Abstract
The quest for a ‘unified field theory’, which aims to integrate gravitational and electromagnetic fields into a single field structure, spanned most of Einstein’s professional life from 1919 until his death in 1955. It is seldom noted that Hans Reichenbach was possibly the only philosopher who could navigate the technical intricacies of the various unification attempts. By analyzing published writings and private correspondences, this paper aims to provide an overview of the Einstein-Reichenbach relationship from the point of view of their evolving attitudes toward the program of unifying electricity and gravitation. The paper concludes that the Einstein-Reichenbach relationship is more complex than usually portrayed. Reichenbach was not only the indefatigable ‘defender’ of relativity theory but also the caustic ‘attacker’ of Einstein’s and others’ attempts at unified field theory. Over the years, Reichenbach managed to provide the first, and possibly only, overall philosophical reflection on the unified field theory program. Thereby, Reichenbach was responsible for bringing to the debate, often for the first time, some of the central issues of the philosophy of space-time physics: (a) the relation between a theory’s abstract geometrical structures (metric, affine connection) and the behavior of physical probes (rods and clocks, free particles, and so on); (b) the question of whether such association should be regarded as a geometrization of physics or a physicalization of geometry; (c) the interplay between geometrization and unification in the context of a field theory.
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Notes
- 1.
For the history of the unified field theory-project, I draw freely from the standard historical literature on the subject Vizgin (1994), Goenner (2004), and Goldstein and Ritter (2003). For an overview of Einstein’s work on the unified field theory, see Sauer (2014); for the philosophical background of Einstein’s search for a unified field theory, see Dongen (2010); on Einstein’s philosophy of science, see Ryckman (2017).
- 2.
- 3.
The affine geometry is the study of parallel lines Weyl (1918c) introduced the expression ‘affine connection’ (affiner Zusammenhang). The term ‘connection’ refers to the possibility of comparison of vectors at close points. However, it is the notion of ‘sameness’ rather than parallelism that holds significance. Thus, some authors, such as Reichenbach (1928b), prefer to use the term ‘displacement’ (Verschiebung), which emphasizes the small coordinate difference \(d{x_{\nu }}\) along which the vector is transferred. Note that ‘displacement’ also refers to the vector \(dx_\nu \). To avoid confusion, the term ‘transfer’ Übertragung has also been used, for example, by Schouten (1922).
- 4.
In Weyl’s (1918a) theory the vector field \({\varphi _\nu }\) determines the change of length of vectors; the curl of \({\varphi _\nu }\) is the length-curvature tensor \({F_{\mu \nu }}\), which satisfy satisfy an identity which looks a lot like Maxwell-Minkowski equations in empty space. Thus, it was very suggestive to interpret \({\varphi _\nu }\) as the electromagnetic four-potential and its curl \({F_{\mu \nu }}\) as the electromagnetic tensor.
- 5.
Einstein raised at least four objections against Weyl’s theory: (1) the so-called ‘measuring rod objection’ (Maßstab-Einwand) (Einstein, 1918) is most famous. Weyl’s theory predicts that the clocks’ ticking rate should depend on the clocks’ prehistory. However, the spectral lines of atoms used as clocks are well-defined; (2) the geodesic equation in Weyl’s theory contains terms proportional to the vector potential \({\varphi _\nu }\). Thus, the electromagnetic four-vector potential affects the motion of uncharged particles; (3) the representation of the Lagrangian is the mere sum of electromagnetic and gravitational components, thus Weyl’s theory does not achieve a proper unification; (4) The field equations derived from this Lagrangian were of the fourth-order in the \({g_{\mu \nu }}\) which, even in the absence of an electromagnetic field, did not reduce to the generally relativistic equations of gravitation, violating the correspondence principle.
- 6.
Einstein to Kaluza, Apr. 21, 1919; CPAE, Vol. 9, Doc. 26; Einstein to Kaluza, Apr. 28, 1919; CPAE, Vol. 9, Doc. 30; Einstein to Kaluza, May 5, 1919; CPAE, Vol. 9, Doc. 35; Einstein to Kaluza, May 14, 1919; CPAE, Vol. 9, Doc. 40; Einstein to Kaluza, May 29, 1919; CPAE, Vol. 9, Doc. 48.
- 7.
These autobiographical notes HR, 044-06-23 were written in 1927.
- 8.
Reichenbach’s habilitation has recently attracted renewed attention (Friedman, 2001). Reichenbach borrowed from Schlick (1918) the idea that physical knowledge is, ultimately (Zuordnung), the process of relating an axiomatically defined mathematical structure to concrete empirical reality (Padovani, 2009). However, Reichenbach attempted to give this insight a ‘Kantian’ twist. According to Reichenbach, in a physical theory, besides the ‘axioms of connections’ (Verknüpfungsaxiome) encoding the mathematical structure of a theory, one needs a special class of physical principles, the ‘axioms of coordination’ (Zuordnungsaxiome), to ensure the univocal coordination of that structure to reality. For the young Reichenbach, the latter axioms are a priori because they are ‘constitutive’ of the object of a physical theory. However, they are not apodeictic or valid for all time. As is well known, Reichenbach would soon abandon the project of a constitutive but relativized a priori. However, he would firmly maintain the separation between the mathematical framework of a theory (the ‘defined side’) and the way it relates to empirical reality (the ‘undefined side’) as an essential feature of his philosophy (Reichenbach, 1920b, 40; tr. 1969 42) as an essential feature of his philosophy.
- 9.
\(d s^{4}=g_{\mu \nu \sigma \tau } d x_{\mu } d x_{\nu } d x_{\sigma } d x_{\tau }\) instead of \(d s^{2}={g_{\mu \nu }} d x_{\mu } d x_{\nu }\) as in Riemannian geometry.
- 10.
In the 1919 edition of the Raum–Zeit–Materie, Weyl included a presentation of his unified field theory. Thus, the ‘Conclusion’ of the book was characterized by even more inspired rhetoric: “physics and geometry coincide with each other” (Weyl, 1919b, 263). The tendency of physicalizing geometry that prevailed among the leading protagonists of the nineteenth century from Gauss to Helmholtz seemed to be superseded by the project of geometrizing physics that ran from Clifford to Einstein: “geometry has not been physics but physics has become geometry” (Weyl, 1919b, 263).
- 11.
Haas (1920).
- 12.
Schlick to Reichenbach, Sep. 25, 1920; HR, 015-63-23 Schlick to Reichenbach, Nov. 26, 1920; HR, 015-63-22; Schlick to Reichenbach, Dec. 11, 1920; HR, 015-63-19; Reichenbach to Schlick, Nov. 29, 1920; Reichenbach to Schlick, Sep. 10, 1920.
- 13.
Reichenbach was confronted with Schlick’s objection that his ‘axioms of coordination’ were nothing but ‘conventions’. Reichenbach initially opposed some resistance. If the coordinating principles are fully arbitrary, he feared, geometry would be empirically meaningless. In Poincaré’s conventionalism, Reichenbach missed a constraint in “the arbitrariness of the principles […], if the principles are combined” Reichenbach to Schlick, Nov. 26, 1920; HR, 015-63-22. Einstein’s famous lecture on ‘geometry and experience’ of the end January of 1921, which was published a few months lter (Einstein, 1921a), seemed to have tipped the scale in Schlick’s favor. Reichenbach (1922a) turned Einstein’s \(G+P\) formula into his \(G + F\) formula, where F is a ‘metric’ or universal force affecting all bodies in the same way. By setting \(F=0\), geometry becomes empirically testable. Thus, Reichenbach could embrace conventionalism without accepting that the propositions of geometry are empirical meaningless.
- 14.
That is on the field equations of the theory which, in turn, can be derived from an ‘action principle’.
- 15.
A different variation of this strategy of ‘doubling the geometry’ was suggested by Eddington (1921a) at about the same time. He considered non-Riemannian geometries as mere ‘graphical representations’ that might serve to organize different theories into a common mathematical framework. The “natural geometry” remains exactly Riemannian (Eddington, 1921a).
- 16.
- 17.
In September 1921, Pauli’s (1921) encyclopedia article on relativity theory was published as part of the fifth volume of the Enzyklopädie der Mathematischen Wissenschaften. In the chapter dedicated to Weyl’s theory, Pauli suggested that Weyl provided two different versions of the theory. In its first version, Weyl’s theory sought to make predictions on the behavior of rods and clocks, just like Einstein’s theory. From this point of view, the theory is empirically meaningful, but inadequate because of the existence of atoms with sharp spectral lines. Later, Weyl renounced this interpretation. The ideal process of the congruent displacement vectors has nothing to do with the real behavior of rods and clocks (Pauli, 1921, 763; tr. 1958, 196). However, in this way, the theory furnishes only “formal, and not physical evidence for a connection between [the] world metric and electricity” (Pauli, 1921, 763; tr. 1958, 196). In this form, Pauli argues, the theory loses its “convincing power [Überzeugungskraft]” (Pauli, 1921, 763; tr. 1958, 196).
- 18.
Reichenbach might have been inspired by Pauli (1921). However, his name is not mentioned.
- 19.
- 20.
In such a theory, the unit of length would be defined as certain number of spacing between the atoms of a cubic crystal system; each of atom, in turn, consists of electrons and protons arranged according to a specific law. A specific solution to the field equations must provide information about all the details of this arrangement. Something similar can be said for the unit of time that correspond to the vibrations of an atom.
- 21.
As one might infer from Reichenbach’s later writings, his point of view might have been similar to that of Pauli. In a long letter to Eddington in September 1923, Pauli insisted that a good theory should start with “the definition of the field quantities used, and how these quantities can be measured” (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45). One of the great achievements of relativity theory was that the coefficients \({g_{\mu \nu }}\) could be measured with rods and clocks. Pauli explained that Weyl attempted to pursue this strategy again but then abandoned this approach (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45). In this way, he produced what Eddington had rightly called a ‘graphical representation’ of the two fields in unified formalism, but not a ‘natural geometry’ found experimentally as in general relativity (see Eddington, 1923, 197). Similarly, in Einstein-Eddington new theory “[t]he quantities [\({\Gamma ^\tau _{\mu \nu }}\)] cannot be measured directly” (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45). The measurable quantities \({g_{\mu \nu }}\) and \({F_{\mu \nu }}\) can be calculated from the \({\Gamma ^\tau _{\mu \nu }}\) only through complicated calculations. Thus, not only we do not have a “ ‘natural geometry’ but also not a ‘natural theory’ ” (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45).
- 22.
As Weyl himself ironically remarked, Einstein undertook “the same purely speculative paths which [he was] earlier always protesting against” (Weyl to Einstein, May 18, 23; CPAE, Vol. 13, Doc. 30; cf. Weyl to Seelig, May 19, 1952, cit. in Seelig, 1960, 274f.).
- 23.
The process of obtaining the venia legendi at another university.
- 24.
Einstein (1925a).
- 25.
In a review of the German translation (Eddington, 1925) of Eddington’s relativity textbook (Eddington, 1923) that came out a few weeks later, Pauli (1926) expressed similar concerns. Without the equivalence principle, the entire geometrization program appeared to Pauli unjustified: “An attempt at an analogous geometrical interpretation of the electromagnetic field faces the difficulty that there is no empirical fact corresponding to the equality of heavy and inert mass, which would make such an interpretation appear ‘natural’ ” (Pauli, 1926). The solution was to avoid any connection between geometry and the behavior of rods and clocks. However, in this way, one could at most produce what Eddington called a ‘graphical representation’. According to Pauli, similar objections could be raised against Einstein’s last work (Einstein, 1925b).
- 26.
- 27.
Cf. Weyl, 1918b, 477. Einstein regarded this as one of the major shortcomings of Weyl’s theory; see Einstein to Besso, Aug. 20, 1918; CPAE, Vol. 8b, Doc. 604, Einstein to Hilbert, Jun. 9, 1919; CPAE, Vol. 9, Doc. 58.
- 28.
- 29.
The Appendix was not included in the English translation Reichenbach (1958).
- 30.
Starting from a general non-symmetric affine connection \({\Gamma ^\tau _{\mu \nu }}\) and imposing the condition that the length of vectors does not change under parallel transport, one can obtain Einstein and Riemann spaces through the “exchangeability of the specializations” (Reichenbach, 1928b, 5). By imposing that the Riemann tensor vanishes, one obtains Einstein space, while imposing that the connection is symmetric yields Riemannian space.
- 31.
See Footnote 17.
- 32.
For Einstein’s earlier ‘logic of discovery’, see Giovanelli (2020).
- 33.
Weyl, whom Einstein had always scolded for his speculative style of doing physics, could relaunch the accusation in a paper (Weyl, 1929) in which he had uncovered the gauge symmetry of the Dirac theory of the electron (Dirac, 1928a,b). “The hour of your revenge has come”, Pauli wrote to Weyl in August: “Einstein has dropped the ball of distant parallelism, which is also pure mathematics and has nothing to do with physics and you can scold him” (Pauli to Weyl, Aug. 26, 1929; WPWB, Doc. 235). As Pauli complained, writing to Einstein’s close friend Paul Ehrenfest, “God seems to have left Einstein entirely!” (Pauli to Ehrenfest, Sep. 29, 1929; WPWB, Doc. 237).
- 34.
It is interesting to notice that one of the reasons that induced Einstein to abandon the theory was not dissimilar to Reichenbach’s criticism: “The main reason for the uselessness of the distant parallelism construction lies, I feel, in that one can attribute absolutely no physical meaning to the ‘straight lines’ of the theory, while the physically meaningful (macroscopic) equations of motion cannot be obtained from it. In other words, the \(h_{s v}\) give rise to no useful representation of the electromagnetic field” (Einstein to Cartan, May 21, 1932; Debever, 1979, A XXXV). Thus, for Einstein, it was legitimate to abandon the physical interpretation of straight lines from the outset if the theory provided a way to derive the laws of motion of the electrons.
- 35.
In a way not dissimilar to Reichenbach, Weyl considered early unified field theories as “merely geometrical dressings (geometrische Einkleidungen) rather than as proper geometrical theories of electricity” (Weyl, 1931, 343).
- 36.
In English in the text.
- 37.
In English in the text.
- 38.
In English in the text.
- 39.
Einstein (1949a).
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Giovanelli, M. (2023). Coordination, Geometrization, Unification: An Overview of the Reichenbach–Einstein Debate on the Unified Field Theory Program. In: Russo Krauss, C., Laino, L. (eds) Philosophers and Einstein's Relativity. Boston Studies in the Philosophy and History of Science, vol 342. Springer, Cham. https://doi.org/10.1007/978-3-031-36498-3_6
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