Einstein's 1923 purely affine field theory, summarized in "The Theory of the Affine Field" (Nature 112, 448–449), takes a general affine connection as the only fundamental field and determines it, together with field equations that reproduce the known laws of gravitation and electromagnetism in first approximation, from a single variational principle. By Einstein's own assessment the theory nevertheless leaves two things unexplained: it provides no account of the structure of charged elementary particles — no singularity-free particle-like solutions representing the electron, and no explanation of the mass difference between positive and negative charge carriers — and it requires the coupling constant β, which ties the electromagnetic current density directly to the potential (iμ=−β gμσfσ\mathfrak{i}^\mu = -\beta\,\mathfrak{g}^{\mu\sigma}f_\sigma), to be practically infinitesimal without saying why. The objective is to determine whether the classical purely affine framework can determine its own charged sources and the magnitude of its coupling, or show that it cannot.

Background

General relativity (1915) describes gravitation as the geometry of a four-dimensional Riemannian spacetime: a metric tensor gμνg_{\mu\nu} determines both the Levi-Civita connection (the Christoffel symbols, which define parallel transport and geodesics) and the Ricci curvature tensor RμνR_{\mu\nu}, which sources gravity through Einstein's field equations. Electromagnetism, by contrast, enters as an additional field fμνf_{\mu\nu} defined on spacetime, and in the years after 1915 several physicists — Weyl, Eddington, Kaluza, and Einstein himself — sought a "unified field theory" in which both interactions would follow from a single geometric structure. Einstein's further ambition was that the elementary charged particles themselves (in 1923 understood as the electron and the positively charged proton, the only known charge carriers) should emerge as singularity-free solutions of the unified field equations rather than being inserted as external sources.

The route Einstein pursued in 1923, following Arthur Eddington, starts from an affine connection Γμνα\Gamma^\alpha_{\mu\nu} — the general linear rule for parallel transport of vectors, forty independent functions per point, of which the Levi-Civita connection is the special metric-compatible case — taken as the sole fundamental field, with no metric postulated in advance. The Ricci tensor built from a general (nonsymmetric) connection is itself nonsymmetric and splits into a symmetric part γμν\gamma_{\mu\nu} and an antisymmetric part ϕμν\phi_{\mu\nu}; Eddington's proposal was to identify these with the gravitational and electromagnetic fields respectively. Eddington, however, gave no law determining the forty connection functions. Einstein's 1923 papers in the Prussian Academy proceedings, summarized in English in his Nature article "The Theory of the Affine Field", supplied such a law through Hamilton's principle (a variational principle) δ∫H dτ=0\delta\int\mathfrak{H}\,d\tau = 0, where the scalar density H\mathfrak{H} is a function only of γμν\gamma_{\mu\nu} and ϕμν\phi_{\mu\nu} and the forty Γμνα\Gamma^\alpha_{\mu\nu} are varied independently. The variation expresses the connection as the Christoffel part of an emergent metric (obtained from the metric density gμν=∂H/∂γμν\mathfrak{g}^{\mu\nu} = \partial\mathfrak{H}/\partial\gamma_{\mu\nu}) plus terms proportional to the electromagnetic current density iμ=∂νfμν\mathfrak{i}^\mu = \partial_\nu\mathfrak{f}^{\mu\nu}, where fμν=∂H/∂ϕμν\mathfrak{f}^{\mu\nu} = \partial\mathfrak{H}/\partial\phi_{\mu\nu} plays the role of the electromagnetic field density. With the simplest choice for the Legendre-transform density, H∗=2α−g−β2fμνfμν\mathfrak{H}^* = 2\alpha\sqrt{-g} - \frac{\beta}{2}f_{\mu\nu}\mathfrak{f}^{\mu\nu} (with α\alpha and β\beta constants), the resulting field equations

Rμν=−κ[(14gμνfστfστ−fμσfνσ)−β26 fμfν]R_{\mu\nu} = -\kappa\left[\left(\tfrac{1}{4}g_{\mu\nu}f_{\sigma\tau}f^{\sigma\tau} - f_{\mu\sigma}f^{\sigma}_{\nu}\right) - \tfrac{\beta^2}{6}\, f_\mu f_\nu\right]

where gμνg_{\mu\nu} is the emergent metric, fμνf_{\mu\nu} the electromagnetic field strength, fμf_\mu the electromagnetic potential, κ\kappa the gravitational coupling, and the term −β26fμfν-\tfrac{\beta^2}{6} f_\mu f_\nu is the documented addition −16iμiν-\tfrac{1}{6} i_\mu i_\nu on the side of the electromagnetic energy-momentum tensor, with the current tied to the potential by iμ=−βfμi_\mu = -\beta f_\mu: these equations unite a gravitational and an electromagnetic energy-momentum term on the right-hand side and reproduce the known laws of gravitation and electromagnetism in first approximation.

Two features of the theory remained unexplained, and Einstein's Nature summary closed by acknowledging them. First, the theory brings — in the wording recorded for the Nature summary — "no enlightenment on the structure of electrons": it exhibits no singularity-free particle-like charged solutions, and Einstein conceded in the companion Academy papers that the theory "may not account for the difference in mass of positive and negative electrons" (both statements as documented, with the German originals, in Goenner's history of unified field theories). Second, the current density is tied to the electromagnetic potential by

iμ=∂νfμν=−β gμσfσ,\mathfrak{i}^\mu = \partial_\nu\mathfrak{f}^{\mu\nu} = -\beta\,\mathfrak{g}^{\mu\sigma}f_\sigma,

so that, in Einstein's words as translated by Goenner, "the extraordinary smallness of 1/λ21/\lambda^2" — Goenner's rendering of the first Academy paper's parametrization of this requirement, in which Eddington's constant λ\lambda must be very large; in the final form of the theory, presented in the third Academy paper and summarized in Nature, the same requirement appears as the practical infinitesimality of the coupling β\beta itself — "implies that finite [fields] are possible only for tiny, almost vanishing current density": unless β\beta is practically infinitesimal, any nonvanishing potential would generate large currents even where no matter is present, contradicting experience. Why β\beta should be so small, the theory does not say. The open question below asks whether the classical purely affine framework can supply what it here lacks.

Problem Statement

Within the classical purely affine unified field theory formulated by Einstein in 1923 — in which the only fundamental field is a general affine connection Γμνα\Gamma^\alpha_{\mu\nu}, the Ricci tensor RμνR_{\mu\nu} built from it splits into a symmetric part γμν\gamma_{\mu\nu} (gravitational) and an antisymmetric part ϕμν\phi_{\mu\nu} (electromagnetic), and the Hamilton principle δ∫H dτ=0\delta\int\mathfrak{H}\,d\tau = 0 with H\mathfrak{H} depending only on γμν\gamma_{\mu\nu} and ϕμν\phi_{\mu\nu} determines both the connection and the field equations

Rμν=−κ[(14gμνfστfστ−fμσfνσ)−β26 fμfν]R_{\mu\nu} = -\kappa\left[\left(\tfrac{1}{4}g_{\mu\nu}f_{\sigma\tau}f^{\sigma\tau} - f_{\mu\sigma}f^{\sigma}_{\nu}\right) - \tfrac{\beta^2}{6}\, f_\mu f_\nu\right]

where gμνg_{\mu\nu} is the emergent metric (with contravariant density gμν=∂H/∂γμν\mathfrak{g}^{\mu\nu} = \partial\mathfrak{H}/\partial\gamma_{\mu\nu}), fμνf_{\mu\nu} the electromagnetic field strength (with density fμν=∂H/∂ϕμν\mathfrak{f}^{\mu\nu} = \partial\mathfrak{H}/\partial\phi_{\mu\nu}), fμf_\mu the electromagnetic potential, κ\kappa the gravitational coupling constant, and β\beta the current–potential coupling constant — the practically infinitesimal proportionality constant in iμ=−βfμi_\mu = -\beta f_\mu, the same constant as in the Legendre-transform density H∗\mathfrak{H}^*, the term −β26fμfν-\tfrac{\beta^2}{6} f_\mu f_\nu being the documented addition −16iμiν-\tfrac{1}{6} i_\mu i_\nu to the electromagnetic energy-momentum tensor — determine whether the theory can account for the structure of charged elementary particles and for the magnitude of its coupling constant β\beta: the two respects in which, by Einstein's own assessment, it brought no enlightenment. Determine, with the original generality of the framework (no restriction to special backgrounds, ansätze, symmetry classes, or parameter ranges beyond those the theory itself imposes), the following:

  1. Particle structure: whether the affine field equations admit singularity-free, particle-like charged solutions that can be identified with the electron and with the positive charge carrier — solutions with finite electromagnetic self-energy, localized charge, and the observed charge-to-mass relationships, including the mass difference between the positive and negative carriers — or a demonstration that no such solutions exist in the framework, which would establish that the theory cannot determine its own matter content.
  2. Smallness of β\beta: the current density is tied to the potential by iμ=∂νfμν=−β gμσfσ\mathfrak{i}^\mu = \partial_\nu\mathfrak{f}^{\mu\nu} = -\beta\,\mathfrak{g}^{\mu\sigma}f_\sigma, so unless β\beta is practically infinitesimal any nonvanishing potential would generate unobserved currents even in vacuum; determine whether the theory itself fixes or explains why β\beta is so small, or prove that its magnitude must remain an external input.

These two questions are linked by the single overarching objective that the 1923 program set for itself: a complete unified field theory in which both the fields and their charged sources, and the constants relating them, are determined by one geometric law.

References

  1. Albert Einstein, "The Theory of the Affine Field", Nature 112 (2812), 448–449 (22 September 1923), DOI 10.1038/112448a0, https://doi.org/10.1038/112448a0
  2. Albert Einstein, "Zur allgemeinen Relativitätstheorie", Sitzungsberichte der Preußischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse, 1923 (V), 32–38, page scans: https://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte/B98WXTVX (first paper of the 1923 affine series)
  3. Albert Einstein, "Bemerkungen zu meiner Arbeit 'Zur allgemeinen Relativitätstheorie'", Sitzungsberichte der Preußischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse, 1923 (XII), 76–77, page scans: https://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte/EVSSPDP8 (second paper of the 1923 affine series)
  4. Albert Einstein, "Zur affinen Feldtheorie", Sitzungsberichte der Preußischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse, 1923 (XVII), 137–140, page scans: https://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte/GBEHMB2D (third paper of the 1923 affine series, presented 31 May 1923)
  5. Hubert F. M. Goenner, "On the History of Unified Field Theories", Living Reviews in Relativity 7, article 2 (2004), DOI 10.12942/lrr-2004-2, https://doi.org/10.12942/lrr-2004-2
  6. A. T. Filippov, "Weyl-Eddington-Einstein affine gravity in the context of modern cosmology", Theoretical and Mathematical Physics 163, 753–767 (2010), DOI 10.1007/s11232-010-0059-6, https://doi.org/10.1007/s11232-010-0059-6
  7. I. V. Kharuk, "Emergent Planck mass and dark energy from affine gravity", arXiv:2002.12178 (2020), https://arxiv.org/abs/2002.12178 (journal version: Theoretical and Mathematical Physics 209, 1423–1436 (2021), DOI 10.1134/S004057792110007X)
  8. Oscar Castillo-Felisola and Jose Perdiguero, "Does the metric play a fundamental role in the building of gravitational models?", arXiv:2306.03729 (2023), https://arxiv.org/abs/2306.03729 (journal version: International Journal of Geometric Methods in Modern Physics 21 (1) (2024), DOI 10.1142/S0219887824300010)
  9. Oscar Castillo-Felisola, Bastian Grez, Manuel Morocho-López, Jose Perdiguero, Aureliano Skirzewski, Jefferson Vaca, and Nicolas Zambra-Gómez, "A polynomial affine model of gravity: after ten years", arXiv:2412.21122 (2024), https://arxiv.org/abs/2412.21122 (journal version: "Polynomial Affine Model of Gravity: After 10 Years", Universe 11 (3), 102 (2025), DOI 10.3390/universe11030102; the journal spells the sixth author's name Jefferson Vaca-Santana)
  10. Tilman Sauer, "Einstein's Unified Field Theory Program", in The Cambridge Companion to Einstein, Cambridge University Press (2014), pp. 281–305, DOI 10.1017/cco9781139024525.011, https://doi.org/10.1017/cco9781139024525.011