot{B}}^{'}} = \pm ({\bar{A}}^{- 1})_{\dot{B}}^{\dot{C}} ξ_{\dot{C}} . \end{matrix}

(78)

The space of 2-spinors can be used as a representation space for the (proper, orthochronous) Lorentz group, with the 2-spinors being the elements of the most simple representation

D^{(1 / 2, 0)}

Higher-order spinors with dotted and undotted indices

S_{C \dots \dot{D} \dots}^{A \dots \dot{B} \dots}

transform correspondingly.

For the raising and lowering of indices now a real, antisymmetric

(2 \times 2)

-matrix

ε

with components

ε^{A B} = δ_{1}^{A} δ_{B}^{2} - δ_{2}^{A} δ_{B}^{1} = ε_{A B}

is needed, such that

\begin{matrix} ξ^{A} = ε^{A B} ξ_{B}, ξ_{A} = ξ^{B} ε_{B A} . \end{matrix}

(79)

Next to a spinor, bispinors of the form

ζ^{A B}, ξ^{A \dot{B}}

, etc. are the simplest quantities (spinors of 2nd order). A vector

X^{k}

can be represented by a bispinor

X^{A \dot{B}}

\begin{matrix} X^{A \dot{B}} = σ_{k}^{A \dot{B}} X^{k}, \end{matrix}

(80)

where

σ_{k}^{A \dot{B}}

(

k = 0, \dots, 3

) is a quantity linking the tangent space of space-time and spinor space.

k

numerates the matrices and

A

\dot{B}

designate rows and columns, then we can chose

σ_{0}^{A \dot{B}}

to be the unit matrix while for the other three indices

σ_{j}^{A \dot{B}}

are taken to be the Pauli matrices. Often the quantity

s_{k}^{A \dot{B}} = \frac{1}{\sqrt{2}} σ_{k}^{A \dot{B}}

is introduced. The reciprocal matrix

s_{A \dot{B}}^{k}

is defined by

\begin{matrix} s_{j}^{A \dot{B}} s_{A \dot{B}}^{k} = δ_{j}^{k}, \end{matrix}

(81)

whereas

\begin{matrix} s_{j}^{A \dot{B}} s^{j C \dot{D}} = ε^{A C} ε^{\dot{B} \dot{D}} . \end{matrix}

(82)

In order to write down spinorial field equations, we need a spinorial derivative,

\begin{matrix} \partial_{A \dot{B}} = s_{A \dot{B}}^{k} \partial_{k} \end{matrix}

(83)

with

\partial_{A \dot{B}} \partial^{A \dot{B}} = \partial_{k} \partial^{k} .

The simplest spinorial equation is the Weyl equation:

\begin{matrix} \partial_{A \dot{B}} ψ^{A} = 0, \dot{B} = 1, 2 . \end{matrix}

(84)

The next simplest spinor equation for two spinors

χ^{\dot{B}}, ψ_{A}

would be

\begin{matrix} \partial_{A \dot{C}} χ^{\dot{C}} = - \frac{2 π}{\sqrt{2} h} m ψ_{A}; \partial^{C \dot{B}} ψ_{C} = \frac{2 π}{\sqrt{2} h} m χ^{\dot{B}}, \end{matrix}

(85)

where

m

is a mass. Equation ( 85 ) is the 2-spinor version of Dirac's equation.

Dirac- or 4-spinors with 4 components

ψ^{k}

k = 1, \dots, 4

, may be constructed from 2-spinors as a direct sum of contravariant undotted and covariant dotted spinors

ψ

and

φ

: For

k = 1, 2

, we enter

ψ^{1}

and

ψ^{2}

; for

k = 3, 4

, we enter

φ_{\dot{1}}

and

φ_{\dot{2}} .

In connection with Dirac spinors, instead of the Pauli-matrices the Dirac

γ

-matrices (

4 \times 4

-matrices) appear; they satisfy

\begin{matrix} γ^{i} γ^{k} + γ^{k} γ^{i} = 2 η^{i k} 1 . \end{matrix}

(86)

The Dirac equation is in 4-spinor formalism [51, 52] :

\begin{matrix} (i γ^{l} \frac{\partial}{\partial x^{l}} + κ) χ = 0, \end{matrix}

(87)

with the 4-component Dirac spinor

χ

. In the first version of Dirac's equation,

α

and

β

-matrices were used, related to the

γ

's by

\begin{matrix} γ^{0} = β, γ^{m} = β α^{m}, m = 1, 2, 3, \end{matrix}

(88)

where the matrices

β

and

α^{m}

are given by

(\begin{matrix} 0 & - σ_{i} \\ σ_{i} & 0 \end{matrix})

(\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})

The generally-covariant formulation of spinor equations necessitates the use of

n

-beins

h_{\hat{ı}}^{k}

, whose internal “rotation” group, operating on the “hatted” indices, is the Lorentz group. The group of coordinate transformations acts on the Latin indices. In Cartan's one-form formalism (cf.

Section 2.1.4 ), the covariant derivative of a 4-spinor is defined by

\begin{matrix} D ψ = d ψ + \frac{1}{4} ω_{\hat{ı} \hat{k}} σ^{\hat{ı} \hat{k}} ψ, \end{matrix}

(89)

where

σ^{\hat{ı} \hat{k}} : = \frac{1}{2} [γ^{\hat{ı}} γ^{\hat{k}}]

Equation ( 89 ) is a special case of the general formula for the covariant derivative of a tensorial form

ψ

, i.e., a vector in some vector space

V

, whose components are differential forms,

\begin{matrix} D ψ = d ψ + ω^{\hat{ı} \hat{k}} ρ_{\hat{ı} \hat{k}} (e_{α}) ψ, \end{matrix}

(90)

where

ρ (e_{α})

is a particular representation of the corresponding Lie algebra in

V

with basis vectors

e_{α} .

For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used^$56$ .

^$53$ The full Lorentz group contains as further elements the temporal and spatial reflections.

^$54$ $X_{A B}^{+} = X_{B A}; A, B = 1, 2 .$

^$55$ 2-component spinors are also called Weyl-spinors.

^$56$ For a contemporary exposition of the use of spinors in space-time, see the book of Penrose and Rindler [255] .

Symmetries

In Section 2.1.1 we briefly met the Lie derivative of a vector field

ℒ_{X}

with respect to the tangent vector

X

defined by

(L_{X} Y)^{k} : = ([X, Y])^{k} = X^{i} \partial_{i} Y^{k} - Y^{i} \partial_{i} X^{k}

. With its help we may formulate the concept of isometries of a manifold, i.e., special mappings, also called “motions”, locally generated by vector fields

X

satisfying

\begin{matrix} ℒ_{X} g_{i k} : = \partial_{k} (g_{i j}) X^{k} + g_{l j} \partial_{i} X^{l} + g_{i l} \partial_{j} X^{l} = 0 . \end{matrix}

(91)

The generators

X

solving Equation ( 91 ) given some metric, form a Lie group

G_{r}

, the group of motions of

M_{D}

. If a group

G_{r}

is prescribed, e.g., the group of spatial rotations

O (3)

, then from Equation ( 91 ) the functional form of the metric tensor having

O (3)

as a symmetry group follows.

A Riemannian space is called (locally) stationary if it admits a timelike Killing vector; it is called (locally) static if this Killing vector is hypersurface orthogonal. Thus if, in a special coordinate system, we take

X^{i} = δ_{0}^{i}

then from Equation ( 91 ) we conclude that stationarity reduces to the condition

\partial_{0} g_{i k} = 0

. If we take

X

to be the tangent vector field to the congruence of curves

x^{i} = x^{i} (u)

, i.e., if

X^{k} = \frac{d x^{k}}{d u}

, then a necessary and sufficient condition for hypersurface-orthogonality is

ε^{i j k l} X_{j} X_{[k, l]} = 0 .

A generalisation of Killing vectors are conformal Killing vectors for which

ℒ_{X} g_{i k} = Φ g_{i k}

with an arbitrary smooth function

Φ

holds. In purely affine spaces, another type of symmetry may be defined:

ℒ_{X} Γ_{i k}^{l} = 0

; they are called affine motions [425] .

2.2 Dynamics

Within a particular geometry, usually various options for the dynamics of the fields (field equations, in particular as following from a Lagrangian) exist as well as different possibilities for the identification of physical observables with the mathematical objects of the formalism. Thus, in general relativity, the field equations are derived from the Lagrangian

ℒ = \sqrt{- g} (R + 2 Λ - 2 κ L_{M}),

where

R (g_{i k})

is the Ricci scalar,

g : = det g_{i k}, Λ

the cosmological constant, and

L_{M}

the matter Lagrangian depending on the metric, its first derivatives, and the matter variables. This Lagrangian leads to the well-known field equations of general relativity,

\begin{matrix} R^{i k} - \frac{1}{2} R g^{i k} = - κ T^{i k}, \end{matrix}

(92)

with the energy-momentum(-stress) tensor of matter

\begin{matrix} T^{i k} : = \frac{2}{\sqrt{- g}} \frac{δ (\sqrt{- g} L_{M})}{δ g_{i k}} \end{matrix}

(93)

and

κ = \frac{8 π G}{c^{4}}

, where

G

is Newton's gravitational constant.

G^{i k} : = R^{i k} - \frac{1}{2} R g^{i k}

is called the Einstein tensor. In empty space, i.e., for

T^{i k} = 0

, Equation ( 92 ) reduces to

\begin{matrix} R^{i k} = 0 . \end{matrix}

(94)

If only an electromagnetic field

F_{i k} = \frac{\partial A_{k}}{\partial x^{i}} - \frac{\partial A_{i}}{\partial x^{k}}

derived from the 4-vector potential

A_{k}

is present in the energy-momentum tensor, then the Einstein–Maxwell equations follow:

\begin{matrix} R^{i k} - \frac{1}{2} R g^{i k} = - κ (F_{i l} F_{k}^{l} + \frac{1}{4} g_{i k} F_{l m} F^{l m}), \nabla_{l} F^{i l} = 0 . \end{matrix}

(95)

The components of the metrical tensor are identified with gravitational potentials. Consequently, the components of the (Levi-Civita) connection correspond to the gravitational “field strength”, and the components of the curvature tensor to the gradients of the gravitational field. The equations of motion of material particles should follow, in principle, from Equation ( 92 ) through the relation

\begin{matrix} \nabla_{l} T^{i l} = 0 \end{matrix}

(96)

implied by it^$57$ . For point particles, due to the singularities appearing, in general this is a tricky task, up to now solved only approximately.

However, the world lines for point particles falling freely in the gravitational field are, by definition, the geodesics of the Riemannian metric. This definition is consistent with the rigourous derivation of the geodesic equation for non-interacting dust particles in a fluid matter description. It is also consistent with all observations.

For most of the unified field theories to be discussed in the following, such identifications were made on internal, structural reasons, as no link-up to empirical data was possible. Due to the inherent wealth of constructive possibilities, unified field theory never would have come off the ground proper as a physical theory even if all the necessary formal requirements could have been satisfied. As an example, we take the identification of the electromagnetic field tensor with either the skew part of the metric, in a “mixed geometry” with metric compatible connection, or the skew part of the Ricci tensor in metric-affine theory, to list only two possibilities. The latter choice obtains likewise in a purely affine theory in which the metric is a derived secondary concept. In this case, among the many possible choices for the metric, one may take it proportional to the variational derivative of the Lagrangian with respect to the symmetric part of the Ricci tensor. This does neither guarantee the proper signature of the metric nor its full rank. Several identifications for the electromagnetic 4-potential and the electric current vector density have also been suggested (cf. below and [142] ).

^$57$ In the early papers on general relativity, Equation ( 96 ) was called a “conservation law” because, in Minkowski space, it implies the conservation laws for energy and linear momentum.

For an arbitrary Riemannian manifold this no longer holds true.

2.3 Number field

Complex fields may also be introduced on a real manifold. Such fields have also been used for the construction of unified field theories, although mostly after the period dealt with here (cf.

Part II, in preparation). In particular, manifolds with a complex fundamental form were studied, e.g., with

g_{i k} = s_{i k} + i a_{i k}

, where

i = \sqrt{- 1}

[96] . Also, geometries based on Hermitian forms were studied [312] . In later periods, hypercomplex numbers, quaternions, and octonions also were used as basic number fields for gravitational or unified theories (cf. Part II, forthcoming).

In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive, e.g., at complex manifolds and so on. In this part of the article we do not need to take into account this generalisation.

2.4 Dimension

Since the suggestions by Nordström and Kaluza [237, 180] , manifolds with

D > 4

have been used for unified field theories. In most of the cases, the additional dimensions were taken to be spacelike; nevertheless, manifolds with more than one direction of time also have been studied.

3 Early Attempts at a Unified Field Theory

3.1 First steps in the development of unified field theories

Even before (or simultaneously with) the introduction and generalisation of the concept of parallel transport and covariant derivative by Hessenberg (1916/17) [159] , Levi-Civita (1917), [203] , Schouten (1918) [293] , Weyl (1918) [397] , and König (1919) [192] , the introduction of an asymmetric metric was suggested by Rudolf Förster^$58$ in 1917. In his letter to Einstein of 11 November 1917, he writes ([320] , Doc. 398, p. 552):

“Perhaps, there exists a covariant 6-vector by which the appearance of electricity is explained and which springs lightly from the $g_{μ ν}$ , not forced into it as an alien element.”^$59$ “Vielleicht findet sich ein kovarianter Sechservektor der das Auftreten der Elektrizität erklärt und ungezwungen aus den $g_{μ ν}$ herauskommt, nicht als fremdes Element herangetragen wird.”

Einstein replied:

“The aim of dealing with gravitation and electricity on the same footing by reducing both groups of phenomena to $g_{μ ν}$ has already caused me many disappointments. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence. But it is easier to suspect something than to discover it.”^$60$ “Das Ziel, Gravitation und Elektromagnetismus einheitlich zu behandeln, indem man beide Phänomengruppen auf die $g_{μ ν}$ zurückführt, hat mir schon viele erfolglose Bemühungen gekostet. Vielleicht sind Sie glücklicher im Suchen. Ich bin fest überzeugt, dass letzten Endes alle Feldgrössen sich als wesensgleich herausstellen werden. Aber leichter ist ahnen als finden.” (16 November 1917 [320] , Vol. 8A, Doc. 400, p. 557)

In his next letter, Förster gave results of his calculations with an asymmetric

g_{μ ν} = s_{μ ν} + a_{μ ν}

, introduced an asymmetric “three-index-symbol” and a possible generalisation of the Riemannian curvature tensor as well as tentative Maxwell's equations and interpretations for the 4-potential

A_{μ}

, and special solutions (28 December 1917) ([320] , Volume 8A, Document 420, pp. 581–587).

Einstein's next letter of 17 January 1918 is skeptical:

“Since long, I also was busy by starting from a non-symmetric $g_{μ ν}$ ; however, I lost hope to get behind the secret of unity (gravitation, electromagnetism) in this way. Various reasons instilled in me strong reservations: [...] your other remarks are interesting in themselves and new to me.”^$61$ “Das Ausgehen von einem nichtsymmetrischen $g_{μ ν}$ hat mich auch schon lange beschäftigt; ich habe aber die Hoffnung aufgegeben, auf diese Weise hinter das Geheimnis der Einheit (Gravitation–Elektromagnetismus) zu kommen. Verschiedene Gründe flössen da schwere Bedenken ein: [...] Ihre übrigen Bemerkungen sind ebenfalls an sich interessant und mir neu.” ([320] , Volume 8B, Document 439, pp. 610–611)

Einstein's remarks concerning his previous efforts must be seen under the aspect of some attempts at formulating a unified field theory of matter by G. Mie [228, 229, 230] ^$62$ , J. Ishiwara, and G. Nordström, and in view of the unified field theory of gravitation and electromagnetism proposed by David Hilbert.

“According to a general mathematical theorem, the electromagnetic equations (generalized Maxwell equations) appear as a consequence of the gravitational equations, such that gravitation and electrodynamics are not really different.”^$63$ “In Folge eines allgem. math. Satzes erscheinen die elektrody. Gl. (verallgemeinerte Maxwellsche) als math. Folge der Gravitationsgl., so dass Gravitation und Elektrodynamik eigentlich garnicht verschiedenes sind.” (letter of Hilbert to Einstein of 13 November 1915 [161] )

The result is contained in (Hilbert 1915, p. 397)^$64$ .

Einstein's answer to Hilbert on 15 November 1915 shows that he had also been busy along such lines:

“Your investigation is of great interest to me because I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism. The hints dropped by you on your postcards bring me to expect the greatest.”^$65$ “Ihre Untersuchung interessiert mich gewaltig, zumal ich mir oft schon das Gehirn zermartert habe, um eine Brücke zwischen Gravitation und Elektromagnetik zu schlagen. Die Andeutungen, welche Sie auf Ihren Karten geben, lassen das Grösste erwarten.” [100]

Even before Förster alias Bach corresponded with Einstein, a very early bird in the attempt at unifying gravitation and electromagnetism had published two papers in 1917, Reichenbächer [269, 268] .

His paper amounts to a scalar theory of gravitation with field equation

R = 0

instead of Einstein's

R_{a b} = 0

outside the electrons. The electron is considered as an extended body in the sense of Lorentz–Poincaré, and described by a metric joined continuously to the outside metric^$66$ :

\begin{matrix} d s^{2} = d r^{2} + r^{2} d φ^{2} + r^{2} {cos}^{2} φ d ψ^{2} + {(1 - \frac{α}{r})}^{2} d x_{0}^{2} . \end{matrix}

(97)

Reichenbächer, at this point, seems to have had a limited understanding of general relativity: He thinks in terms of a variable velocity of light; he equates coordinate systems and reference systems, and apparently considers the transition from the Minkowskian to a non-flat metric as achieved by a coordinate rotation, a “Drehung gegen den Normalzustand” (“rotation with respect to the normal state”) ([269] , p. 137). According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:

“The disturbance, which is generated by the electrons and which forces us to adopt a coordinate system different from the usual one, is interpreted as the electromagnetic six-vector, as is known.”^$67$ “Die Störung, die durch die Elektronen erzeugt wird und uns also zur Annahme eines von dem gewöhnlichen abweichenden Weltkoordinatensystem zwingt, wird nun bekanntlich als der elektromagnetische Sechservektor aufgefasst.” ([269] , p. 136)

By his “coordinate rotation”, or, as he calls it in ([268] , p. 174), “electromagnetic rotation”, he tries to geometrize the electromagnetic field. As Weyl's remark in Raum–Zeit–Materie ([398] , p. 267, footnote 30) shows, he did not grasp Reichenbächer's reasoning; I have not yet understood it either. Apparently, for Reichenbächer the metric deviation from Minkowski space is due solely to the electromagnetic field, whereas gravitation comes in by a single scalar potential connected to the velocity of light. He claims to obtain the same value for the perihelion shift of Mercury as Einstein ([268] , p. 177). Reichenbächer was slow to fully accept general relativity; as late as in 1920 he had an exchange with Einstein on the foundations of general relativity [270, 69] .

After Reichenbächer had submitted his paper to Annalen der Physik and seemingly referred to Einstein,

“Planck was uncertain to which of Einstein's papers Reichenbächer appealed. He urged that Reichenbächer speak with Einstein and so dissolve their differences. The meeting was amicable. Reichenbächer's paper appeared in 1917 as the first attempt at a unified field theory in the wake of Einstein's covariant field equations.” ([261] , p. 208)

In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [125] . Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [273] .

^$58$ Förster published under a nom de plume “R. Bach” because his employer Krupp did not like his employees using their free time on something as academical as research in gravitation and unified field theory. Bach wrote also about Weyl's theory (Bach 1921) [2] .

^$62$ In his textbook, M. von Laue presented and discussed Mie's theory [387] .

^$64$ See also the Diploma thesis by König [190] . In it it is shown that from the divergence relation $T^{μ ν};_{ν} = 0$ and the most general Lagrangian $L (u, v)$ with $u = F_{μ ν} F^{μ ν} a n d v =^{*} F_{μ ν} F^{μ ν}$ , the field equations follow only for the generic case of full rank of the electromagnetic field tensor $F_{μ ν}$ .

^$66$ Reichenbächer's solution is a special case of a huge number of spherically symmetric solutions of $R = 0$ given in Goenner and Havas 1980 [143] . Reichenbächer published 29 papers between 1917 and 1930.

3.2 Early disagreement about how to explain elementary particles by field theory

In his book on Einstein's relativity, Max Born, in 1920, had asked about the forces hindering “an electron or an atom” to disintegrate.

“Now, these objects are tremendous concentrations of energy in the smallest place; therefore, they will house huge curvatures of space or, in other words, gravitational fields. The idea that they keep together the dispersing electrical charges lies close at hand.”^$68$ “Nun sind diese Gebilde ungeheure Anhäufungen von Energie auf kleinsten Räumen; daher werden sie gewaltige Raumkrümmungen, oder mit anderen Worten, Gravitationsfelder, in sich bergen. Der Gedanke liegt nahe, dass diese es sind, die die auseinanderstrebenden elektrischen Ladungen zusammenhalten.” ([17] , p. 235)

Thus, the idea of a program for building the extended constituents of matter from the fields the source of which they are, was very much alive around 1920. However, Pauli's remark after Weyl's lecture in Bad Nauheim (86. Naturforscherversammlung, 19–25 September 1920) [244] showed that not everybody was a believer in it. He claimed that in bodies smaller than those carrying the elementary charge (electrons), an electric field could not be measured. There was no point of creating the “interior” of such bodies with the help of an electric field. Pauli:

“None of the present theories of the electron, also not Einstein's (Einstein 1919 [68] ), up to now did achieve solving satisfactorily the problem of the electrical elementary quanta; it seems obvious to look for a deeper reason for this failure. I wish to see this reason in the fact that it is altogether not permitted to describe the electromagnetic field in the interior of an electron as a continuous space function. The electrical field is defined as the force on a charged test particle, and if no smaller test particles exist than the electron (vice versa the nucleus), the concept of electrical field at a certain point in the interior of the electron – with which all continuum theories are working – seems to be an empty fiction, because there are no arbitrarily small measures. Therefore, I'd like to ask Mr. Einstein whether he approves of the opinion that a solution of the problem of matter may be expected only from a modification of our perception of space (perhaps also of time) and of electricity in the sense of atomism, or whether he thinks that the mentioned reservations are unconvincing and is of the opinion that the fundaments of continuum theory must be upheld.”^$69$ “Keiner der bisherigen Theorien des Elektrons, auch nicht der Einsteinschen (Einstein 1919 [68] ) ist es bisher gelungen, das Problem der elektrischen Elementarquante befriedigend zu lösen, und es liegt nahe, nach einem tieferen Grund für diesen Mißerfolg zu suchen.
Ich möchte nun diesen Grund darin suchen, daß es überhaupt unstatthaft ist, das elektrische Feld im Innern des Elektrons als stetige Raumfunktion zu beschreiben. Die elektrische Feldstärke ist definiert als die Kraft auf einen geladenen Probekörper und, wenn es keine kleineren Probekörper gibt als das Elektron (bzw den N-Kern), scheint der Begriff der elektrischen Feldstärke in einem bestimmten Punkt im Innern des Elektrons, mit welchem alle Kontinuumstheorien operieren, eine leere, inhaltslose Fiktion zu sein, da es keine beliebig kleinen Maßstäbe gibt. Ich möchte deshalb Herrn Einstein fragen, ob er der Auffassung zustimmt, daß man die Lösung des Problems der Materie nur von einer Modifikation unserer Vorstellungen vom Raum (vielleicht auch von der Zeit) und vom elektrischen Felde im Sinne des Atomismus erwarten darf, oder ob er die angeführten Bedenken nicht für stichhaltig hält und die Ansicht vertritt, daß man an den Grundlagen der Kontinuumstheorie festhalten muß.”

Pauli referred to Einstein's paper about elementary particles and field theory in which he had exchanged his famous field equations for traceless equations with the electromagnetic field tensor as a source. Einstein's answer is tentative and evasive: We just don't know yet^$70$ .

“With the progressing refinement of scientific concepts, the manner by which concepts are related to (physical) events becomes ever more complicated. If, in a certain stage of scientific investigation, it is seen that a concept can no longer be linked with a certain event, there is a choice to let the concept go, or to keep it; in the latter case, we are forced to replace the system of relations among concepts and events by a more complicated one. The same alternative obtains with respect to the concepts of timeand space-distances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me.”^$71$ “Mit fortschreitender Verfeinerung des wissenschaftlichen Begriffssystems wird die Art und Weise der Zuordnung der Begriffe von den Erlebnissen immer komplizierter. Hat man in einem gewissen Studium der Wissenschaft gesehen, daß einem Begriff ein bestimmtes Erlebnis nicht mehr zugeordnet werden kann, so hat man die Wahl, ob man den Begriff fallen lassen oder ihn beibehalten will; in letzterem Fall ist man aber gezwungen, das System der Zuordnung der Begriffe zu den Erlebnissen durch ein komplizierteres zu ersetzen. Vor dieser Alternative sind wir auch hinsichtlich der Begriffe der zeitlichen und räumlichen Entfernung gestellt. Die Antwort kann nach meiner Ansicht nur nach Zweckmäßigkeitsgründen gegeben werden; wie sie ausfallen wird, erscheint mir zweifelhaft.”

In the same discussion Gustav Mie came back to Förster's idea of an asymmetric metric but did not like it

“[...] that an antisymmetric tensor was added to the symmetric tensor of the gravitational potential, which represented the six-vector of the electromagnetic field. But a more precise reasoning shows that in this way no reasonable world function is obtained.”^$72$ “[...] dass man dem symmetrischen Tensor des Gravitationspotentials einen antisymmetrischen Tensor hinzufügte, der den Sechservektor des elektromagnetischen Feldes repräsentierte.
Aber eine genauere Überlegung zeigt, dass man so zu keiner vernünftigen Weltfunktion kommt.”

It is to be noted that Weyl, at the end of 1920, already had given up on a possible field theory of matter:

“Finally I cut loose firmly from Mie's theory and arrived at another position with regard to the problem of matter. To me, field physics no longer appears as the key to reality; in contrary, the field, the ether, for me simply is the totally powerless transmitter of causations, yet matter is a reality beyond the field and causes its states.”^$73$ “Endlich habe ich mich gründlich von der Mieschen Theorie losgemacht und bin zu einer anderen Stellung zum Problem der Materie gelangt. Die Feldphysik erscheint mir keineswegs mehr als der Schlüssel zu der Wirklichkeit; sondern das Feld, der Äther, ist mir nur noch der in sich selbst völlig kraftlose Übermittler der Wirkungen, die Materie aber eine jenseits des Feldes liegende und dessen Zustände verursachende Realität.” (letter of Weyl to F. Klein on 28 December 1920, see [292] , p. 83)

In the next year, Einstein had partially absorbed Pauli's view but still thought it to be useful to apply field theory to the constituents of matter:

“The physical interpretation of geometry (theory of the continuum) presented here, fails in its direct application to spaces of submolecular scale. Yet it retains part of its meaning also with regard to questions concerning the constitution of elementary particles. Because one may try to ascribe to these field concepts [...] a physical meaning even if a description of the electrical elementary particles which constitute matter is to be made. Only success can decide whether such a procedure finds its justification [...].”^$74$ “Die hier vertretene physikalische Interpretation der Geometrie (Kontinuumstheorie) versagt zwar bei ihrer unmittelbaren Anwendung auf Räume von submolekularer Grössenordnung. Einen Teil ihrer Bedeutung behält sie indessen auch noch den Fragen der Konstitution der Elementarteilchen gegenüber. Denn man kann versuchen, denjenigen Feldbegriffen [...] auch dann physikalische Bedeutung zuzuschreiben, wenn es sich um die Beschreibung der elektrischen Elementarteilchen handelt, die die Materie konstituieren. Nur der Erfolg kann über die Berechtigung eines solchen Verfahrens entscheiden [...].” [70]

During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:

“In the program, Mr. Einstein expressed during his two talks given in November 1929 at the Institut Henri Poincaré, he wished to search for the physical laws in solutions of his equations without singularities – with matter and the electromagnetic field thus being continuous. Let us move into the field chosen by him without too much surprise to see him apparently follow a road opposed to the one successfully walked by the contemporary physicists.”^$75$ “Dans le programme qu'a esquissé M. Einstein dans ses deux conférences faites en novembre 1929 à l'institut Henri Poncaré, il voulait chercher les lois physiques dans les solutions sans singularité de ses equations, la matière et l'electricité n'existant donc qu'à l'état continu. Placons-nous sur le terrain choisi par lui, sans trop nous étonner de le voir suivre en apparence une voie opposée à celle suivi avec succès par les physiciens contemporains.” ([34] , p. 17 (1178))

^$70$ For the evolution of Einstein's position vis-a-vis continuum theory and short-comings of it, cf. J. Stachel [328] .

4 The Main Ideas for Unification between about 1918 and 1923

After 1915, Einstein first was busy with extracting mathematical and physical consequences from general relativity (Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves). Although he kept thinking about how to find elementary particles in a field theory [68] and looked closer into Weyl's theory [70] , at first he only reacted to the new ideas concerning unified field theory as advanced by others. The first such idea after Förster's, of course, was Hermann Weyl's gauge approach to gravitation and electromagnetism, unacceptable to Einstein and to Pauli for physical reasons [245, 291] .

Next came Kaluza's five-dimensional unification of gravitation and electromagnetism, and Eddington's affine geometry.

4.1 Weyl's theory

4.1.1 The geometry

Weyl's fundamental idea for generalising Riemannian geometry was to note that, unlike for the comparison of vectors at different points of the manifold, for the comparison of scalars the existence of a connection is not required. Thus, while lengths of vectors at different points can be compared without a connection, directions cannot. This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:

“If we make no further assumption, the points of a manifold remain totally isolated from each other with regard to metrical structure. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given.”^$76$ “Machen wir keine weitere Voraussetzung, so bleiben die einzelnen Punkte der Mannigfaltigkeit in metrischer Hinsicht vollständig gegeneinander isoliert. Ein metrischer Zusammenhang von Punkt zu Punkt wird erst dann in sie hineingetragen, wenn ein Prinzip der Übertragung der Längeneinheit von einem Punkte P zu seinem unendlich benachbarten vorliegt.”

In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance.

“However, the possibility of such a comparison `at a distance' in no way can be admitted in a pure infinitesimal geometry.”^$77$ “Die Möglichkeit einer solchen `ferngeometrischen' Vergleichung kann aber in einer reinen Infinitesimalgeometrie durchaus nicht zugestanden werden.” ([397] , p. 397)

In order to invent a purely “infinitesimal” geometry, Weyl introduced the 1-dimensional, Abelian group of gauge transformations,

\begin{matrix} g \to \bar{g} : = λ g, \end{matrix}

(98)

besides the diffeomorphism group (coordinate transformations). At a point, Equation ( 98 ) induces a local recalibration of lengths

l

while preserving angles, i.e.,

δ l = λ l

. If the non-metricity tensor is assumed to have the special form

Q_{i j k} = Q_{k} g_{i j}

, with an arbitrary vector field

Q_{k}

, then as we know from Equation ( 57 ), with regard to these gauge transformations

\begin{matrix} Q_{k} \to Q_{k} + \partial_{k} σ . \end{matrix}

(99)

We see a striking resemblance with the electromagnetic gauge transformations for the vector potential in Maxwell's theory. If, as Weyl does, the connection is assumed to be symmetric (i.e., with vanishing torsion), then from Equation ( 42 ) we get

\begin{matrix} Γ_{i j}^{k} = {_{i j}^{k}} + \frac{1}{2} (δ_{i}^{k} Q_{j} + δ_{j}^{k} Q_{i} - g_{i j} g^{k l} Q_{l}) . \end{matrix}

(100)

Thus, unlike in Riemannian geometry, the connection is not fully determined by the metric but depends also on the arbitrary vector function

Q_{i}

, which Weyl wrote as a linear form

d Q = Q_{i} d x^{i}

With regard to the gauge transformations ( 98 ),

Γ_{i j}^{k}

remains invariant. From the 1-form

d Q

, by exterior derivation a gauge-invariant 2-form

F = F_{i j} d x^{i} \land d x^{j}

with

F_{i j} = Q_{i, j} - Q_{j, i}

follows.

It is named “Streckenkrümmung” (“line curvature”) by Weyl, and, by identifying

Q

with the electromagnetic 4-potential, he arrived at the electromagnetic field tensor

F

Let us now look at what happens to parallel transport of a length, e.g., the norm

| X |

of a tangent vector along a particular curve

C

with parameter

u

to a different (but infinitesimally neighbouring) point:

\begin{matrix} \frac{d | X |}{d u} = | X |_{∥ k} X^{k} = (σ - \frac{1}{2} Q_{k} X^{k}) | X | . \end{matrix}

(101)

By a proper choice of the curve's parameter, we may write ( 101 ) in the form

d | X | = - Q_{k} X^{k} | X | d u

and integrate along

C

to obtain

| X | = \int exp (- Q_{k} X^{k} d u)

. If

X

is taken to be tangent to

C

, i.e.,

X^{k} = \frac{d x^{k}}{d u}

, then

\begin{matrix} | \frac{d x^{k}}{d u} | = \int exp (- Q_{k} (x) d x^{k}), \end{matrix}

(102)

i.e., the length of a vector is not integrable; its value generally depends on the curve along which it is parallely transported. The same holds for the angle between two tangent vectors in a point (cf.

Equation ( 44 )). For a vanishing electromagnetic field, the 4-potential becomes a gradient (“pure gauge”), such that

Q_{k} d x^{k} = \frac{\partial ω}{\partial x^{k}} d x^{k} = d ω

, and the integral becomes independent of the curve.

Thus, in Weyl's connection ( 100 ), both the gravitational and the electromagnetic fields, represented by the metrical field

g

and the vector field

Q

, are intertwined. Perhaps, having in mind Mie's ideas of an electromagnetic world view and Hilbert's approach to unification, in the first edition of his book, Weyl remained reserved:

“Again physics, now the physics of fields, is on the way to reduce the whole of natural phenomena to one single law of nature, a goal to which physics already once seemed close when the mechanics of mass-points based on Newton's Principia did triumph. Yet, also today, the circumstances are such that our trees do not grow into the sky.”^$78$ “Wieder ist die Physik, heute die Feldphysik, auf dem Wege, die Gesamtheit der Naturerscheinungen auf ein einziges Naturgesetz zurückzuführen, ein Ziel, dem sie schon einmal, als die durch Newtons Principia begründete mechanische Massenpunkt-Physik ihre Triumphe feierte, nahe zu sein. Doch ist auch heute dafür gesorgt, dass unsere Bäume nicht in den Himmel wachsen.” ([396] , p. 170; preface dated “Easter 1918”)

However, a little later, in his paper accepted on 8 June 1918, Weyl boldly claimed:

“I am bold enough to believe that the whole of physical phenomena may be derived from one single universal world-law of greatest mathematical simplicity.”^$79$ “Ich bin verwegen genug, zu glauben, dass die Gesamtheit der physikalischen Erscheinungen sich aus einem einzigen universellen Weltgesetz von höchster mathematischer Einfachheit herleiten lässt.” ([397] , p. 385, footnote 4)

The adverse circumstances alluded to in the first quotation might be linked to the difficulties of finding a satisfactory Lagrangian from which the field equations of Weyl's theory can be derived.

Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2.1.5 ^$80$ . As the Lagrangian

ℒ = \sqrt{- g} L

must have gauge-weight

w = 0

, we are looking for a scalar

L

of gauge-weight

- 2

. Weitzenböck^$81$ has shown that the only possibilities quadratic in the curvature tensor and the line curvature are given by the four expressions [391]

\begin{matrix} (K_{i j} g^{i j})^{2}, K_{i j} K_{k l} g^{i k} g^{j l}, K_{j k l}^{i} K_{i m n}^{j} g^{k m} g^{l n}, F_{i j} F_{k l} g^{i k} g^{j l} . \end{matrix}

(103)

While the last invariant would lead to Maxwell's equations, from the invariants quadratic in curvature, in general field equations of fourth order result.

Weyl did calculate the curvature tensor formed from his connection ( 100 ) but did not get the correct result^$82$ ; it is given by Schouten ([309] , p. 142) and follows from Equation ( 51 ):

\begin{matrix} K_{j k l}^{i} = R_{j k l}^{i} + Q_{j; [k} δ_{l]}^{i} + δ_{j}^{i} Q_{[l; k]} - Q_{; [k}^{i} g_{l] j} + \frac{1}{2} (- δ_{[l}^{i} Q_{k]} Q_{j} + Q_{[k} g_{l] j} Q^{i} - δ_{[k}^{i} g_{l] j} Q_{r} Q^{r}) . \end{matrix}

(104)

If the metric field

g

and the 4-potential

Q_{i} \equiv A_{i}

are varied independently, from each of the curvature-dependent scalar invariants we do get contributions to Maxwell's equations.

Perhaps Bach (alias Förster) was also dissatisfied with Weyl's calculations: He went through the entire mathematics of Weyl's theory, curvature tensor, quadratic Lagrangian field equations and all; he even discussed exact solutions. His Lagrangian is given by

ℒ = \sqrt{g} (3 W_{4} - 6 W_{3} + W_{2})

, where the invariants are defined by

W_{4} : = S_{p q i k} S^{p q i k}, W_{3} : = g^{i k} g^{l m} F_{i k q}^{p} F_{l m p}^{q}, W_{2} : = g^{i k} F_{i k p}^{p}

with

\begin{matrix} S_{[p q] [i k]} & : = & \frac{1}{4} (F_{p q i k} - F_{q p i k} + F_{i k p q} - F_{k i p q}), \end{matrix}

\begin{matrix} F_{p q i k} & = & R_{p q i k} + \frac{1}{2} (g_{p q} f_{k i} + g_{p k} f_{q; i} + g_{q i} f_{p; k} - g_{p i} f_{q; k} - g_{q k} f_{p; i}) \end{matrix}

\begin{matrix} + \frac{1}{2} (f_{q} g_{p; [k} f_{i]} + f_{p} g_{q; [k} f_{i]} + g_{p [i} g_{k] q} f_{r} f^{r}), \end{matrix}

where

R_{p q i k}

is the Riemannian curvature tensor,

f_{i k} = f_{i, k} - f_{k, i}

, and

f_{i}

is the electromagnetic 4-potential [2] .

^$80$ In order to distinguish it from the “coordinate-weight” connected with relative tensors [25] . In his book, Weyl just uses the expression “weight”.

^$81$ Roland Weitzenböck (1885–1955). Studied mathematics at the University of Vienna where he obtained his doctoral degree in 1910. Became a professional officer during the First World War. He obtained professorships in Graz and Vienna and, in 1921, at the University of Amsterdam. He specialised in the theory of invariants (cf. [155] ).

^$82$ See Section IV, pp. 403–404 in [397] . Weyl's expression violates Equation ( 38 ): His curvature tensor allows for a trace $V_{k l} \neq 0 .$ In fact $V_{k l} = α Q_{[l; k]}$ is proportional to the electromagnetic field tensor.

4.1.2 Physics

While Weyl's unification of electromagnetism and gravitation looked splendid from the mathematical point of view, its physical consequences were dire: In general relativity, the line element

d s

had been identified with spaceand time intervals measurable by real clocks and real measuring rods.

Now, only the equivalence class

{λ g_{i k} | λ a r b i t r a r y}

was supposed to have a physical meaning: It was as if clocks and rulers could be arbitrarily “regauged” in each event, whereas in Einstein's theory the same clocks and rulers had to be used everywhere. Einstein, being the first expert who could keep an eye on Weyl's theory, immediately objected, as we infer from his correspondence with Weyl. In spring 1918, the first edition of Weyl's famous book on differential geometry, special and general relativity Raum–Zeit–Materie appeared, based on his course in Zürich during the summer term of 1917 [396] . Weyl had arranged that the page proofs be sent to Einstein. In communicating this on 1 March 1918, he also stated that

“As I believe, during these days I succeeded in deriving electricity and gravitation from the same source. There is a fully determined action principle, which, in the case of vanishing electricity, leads to your gravitational equations while, without gravity, it coincides with Maxwell's equations in first order. In the most general case, the equations will be of 4th order, though.”^$83$ “Diese Tage ist es mir, wie ich glaube, gelungen, Elektrizität und Gravitation aus einer gemeinsamen Quelle herzuleiten.
Es ergibt sich ein völlig bestimmtes Wirkungsprinzip, das im elektrizitätsfreien Fall auf Ihre Gravitationsgleichungen führt, im gravitationsfreien dagegen Gleichungen ergibt, die in erster Näherung mit den Maxwellschen übereinstimmen. Im allgemeinsten Fall werden die Gleichungen allerdings 4. Ordnung.”

He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy ([320] , Volume 8B, Document 472, pp. 663–664). At the end of March, Weyl visited Einstein in Berlin, and finally, on 5 April 1918, he mailed his note to him for the Berlin Academy. Einstein was impressed: In April 1918, he wrote four letters and two postcards to Weyl on his new unified field theory – with a tone varying between praise and criticism. His first response of 6 April 1918 on a postcard was enthusiastic:

“Your note has arrived. It is a stroke of genious of first rank. Nevertheless, up to now I was not able to do away with my objection concerning the scale.”^$84$ “Ihre Abhandlung ist gekommen. Es ist ein Genie-Streich ersten Ranges. Allerdings war ich nicht imstande, meinen Massstab-Einwand zu erledigen.” ([320] , Volume 8B, Document 498, 710)

Einstein's “objection” is formulated in his “Addendum” (“Nachtrag”) to Weyl's paper in the reports of the Academy, because Nernst had insisted on such a postscript. There, Einstein argued that if light rays would be the only available means for the determination of metrical relations near a point, then Weyl's gauge would make sense. However, as long as measurements are made with (infinitesimally small) rigid rulers and clocks, there is no indeterminacy in the metric (as Weyl would have it): Proper time can be measured. As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i.e., the frequencies would depend on the location of the emitter. He concluded with the words

“Regrettably, the basic hypothesis of the theory seems unacceptable to me, [of a theory] the depth and audacity of which must fill every reader with admiration.”^$85$ “[...] scheint mir die Grundhypothese der Theorie leider nicht annehmbar, deren Tiefe und Kühnheit aber jeden Leser mit Bewunderung erfüllen muss.” ([395] , Addendum, p. 478)

Einstein's remark concerning the path-dependence of the frequencies of spectral lines stems from the path-dependency of the integral ( 102 ) given above. Only for a vanishing electromagnetic field does this objection not hold.

Weyl answered Einstein's comment to his paper in a “reply of the author” affixed to it. He doubted that it had been shown that a clock, if violently moved around, measures proper time

\int d s

. Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:

“The most plausible assumption that can be made for a clock resting in a static field is this: that it measure the integral of the $d s$ normed in this way [i.e., as in Einstein's theory]; the task remains, in my theory as well as in Einstein's, to derive this fact by a dynamics carried through explicitly.”^$86$ “Die plausibelste Annahme, die man über ein im statischen Feld ruhende Uhr machen kann, ist die, dass sie das Integral des so normierten [d. h. so wie in der Einsteinschen Theorie] $d s$ misst; es bleibt in meiner wie in der Einsteinschen Theorie die Aufgabe $^{1}$ , diese Tatsache aus einer explizit durchgeführten Dynamik abzuleiten.” ([395] , p. 479)

Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i.e., to give a theory of clocks and rulers within general relativity. Presumably, such a theory would have to include microphysics. In a letter to his former student Walter Dällenbach, he wrote (after 15 June 1918):

“[Weyl] would say that clocks and rulers must appear as solutions; they do not occur in the foundation of the theory. But I find: If the $d s$ , as measured by a clock (or a ruler), is something independent of pre-history, construction and the material, then this invariant as such must also play a fundamental role in theory. Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist. [...] In any case, I am as convinced as Weyl that gravitation and electricity must let themselves be bound together to one and the same; I only believe that the right union has not yet been found.”^$87$ “[Weyl] würde sagen, Uhren und Massstäbe müssten erst als Lösungen auftreten; im Fundament der Theorie kommen sie nicht vor. Aber ich finde: Wenn das mit einer Uhr (bzw.
einem Massstab) gemessene $d s$ ein von der Vorgeschichte, dem Bau und dem Material Unabhängiges ist, so muss diese Invariante als solche auch in der Theorie eine ganz fundamentale Rolle spielen. Wenn aber die Art des wirklichen Naturgeschehens nicht so wäre, so gäbe es keine Spektrallinien und keine wohldefinierten chemischen Elemente. [...] Jedenfalls bin ich mit Weyl überzeugt, dass Gravitation und Elektrizität zu einem Einheitlichen sich verbinden lassen müssen, nur glaube ich, dass die richtige Verbindung noch nicht gefunden ist.” ([320] , Volume 8B, Document 565, 803)

Another famous theoretician who could not side with Weyl was H. A. Lorentz; in a paper on the measurement of lengths and time intervals in general relativity and its generalisations, he contradicted Weyl's statement that the world-lines of light-signals would suffice to determine the gravitational potentials [210] .

However, Weyl still believed in the physical value of his theory. As further “extraordinarily strong support for our hypothesis of the essence of electricity” he considered the fact that he had obtained the conservation of electric charge from gauge-invariance in the same way as he had linked with coordinate-invariance earlier, what at the time was considered to be “conservation of energy and momentum”, where a non-tensorial object stood in for the energy-momentum density of the gravitational field ([398] , pp. 252–253).

Moreover, Weyl had some doubts about the general validity of Einstein's theory which he derived from the discrepancy in value by 20 orders of magniture of the classical electron radius and the gravitational radius corresponding to the electron's mass ([397] , p. 476; [151] ).

4.1.3 Reactions to Weyl's theory I: Einstein and Weyl

There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [320] . We subsume some of the relevant discussions.

Even before Weyl's note was published by the Berlin Academy on 6 June 1918, many exchanges had taken place between him and Einstein. On a postcard to Weyl on 8 April 1918, Einstein reaffirmed his admiration for Weyl's theory, but remained firm in denying its applicability to nature. Weyl had given an argument for dimension 4 of space-time that Einstein liked: As the Lagrangian for the electromagnetic field

F_{i k} F^{i k}

is of gauge-weight

- 2

and

\sqrt{- g}

has gauge-weight

D / 2

in an

M_{D}

, the integrand in the Hamiltonian principle

\sqrt{- g} F_{i k} F^{i k}

can have weight zero only for

D = 4

: “Apart from the [lacking] agreement with reality it is in any case a grandiose intellectual performance”^$88$ “Abgesehen von der Übereinstimmung mit der Wirklichkeit ist es jedenfalls eine grandiose Leistung des Gedanken.” ([320] , Vol. 8B, Doc. 499, 711). Weyl did not give in:

“Your rejection of the theory for me is weighty; [...] But my own brain still keeps believing in it. And as a mathematician I must by all means hold to [the fact] that my geometry is the true geometry `in the near', that Riemann happened to come to the special case $F_{i k} = 0$ is due only to historical reasons (its origin is the theory of surfaces), not to such that matter.”^$89$ “Ihre Ablehnung der Theorie fällt für mich schwer ins Gewicht; [...] Aber mein eigenes Hirn bewahrt noch den Glauben an sie. Und daran muss ich als Mathematiker durchaus festhalten: Meine Geometrie ist die wahre Nahegeometrie, dass Riemann nur auf den Spezialfall $F_{i k} = 0$ geriet, hat lediglich historische Gründe (Entstehung aus der Flächentheorie), keine sachlichen.” ([320] , Volume 8B, Document 544, 767)

After Weyl's next paper on “pure infinitesimal geometry” had been submitted, Einstein put forward further arguments against Weyl's theory. The first was that Weyl's theory preserves the similarity of geometric figures under parallel transport, and that this would not be the most general situation (cf. Equation ( 49 )). Einstein then suggested the affine group as the more general setting for a generalisation of Riemannian geometry ([320] , Vol. 8B, Doc. 551, 777). He repeated this argument in a letter to his friend Michele Besso from his vacations at the Baltic Sea on 20 August 1918, in which he summed up his position with regard to Weyl's theory:

“[Weyl's] theoretical attempt does not fit to the fact that two originally congruent rigid bodies remain congruent independent of their respective histories. In particular, it is unimportant which value of the integral $\int φ_{ν} d x_{ν}$ is assigned to their world line.
Otherwise, sodium atoms and electrons of all sizes would exist. But if the relative size of rigid bodies does not depend on past history, then a measurable distance between two (neighbouring) world-points exists. Then, Weyl's fundamental hypothesis is incorrect on the molecular level, anyway. As far as I can see, there is not a single physical reason for it being valid for the gravitational field. The gravitational field equations will be of fourth order, against which speaks all experience until now [...].”^$90$ “[Weyl's] theoretischer Versuch, passt nicht zu der Thatsache, dass zwei ursprünglich kongruente feste Körper auch kongruent bleiben, unabhängig davon, welche Schicksale sie durchmachen. Insbesondere hat es keine Bedeutung, welcher Wert des Integrals $\int φ_{ν} d x_{ν}$ ihrer Weltlinie zukommt. Sonst würde es Natrium-Atome und Elektronen in allen Grössen geben müssen. Wenn aber die relative Grösse starrer Körper von der Vorgeschichte unabhängig ist, dann gibt es einen messbaren Abstand zweier (benachbarter) Weltpunkte. Dann ist die Weylsche Grundannahme jedenfalls nicht richtig für das molekulare. Dafür, dass sie für das Gravitationsfeld zutreffe, spricht, soweit ich sehe, kein einziger physikalischer Grund. Dagegen aber spricht, dass die Feldgleichungen der Gravitation von vierter Ordnung werden, wofür die bisherige Erfahrung keinerlei Anhalts bietet [...].” ([98] , p. 133)

Einstein's remark concerning “affine geometry” is referring to the affine geometry in the sense it was introduced by Weyl in the 1st and 2nd edition of his book [396] , i.e., through the affine group and not as a suggestion of an affine connexion.

From Einstein's viewpoint, in Weyl's theory the line element

d s

is no longer a measurable quantity – the electromagnetical 4-potential never had been one. Writing from his vacations on 18 September 1918, Weyl presented a new argument in order to circumvent Einstein's objections. The quadratic form

R g_{i k} d x^{i} d x^{k}

is an absolute invariant, i.e., also with regard to gauge transformations (gauge weight 0). If this expression would be taken as the measurable distance in place of

d s

, then

“[...] by the prefixing of this factor, so to speak, the absolute norming of the unit of length is accomplished after all”^$91$ “[...] durch Vorsetzen dieses Faktors wird dann doch sozusagen die absolute Normierung der Längeneinheit vollzogen.” ([320] , Volume 8B, Document 619, 877–879)

Einstein was unimpressed:

“But the expression $R g_{i k} d x^{i} d x^{k}$ for the measured length is not at all acceptable in my opinion because $R$ is very dependent on the matter density. A very small change of the measuring path would strongly influence the integral of the square root of this quantity.”^$92$ “Der Ausdruck $R g_{i k} d x^{i} d x^{k}$ für die gemessene Länge ist aber, wenn man für $R$ die Krümmungsinvariante nimmt, nach meiner Meinung keineswegs akzeptabel, weil $R$ sehr abhängig ist von der materiellen Dichte. Eine ganz kleine Änderung des Messweges würde das Integral der Quadratzwurzel dieser Grösse sehr stark beeinflussen.”

Einstein's argument is not very convincing:

g_{i k}

itself is influenced by matter through his field equations; it is only that now

R

is algebraically connected to the matter tensor. In view of the more general quadratic Lagrangian needed in Weyl's theory, the connection between

R

and the matter tensor again might become less direct. Einstein added:

“Of course I know that the state of the theory as I presented it is not satisfactory, not to speak of the fact that matter remains unexplained. The unconnected juxtaposition of the gravitational terms, the electromagnetic terms, and the $λ$ -terms undeniably is a result of resignation.[...] In the end, things must arrange themselves such that action-densities need not be glued together additively.”^$93$ “Natürlich weiss ich, dass der Zustand der Theorie, wie ich ihn hingestellt habe, ein nicht befriedigender ist, abgesehen davon, dass die Materie unerklärt bleibt. Die zusammenhanglose Nebeneinandersetzung der Gravitationsglieder, der elektromagnetischen Glieder und der $λ$ -Glieder ist unleugbar ein Produkt der Resignation. [...] Endlich muss es so herauskommen, dass man nicht Wirkungsdichten additiv aneinander kleben muss.” ([320] , Volume 8B, Document 626, 893–894)

The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.

^$88$ orgQuoteDe

4.1.4 Reactions to Weyl's theory II: Schouten, Pauli, Eddington, and others

Sommerfeld seems to have been convinced by Weyl's theory, as his letter to Weyl on 3 June 1918 shows:

“What you say here is really marvelous. In the same way in which Mie glued to his consequential electrodynamics a gravitation which was not organically linked to it, Einstein glued to his consequential gravitation an electrodynamics (i.e., the usual electrodynamics) which had not much to do with it. You establish a real unity.”^$94$ “Was Sie da sagen, ist wirklich wundervoll. So wie Mie seiner konsequenten Elektrodynamik eine Gravitation angeklebt hatte, die nicht organisch mit jener zusammenhing, ebenso hat Einstein seiner konsequenten Gravitation eine Elektrodynamik (d.h. die gewöhliche Elektrodynamik) angeklebt, die mit jener nicht viel zu tun hatte. Sie stellen eine wirkliche Einheit her.” [326]

Schouten, in his attempt in 1919 to replace the presentation of the geometrical objects used in general relativity in local coordinates by a “direct analysis”, also had noticed Weyl's theory.

In his “addendum concerning the newest theory of Weyl”, he came as far as to show that Weyl's connection is gauge invariant, and to point to the identification of the electromagnetic 4-potential.

Understandably, no comments about the physics are given ([294] , pp. 89–91).

In the section on Weyl's theory in his article for the Encyclopedia of Mathematical Sciences, Pauli described the basic elements of the geometry, the loss of the line-element

d s

as a physical variable, the convincing derivation of the conservation law for the electric charge, and the too many possibilities for a Lagrangian inherent in a homogeneous function of degree 1 of the invariants ( 103 ).

As compared to his criticism with respect to Eddington's and Einstein's later unified field theories, he is speaking softly, here. Of course, as he noted, no progress had been made with regard to the explanation of the constituents of matter; on the one hand because the differential equations were too complicated to be solved, on the other because the observed mass difference between the elementary particles with positive and negative electrical charge remained unexplained. In his general remarks about this problem at the very end of his article, Pauli points to a link of the asymmetry with time-reflection symmetry (see [245] , pp. 774–775; [243] ). For Einstein, this criticism was not only directed against Weyl's theory

“but also against every continuum-theory, also one which treats the electron as a singularity. Now as before I believe that one must look for such an overdetermination by differential equations that the solutions no longer have the character of a continuum.
But how?” ([102] , p. 43)

In a letter to Besso on 26 July 1920, Einstein repeated an argument against Weyl's theory which had been removed by Weyl – if only by a trick to be described below; Einstein thus said:

“One must pass to tensors of fourth order rather than only to those of second order, which carries with it a vast indeterminacy, because, first, there exist many more equations to be taken into account, second, because the solutions contain more arbitrary constants.”^$95$ “Man muss zu Tensoren übergehen die 4. Ordnung sind statt nur zweiter Ordnung, was eine weitgehende Unbestimmtheit der Theorie mit sich bringt, erstens weil es bedeutend mehr Gleichungen gibt, die in Betracht kommen, zweitens, weil die Lösungen mehr willkürliche Konstanten enthalten.” ([98] , p. 153)

In his book “Space, Time, and Gravitation”, Eddington gave a non-technical introduction into Weyl's “welding together of electricity and gravitation into one geometry”. The idea of gauging lengths independently at different events was the central theme. He pointed out that while the fourfold freedom in the choice of coordinates had led to the conservation laws for energy and momentum, “in the new geometry is a fifth arbitrariness, namely that of the selected gauge-system. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge.” One natural gauge was formed by the “radius of curvature of the world”; “the electron could not know how large it ought to be, unless it had something to measure itself against” ([55] , pp. 174, 173, 177).

As Eddington distinguished natural geometry and actual space from world geometry and conceptual space serving for a graphical representation of relationships among physical observables, he presented Weyl's theory in his monograph “The mathematical theory of relativity”

“from the wrong end – as its author might consider; but I trust that my treatment has not unduly obscured the brilliance of what is unquestionably the greatest advance in the relativity theory after Einstein's work.” ([57] , p. 198)

Of course, “wrong end” meant that Eddington took Weyl's theory such

“that his non-Riemannian geometry is not to be applied to actual space-time; it refers to a graphical representation of that relation-structure which is the basis of all physics, and both electromagnetic and metrical variables appear in it as interrelated.” ([57] , p. 197)

Again, Eddington liked Weyl's natural gauge encountered in Section 4.1.5 , which made the curvature scalar a constant, i.e.,

K = 4 λ

; it became a consequence of Eddington's own natural gauge in his affine theory,

K_{i j} = λ g_{i j}

(cf. Section 4.3 ). For Eddington, Weyl's theory of gauge-transformation was a hybrid:

“He admits the physical comparison of length by optical methods [...]; but he does not recognise physical comparison of length by material transfer, and consequently he takes $λ$ to be a function fixed by arbitrary convention and not necessarily a constant.” ([57] , pp. 220–221)

In the depth of his heart Weyl must have kept a fondness for his idea of “gauging ” a field all during the decade between 1918 and 1928. As he had abandoned the idea of describing matter as a classical field theory since 1920, the linking of the electromagnetic field via the gauge idea could only be done through the matter variables. As soon as the new spinorial wave function (“matter wave”) in Schrödinger's and Dirac's equations emerged, he adapted his idea and linked the electromagnetic field to the gauging of the quantum mechanical wave function [407, 408] . In October 1950, in the preface for the first American printing of the English translation of the fourth edition of his book Space, Time, Matter from 1922, Weyl clearly expressed that he had given up only the particular idea of a link between the electromagnetic field and the local calibration of length:

“While it was not difficult to adapt also Maxwell's equations of the electromagnetic field to this principle [of general relativity], it proved insufficient to reach the goal at which classical field physics is aiming: a unified field theory deriving all forces of nature from one common structure of the world and one uniquely determined law of action.[...] My book describes an attempt to attain this goal by a new principle which I called gauge invariance. (Eichinvarianz). This attempt has failed.” ([410] , p. V)

4.1.5 Reactions to Weyl's theory III: Further research

Pauli, still a student, and with his article for the Encyclopedia in front of him, pragmatically looked into the gravitational effects in the planetary system, which, as a consequence of Einstein's field equations, had helped Einstein to his fame. He showed that Weyl's theory had, for the static case, as a possible solution a constant Ricci scalar; thus it also admitted the Schwarzschild solution and could reproduce all desired effects [243, 242] .

Weyl himself continued to develop the dynamics of his theory. In the third edition of his Space–Time–Matter [398] , at the Naturforscherversammlung in Bad Nauheim in 1920 [399] , and in his paper on “the foundations of the extended relativity theory” in 1921 [402] , he returned to his new idea of gauging length by setting

R = λ = c o n s t .

(cf. Section 4.1.3 ); he interpreted

λ

to be the “radius of curvature” of the world. In 1919, Weyl's Lagrangian originally was

ℒ = \frac{1}{2} \sqrt{g} K^{2} + β F_{i k} F^{i k}

together with the constraint

K = 2 λ,

with constant

λ

([398] , p. 253). As an equivalent Lagrangian Weyl gave, up to a divergence^$96$

\begin{matrix} ℒ = \sqrt{g} (R + α F_{i k} F^{i k} + \frac{1}{4} (2 λ - 3 φ_{l} φ^{l})), \end{matrix}

(105)

with the 4-potential

φ_{l}

and the electromagnetic field

F_{i k}

. Due to his constraint, Weyl had navigated around another problem, i.e., the formulation of the Cauchy initial value problem for field equations of fourth order: Now he had arrived at second order field equations. In the paper in 1921, he changed his Lagrangian slightly into

\begin{matrix} ℒ = \sqrt{g} (R + α F_{i k} F^{i k} + \frac{ε^{2}}{4} (1 - 3 φ_{l} φ^{l})), \end{matrix}

(106)

with

ε

a factor in Weyl's connection ( 100 ),

\begin{matrix} Γ_{i j}^{k} = {_{i j}^{k}} + \frac{ε}{2} (δ_{i}^{k} φ_{j} + δ_{j}^{k} φ_{i} - g_{i j} g^{k l} φ_{l}) . \end{matrix}

(107)

In both presentations, he considered as an advantage of his theory:

“Moreover, this theory leads to the cosmological term in a uniform and forceful manner, [a term] which in Einstein's theory was introduced ad hoc”^$97$ “Ausserdem führt dies Theorie auf einheitliche Weise und zwingend zu dem kosmologischen Gliede, das bei Einstein nur eine ad hoc gemachte Annahme war [...].” ([402] , p. 474)

Reichenbächer seemingly was unhappy about Weyl's taking the curvature scalar to be a constant before the variation; in the discussion after Weyl's talk in 1920, he inquired whether one could not introduce Weyl's “natural gauge” after the variation of the Lagrangian such that the field equations would show their gauge invariance first ([399] , p. 651). Eddington criticised Weyl's choice of a Lagrangian as speculative:

“At the most we can only regard the assumed form of action [...] as a step towards some more natural combination of electromagnetic and gravitational variables.” ([57] , p. 212)

The changes, which Weyl had introduced in the 4th edition of his book [401] , and which, according to him, were of fundamental importance for the understanding of relativity theory, were discussed by him in a further paper [400] . In connection with the question of whether, in general relativity, a formulation might be possible such that “matter whose characteristical traits are charge, mass, and motion generates the field”, a question which was considered as unanswered by Weyl, he also mentioned a publication of Reichenbächer [271] . For Weyl, knowledge of the charge and mass of each particle, and of the extension of their “world-channels” were insufficient to determine the field uniquely. Weyl's hint at a solution remains dark; nevertheless, for him it meant

“to reconciliate Reichenbächer's idea: matter causes a `deformation' of the metrical field and Einstein's idea: inertia and gravitation are one.” ([400] , p. 561, footnote)

Although Einstein could not accept Weyl's theory as a physical theory, he cherished “its courageous mathematical construction” and thought intensively about its conceptual foundation:

This becomes clear from his paper “On a complement at hand of the bases of general relativity” of 1921 [71] . In it, he raised the question whether it would be possible to generate a geometry just from the conformal invariance of Equation ( 9 ) without use of the conception “distance”, i.e., without using rulers and clocks. He then embarked on conformal invariants and tensors of gauge-weight 0, and gave the one formed from the square of Weyl's conformal curvature tensor ( 59 ), i.e.

C_{j k l}^{i} C_{i}^{j k l} .

His colleague in Vienna, Wirtinger^$98$ , had helped him in this^$99$ . Einstein's conclusion was that, by writing down a metric with gauge-weight 0, it was possible to form a theory depending only on the quotient of the metrical components. If

J

has gauge-weight

- 1

, then

J g_{i k}

is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation

\begin{matrix} J = J_{0} = c o n s t . \end{matrix}

(108)

would have to be solved.

Eisenhart wished to partially reinterpret Weyl's theory: In place of putting the vector potential equal to Weyl's gauge vector, he suggested to identify it with

\frac{- F_{k}^{i} J^{k}}{μ}

, where

J^{i}

is the electrical 4-current vector (-density) and

μ

the mass density. He referred to Weyl, Eddington's book, and to Pauli's article in the Encyclopedia of Mathematical Sciences [115] .

Einstein's rejection of the physical value of Weyl's theory was seconded by Dienes^$100$ , if only with a not very helpful argument. He demanded that the connection remain metric-compatible from which, trivially, Weyl's gauge-vector must vanish. Dienes applied the same argument to Eddington's generalisation of Weyl's theory [49] . Other mathematicians took Weyl's theory at its face value and drew consequences; thus M. Juvet calculated Frenet's formulas for an “

n

-èdre” in Weyl's geometry by generalising a result of Blaschke for Riemannian geometry [179] . More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. John's College in Cambridge, M. H. A. Newman [236] .

In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge. Newman applied his scheme to a variational principle with Lagrangian

K^{2}

and concluded:

“The part independent of the `electrical' vector $φ_{i}$ is found to be $K_{i j} - \frac{1}{4} K g_{i j}$ , a tensor which has been considered by Einstein from time to time in connection with the theory of gravitation.” ([236] , p. 623)

After the Second World War, research following Weyl's classical geometrical approach with his original 1-dimensional Abelian gauge-group was resumed. The more important development, however, was the extension to non-Abelian gauge-groups and the combination with Kaluza's idea.

We shall discuss these topics in Part II of this article. The shift in Weyl's interpretation of the role of the gauging from the link between gravitation and electromagnetism to a link between the quantum mechanical state function and electromagnetism is touched on in Section 7 .

^$96$ In fact, from Equation ( 104 ) we obtain $K = R + 3 Q_{l} Q^{l} + 3 Q_{; j}^{j}$ .

^$98$ Wilhelm Wirtinger (1865–1945). Born in Ybbs, Austria. Studied mathematics at the University of Vienna. Received his doctorate in 1887, and continued his studies at the Universities of Berlin and of Göttingen. From 1895 a full professor at the University of Vienna, but accepted professorship at University of Innsbruck, returning to Vienna only in 1905. Wrote an important paper on the general theta function and had an exceptional range in mathematics (function theory, algebra, number theory, plane geometry, theory of invariants).

^$99$ In the same year, Wirtinger sent in a paper on relativity theory published only in 1922 [421] .

^$100$ Paul Dienes (1982–1952). Born in Tokaj, Hungary. Studied mathematics. From 1929–1945 Reader, and from 1945–48 Professor at Birkbeck College, University of London.

4.2 Kaluza's five-dimensional unification

What is now called Kaluza–Klein theory in the physics community is a mixture of quite different contributions by both scientists^$101$ . Kaluza's idea of looking at four spatial and one time dimension originated in or before 1919; by then he had communicated it to Einstein:

“The idea of achieving [a unified field theory] by means of a five-dimensional cylinder world never dawned on me. [...] At first glance I like your idea enormously.” (letter of Einstein to Kaluza of 21 April 1919)

This remark is surprising because Nordström had suggested a five-dimensional unification of his scalar gravitational theory with electromagnetism five years earlier [237] , by embedding space-time into a five-dimensional world in quite the same way as Kaluza did. In principle, Einstein could have known Nordström's work. In the same year 1914, he and Fokker had given a covariant formulation of Nordström's pure (scalar) theory of gravitation [103] . In a subsequent letter to Kaluza of 5 May 1919 Einstein still was impressed: “The formal unity of your theory is startling.” However, on 29 May 1919, Einstein became somewhat reserved^$102$ :

“I respect greatly the beauty and boldness of your idea. But you understand that, in view of the existing factual concerns, I cannot take sides as planned originally.”^$103$ “Ich habe grossen Respekt vor der Schönheit und Kühnheit Ihrer Gedanken. Aber Sie begreifen ja, dass ich bei den obwaltenden sachlichen Bedenken nicht in der ursprünglich geplanten Weisedafür Partei nehmen kann.”

Kaluza's paper was communicated by Einstein to the Academy, but for reasons unknown was published only in 1921 [180] . Kaluza's idea was to write down the Einstein field equations for empty space in a five-dimensional Riemannian manifold with metric

g_{α β}

, i.e.,

R_{α β} = 0

α, β = 1, \dots, 5

, where

R_{α β}

is the Ricci tensor of

M_{5}

, and to look at small deviations

γ

from Minkowski space:

g_{α β} = - δ_{α β} + γ_{α β}

.^$104$ . In order to obtain a theory in space-time, he assumed the so-called “cylinder condition”

\begin{matrix} g_{α β},_{5} = 0, \end{matrix}

(109)

equivalent to the existence of a spacelike translational symmetry (Killing vector). Equation ( 109 ) is used for all “functions of state” (Zustandsgrössen), i.e., also for the matter variables. Kaluza did not normalize the Killing vector to a constant, i.e., he kept

\begin{matrix} g_{55} \neq c o n s t . \end{matrix}

(110)

Equation ( 110 ) is called the “sharpened cylinder condition” by some authors including Einstein. Of the 15 components of

g_{α β}

, five had to get a new physical interpretation, i.e.

g_{α 5}

and

g_{55}

; the components

g_{i k}

i, k = 1, . . ., 4

, were to describe the gravitational field as before; Kaluza took

g_{i 5}

proportional to the electromagnetic vector potential

A_{i}

. The component

g_{55}

turned out to be a (scalar) gravitational potential which, in the static case, satisfies the equation

\begin{matrix} \nabla^{2} g_{55} = - κ μ_{0}, \end{matrix}

(111)

with the constant matter density

μ_{0}

Kaluza also showed that the geodesics of the five-dimensional space reduce to the equations of motion for a charged point particle in space-time, if a weakness assumption is made for the components of the 5-velocity

u^{α}

u^{1}, u^{2}, u^{3}, u^{5} ≪ 1

u^{4} ≃ 1

. The Lorentz force appears augmented by an additional term containing

g_{55}

of the order

(\frac{u}{c})^{2}

which thus may be neglected. From the fifth equation of motion Kaluza concluded that the fifth component of momentum

p_{5} \sim e

, with

e

being the particles' electric charge (up to a constant of proportionality). From the equations of motion, charge conservation also followed in Kaluza's linear approximation. Kaluza was well aware that his theory broke down if applied to elementary particles like electrons or protons, and speculated about an escape in which gravitation had to be considered as some “difference effect”, and the gravitational constant given “a statistical meaning”. For him, any theory claiming universal validity was endangered by quantum theory, anyway.

From the cylinder condition, a grave objection toward Kaluza's approach results: Covariance with regard to the diffeomorphism group of

M_{5}

is destroyed. The remaining covariance group

G_{5}

is given by

\begin{matrix} x^{5^{'}} = x^{5} + f (x^{k}), x^{l^{'}} = x^{l} (x^{m}), k, l, m = 1, . . ., 4 . \end{matrix}

(112)

The objects transforming properly under ( 112 ) are: the scalar

g_{5^{'} 5^{'}} = g_{55}

, the vector-potential

g_{5^{'} k^{'}} = g_{5 l} \frac{\partial x^{l}}{\partial x^{k^{'}}} + g_{55} \frac{\partial x^{5}}{\partial x^{k^{'}}}

, and the projected metric

\begin{matrix} g_{i^{'} k^{'}} - \frac{g_{i^{'} 5^{'}} g_{k^{'} 5^{'}}}{g_{5^{'} 5^{'}}} = (g_{l m} - \frac{g_{l 5} g_{m 5}}{g_{55}}) \frac{\partial x^{l}}{\partial x^{i^{'}}} \frac{\partial x^{m}}{\partial x^{k^{'}}} . \end{matrix}

(113)

Klein identified the group; however, he did not comment on the fact that now further invariants are available for a Lagrangian, but started right away from the Ricci scalar of

M_{5}

[184] . The group

G_{5}

is isomorphic to the group

H_{5}

of transformations for five homogeneous coordinates

X^{μ^{'}} = f^{μ} (X^{ν})

with

f^{ν}

homogeneous functions of degree 1. Here, contact is made to the projective formulation of Kaluza's theory (cf. “projective geometry” in Sections 2.1.3 and 6.3.2 ).

While towards the end of May 1919 Einstein had not yet fully supported the publication of Kaluza's manuscript, on 14 October 1921 he thought differently:

“I am having second thoughts about having kept you from the publication of your idea on the unification of gravitation and electricity two years ago. I value your approach more than the one followed by H. Weyl. If you wish, I will present your paper to the Academy after all.”^$105$ “Ich mache mir Gedanken darüber, dass ich Sie vor zwei Jahren von der Publikation Ihrer Idee über die Vereinigung von Gravitation und Elektrizität abgehalten habe. Ihr Weg scheint mir jedenfalls mehr für sich zu haben als der von H. Weyl beschrittene. Wenn Sie wollen, lege ich Ihre Arbeit doch der Akademie vor [...].” (letter from Einstein to Kaluza reprinted in [47] , p. 454)

It seems that at some point Einstein had set his calculational aide Grommer^$106$ to work on regular spherically symmetric solutions of Kaluza's theory. This led to a joint publication which was submitted just one month after Einstein had finally presented a rewritten manuscript of Kaluza's to the Berlin Academy [104] .

The negative result of his own paper, i.e., that no non-singular, statical, spherically symmetric exact solution exists, did not please Einstein. He also thought that Kaluza's assumption of general covariance in the five-dimensional manifold had no support from physics; he disliked the preference of the fifth coordinate due to Equation ( 109 ) which seemed to contradict the equivalence of all five coordinates used by Kaluza in the construction of the field equations [104] . In any case, apart from an encouraging letter to Kaluza in 1925 in which he called Kaluza's idea the only serious attempt at unified field theory besides the Weyl–Eddington approach, Einstein kept silent on the five-dimensional theory until 1926.

^$101$ A detailed investigation of this subject will appear soon, and will be included in the next update of this article [144] .

^$102$ Some letters of Einstein to Kaluza are quoted by Pais ([240] , p. 330), some are reprinted as facsimile in the Proceedings of the Erice Summer School of Cosmology and Gravitation 1982 [47] , p. 452.

^$104$ Greek lettered indices run in $M_{5}$ , Latin ones in space time; $x^{4}$ is the time coordinate, $x^{5}$ the new spacelike coordinate.

^$106$ Jakob Grommer (1879–1933). Born near Brest, then in Russia. First a Talmud student with a keen interest in mathematics. Came to Göttingen to study mathematics and obtained his Ph.D. there. Worked with Einstein for at least a decade (1917–1927) as his calculational assistant. He held a university position in Minsk from 1929 on and later became a member of the Belorussian Academy of Sciences. From his youth he was inflicted with elephantiasis.

4.3 Eddington's affine theory

4.3.1 Eddington's paper

The third main idea that emerged was Eddington's suggestion to forego the metric as a fundamental concept and start right away with a (general) connection, which he then restricted to a symmetric one

Γ

in order to avoid an “infinitely crinkled” world [56] . His motivation went beyond the unification of gravitation and electromagnetism:

“In passing beyond Euclidean geometry, gravitation makes its appearance; in passing beyond Riemannian geometry, electromagnetic force appears; what remains to be gained by further generalisation? Clearly, the non-Maxwellian binding forces which hold together an electron. But the problem of the electron must be difficult, and I cannot say whether the present generalisation succeeds in providing the material for its solution” ([56] , p. 104)

In the first, shorter, part of two, Eddington describes affine geometry; in the second he relates mathematical objects to physical variables. He distinguishes the affine geometry as the “geometry of the world-structure” from Riemannian geometry as “the natural geometry of the world”. He starts by calculating both the curvature and Ricci tensors from the symmetric connection according to Equation ( 39 ). The Ricci tensor

K^{i j} (Γ) : =^{*} G^{i j}

is asymmetric^$107$ ,

\begin{matrix} ^{*} G_{k l} = R_{k l} + F_{k l}, \end{matrix}

(114)

with

R_{k l} (Γ)

being the symmetric and

F_{k l} (Γ)

the antisymmetric part. According to Equation ( 31 )

F_{k l}

derives from a “vector potential”, i.e.,

F_{k l} = \partial_{k} Γ_{l} - \partial_{l} Γ_{k}

with

Γ_{l} : = Γ_{l r}^{r}

, such that an immediate physical identification of

F_{k l}

with the electromagnetic field tensor is at hand. With half of Maxwell's equations being satisfied automatically, the other half is used to define the electric charge current

j^{k}

j^{l} : = F_{∥ k}^{l k}

. By this, Eddington claims to guarantee charge conservation:

“The divergence of $j^{k}$ will vanish identically if $j^{k}$ is itself the divergence of any antisymmetrical contravariant tensor.” ([62] , p. 223; cf. also [56] , p. 113)

Now, by Equation ( 25 ),

\begin{matrix} F_{∥ [l ∥ k]}^{l k} = K_{[r k]} F^{r k} + S_{j k}^{s} \nabla_{s} F^{j k} . \end{matrix}

(115)

For a symmetric connection thus, unlike in Riemannian geometry,

\begin{matrix} j_{∥ k}^{k} = F_{∥ [l ∥ k]}^{l k} = F_{r k} F^{r k} \neq 0 . \end{matrix}

(116)

However, for a tensor density, due to Equation ( 16 ) we obtain

\begin{matrix} {\hat{j}}_{∥ k}^{k} = {\hat{F}}_{∥ [l ∥ k]}^{l k} = \frac{1}{2} (V_{r k} + 2 K_{[r k]}) {\hat{F}}^{r k} + S_{j k}^{s} \nabla_{s} {\hat{F}}^{j k}, \end{matrix}

(117)

and thus for a torsionless connection (cf. Equation ( 38 ))

j_{∥ k}^{k} = 0 .

Eddington introduces the metrical tensor by the definition

\begin{matrix} λ g_{k l} = R_{k l}, \end{matrix}

(118)

“introducing a universal constant

λ

, for convenience, in order to remain free to use the centimetre instead of the natural unit of length”. This is called “Einstein's gauge” by Eddington; he is delighted that

“Our gauging-equation is therefore certainly true wherever light is propagated, i.e., everywhere inside the electron. Who shall say what is the ordinary gauge inside the electron?” ([56] , p. 114)

While this remark certainly is true, there is no guarantee in Eddington's approach that

g_{k l}

thus defined is a Lorentzian metric, i.e., that it could describe light propagation at all. Only connections leading to a Lorentz metric can be used if a physical interpretation is wanted. Note also, that the interpretation of

R_{k l}

as the metric implies that

det R_{k l} \neq 0 .

We must read Equation ( 118 ) as giving

g_{k l} (Γ)

if the only basic variable in affine geometry, i.e., the connection

Γ_{i j}^{k}

, has been determined by help of some field equations. Thus, in general,

g_{k l}

is not metric-compatible; in order to make it such, we are led to the differential equations

R_{i j ∥ k} = 0

for

Γ_{i j}^{k}

, an equation not considered by Eddington. In the absence of an electromagnetic field, Equation ( 118 ) looks like Einstein's vacuum field equation with cosmological constant. In principle, now a fictitious “Riemannian” connection (the Christoffel symbol) can be written down which, however, is a horribly complicated function of the affine connection – as the only fundamental geometrical quantity available. This is due to the expression for the inverse of the metric, a function cubic in

R_{k l}

. Eddington's affine theory thus can also be seen as a bi-connection theory.

Note also that Eddington does not explicitly say how to obtain the contravariant form of the electromagnetic field

F^{i j}

from

F_{i j}

; we must assume that he thought of raising indices with the complicated inverse metric tensor.

In connection with cosmological considerations, Eddington cherished the

λ

-term in Equation ( 118 ):

“I would as soon think of reverting to Newtonian theory as of dropping the cosmic constant.” ([61] , p. 35)

Now, Eddington was able to identify the energy-momentum tensor

T^{i k}

of the electromagnetic field by decomposing the Ricci tensor

K^{i j}

formed from Equation ( 51 ) into a metric part

R_{i k}

and the rest. The energy-momentum tensor

T^{i k}

of the electromagnetic field is then defined by Einstein's field equations with a fictitious cosmological constant

κ T^{i k} : = G^{i k} - \frac{1}{2} g^{i k} (G - 2 λ)

Although Eddington's interest did not rest on finding a proper set of field equations, he nevertheless discussed the Lagrangian

ℒ = \sqrt{- g}^{*} G^{i k}

^{*} G_{i k}

, and showed that a variation with regard to

g_{i k}

did not lead to an acceptable field equation.

Eddington's main goal in this paper was to include matter as an inherent geometrical structure:

“What we have sought is not the geometry of actual space and time, but the geometry of the world-structure which is the common basis of space and time and things.” ([56] , p. 121)

By “things” he meant

(1) the energy-momentum tensor of matter, i.e., of the electromagnetic field,
(2) the tensor of the electromagnetic field, and
(3) the electric charge-and-current vector.

His aim was reached in the sense that all three quantities were fixed entirely by the connection; they could no longer be given from the outside. As to the question of the electron, it is seen as “a region of abnormal world-curvature”, i.e., of abnormally large curvature.

While Pauli liked Eddington's distinction between “natural geometry” and “world geometry” – with the latter being only “a graphical representation” of reality – he was not sure at all whether “a point of view could be taken from which the gravitational and electromagnetical fields appear as union”. If so, then it must be a purely phenomenological one without any recourse to the nature of the charged elementary particles (cf. his letter to Eddington quoted below). Lorentz did not like the large number of variables in Eddington's theory; there were 4 components of the electromagnetic potential, 10 components of the metric and 40 components of the connection:

“It may well be asked whether after all it would not be preferable simply to introduce the functions that are necessary for characterising the electromagnetic and gravitational fields, without encumbering the theory with so great a number of superfluous quantities.” ([210] , p. 382)

^$107$ We use some of Eddington's notation. His notation representing the covariant derivative by a lower index is highly ambiguous, though, and will be avoided. $^{*} G_{k l}$ is not a dual.

4.3.2 Einstein's reaction and publications

Eddington's publication early in 1921, generalising Einstein's and Weyl's theories started a new direction of research both in physics and mathematics. At first, Einstein seems to have been reserved (cf. his letters to Weyl in June and September 1921 quoted by Stachel in his article on Eddington and Einstein ([329] , pp. 453–475; here p. 466)), but one and a half years later he became attracted by Eddington's idea. To Bohr, Einstein wrote from Singapore on 11 January 1923:

“I believe I have finally understood the connection between electricity and gravitation.
Eddington has come closer to the truth than Weyl.” ([138] , p. 274)

He now tried to make Eddington's theory work as a physical theory; Eddington had not given field equations:

“I must absolutely publish since Eddington's idea must be thought through to the end.” (letter of Einstein to Weyl of 23 May 1923; cf. [240] , p. 343)

And a few days later, he was still intrigued about this sort of unified field theory, in particular about its elusiveness:

“[...] Over it lingers the marble smile of inexorable nature, which has bestowed on us more longing than brains.”^$108$ “[...] Darüber steht das marmorne Lächeln der unerbittlichen Natur, die uns mehr Sehnsucht als Geist verliehen hat.” (letter of Einstein to Weyl of 26 May 1923; cf.[240] , p. 343)

And indeed Einstein published fast, even while still on the steamer returning from Japan through Palestine and Spain: The paper of February 1923 in the reports of the Berlin Academy carries, as location of the sender, the ship “Haruna Maru” of the Japanese Nippon Yushen Kaisha line^$109$ [75] .

“In past years, the wish to understand the gravitational and electromagnetic field as one in essence has dominated the endeavours of theoreticians. [...] From a purely logical point of view only the connection should be used as a fundamental quantity, and the metric as a quantity derived thereof [...] Eddington has done this.”^$110$ “Der Wunsch, das Gravitationsfeld und das elektromagnetische Feld als Wesenseinheit zu begreifen, beherrscht in den letzten Jahren das Streben der Theoretiker. [...] Von einem logisch einleuchtenden Standpunkt her sollte nur die Konnektion als fundamentale Grösse benutzt werden und die Metrik eine daraus abgeleitete Grösse sein. [...] Dies that Eddington.” ([75] , p. 32)

Like Eddington, Einstein used a symmetric connection and wrote down the equation^$111$

\begin{matrix} λ^{2} K_{k l} = g_{k l} + φ_{k l}, \end{matrix}

(119)

where

g_{k l} = g_{(k l)}

and

φ_{k l} = φ_{[k l]}

, and

λ

is a “large number”. By this, the metric was defined as the symmetric part of the Ricci tensor. Due to

\begin{matrix} φ_{k l} = \frac{1}{2} (\frac{\partial Γ_{k j}^{j}}{\partial x^{l}} - \frac{\partial Γ_{l j}^{j}}{\partial x^{k}}) \end{matrix}

(120)

one half of Maxwell's equations is satisfied if

φ_{k l}

is taken to be the electromagnetic field tensor.

Let us note, however, that while

Γ_{k j}^{j}

transforms inhomogeneously, its transformation law

Γ_{k^{'} j^{'}}^{j^{'}} = Γ_{l m}^{m} \frac{\partial x^{l}}{\partial x^{k^{'}}} + \frac{\partial^{2} x^{l^{'}}}{\partial x^{k^{'}} \partial x^{m^{'}}} \cdot \frac{\partial x^{m^{'}}}{\partial x^{l^{'}}}

is not exactly the same as that of the electric 4-potential under gauge transformations.

For a Lagrangian, Einstein used

ℒ = 2 \sqrt{- det K_{i j}}

; he claims that for vanishing electromagnetic field the vacuum field equations of general relativity, with the cosmological term included, hold.

Einstein varied with regard to

g_{k l}

and

φ_{k l}

, not, as one might have expected, with regard to the connection

Γ_{k j}^{j}

. If

{\hat{f}}^{k l} : = \frac{δ ℒ}{δ φ_{k l}}

, then the electric current density

j^{l}

is defined by

{\hat{j}}^{l} : = \frac{\partial {\hat{f}}^{l k}}{\partial x^{k}} .

{\hat{f}}^{k l}

is interpreted as “the contravariant tensor of the electromagnetic field”.

The field equations are obtained from the Lagrangian by variation with regard to the connection

Γ_{k j}^{l}

and are (Einstein worked in space-time)

\begin{matrix} {\hat{s}}_{∥ m}^{k l} + \frac{1}{3} δ_{m}^{k} {\hat{j}}^{l} + \frac{1}{3} δ_{m}^{l} {\hat{j}}^{k} = 0, 3 {\hat{s}}_{∥ l}^{k l} + 5 {\hat{j}}^{k} = 0, \end{matrix}

(121)

with the definition of the current density

{\hat{j}}^{k}

given before, and

{\hat{s}}^{k l} = \frac{δ ℒ}{δ g_{k l}}

. Besides

{\hat{s}}^{k l}

, Einstein also uses

s^{k l}

introduced by

\begin{matrix} {\hat{s}}^{k l} = s^{k l} \sqrt{- det s_{k l}}, s^{k l} s_{m l} = δ_{m}^{k} . \end{matrix}

(122)

From Equation ( 121 ) the connection can be obtained. If

{\hat{j}}^{l} = \sqrt{- det s_{k l}} j^{l}

, and

j_{k} = s_{k l} j^{l}

, then the affine connection may formally be expressed by

\begin{matrix} Γ_{k j}^{l} = \frac{1}{2} s^{l r} (\frac{\partial s_{k r}}{\partial x^{j}} + \frac{\partial s_{j r}}{\partial x^{k}} - \frac{\partial s_{k j}}{\partial x^{r}}) - \frac{1}{2} s_{k j} {\hat{j}}^{l} + \frac{1}{3} δ_{(k}^{l} {\hat{j}}_{j)} \end{matrix}

(123)

This equation is an identity if a solution of the field equations ( 121 ) is inserted. From Equation ( 123 ),

\begin{matrix} Γ_{k j}^{j} = \frac{\partial}{\partial x^{k}} \sqrt{det s_{l r}} + \frac{1}{3} j_{k} . \end{matrix}

(124)

If no electromagnetic field is present,

{\hat{s}}^{k l}

reduces to

{\hat{s}}^{k l} = g^{k l} \sqrt{- det g_{k l}}

; the definition of the metric

g_{i j}

in Equation ( 119 ) is reinterpreted by Einstein as giving his vacuum field equation with cosmological constant

λ^{- 2}

. In order that this makes sense, the identifications in Equation ( 119 ) are always to be made after the variation of the Lagrangian is performed.

For non-vanishing electromagnetic field, due to Equation ( 124 ) the Equation ( 120 ) now becomes

\begin{matrix} {\hat{φ}}_{k l} = \frac{1}{6} (\frac{\partial {\hat{j}}_{k}}{\partial x^{l}} - \frac{\partial {\hat{j}}_{l}}{\partial x^{k}}), \end{matrix}

(125)

which means that for vanishing current density no electromagnetic field is possible. Einstein concluded:

“But the extraordinary smallness of $\frac{1}{λ^{2}}$ implies that finite $φ_{k l}$ are possible only for tiny, almost vanishing current density. Except for singular positions, the current density is practically vanishing.”^$112$ “Aber die ausserordentliche Kleinheit von $\frac{1}{λ^{2}}$ bringt es mit sich, dass endliche $φ_{k l}$ nur bei winzigen, praktisch verschwindenden kovarianten Stromdichten möglich sind. Singuläre Stellen ausgenommen verschwindet praktisch also die Stromdichte.”

Einstein went on to show that Maxwell's vacuum equations are holding in first order approximation.

Up to the same order,

{\hat{f}}^{k l} ≃ {\hat{φ}}_{k l}

. In general however,

{\hat{φ}}_{k l} \neq s_{k m} s_{l n} {\hat{f}}^{n m} .

Also, the geometrical theory presented here is energetically closed, i.e., the current density

{\hat{j}}_{l}

cannot be given arbitrarily as in the usual Maxwell theory with external sources.

Einstein was not sure whether “electrical elementary elements”, i.e., nonsingular electrons, are possible in this theory; they might be. He found it remarkable “[...] that, according to this theory, positive and negative electricity cannot differ just in sign”^$113$ “[...] dass nach dieser Theorie die positive und die negative Elektrizität keineswegs bloss dem Vorzeichen nach verschieden sein können.” ([75] , p. 38). His final conclusion was:

“that EDDINGTON'S general idea in context with the Hamiltonian principle leads to a theory almost free of ambiguities; it does justice to our present knowledge about gravitation and electricity and unifies both kinds of fields in a truly accomplished manner.”^$114$ “dass EDDINGTONS allgemeiner Gedanke in Verbindung mit dem Hamiltonschen Prinzip zu einer von Willkür fast freien Theorie führt, welche unserem bisherigen Wissen über Gravitation und Elektrizität gerecht wird und beide Feldarten in wahrhaft vollendeter Weise vereinigt.” ([75] , p. 38)

Until the end of May 1923, two further publications followed in which Einstein elaborated on the theory. In the second paper, he exchanged the Lagrangian

ℒ = 2 \sqrt{- det K_{i j}}

for a new one, i.e., for

ℒ = - 2 \sqrt{det K_{i j}} + \hat{R} - \frac{1}{6} {\hat{s}}^{l m} i_{l} i_{m},

where

{\hat{ı}}^{k} = i^{k} \sqrt{det s_{k l}}

ℒ

is to be varied with respect to

{\hat{s}}^{k l}

and

{\hat{f}}^{k l}

. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skew-symmetric part, respectively, of

\begin{matrix} K_{j k} = R_{j k} + \frac{1}{6} (\frac{\partial i_{j}}{\partial x^{k}} - \frac{\partial i_{k}}{\partial x^{j}} + i_{j} i_{k}) . \end{matrix}

(126)

Although the theory offered, for every solution with positive charge, also a solution with negative charge, the masses in the two cases were the same. However, the only known particle with positive charge at the time (what is now called the proton) had a mass greatly different from the particle with negative charge, the electron. Einstein noted:

“Therefore, the theory may not account for the difference in mass of positive and negative electrons.”^$115$ “Die Theorie vermag also jedenfalls von der Verschiedenheit der Masse der positiven und negativen Elektronen keine Rechenschaft zu geben.” ([72] , p. 77)

In the third paper [74] , apart from changing notations^$116$ , Einstein set

λ = 1

. He also dropped the assumption ( 119 ) and replaced it by allowing his Lagrangian (Hamiltonian)

\hat{H}

to be a function of the two independent variables,

\begin{matrix} γ_{i j} = K_{(i j)}, φ_{i j} = K_{[i j]} . \end{matrix}

(127)

The logic of the subsequent derivations in his paper is quite involved. The first step consisted in the definition of tensor densities

\begin{matrix} {\hat{g}}^{k l} : = \frac{δ \hat{H}}{δ γ_{k l}}, {\hat{f}}^{k l} : = \frac{δ \hat{H}}{δ φ_{k l}} . \end{matrix}

(128)

In the second step, the variations

δ γ_{k l}

and

δ φ_{k l}

were expressed by

δ Γ_{i k}^{l}

via ( 127 ) and inserted into

δ \hat{H} = 0

. The ensuing equation could be solved for

Γ_{i k}^{l}

and led to Equation ( 123 ). In the third step, the Lagrangian

{\hat{H}}^{*}

is taken as a functional of the variables introduced in the first step, i.e., of

{\hat{g}}^{k l}, {\hat{f}}^{k l}

such that in place of Equation ( 128 ) the relations

\begin{matrix} γ_{k l} : = \frac{δ \hat{H} *}{δ {\hat{g}}^{k l}}, φ_{k l} : = \frac{δ \hat{H} *}{δ {\hat{f}}^{k l}} . \end{matrix}

(129)

hold. Einstein then took “the expression most natural vis-a-vis our present knowledge”, i.e.,

{\hat{H}}^{*} = 2 α \sqrt{- g} - \frac{β}{2} f_{i k} {\hat{f}}^{i k}

. By using both Equation ( 127 ) and Equation ( 129 ), Einstein obtained the Einstein–Maxwell equations augmented by a term

- \frac{1}{6} i_{k} i_{l}

on the side of the energy-momentum tensor of the electromagnetic field and Equation ( 125 ) with a changed l.h.s. now reading

- β f_{i k}

After a field rescaling, he then took a third expression to become his Lagrangian

\bar{H} = \sqrt{- g} [R - 2 α + κ (\frac{1}{2} f_{i j} f^{i j}) - \frac{1}{β} i_{l} i^{l}],

where

α

and

β

are arbitrary constants, and

κ

is the gravitational constant.

i_{l}

is defined to be proportional to the electromagnetic 4-potential

f_{k}

, i.e.,

\frac{1}{β} i_{l} = - f_{k},

and

f_{i j}

corresponds to

φ_{i j}, \sqrt{- g} f^{i j}

{\hat{f}}^{i j} .

After the field equations had been obtained by this longwinded procedure, it became obvious that they could also be derived from

\bar{H} = H

taken as an “effective” Lagrangian varied with respect to

g_{i k}

and

f_{i k}

. In Einstein's words: “

R

is the Riemannian curvature scalar formed from

g_{i j}

”^$117$ “

R

bedeutet hierbei den aus den

g_{i j}

gebildeten RIEMANNschen Krümmungsskalar”. In the third paper as well, Einstein's desire to create a unified field theory satisfying all his criteria still was not fulfilled: His equations, again, did not give a singularity-free electron. In a paper on Hilbert's vision of a unified science, Sauer and Majer recently have found out from lectures of Hilbert given in Hamburg and Zürich in 1923, that Hilbert considered Einstein's work in affine theory a return to his own results of 1915 by “[...] a colossal detour via Levi-Civita, Weyl, Schouten, Eddington [...]” [214] . It seems that, in this evaluation, Hilbert was influenced by Einstein's proportionality between the 4-potential and the electrical current which Hilbert had assumed as early as in 1915 [160] ^$118$ .

^$109$ There even exist a manuscript dating from Einstein's stop-over in Singapore; it is incomplete and of little importance [222] .

^$111$ In place of our $K_{i k}$ Einstein used $R_{i k}$ .

^$113$ orgQuoteDe

^$116$ He exchanged $g_{i k}$ by $γ_{i k}$ , $R_{i k}$ by $r_{i k}$ .

^$117$ orgQuoteDe

^$118$ This proportionality is about the simplest assumption one can make; the equation $\nabla_{l} F^{k l} \sim A^{k}$ corresponds, in the case of the gravitational field, to the equation $R_{i k} = λ g_{i k}$ leading to Einstein spaces.

4.3.3 Comments by Einstein's colleagues

While, in the meantime, mathematicians had taken over the conceptual development of affine theory, some other physicists, including the perpetual pièce de resistance Pauli, kept a negative attitude:

“[...] I now do not at all believe that the problem of elementary particles can be solved by any theory applying the concept of continuously varying field strengths which satisfy certain differential equations to regions in the interior of elementary particles. [...] The quantities $Γ_{ν α}^{μ}$ cannot be measured directly, but must be obtained from the directly measured quantities by complicated calculational operations. Nobody can determine empirically an affine connection for vectors at neighbouring points if he has not obtained the line element before. Therefore, unlike you and Einstein, I deem the mathematician's discovery of the possibility to found a geometry on an affine connection without a metric as meaningless for physics, in the first place.”^$119$ “[...] Ich glaube nun überhaupt nicht, dass dieses Problem der elektrischen Elementarteilchen von irgend einer Theorie gelöst werden kann, die den Begriff der kontinuierlich variierenden Feldstärken, die gewissen Differentialgleichungen genügen, auf die Gebiete im Innern der Elementarteilchen anwendet. [...] Die Grössen $Γ_{ν α}^{μ}$ können nicht direkt gemessen werden, sondern müssen aus den direkt gemessenen Grössen erst durch komplizierte Rechenoperationen gewonnen werden. Niemand kann empirisch einen affinen Zusammenhang zwischen Vektoren in benachbarten Punkten feststellen, wenn er nicht vorher bereits das Linienelement ermittelt hat. Deswegen halte ich im Gegensatz zu Ihnen und Einstein die Erfindung der Mathematiker, dass man auch ohne Linienelement auf einen affinen Zusammenhang eine Geometrie gründen kann, zunächst für die Physik bedeutungslos.” (Pauli to Eddington on 20 September 1923; [250] , pp. 115–119)

Also Weyl, in the 5th edition of Raum–Zeit–Materie ([398] , Appendix 4), in discussing “world-geometric extensions of Einstein's theory”, found Eddington's theory not convincing. He criticised a theory that keeps only the connection as a fundamental building block for its lack of a guarantee that it would also house the conformal structure (light cone structure). This is needed for special relativity to be incorporated in some sense, and thus must be an independent fundamental input [405] .

Likewise, Eddington himself did not appreciate much Einstein's followership. In Note 14, § 100 appended to the second edition of his book, he laid out Einstein's theory but not without first having warned the reader:

“The theory is intensely formal as indeed all such action-theories must be, and I cannot avoid the suspicion that the mathematical elegance is obtained by a short cut which does not lead along the direct route of real physical progress. From a recent conversation with Einstein I learn that he is of much the same opinion.” ([62] , pp. 257–261)

In fact, when Eddington's book was translated into German in 1925 [58] , Einstein wrote an appendix to it in which he repeated, with minor changes, the results of his last paper on the affine theory.

His outlook on the state of the theory now was rather bleak:

“For me, the final result of this consideration regrettably consists in the impression that the deepening of the geometrical foundations by Weyl–Eddington is unable to bring progress for our physical understanding; hopefully, future developments will show that this pessimistic opinion has been unjustified.”^$120$ “Für mich besteht das Endergebnis dieser Betrachtung leider in dem Eindruck, dass uns die Weyl–Eddingtonsche Vertiefung der geometrischen Grundlagen keinen Fortschritt der physikalischen Erkenntnis zu bringen vermag; hoffentlich wird die künftige Entwicklung zeigen, dass diese pessimistische Meinung unberechtigt gewesen ist.” ([58] , p. 371)

An echo of this can be found in Einstein's letter to Besso of 5 June 1925:

“I am firmly convinced that the entire chain of thought Weyl–Eddington–Schouten does not lead to something useful in physics, and I now have found another, physically better founded approach. To me, the quantum-problem seems to require something like a special scalar, for the introduction of which I have found a plausible way.”^$121$ “Ich bin fest überzeugt, dass die ganze Gedanken-Reihe Weyl–Eddington–Schouten zu nichts physikalisch brauchbarem führt und habe jetzt eine andere Spur gefunden, die mehr physikalisch fundiert ist. Das Quanten-Problem scheint mir etwas wie einen besonderen Skalar zu verlangen, für dessen Einführung ich einen plausiblen Weg gefunden habe.” ([98] , p. 204)

This remark shows that Einstein must have taken some notice of Schouten's work in affine geometry.

What the “special scalar” was, remains an open question.

4.3.4 Overdetermination of partial differential equations and elementary particles

Einstein spent much time in thinking about the “quantum problem”, as he confessed to Born:

“I do not believe that the theory will be able to dispense with the continuum. But I fail to succeed in giving my pet idea a tangible form: to understand the quantum-structure through an overdetermination by differential equations.”^$122$ “Ich glaube nicht, dass die Theorie das Kontinuum wird entbehren können. Es will mir aber nicht gelingen, meiner Lieblingsidee, die Quantenstruktur aus einer Überbestimmung durch Differentialgleichungen zu verstehen, greifbare Gestalt zu geben.” ([102] , pp. 48–49)

In a paper from December 1923, Einstein not only stated clearly the necessary conditions for a unified field theory to be acceptable to him, but also expressed his hope that this technique of “overdetermination” of systems of differential equations could solve the “quantum problem”.

“According to the theories known until now the initial state of a system may be chosen freely; the differential equations then give the evolution in time. From our knowledge about quantum states, in particular as it developed in the wake of Bohr's theory during the past decade, this characteristic feature of theory does not correspond to reality.
The initial state of an electron moving around a hydrogen nucleus cannot be chosen freely; its choice must correspond to the quantum conditions. In general: not only the evolution in time but also the initial state obey laws.”^$123$ “Nach den bisherigen Theorien kann der Anfangszustand eines Systems frei gewählt werden; die Differentialgleichungen liefern dann die zeitliche Fortsetzung. Nach unserem Wissen über die Quantenzustände, wie es sich insbesondere im Anschluss an die BOHRsche Theorie im letzten Jahrzehnt entwickelt hat, entspricht dieser Zug der Theorie nicht der Wirklichkeit. Der Anfangszustand eines um einen Wasserstoffkern bewegten Elektrons kann nicht frei gewählt werden, sondern diese Wahl muss den Quantenbedingungen entsprechen. Allgemein: nicht nur die zeitliche Fortsetzung, sondern auch der Anfangszustand unterliegt Gesetzen.” ([73] , pp. 360–361)

He then ventured the hope that a system of overdetermined differential equations is able to determine

“also the mechanical behaviour of singular points (electrons) in such a way that the initial states of the field and of the singular points are subjected to constraints as well.
[...] If it is possible at all to solve the quantum problem by differential equations, we may hope to reach the goal in this direction.”

We note here Einstein's emphasis on the very special problem of the quantum nature of elementary particles like the electron, as compared to the general problem of embedding matter fields into a geometrical setting.

One of the crucial tests for an acceptable unified field theory for him now was:

“The system of differential equations to be found, and which overdetermines the field, in any case must admit this static, spherically symmetric solution which describes, respectively, the positive and negative electron according to the equations given above [i.e the Einstein–Maxwell equations].”^$124$ “Das gesuchte Gleichungssystem, welches das Feld überbestimmt, muss jedenfalls jene statische, kugelsymmetrische Lösung zulassen, welche gemäss obigen Gleichungen [i.e., the Einstein–Maxwell equations] das positive bzw. negative Elektron beschreibt.”

This attitude can also be found in a letter to M. Besso from 5 January 1924:

“The idea I am wrestling with concerns the understanding of the quantum facts; it is:
overdetermination of the laws by more field equations than field variables. In such a way, the un-ambiguity of the initial conditions ought to be understood without leaving field theory. [...] The equations of motion of material points (electrons) will be given up totally; their motion ought to be co-determined by the field laws.”^$125$ “Die Idee, mit der ich mich herumschlage, betrifft das Verstehen der Quantentheorie und heisst: Überbestimmung der Gesetze durch mehr Gleichungen als Feldvariable. So soll die Nichtwillkürlichkeit der Anfangsbedingungen begriffen werden, ohne die Feldtheorie zu verlassen.[...] Die Bewegungsgleichungen materieller Punkte (Elektronen) wird ganz aufgegeben; das motorische Verhalten der letzteren soll durch die Feldgesetze mitbestimmt werden.” ([98] , p. 197)

In his answer, Besso asked for more information concerning the quantum aspect of the concept of “overdetermination”, because:

“On the one hand, this seems to be connected only formally with a field theory; on the other, it has not yet dawned on me how in this manner something corresponding to the discrete quantum orbits may be reached.”^$126$ “Einerseits scheint das nur noch formell etwas mit einer Feldtheorie gemeinsam zu haben; und andererseits schimmert mir noch nicht, wie auf diesem Wege etwas den diskreten Quantenbahnen entsprechendes zu erreichen ist.” ([98] , p. 199)

5 Differential Geometry's High Tide

In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [336] :

(1) The generalisation of parallel transport in the sense of Levi-Civita and Weyl. Schouten is the leading figure in this approach [299] .
(2) The “geometry of paths” considering the lines of constant direction for a connection – with the proponents Veblen, Eisenhart [121, 113, 114, 373] , J. M. Thomas [347] , and T. Y. Thomas^$127$ [348, 346] .
Here, only symmetric connections can appear.
(3) The idea of mapping a manifold at one point to a manifold at a neighbouring point is central (affine, conformal, projective mappings). The names of König [191] and Cartan [27, 301] are connected with this program.

In his assessment, Eisenhart [120] adds to this all the geometries whose metric is

“based upon an integral whose integrand is homogeneous of the first degree in the differentials. Developments of this theory have been made by Finsler, Berwald, Synge, and J. H. Taylor. In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics. Affine properties of these spaces are obtained from a natural generalisation of the definition of Levi-Civita for Riemannian spaces.” ([120] , p. V)

In fact, already in May 1921 Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [296, 295] . In the first he wrote:

“Motivated by relativity theory, differential geometry received a totally novel, simple and satisfying foundation; I just refer to G. Hessenberg's `Vectorial foundation...', Math. Ann. 78, 1917, S. 187–217 and H. Weyl, Raum–Zeit–Materie, 2. Section, Leipzig 1918 (3. Aufl. Berlin 1920) as well as `Reine Infinitesimalgeometrie' etc.^$128$ . [...] In the present investigation all 18 different linear connections are listed and determined in an invariant manner. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field [...].”^$129$ “Durch die Relativitätstheorie veranlasst, hat die Differentialgeometrie eine ganz neue, einfache und befriedigende Begründung erfahren; ich nenne nur G. Hessenberg `Vektorielle Begründung...', Math. Ann. 78, 1917, S. 187–217 und H. Weyl, Raum–Zeit–Materie, 2. Kap., Leipzig 1918 (3. Aufl. Berlin 1920) sowie `Reine Infinitesimalgeometrie' etc. [...] In der vorliegenden Untersuchung sind nun alle achtzehn verschiedenen Arten der linearen Übertragung vollständig aufgezählt und in invarianter Weise festgelegt. Die allgemeinste Übertragung wird durch zwei Felder dritten Grades, ein Tensorfeld zweiten Grades und ein Vektorfeld charakterisiert, [...].” ([296] , p. 57)

The fields referred to are the torsion tensor

S_{i j}^{k}

, the tensor of non-metricity

Q_{i j}^{k}

, the metric

g_{i j}

, and the tensor

C_{i j}^{k}

which, in unified field theory, was rarely used. It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1-form not by the connection

L_{i j}^{k}

in Equation ( 13 ), but by

\begin{matrix} {\overset{+}{ω}}_{i} = \frac{\partial ω_{i}}{\partial x^{k}} -' L_{k i}^{j} ω_{j}, \end{matrix}

(130)

with

' L \neq L

. In fact

\begin{matrix} C_{i j}^{k} : = L_{i j}^{k} -' L_{i j}^{k} . \end{matrix}

(131)

In the first paper, Schouten had considered only the special case

C_{i j}^{k} = C_{i} δ_{j}^{k}

^$130$ .

Furthermore, on p. 57 of [296] we read:

“The general connection for $n = 4$ at least theoretically opens the door for an extension of Weyl's theory. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression.”^$131$ “Die allgemeinen Übertragungen für $n = 4$ eröffnen für die Physik wenigstens theoretisch die Möglichkeit einer Erweiterung der Weylschen Theorie. Für eine solche Erweiterung ist eine invariante Festlegung der Übertragung notwendig, da eine physische Erscheinung nur mit einem invarianten Ausdruck korrespondieren kann.”

Through footnote 5 on the same page we learn the pedagogical reason why Schouten did not use the `direct' method [293, 335] in his presentation, but rather a coordinate dependent formalism^$132$ :

“As the results of the present investigation might be of interest for a wider circle of mathematicians, and also for a number of physicists [...].”^$133$ “Da die Resultate der vorliegenden Arbeit aber für weitere Kreise von Mathematikern und auch für manche Physiker interessant sein dürften [...].”

At the end of the first paper we can find a section “Eventual importance of the present investigation for physics” (p. 79–81) and the confirmation that during the proofreading Schouten received Eddington's paper ([56] , accepted 19 February 1921). Thus, while Einstein and Weyl influenced Eddington, Schouten apparently did his research without knowing of Eddington's idea.

Einstein, perhaps, got to know Schouten's work only later through the German translation of Eddington's book where it is mentioned ([58] , p. 319), and to which he wrote an addendum, or, more directly, through Schouten's book on the Ricci calculus, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, in the same famous yellow series of Springer Verlag [299] .

On the other hand, Einstein's papers following Eddington's [75, 72] inspired Schouten to publish on a theory with vector torsion that tried to remedy a problem Einstein had noted in his papers, i.e., that no electromagnetic field could be present in regions of vanishing electric current density.

According to Schouten

“[...] we see that the electromagnetic field only depends on the curl of the electric current vector, so that the difficulty arises that the electromagnetic field cannot exist in a place with vanishing current density. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical.” ([299] , p. 850)

Schouten criticised Einstein's argument for using a symmetric connection^$134$ as unfounded (cf. Equation ( 15 )). He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:

“We will not consider the most general case, but the semi-symmetric case in which the alternating part of the parameters has the form: $1 / 2 (Γ_{μ λ}^{^{'} ν} - Γ_{λ μ}^{^{'} ν}) = 1 / 2 (S_{λ} δ_{μ}^{ν} - S_{μ} δ_{λ}^{ν}),$ in which $S_{λ}$ is a general covariant vector.” ([297] , p. 851)

The affine connection

Γ^{^{'}}

can then be decomposed as follows:

\begin{matrix} Γ_{j k}^{^{'} l} = Λ_{j k}^{l} + S_{[j} δ_{k]}^{l} . \end{matrix}

(132)

Hence, besides the covariant derivative

\nabla^{^{'}}

following from use of

Γ_{j k}^{^{'} l}

, in his calculations Schouten also introduced a covariant derivative

\nabla^{*}

formed with

Λ_{j k}^{l}

. Schouten's point of departure for the field equations is Einstein's first Lagrangian

ℒ = \sqrt{det K_{i j}}

and, consequently, his field equations were the same as Einstein's apart from additional terms in vector torsion. Also, Schouten's definition of some of the observables is different; For example, the electromagnetic field tensor unlike in Equation ( 125 ) is now

\begin{matrix} {\hat{F}}_{k l} = \frac{1}{6} (\frac{\partial {\hat{ı}}_{k}}{\partial x^{l}} - \frac{\partial {\hat{ı}}_{l}}{\partial x^{k}}) - (\frac{\partial {\hat{S}}_{k}}{\partial x^{l}} - \frac{\partial {\hat{S}}_{l}}{\partial x^{k}}), \end{matrix}

(133)

where

i^{k} : = \nabla_{l}^{*} f^{k l} - P_{l} f^{k l}

, and

P_{k} : = - \frac{\partial}{\partial x^{k}} (log \sqrt{- det K_{i j}}) + Λ_{l k}^{l}

. On the same topic, Schouten wrote a paper with Friedman in Leningrad [141] . A similar, but less detailed, classification of connections than Schouten's has also been given by Cartan. He relied on the curvature, torsion and homothetic curvature 2-forms ([30] , Section III; cf. also Section 2.1.4 ). In 1925, Eyraud^$135$ came back to Schouten's paper [297] and proved that his connection can be mapped projectively and conformally on a Riemannian space [123, 122] .

Other mathematicians were also stimulated by Einstein's use of differential geometry in his general relativity and, particularly, by the idea of unified field theory. Examples are Eisenhart and Veblen, both in Princeton, who developed the “geometry of paths”^$136$ under the influence of papers by Weyl, Eddington, and Einstein [121, 116, 383] . In Eisenhart's paper, we may read that

“Einstein has said (in Meaning of Relativity ) that `a theory of relativity in which the gravitational field and the electromagnetic field enter as an essential unity' is desirable and recently has proposed such a theory.” ([116] , pp. 367–368)

and

“His geometry also is included in the one now proposed and it may be that the latter, because of its greater generality and adaptability will serve better as the basis for the mathematical formulation of the results of physical experiments.” ([116] , p. 369)

The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad. Seven years after Schouten's classification of connections, Fréedericksz of Leningrad – known better for his contributions to the physics of liquid crystals – put forward a classification of his own by using both the connection and the curvature tensor [137] .

^$130$ The 18 possibilities were numbered by Schouten as I, . . . , VI a–c; he mentioned 5 examples: Einstein (VI c), Hessenberg (VI a), Weyl (IV c) and (II c) (the latter also corresponds to Eddington's choice) as well as König (II a) (cf. also [295] ).

^$132$ A summary of Schouten's papers from 1922 is given in [299] .

^$134$ Einstein had wished to avoid the distinction between $\overset{+}{\nabla}$ and $\bar{\nabla}$ .

^$135$ Henri Eyraud (1892–1994). Studied mathematics at the University of Paris and received his doctorate in 1926 with a thesis on “Metrical spaces and physico-geometrical theories”. From 1930 professor of mathematics at the University of Lyon and director of the Institute of “Financial and Assurance-Sciences”. Perhaps he considered his papers on the geometry of unified field theory as a sin of his youth:

In Poggendorff, among the 33 papers listed, all are from his later main interest.

^$136$ The geometry of paths involves a change of connection that preserves the geodesics when vectors are displaced along themselves.

6 The Pursuit of Unified Field Theory by Einstein and His Collaborators

6.1 Affine and mixed geometry

Already in July 1925 Einstein had laid aside his doubts concerning “the deepening of the geometric foundations”. He modified Eddington's approach to the extent that he now took both a non-symmetric connection and a non-symmetric metric, i.e., dealt with a mixed geometry (metric-affine theory):

“[...] Also, my opinion about my paper which appeared in these reports [i.e., Sitzungsberichte of the Prussian Academy, Nr. 17, p. 137, 1923], and which was based on Eddington's fundamental idea, is such that it does not present the true solution of the problem. After an uninterrupted search during the past two years I now believe to have found the true solution.”^$137$ “[...] Auch von meiner in diesen Sitzungsberichten (Nr. 17, p. 137 1923) erschienenen Abhandlung, welche ganz auf Eddingtons Grundgedanke basiert war, bin ich der Ansicht, dass sie die wahre Lösung des Problems nicht gibt.
Nach unablässigem Suchen in den letzten zwei Jahren glaube ich nun die wahre Lösung gefunden zu haben.” ([76] , p. 414)

As in general relativity, he started from the Lagrangian

ℒ = {\hat{g}}^{i k} R_{i k}

, but now with

{\hat{g}}^{i k}

and the connection

Γ_{k j}^{l}

being varied separately as independent variables. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:

\begin{matrix} - \frac{\partial g_{i k}}{\partial x^{l}} + g_{r k} Γ_{i l}^{r} + g_{i r} Γ_{l k}^{r} + g_{i k} φ_{l} + g_{i l} φ_{k} = 0, R_{i k} = 0, \end{matrix}

(134)

i.e.,

64 + 16

equations for the same number of variables.

φ_{k}

is an arbitrary covariant vector. The asymmetric

g_{i k}

is related to

{\hat{g}}^{l m}

\begin{matrix} {\hat{g}}_{i r} {\hat{g}}^{j r} = {\hat{g}}_{r i} {\hat{g}}^{r j} = δ_{i}^{j}, {\hat{g}}_{i k} = \frac{g_{i k}}{\sqrt{- g}} . \end{matrix}

(135)

The three equations ( 134 ) and

\begin{matrix} \frac{\partial {\hat{g}}^{i k}}{\partial x^{k}} - \frac{\partial {\hat{g}}^{k i}}{\partial x^{k}} = 0, R_{i k} = 0, \end{matrix}

(136)

were the result of the variation. In order to be able to interpret the symmetric part of

g_{i k}

as metrical tensor and its anti(skew)-symmetric part as the electromagnetic field tensor, Einstein put

φ_{k} = 0,

i.e., overdetermined his system of partial differential equations. However, he cautioned:

“However, for later investigations (e.g., the problem of the electron) it is to be kept in mind that the HAMILTONian principle does not provide an argument for putting $φ_{k}$ equal to zero.”^$138$ “Man wird jedoch für spätere Untersuchungen (z. B. Problem des Elektrons) im Sinne behalten müssen, dass das HAMILTONsche Prinzip für das Verschwinden der $φ_{k}$ keinen Anhaltspunkt liefert.”

In comparing Equation ( 134 ) with

φ_{k} = 0

and Equation ( 47 ), we note that the expression does not seem to correspond to a covariant derivative due to the

+

sign where a

-

sign is required.

But this must be due to either a calculational error, or to a printer's typo because in the paper of J. M. Thomas following Einstein's by six months and showing that Einstein's

“new equations can be obtained by direct generalisation of the equations of the gravitational field previously given by him. The process of generalisation consists in abandoning assumptions of symmetry and in adopting a definition of covariant differentiation which is not the usual one, but which reduces to the usual one in case the connection is symmetric.” ([345] , p. 187)

J. M. Thomas wrote Einstein's Equation ( 134 ) in the form

\begin{matrix} g_{i k / l} = g_{i k} φ_{l} + g_{i l} φ_{k}, φ_{l} = - \frac{2}{n - 1} Ω_{r l}^{r}, \end{matrix}

(137)

with

Ω

being the skew-symmetric part of the asymmetric connection

H_{i j}^{k} = Γ_{(i j)}^{k} + Ω_{[i j]}^{k}

, and

g_{i j}

being the symmetric part of the asymmetric metric

h_{i j} = g_{(i j)} + ω_{[i j]}

. The two covariant derivatives introduced by J. M. Thomas are

g_{i j, k} = \frac{\partial g_{i j}}{\partial x^{k}} - g_{r j} Γ_{i k}^{r} - g_{i r} Γ_{j k}^{r}

and

h_{i j / k} = \frac{\partial h_{i j}}{\partial x^{k}} - h_{r j} H_{i k}^{r} - h_{i r} H_{j k}^{r}

J. M. Thomas then could reformulate Equation ( 137 ) in the form

\begin{matrix} g_{i j / l} = g_{[r i]} Ω_{j l}^{r} + g_{[i r]} Ω_{l j}^{r}, \end{matrix}

(138)

and derive the result

\begin{matrix} g_{i j, l} + g_{j l, i} + g_{l i, j} = 0 \end{matrix}

(139)

(see [345] , p. 189).

After having shown that his new theory contains the vacuum field equations of general relativity for vanishing electromagnetic field, Einstein then proved that, in a first-order approximation, Maxwell's field equations result cum grano salis : Instead of

F_{i k, l} + F_{l i, k} + F_{k l, i} = 0

he only obtained

\sum \frac{\partial}{\partial x^{l}} (F_{i k, l} + F_{l i, k} + F_{k l, i}) = 0

This was commented on in a paper by Eisenhart who showed “more particularly what kind of linear connection Einstein has employed” and who obtained “in tensor form the equations which in this theory should replace Maxwell's equations.” He then pointed to some difficulty in Einstein's theory: When identification of the components of the antisymmetric part

φ_{i j}

of the metric

a_{i j} = g_{i j} + φ_{i j}

with the electromagnetic field is made in first order,

“they are not the components of the curl of a vector as in the classical theory, unless an additional condition is added.” ([119] , p. 129)

Toward the end of the paper Einstein discussed time-reversal; according to him, by it the sign of the magnetic field is changed, while the sign of the electric field vector is left unchanged^$139$ . As he wanted to obtain charge-symmetric solutions from his equations, Einstein now proposed to change the roles of the magnetic fields and the electric fields in the electromagnetic field tensor. In fact, the substitutions

\tilde{E} \to \tilde{B}

and

\tilde{B} \to - \tilde{E}

leave invariant Maxwell's vacuum field equations (duality transformations)^$140$ . Already Pauli had pointed to time-reflection symmetry in relation with the problem of having elementary particles with charge

\pm e

and unequal mass ([245] , p. 774).

At first, Einstein seems to have been proud about his new version of unified field theory; he wrote to Besso on 28 July 1925 that he would have liked to present him “orally, the egg laid recently, but now I do it in writing”, and then explained the independence of metric and connection in his mixed geometry. He went on to say:

“If the assumption of symmetry^$141$ is dropped, the laws of gravitation and Maxwell's field laws for empty space are obtained in first approximation; the antisymmetric part of ${\hat{g}}^{i k}$ is the electromagnetic field. This is surely a magnificent possibility which likely corresponds to reality. The question now is whether this field theory is consistent with the existence of quanta and atoms.
In the macroscopic realm, I do not doubt its correctness.”^$142$ “Lässt man die Voraussetzung der Symmetrie fallen, so erhält man in erster Näherung die Gesetze der Gravitation und die Maxwell'schen Feldgesetze für den leeren Raum, wobei der antisymmetrische Teil der ${\hat{g}}^{i k}$ das elektromagnetische Feld ist. Dies ist doch eine prachtvolle Möglichkeit, die doch der Realität entsprechen dürfte. Nun ist die Frage, ob diese Feldtheorie mit der Existenz der Atome und Quanten vereinbar ist. Im Makroskopischen zweifle ich nicht an ihrer Richtigkeit.” ([98] , p. 209)

We have noted before that a similar suggestion within a theory with a geometry built from an asymmetric metric had been made, in 1917, by Bach alias Förster. Yet, in the end, also this novel approach did not convince Einstein. Soon after the publication discussed, he found his argument concerning charge symmetric solutions not to be helpful. The link between the occurrence of solutions with both signs of the charge with time-symmetry of the field equations induced him to doubt, if only for a moment, whether the endeavour of unifying electricity and gravitation made sense at all:

“To me, the insight seems to be important that an explanation of the dissimilarity of the two electricities is possible only if time is given a preferred direction, and if this is taken into account in the definition of the decisive physical quantities. In this, electrodynamics is basically different from gravitation; therefore, the endeavour to melt electrodynamics with the law of gravitation into one unity, to me no longer seems to be justified.”^$143$ “Wesentlich scheint mir die Erkenntnis zu sein, dass eine Erklärung der Ungleichartigkeit der beiden Elektrizitäten nur möglich ist, wenn man der Zeit eine Ablaufrichtung zuschreibt und diese bei der Definition der massgebenden physikalischen Grössen heranzieht. Hierin unterscheidet sich die Elektrodynamik von der Gravitation; deshalb erscheint mir auch das Bestreben, die Elektrodynamik mit dem Gravitationsgesetz zu einer Einheit zu verschmelzen, nicht mehr gerechtfertigt.” [77]

In a paper dealing with the field equations

\begin{matrix} R_{i k} - \frac{R}{4} g_{i k} = - κ T_{i k}, \end{matrix}

(140)

which had been discussed earlier by Einstein [68] , and to which he came back now after Rainich^$144$ 's insightful paper into the algebraic properties of both the curvature tensor and the electromagnetic field tensor ([262, 263, 264, 265] ), Einstein indicated that he had lost hope in the extension of Eddington's affine theory:

“That the equations ( 140 ) have received only little attention is due to two circumstances.
First, the attempts of all of us were directed to arrive, along the path taken by Weyl and Eddington or a similar one, at a theory melting into a formal unity the gravitational and electromagnetic fields; but by lasting failure I now have laboured to convince myself that truth cannot be approached along this path.”^$145$ “Dass die Gleichungen ( 140 ) noch wenig Beachtung gefunden haben, liegt an zwei Umständen. Erstens nämlich waren unser aller Bestrebungen darauf gerichtet, auf dem von Weyl und Eddington eingeschlagenen oder einem ähnlichen Weg zu einer Theorie zu gelangen, die das Gravitationsfeld und das elektromagnetische Feld zu einer formalen Einheit verschmilzt; durch mannigfache Misserfolge habe ich mich aber nun zu der Überzeugung durchgerungen, dass man auf diesem Wege der Wahrheit nicht näher kommt.” (Einstein's italics; [78] , p. 100)

The new field equation was picked up by R. N. Sen of Kalkutta who calculated “the energy of an electric particle” according to it [322] .

In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas 1925 in words similar to those in his letter in June:

“Regrettably, I had to throw away my work in the spirit of Eddington. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl–Eddington. The equations $R_{i k} - \frac{1}{4} R g_{i k} = - κ T_{i k} e l e c t r o m a g n e t i c$ I take as the best we have nowadays. They are 9 equations for the 14 variables $g_{i k}$ and $γ_{i k}$ . New calculations seem to show that these equations yield the motion of the electrons. But it appears doubtful whether there is room in them for the quanta.”^$146$ “Meine Arbeit im Sinne Eddington's habe ich leider verwerfen müssen. Ueberhaupt bin ich jetzt überzeugt, dass mit dem Weyl–Eddington'schen Gedanken-Komplex leider nichts zu machen ist. Ich halte die Gleichung $R_{i k} - \frac{1}{4} R g_{i k} = - κ T_{i k} e l e k t r o m a g n e t i s c h$ [cf. Equation ( 140 )] für das beste, was wir heute haben. Es sind 9 Gleichungen für die 14 Grössen $g_{i k}$ und $γ_{i k}$ . Aus den neuen Rechnungen scheint sich zu ergeben, dass diese Gleichungen die Bewegung der Elektronen liefern. Aber es erscheint zweifelhaft, ob die Quanten darin Platz haben.” ([98] , p. 216)

According to the commenting note by Tonnelat, the 14 variables are given by the 10 components of the symmetric part

g_{(i k)}

^$147$ of the metric and the 4 components of the electromagnetic vector potential “the rotation of which are formed by the

γ_{[i k]}

”^$148$ “dont les

γ_{[i k]}

forment le rotationel”.

But even “the best we have nowadays” did not satisfy Einstein; half a year later, he expressed his opinion in a letter to Besso:

“Also, the equation put forward by myself^$149$ , $R_{i k} = g_{i k} f_{l m} f^{l m} - \frac{1}{2} f_{l} f_{k m} g^{l m}$ gives me little satisfaction. It does not allow for electrical masses free from singularities.
Moreover, I cannot bring myself to gluing together two items (as the l.h.s. and the r.h.s. of an equation) which from a logical-mathematical point of view have nothing to do with each other.”^$150$ “Auch die ja von mir selbst aufgestellte Gleichung $R_{i k} = g_{i k} f_{l m} f^{l m} - \frac{1}{2} f_{l} f_{k m} g^{l m}$ befriedigt mich wenig. Sie lässt keine singularitätenfreien elektrischen Massen zu.
Ferner kann ich mich nicht dazu entschliessen, zwei Sachen zusammenzuleimen (wie die rechte und die linke Seite einer Gleichung), die logisch-mathematisch nichts miteinander zu schaffen haben.” ([98] , p. 230)

^$139$ Some extended discussion about the transformation of the field components with regard to time-reversal exist, in which two differing points of view are expressed (cf. [169] , pp. 91–92, which corresponds to Einstein's view, and [390] ).

^$140$ Einstein's proof that charge-symmetric solutions with the same mass are unavoidable, although to him a rather negative feature of his unified field theories, later was interpreted as Einstein's discovery of the concept of antimatter ([356, 371] , p. 78, footnote 44). To me, this seems to be a case of whiggish historical hindsight. According to Bargmann, Einstein's lasting result is that he pointed out the importance of the discrete symmetry operations [6] .

^$144$ George Yuri Rainich (1886–1968). Of Russian origin. He studied mathematics at universities in Odessa, Göttingen, and Munich, taking his final exam at the University of Kazan in 1913. He then taught at Kazan and Odessa until 1922, when he came to the United States of America. He was a Johnston Scholar at Johns Hopkins University from 1923–1926 and then Professor of Mathematics at the University of Michigan in Ann Arbor, U.S.A.

^$147$ It remains unclear how these $γ_{[i k]}$ are embedded into the theory, possibly in the sense of Rainich. Einstein's paper practically excludes that they form the antisymmetric part of an asymmetric metric tensor.

^$148$ orgQuoteFr

6.2 Further work on (metric-) affine and mixed geometry

Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school. Thus J. M. Thomas, after having given a review of Weyl's, Einstein's, and Schouten's approaches, said about his own work:

“I show in the present paper that his [Einstein's] new equations can be obtained by a direct generalisation of the equations of the gravitational field previously given by him [ $g_{i j; k} = 0; R_{i j} = 0$ ]. [...] In the final section I show that the adoption of the ordinary definition of covariant differentiation leads to a geometry which includes as a special case that proposed by Weyl as a basis for the electric theory; further that the asymmetric connection for this special case is of the type adopted by Schouten for the geometry at the basis of his electric theory.” ([345] , p. 187)

We met J. M. Thomas' paper before in section 6.1 .

During the period considered here, a few physicists followed the path of Eddington and Einstein. One who had absorbed Eddington's and Einstein's theories a bit later was Infeld^$151$ of Warsaw^$152$ . In January 1928, he followed Einstein by using an asymmetric metric the symmetric part

γ_{i k}

of which stood for the gravitational potential, the skew-symmetric part

φ_{i k}

for the electromagnetic field. However, he set the non-metricity tensor (of the symmetric part

γ

of the metric)

Q_{i j}^{k} = 0

, and assumed for the skew-symmetric part

φ

\begin{matrix} \nabla_{l} φ_{i j} = J_{i j l}, \end{matrix}

(141)

with an arbitrary tensor

J_{i j l}

. The electric current vector then is defined by

J^{i} = J_{l}^{i l}

where the indices, as I assume, are moved with

γ_{i k}

. In a weak-field approximation for the metric, Infeld's connection turned out to be

L_{i k}^{l} = {_{i k}^{l}} + \frac{1}{2} (φ_{i, k}^{l} + φ_{k, i}^{l} + δ^{l s} φ_{i k, s}) .

For field equations Infeld postulated the (generalised) Einstein field equations in empty space,

K_{i j} = 0

. He showed that, in first approximation, he got what is wanted, i.e., Einstein's and Maxwell's equations [165] .

Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as

\begin{matrix} L_{i k}^{l} = {_{i k}^{l}} + \frac{α}{2} (φ_{i, k}^{l} + φ_{k, i}^{l} + g^{l s} φ_{i k, s}), \end{matrix}

(142)

where

α

is “an extremely small numerical factor”. By neglecting terms

\sim α^{2}

he could gain both Einstein's field equation in empty space ( 94 ) and Maxwell's equation, if the electric current vector is identified with

α^{- 1} (L_{i l}^{l} - L_{l i}^{l}) .

Thus, he is back at vector torsion treated before by Schouten [297] .

The Japanese physicist Hattori embarked on a metric-affine geometry derived purely from an asymmetric metrical tensor

h_{i k} = g_{(i k)} + f_{[i k]}

. He defined an affine connection

\begin{matrix} L_{i k}^{j} = {_{i k}^{j}} + g^{j l} (f_{l i, k} + f_{l i, k} - f_{i k, l}), \end{matrix}

(143)

where

g^{i l} g_{l k} = δ_{k}^{i}

, and the Christoffel symbol is formed from g. The electromagnetic field was not identified with

f_{i k}

by Hattori, but with the skew-symmetric part of the (generalised) Ricci tensor formed from

L_{i k}^{j}

. By introducing the tensor

f_{i j k} : = \frac{\partial f_{i j}}{\partial x^{k}} + \frac{\partial f_{j k}}{\partial x^{i}} + \frac{\partial f_{k i}}{\partial x^{j}}

, he could write the (generalised) Ricci tensor as

\begin{matrix} K_{i k} = R_{i k} + \frac{1}{4} f_{i m}^{n} f_{k n}^{m} - \nabla_{l} f_{i k}^{l}, \end{matrix}

(144)

where the covariant derivative

\nabla

is formed with the Levi-Civita connection of

g_{i j} .

The electromagnetic field tensor

F_{i k}

now is introduced through a tensor potential by

F_{i k} : = \nabla_{l} f_{i k}^{l}

and leads to half of “Maxwell's” equations. In the sequel, Hattori started from a Lagrangian

ℒ = (g^{i k} + α^{2} F^{i k}) K_{i k}

with the constant

α^{2}

and varied, alternatively, with respect to

g_{i j}

and

f_{i j}

. He could write the field equations in the form of Einstein's, with the energy-momentum tensor of the electromagnetic field

F_{i k}

and a “matter” tensor

M^{i k}

on the r.h.s.,

M^{i k}

being a complicated, purely geometrical quantity depending on

K_{i k}, K, f_{i k l}

, and

F_{i k l}

F_{i k l}

is formed from

F_{i k}

f_{i k l}

from

f_{i k}

. From the variation with regard to

f_{i k}

, in addition to Maxwell's equation, a further field equation resulted, which could be brought into the form

\begin{matrix} F^{i k} = \frac{2}{3} \nabla_{l} F^{i k l}, \end{matrix}

(145)

i.e.,

f_{i k l} \sim F_{i k l}

. Hattori's conclusion was:

“The preceding equation shows that electrical charge and electrical current are distributed wherever an electromagnetic field exists.”^$153$ “Die obige Gleichung zeigt, dass sich die elektrische Ladung und der elektrische Strom überall verteilen, wo das elektromagnetische Feld existiert.”

Thus, the same problem obtained as in Einstein's theory: A field without electric current or charge density could not exist [154] ^$154$ .

Infeld quickly reacted to Hattori's paper by noting that Hattori's voluminous calculations could be simplified by use of Schouten's Equation ( 39 ) of Section 2.1.2 . As in Hattori's theory two connections are used, Infeld criticised that Hattori had not explained what his fundamental geometry should be: Riemannian or non-Riemannian? He then gave another example for a theory allowing the identification of the electromagnetic field tensor with the antisymmetric part of the Ricci tensor: He displayed again the well-known connection with vector torsion used by Schouten [297] without referring to Schouten's paper [164] . He also claimed that Hattori's Equation ( 145 ) is the same as the one that had been deduced from Eddington's theory by Einstein in the Appendix to the German translation of Eddington's book ([58] , p. 367). All in all, Infeld's critique tended to deny that Hattori's theory was more general than Einstein's, and to point out

“that the problem of generalising the theory of relativity cannot be solved along a purely formal way. At first, one does not see how a choice can be made among the various non-Riemannian geometries providing us with the gravitational and Maxwell's equations. The proper world geometry which ought to lead to a unified theory of gravitation and electricity can only be found by an investigation of its physical content.”^$155$ “das Problem der Verallgemeinerung der Relativitätstheorie nicht auf rein formalem Wege gelöst werden kann. Man sieht zunächst nicht, wie die Wahl zwischen den verschiedenen nicht-Riemannschen Geometrien, die uns die Gravitationund die Maxwellschen Gleichungen ergeben, zu treffen ist. Die eigentliche Weltgeometrie, die zu einer einheitlichen Theorie von Gravitation und Elektrizität führen soll, kann nur durch Untersuchung ihres physikalischen Inhalts gefunden werden.” ([164] , p. 811)

Infeld could as well have applied this admonishment to his own unified field theory discussed above. Perhaps, he became irritated by comparing his expression for the connection ( 142 ) with Hattori's ( 145 ).

In June 1931, von Laue submitted a paper of the Genuese mathematical physicist Paolo Straneo to the Berlin Academy [330] . In it Straneo took note of Einstein's teleparallel geometry, but decided to take another route within mixed geometry; he started with a symmetric metric and the asymmetric connection

\begin{matrix} L_{i k}^{j} = {_{i k}^{j}} + 2 δ_{i}^{j} ψ_{k} \end{matrix}

(146)

with both non-vanishing curvature tensor

K_{j k l}^{i} = R_{j k l}^{i} + 2 δ_{j}^{i} (\frac{\partial ψ_{l}}{\partial x^{k}} - \frac{\partial ψ_{k}}{\partial x^{l}})

and torsion

S_{i k}^{j} = 2 δ_{[i}^{j} ψ_{k]}

. Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [297, 141] ) without referring to him. The field equations Straneo wrote down, i. e.

\begin{matrix} K_{i k} - \frac{1}{2} K g_{i k} = - κ T_{i k} + Ψ_{i k}, \end{matrix}

(147)

where

T_{i k}

is the symmetric and

Ψ_{i k}

the antisymmetric part of the l.h.s., do not fulfill Einstein's conception of unification: Straneo kept the energy-momentum tensor of matter as an extraneous object (including the electromagnetic field) as well as the electric current vector. The antisymmetric part of ( 147 ) just is

Ψ_{i k} = (\frac{\partial ψ_{l}}{\partial x^{k}} - \frac{\partial ψ_{k}}{\partial x^{l}})

; thus

Ψ_{i k}

is identified with the electromagnetic field tensor, and the electric current vector

J^{i}

defined by

Ψ_{l}^{i l} = J^{i}

. Straneo wrote further papers on the subject [331, 332] .

By a remark of Straneo, that auto-parallels and geodesics have to be distinguished in an affine geometry, the Indian mathematician Kosambi^$156$ felt motivated to approach affine geometry from the system of curves solving

{\ddot{x}}^{i} + α^{i} (x, \dot{x}, t)

with an arbitrary parameter

t

. He then defined two covariant “vector-derivations” along an arbitrary curve and arrived at an (asymmetric) affine connection. By this, he claimed to have made superfluous the five-vectors of Einstein and Mayer^$157$ [106] . This must be read in the sense that he could obtain the Einstein–Mayer equations from his formalism without introducing a connecting quantity leading from the space of 5-vectors to space-time [194] .

Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning. Due to this complication – for example even a condition of metric compatibility would not have the physical meaning of the conservation of the norm of an angle between vectors under parallel transport, and the further difficulty that much of the formalism was very clumsy to manipulate; essential work along this line was done only much later in the 10940s and 1950s (Einstein, Einstein and Strauss, Schrödinger, Lichnerowicz, Hlavaty, Tonnelat, and many others). In this work a generalisation of the equation for metric compatibility, i.e., Equation ( 47 ), will play a central role. The continuation of this research line will be presented in Part II of this article.

^$151$ Leopold Infeld (1889–1968). Born in Cracow, Poland. Studied at the University of Cracow and received his doctorate in 1923. After teaching in Lwow/Lemberg, he became professor of applied mathematics at the University of Toronto in 1938. Worked on unitary field theory and quantum electrodynamics, with van der Waerden on spinors, worked with Born on non-linear electrodynamics, and with Einstein on equations of motion (“EIH paper”).

^$152$ For the correspondence between Einstein and Infeld, cf. J. Stachel's essay in [329] , pp. 477–497.

^$154$ I have not yet been able to read the contributions from other Japanese authors [196, 163, 181] .

^$156$ Damodar D. Kosambi (1907–1966). Of Indian origin; born in Goa he moved to America in 1918 with his learned father and graduated from Harvard University in 1926 in mathematics, history and languages. Taught at the Muslim University of Aligarh and, from 1932, at Ferguson College, Pune. Mathematician, historian, and Sanskrit scholar.

^$157$ Walther Mayer (1887–1948).

Studied mathematics at the Federal Institute of Technology in Zürich and at the University of Vienna where he wrote his dissertation and became a Privatdozent (lecturer) with the title “professor”. He had made himself a name in topology (“Mayer–Vietoris sequences”), and worked also in differential geometry (well-known textbook “Duschek–Mayer” on differential geometry). In 1929 he became Einstein's assistant with the explicit understanding that he work with him on distant parallelism. It seems that Mayer was appreciated much by Einstein and, despite being in his forties, did accept this role as a collaborator of Einstein. After coming to Princeton with Einstein in 1933, he got a position at the Mathematical Institute of Princeton University and became an associate of the Institute for Advanced Study. Wrote a joint paper with T. Thomas on “Field of parallel vectors in nonanalytic manifolds in the large.” Mayer died in 1948.

6.3 Kaluza's idea taken up again

6.3.1 Kaluza: Act I

Einstein became interested in Kaluza's theory again due to O. Klein's paper concerning a relation between “quantum theory and relativity in five dimensions” (see Klein 1926 [184] , received by the journal on 28 April 1926). Einstein wrote to his friend and colleague Paul Ehrenfest on 23 August 1926: “Subject Kaluza, Schroedinger, general relativity”, and, again on 3 September 1926:

“Klein's paper is beautiful and impressive, but I find Kaluza's principle too unnatural.” However, less than half a year later he had completely reversed his opinion:

“It appears that the union of gravitation and Maxwell's theory is achieved in a completely satisfactory way by the five-dimensional theory (Kaluza–Klein–Fock).” (Einstein to H. A. Lorentz, 16 February 1927)

On the next day (17 February 1927), and ten days later Einstein was to give papers of his own in front of the Prussian Academy in which he pointed out the gauge-group, wrote down the geodesic equation, and derived exactly the Einstein–Maxwell equations – not just in first order as Kaluza had done [79, 80] . He came too late: Klein had already shown the same before [184] . Einstein himself acknowledged indirectly that his two notes in the report of the Berlin Academy did not contain any new material. In his second communication, he added a postscript:

“Mr. Mandel brings to my attention that the results reported by me here are not new.
The entire content can be found in the paper by O. Klein.”^$158$ “Herr Mandel macht mich darauf aufmerksam, dass die von mir hier mitgeteilten Ergebnisse nicht neu sind. Der ganze Inhalt findet sich in der Arbeit von O. Klein.”

He then referred to the papers of Klein [184, 185] and to “Fochs Arbeit” which is a paper by Fock^$159$ 1926 [129] , submitted three months later than Klein's paper. That Klein had published another important clarifying note in Nature, in which he closed the fifth dimension, seems to have escaped Einstein^$160$ [183] . Unlike in his paper with Grommer, but as in Klein's, Einstein, in his notes, applied the “sharpened cylinder condition”, i.e., dropped the scalar field. Thus, the three of them had no chance to find out that Kaluza had made a mistake:

For

g_{55} \neq c o n s t .

, even in first approximation the new field will appear in the four-dimensional Einstein–Maxwell equations ([144] , p. 5).

Mandel^$161$ of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. Klein's results [215] .

In a footnoote, Mandel stated that he had learned of Kaluza's (whom he spelled “Kalusa”) paper only through Klein's article. He started by embedding space-time as a hypersurface

x^{5} = c o n s t .

into

M_{5}

, and derived the field equations in space-time by assuming that the five-dimensional curvature tensor vanishes; by this procedure he obtained also a matter-energy tensor “closely linked to the second fundamental form of this hypersurface”. From the geodesics in

M_{5}

he derived the equations of motion of a charged point particle. One of the two additional terms appearing besides the Lorentz force could be removed by a weakness assumption; as to the second, Mandel opinioned

“that the experimental discovery of the second term appears difficult, yet perhaps not entirely impossible.” ([215] , p. 145)

As to Fock's paper, it is remarkable because it contains, in nuce, the coupling of the Schrödinger wave function

ψ

and the electromagnetic potential by the gauge transformation

ψ = ψ_{0} e^{2 π i p / h}

, where

h

is Planck's constant and

p

“a new parameter with the unit of the quantum of action” [129] .

In Fock's words:

“The importance of the additional coordinate parameter $p$ seems to lie in the fact that it causes the invariance of the equations [i.e., the relativistic wave equations] with respect to addition of an arbitrary gradient to the 4-potential.”^$162$ “Die Bedeutung des überzähligen Koordinatenparameters scheint nämlich gerade darin zu liegen, dass er die Invarianz der Gleichungen [i.e., the relativistic wave equations] in bezug auf die Addition eines beliebigen Gradienten zum Viererpotential bewirkt.” ([129] , p. 228)

Fock derived the general relativistic wave equation and the equations of motion of a charged point particle; the latter is identified with the null geodesics of

M_{5}

. Neither Mandel nor Fock used the “sharpened cylinder condition” ( 110 ).

A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation

\begin{matrix} a^{i k} (\frac{\partial^{2} U}{\partial x^{i} \partial x^{k}} - {_{r}^{i k}} \frac{\partial U}{\partial x^{r}}) = 0, i, k, = 1, \dots, 5 \end{matrix}

(148)

and by neglecting the gravitational field, he arrived at the four-dimensional Schrödinger equation after insertion of the quantum mechanical differential operators

- \frac{i h}{2 π} \frac{\partial}{\partial x^{i}}

. It was Klein's papers and the magical lure of a link between classical field theory and quantum theory that raised interest in Kaluza's idea – seven years after Kaluza had sent his manuscript to Einstein. Klein acknowledged Mandel's contribution in his second paper received on 22 October 1927, where he also gave further references on work done in the meantime, but remained silent about Einstein's papers [188] . Likewise, Einstein did not comment on Klein's new idea of “dimensional reduction” as it is now called and which justifies Klein's name in the “Kaluza–Klein” theories of our time.

By this, the reduction of five-dimensional equations (as e.g., the five-dimensional wave equation) to four-dimensional equations by Fourier decomposition with respect to the new 5th spacelike coordinate

x^{5}

, taken as periodic with period

L

, is understood:

ψ (x, x^{5}) = \frac{1}{\sqrt{L}} Σ_{n} ψ_{n} (x), e^{i n x^{5} / R^{5}}

with an integer

n

. Klein had only the lowest term in the series. The 5th dimension is assumed to be a circle, topologically, and thus gets a finite linear scale: This is at the base of what now is called “compactification”. By adding to this picture the idea of de Broglie waves, Klein brought in Planck's constant and determined the linear scale of

x^{5}

to be unmeasurably small (

\sim 10^{- 30}

From this, the possibility of “forgetting” the fifth dimension arose which up to now has not been observed.

In his papers, Einstein took over Klein's condition

g_{55} = 1

, which removed the additional scalar field admitted by the theory. It was Reichenbächer who apparently first tried to perform the projection into space-time of the most general five-dimensional metric, and without using the cylinder condition ( 109 ):

“Now, a rather laborious calculation of the five-dimensional curvature quantities in terms of a four-dimensional submanifold contained in it has shown to me also in the general case ( $g_{55} \neq c o n s t .$ , dependence of the components of the f u n d a m e n t a l [tensor] of $x^{5}$ is admitted ) that the c h a r a c t e r i s t i c properties of the field equations are then conserved as well, i.e., they keep the form $R^{i k} - \frac{1}{2} g^{i k} R = T^{i k}, \frac{\partial \sqrt{g} F^{i k}}{\partial x^{k}} = s^{i},$ only the $T^{i k}$ contain further terms besides the electromagnetic energy tensor $S^{i k}$ , and the quantities collected in $s^{i}$ do not vanish. [...] The appearance of the new terms on the right hand sides could even be welcomed in the sense that now the field equations are obtained not only for a field point free of matter and charge^$163$ .”^$164$ “Nun hat mir die allerdings reichlich mühsehlige Umrechnung der fünfdimensionalen Krümmungsgrössen auf eine in ihr enthaltene vierdimensionale Untermannigfaltigkeit auch im allgemeinen Falle ( $g_{55} \neq c o n s t .$ , Abhängigkeit der F u n d a m e n t a l komponenten auch von $x^{5}$ zugelassen ) gezeigt, dass die w e s e n t l i c h e n Eigenschaften der Feldgleichungen auch dann erhalten bleiben, d.h. diese behalten die Gestalt: $R^{i k} - \frac{1}{2} g^{i k} R = T^{i k}, \frac{\partial \sqrt{g} F^{i k}}{\partial x^{k}} = s^{i},$ nur enthalten die $T^{i k}$ ausser den Komponenten $S^{i k}$ des elektromagnetischen Energietensors noch weitere Glieder, und die zu $s^{i}$ zusammengefassten Grössen verschwinden nicht.
[...] Man könnte sogar das Auftreten der neuen Glieder auf den rechten Seiten von dem Standpunkt aus begrüssen, dass nunmehr die Feldgleichungen nicht nur für einen materieund ladungsfreien Feldpunkt geliefert werden.” ([275] , p. 426)

Here, in nuce, is already contained what more than a decade later Einstein and Bergmann worked out in detail [101] .

It is likely that Reichenbächer had been led to this excursion into five-dimensional space, an idea which he had rejected before as unphysical, because his attempt to build a unified field theory in space-time through the ansatz for the metric

γ_{i k} = g_{i k} - ε^{2} φ_{i} φ_{k}

with

φ_{k}

the electromagnetic 4-potential, had failed. Beyond incredibly complicated field equations nothing much had been gained [274] . Reichenbächer's ansatz is well founded: As we have seen in Section 4.2 , due to the violation of covariance in

M_{5}

γ_{i k}

transforms as a tensor under the reduced covariance group.

Even L. de Broglie became interested in Kaluza's “bold but very beautiful theory” and rederived Klein's results his way [44] , but not without getting into a squabble with Klein, who felt misunderstood [187, 45] .

He also suggested that one should not accept the cylinder condition, a suggestion looked into by Darrieus who introduced an electrical 5-potential and 5-current, and deduced Maxwell's equations from the five-dimensional homogeneous wave equation and the five-dimensional equation of continuity [41] .

In 1929 Mandel tried to “axiomatise” the five-dimensional theory: His two axioms were the cylinder condition ( 109 ) and its sharpening, Equation ( 110 ). He then weakened the second assumption by assuming that “an objective meaning does not rest in the

g_{i k}

proper, but only in their quotients”, an idea he ascribed to O. Klein and Einstein. He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [219] .

Klein's lure lasted for some years. In 1930, N. R. Sen claimed to have investigated the “Kepler-problem for the five-dimensional wave equation of Klein”. What he did was to calculate the energy levels of the hydrogen atom (as a one particle-system) with the general relativistic wave equation in space-time ( 148 ) with

a^{i k} = γ^{i k},

where

γ^{i k} = g_{i k} + α^{2} γ_{55}

is the metric on space-time following from the 5-metric

γ_{α β}

d x^{5} = 0

. For

g_{i k}

he took the Reissner–Nordström solution and did not obtain a discrete spectrum [323] . He continued his approach by trying to solve Schrödinger's wave equation [324]

g^{i k} (\frac{\partial^{2} u}{\partial x^{i} \partial x^{k}} - {_{r}^{i k}} \frac{\partial u}{\partial x^{r}}) = - \frac{4 π^{2}}{h^{2}} m_{0}^{2} c^{2} u .

Presently, the different contributions of Kaluza and O. Klein are lumped together by most physicists into what is called “Kaluza–Klein theory”. An early criticism of this unhistorical attitude has been voiced in [209] .

^$159$ Vladimir Aleksandrovich Fo(c)k (1898–1974). Born in St. Petersburg (renamed later Petrograd and Leningrad). Studied at Petrograd University and spent his whole carrier at this University. Member of the USSR Academy of Sciences. Fundamental contributions to quantum theory (Fock space, Hartree–Fock method); also worked in and defended general relativity.

^$160$ The correspondence is taken from Pais [240] who, in his book, expresses his lack of understanding as to why Einstein published these two papers at all.

^$161$ Heinrich Mandel (1898– ). From 1928 lecturer at the University of Leningrad, and from 1931 research work at the Physics Institute of this university.

6.3.2 Kaluza: Act II

Four years later, Einstein returned to Kaluza's idea. Perhaps, he had since absorbed Mandel's ideas which included a projection formalism from the five-dimensional space to space-time [215, 216, 217, 218] .

In a paper with his assistant Mayer, Einstein now presented Kaluza's approach in the form of an implicit projective four-dimensional theory, although he did not mention the word “projective” [106] :

“Psychologically, the theory presented here connects to Kaluza's well-known theory; however, it avoids extending the physical continuum to one of five dimensions.”^$165$ “Die hier dargestellte Theorie knüpft psychologisch an die bekannte Theorie von KALUZA an, vermeidet es aber, das physikalische Kontinuum zu einem solchen von fünf Dimensionen zu erweitern.”

In the eyes of Einstein, by avoiding the artificial cylinder condition ( 109 ), the new method removed a serious objection to Kaluza's theory.

Another motivation is also put forward: The linearity of Maxwell's equations “may not correspond to reality”; thus, for strong electromagnetic fields, Einstein expected deviations from Maxwell's equations. After a listing of all the shortcomings of Kaluza's theory, the new approach is introduced:

At every event a five-dimensional vector space

V_{5}

is affixed to space-time

V_{4}

, and “mixed” tensors

γ_{ι}^{k}

are defined linking the tangent space of space-time

V_{4}

with a

V_{5}

such that

\begin{matrix} g_{ι κ} γ_{l}^{ι} γ_{m}^{κ} = g_{l m}, \end{matrix}

(149)

where

g_{l m}

is the metric tensor of

V_{4}

, and

g_{ι κ}

a non-singular, symmetric tensor on

V_{5}

with

ι, κ = 1, \dots, 5

, and

k, l = 1, \dots, 4

^$166$ . Indices are raised and lowered with the metrics of

V_{5}

V_{4}

, respectively. There exists a “preferred direction of

V_{5}

” defined by

γ_{ι}^{k} A^{ι} = 0

, and which is the normal to a “preferred plane”

γ_{ι}^{k} ω_{k} = 0

^$167$ . A consequence then is

\begin{matrix} γ_{σ k} γ^{τ k} = γ_{σ}^{k} γ_{k}^{τ} = δ_{σ}^{τ} - A_{σ} A^{τ} . \end{matrix}

(150)

A covariant derivative for five-vectors in

V_{4}

is defined with a “three-index-symbol”

Γ_{π l}^{ι}

with two indices in

V_{5}

, and one in

V_{4}

standing in for the connection coefficients:

\begin{matrix} {\overset{+}{\nabla}}_{l} X^{ι} = \frac{\partial X^{ι}}{\partial x^{l}} + Γ_{π}^{ι} X^{π} . \end{matrix}

(151)

The covariant derivative of 4-vectors is defined as usual,

\begin{matrix} \nabla_{l} X^{i} = \frac{\partial X^{i}}{\partial x^{l}} + {_{j l}^{i}} X^{j}, \end{matrix}

(152)

where

{_{j l}^{i}}

is calculated from the metric of

V_{4}

as given in Equation ( 149 ). Both covariant derivatives are abbreviated by the same symbol

A_{; k}

. The covariant derivative of tensors with both indices referring to

V_{5}

and those referring to

V_{4}

, is formed correspondingly. In this context, Einstein and Mayer mention an extension of absolute differential calculus by “WAERDEN and BARTOLOTTI” without giving any reference to their respective papers. They may have had in mind van der Waerden's [368] and Bortolotti^$168$ 's [22] papers. The autoparallels of

V_{5}

lead to the exact equations of motion of a charged particle, not the geodesics of

V_{4}

Einstein and Mayer made three basic assumptions:

\begin{matrix} g_{ι κ; l} & = & 0, \end{matrix}

\begin{matrix} γ_{ι; l}^{k} & = & A^{ι} F_{k l}, \end{matrix}

(153)

\begin{matrix} F_{k l} & = & - F_{l k}, \end{matrix}

where

A^{ι}

is the preferred direction and

F_{k l}

an arbitrary 2-form, later to be interpreted as the electromagnetic field tensor. From them

A_{σ; l} = γ_{σ}^{k} F_{l k}

follows. They also noted that a symmetric tensor

F_{k l}

could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of

V_{4}

into

V_{5}

Einstein and Mayer introduced what they called “Fünferkrümmung” (5-curvature) via the three-index symbol given above by

It is related to the Riemannian curvature

R_{m l k}^{r}

V_{4}

and

From ( 154 ), by transvection with

γ^{τ k}

, the 5-curvature itself appears:

By contraction,

P_{ι k} : = γ_{τ}^{r} P_{ι r k}^{τ}

and

P : = γ^{ι k} P_{ι k}

. Two new quantities are introduced:

(1) $U_{ι k} : = P_{ι k} - \frac{1}{4} (P + R)$ , where R is the Ricci scalar of the Riemannian curvature tensor of $V_{4}$ , and
(2) the tensor $N_{k l m} : = F_{{k l; m}}$ ^$169$ .

It turns out that

P = R - F_{k p} F^{k p}

The field equations put forward in the paper by Einstein and Mayer now are

and turn out to be exactly the Einstein–Maxwell vacuum field equations. Thus, by another formalism, Einstein and Mayer rederived what Klein had obtained in his first paper on Kaluza's theory [184] .