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Pascual Jordan's "Erweiterte Gravitationstheorie" - A Historical Analysis of its Mathematical Framework

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Pith's one-line read Pascual Jordan's 1952 textbook defined the covariant derivative axiomatically, as an operation fixed by five properties and independent of parallel transport; this paper argues that this anticipates modern connection theory and was…

desk verdict A useful first English map of Jordan's projective formalism with a solid historical claim at its core, but the 'first axiomatic definition' priority claim needs to be reined in or explicitly defended. read the letter →

arxiv 2507.23758 v1 pith:OWS47I6Q submitted 2025-07-31 math.HO physics.hist-ph

Bernadette Lessel This is my paper · ORCID
classification math.HOphysics.hist-ph MSC 01A6053-0353B05
keywords PascualJordancovariantderivativeprojectiverelativityaxiomaticdefinitionconnectionhistoryofdifferentialgeometryErweiterteGravitationstheorievariablegravitationalconstant
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a definition buried in Pascual Jordan's 1952 textbook 'Schwerkraft und Weltall' deserves a place in the history of differential geometry. Jordan introduced the covariant derivative not through coordinates and parallel transport, but as a tensor operation characterized by five axioms, in the style of a modern connection. The paper insists this was not an idle axiomatization: Jordan needed a derivative that still works in five-dimensional homogeneous projective coordinates, where no coordinate system is plane, and the axiomatic characterization made the extension routine. It also concedes that Jordan was not absolutely the first (Walther Mayer published a similar axiomatic definition in 1930) but contends that Jordan was the first in the relativity literature and probably did not know Mayer's book. The episode matters because it shows a concrete physical program, making the gravitational 'constant' variable, pushing the mathematical formalism toward the operator viewpoint that later became standard.

What carries the argument

The load-bearing object is Jordan's five-axiom characterization of covariant differentiation, together with the three-bar 'Kongruenzdifferentiation' built on it. The axioms fix the covariant derivative as the unique tensor operation satisfying the Leibniz-type sum and product rules, commutation with contraction, metric compatibility $g_{kl||j}=0$, and the gradient and rotation properties; the constructive definition via 'plane' coordinate systems supplies existence and uniqueness of a coordinate-free derivative. This machinery matters because in projective relativity the five homogeneous coordinates $X^\mu$ cannot be made locally flat, so only the axiomatic characterization survives the passage from four to five dimensions. The derived operation of Kongruenzdifferentiation, denoted by three bars, restores the property that reduction of a five-dimensional projector derivative equals the four-dimensional covariant derivative of the reduced tensor. That chain of definitions is what lets Jordan rewrite the Einstein-Maxwell system in five-dimensional projective form with $J=1$, and then his extension drops $J=1$ to make the gravitational 'constant' a variable scalar field.

What would settle it

A systematic search of differential-geometry and unified-field-theory literature from 1900 to 1952 that finds an axiomatic, parallel-transport-free definition of the covariant derivative earlier than Walther Mayer's 1930 textbook, or finds one in the relativity literature before Jordan's 1952 book, would falsify the paper's priority claim.

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Extended reading notes

Core claim

The central discovery is that Jordan's covariant derivative is 'given in an operational, axiomatic way, which is completely uncommon at the time and which resembles very much the contemporary definition of a connection.' Jordan first builds a constructive definition: a tensor's covariant derivative at $P$ is its ordinary partial derivative in a coordinate system that is 'plane at $P$' (metric components constant to first order), transferred to all other systems by transformation rules. He then lists five properties of this operation, including sum and product rules, interchange with contraction, $g_{kl||j}=0$, and gradient and rotation behavior, and declares that any operation with these five properties is the covariant derivative. The paper shows why such a definition was needed: in the five-dimensional projective space with homogeneous coordinates $X^\mu$, the Euler homogeneity condition for projectors prevents the metric from being plane anywhere, so the constructive definition cannot be carried over, but the axioms can. On this basis Jordan defines covariant differentiation of projectors and a 'Kongruenzdifferentiation' that makes reduction from five to four dimensions well behaved. The paper is explicit that Walther Mayer gave a similar axiomatization in 1930; its claim is that Jordan arrived at the same idea independently and was first to bring it into the relativity literature.

Load-bearing premise

The priority claim depends on the assumption that the survey of earlier literature is complete: that Walther Mayer's 1930 textbook is the only axiomatic definition before Jordan and that no one in the relativity literature before 1952 gave one; the paper asserts this with limited citations rather than demonstrating it by an exhaustive search.

Editorial extensions

If this is right

  • Jordan's textbook becomes evidence that the operator-style, axiom-based definition of a connection appeared in the relativity literature years before it became the standard way to present connections.
  • The five axioms are what make the five-dimensional projective extension tractable, so the book's mathematical claim and its historical claim stand or fall together.
  • If the priority claim holds, histories of modern differential geometry should acknowledge an application-driven, physics-motivated axiomatization alongside the better-known pure-mathematical ones.
  • Within the same formalism, treating $J=g_{\mu\nu}X^\mu X^\nu$ as a free field rather than fixing $J=1$ yields a concrete variational principle in which the gravitational 'constant' $\chi$ varies as a scalar field.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether Walther Mayer's 1930 axioms and Jordan's 1952 axioms are genuinely the same mathematical object; a direct comparison of their axiom lists would sharpen or weaken the priority claim.
  • Jordan's Kongruenzdifferentiation reads naturally as a connection on a projective frame bundle; translating it into modern bundle language would make the continuity with later gauge theory visible, a step the paper does not take.
  • Jordan's axiomatic style may be a personal trait extending into his earlier work with Wigner on quantum-theoretic foundations, but the paper only notes this in a footnote; testing that hypothesis would require a broader survey of his publications.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper argues that Pascual Jordan's 1952 textbook "Schwerkraft und Weltall" contains an axiomatic definition of the covariant derivative—an operator characterized by five properties (additivity, the Leibniz rule, commutation with contraction, metric compatibility, and agreement with ordinary derivatives on gradients and rotations)—which closely resembles modern definitions of a connection. The paper claims that Jordan was the first in the relativity literature to give such an axiomatic definition, with only Walther Mayer earlier in the mathematical literature. It situates Jordan's work within the history of the covariant derivative, parallel transport, and connection in differential geometry, and within the development of projective relativity, particularly the Princeton school of Veblen and Eisenhart. The author further contends that Jordan's axiomatization was motivated by his five-dimensional projective extension of Einstein-Maxwell theory with a variable gravitational constant, and she provides an English summary of Jordan's formalism as an aid to non-German-speaking readers.

Significance. If the priority claim is correct, the paper would establish Jordan's 1952 textbook as an important anticipation of modern connection formalism, developed independently for physical applications. The paper's strength is its careful reading of primary sources with page citations, its explicit differentiation between Jordan's complete axiomatic characterization and Schouten's earlier list of properties, and its transparent acknowledgment of speculation (e.g., about Jordan's likely ignorance of Mayer's work). It also provides a valuable English-language summary of a German textbook that influenced the post-war Hamburg relativity group. The historical narrative connecting projective geometry, the Princeton school, and unified field theory is informative and fills a gap in the English-language historiography. The central claim, however, rests on a priority assertion that the paper does not fully defend, as detailed below.

major comments (2)
  1. [Section 2, footnote 5, and Section 5 (quotation from [1, p.34])] The paper's central priority claim—that Jordan was the first in the relativity literature to give an axiomatic definition of the covariant derivative, with only Walther Mayer earlier in any literature—is contradicted, or at least under-justified, by the quotation the paper itself reports. In Section 5, Jordan is quoted as saying that Weyl, Eddington, and Schrödinger had already 'postulated an operation of the covariant (or affine) derivative,' and the paper goes on to discuss which of Jordan's Axioms III and V these authors retained or abandoned. The paper never defines a criterion that would exclude Eddington and Schrödinger from the category of 'axiomatic definition' while including Mayer. Since the novelty claim is load-bearing for the paper's thesis, the author should either supply a precise characterization of what counts as an axiomatic definition (e.g., a complete set of axioms that uniquely determines the operation, independent of a constructive definition) and show explicitly why Eddington's and Schrödinger's treatments fail it, or she should soften the priority claim to one about the specific form of Jordan's axiomatization. Without this, the strongest historical claim is unsupported.
  2. [Section 2, footnote 5] The statement that 'in the entire literature, including mathematical literature, covering Riemannian geometry, only Walther Mayer was earlier' is asserted without a systematic survey. The footnote itself hedges with 'It seems that...,' but the main text states unqualifiedly that Jordan was 'the first in the relativity literature.' A claim of this universality requires either a broader literature search or a more modest formulation restricted to the sources examined. The omission is significant because the priority claim is the paper's primary historiographic contribution, and the reader cannot verify exhaustiveness from the cited references alone.
minor comments (4)
  1. [Section 2.4] The assertion that the projective field equations (15)–(16) are equivalent to the Einstein-Maxwell system (5) is stated without derivation or a direct citation to the relevant pages of [1]. A brief indication of the steps (e.g., how λ eliminates itself and how the reduction formulas for the Ricci tensor are used) would make the summary more self-contained and easier to check.
  2. [Section 3] Several typographical errors appear throughout the manuscript: 'submitted 1988' should be 'submitted 1898' (the Ricci–Levi-Civita treatise), 'pionieering' should be 'pioneering', 'conscientously' should be 'conscientiously', 'hown' should be 'own' (in the introduction), and 'wherfore' should be 'wherefore' or 'therefore'. A careful proofread is needed.
  3. [Section 4.3] The projective differentiation formula (24) is introduced with little explanation of the notation (e.g., the meaning of the index 0 in Aα,0). A brief gloss, even in a footnote, would aid readers unfamiliar with Veblen's projective tensor calculus and would make the historical discussion more accessible.
  4. [Section 3] In the discussion of Schouten's 'Übertragung,' the paper correctly observes that Schouten's list of properties is not an axiomatic definition because Schouten does not prove completeness or treat the covariant derivative as an independent notion. This point is useful, but it would strengthen the paper to refer back to this distinction when discussing Eddington and Schrödinger in Section 5, in connection with the priority claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the historical argument is anchored in external primary and secondary sources, and the only evidentiary weakness is a non-circular priority claim.

full rationale

This paper is a work of historical analysis rather than a derivation, so the usual circularity patterns (fitted input called prediction, self-citation load-bearing, uniqueness imported from the authors, ansatz smuggled in via citation) do not apply. The central claim about Jordan's axiomatic definition of the covariant derivative is supported by direct quotation from Jordan's 1952 textbook and by comparison with Mayer, Schouten, Veblen, and others; it is not defined into existence by the paper itself. The only potentially load-bearing assertion is footnote 5's claim that Walther Mayer was the only earlier axiomatic definition 'in the entire literature' and the associated claim that Jordan was the first in the relativity literature. The paper also quotes Jordan saying that Weyl, Eddington, and Schrödinger already 'postulated an operation of the covariant (or affine) derivative' and discussed which axioms they retained or abandoned. This creates a tension in the historical priority argument, and a reader could reasonably ask for a fuller defense of the distinction between 'postulated an operation' and 'axiomatic definition.' However, that tension is an evidentiary and interpretive weakness, not a circular step: the paper does not assume the conclusion in its premises, and it explicitly reports the primary evidence that could be used against its own framing. The paper also openly marks its own uncertainties, e.g., 'It is still unclear, how Jordan became aware of projective geometry in the first place,' which further shows that the interpretation is not being forced by construction. No equation, prediction, or first-principles result in the paper reduces to its own input, and no load-bearing claim is supported only by a self-citation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim is historical, so the ledger contains domain assumptions about source fidelity and survey completeness, rather than fitted parameters or invented entities. The only quasi-technical quantity J in the summary is Jordan's, not introduced by this paper.

assumptions (3)
  • domain assumption Primary sources (Jordan 1952; Mayer and Duschek 1930) are quoted and translated accurately.
    The author's central claims about Jordan's definition and Mayer's earlier definition depend on faithful reading of these German texts.
  • domain assumption Secondary historical accounts (Reich, Scholz, Ritter) are reliable guides to the history of the covariant derivative and projective relativity.
    Sections 3 and 4 synthesize existing historiography; errors there would propagate.
  • domain assumption The author's survey of axiomatic definitions before 1952 is complete enough to support priority claims.
    Asserted via 'it seems' in footnote 5; not demonstrated.

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Cite this review

Pith. "Pith review of Pascual Jordan's "Erweiterte Gravitationstheorie" - A Historical Analysis of its Mathematical Framework." pith.science (2026). https://pith.science/paper/OWS47I6Q

@misc{pith2026250723758,
  author       = {Pith},
  title        = {Pith review of: Pascual Jordan's "Erweiterte Gravitationstheorie" - A Historical Analysis of its Mathematical Framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OWS47I6Q}},
  note         = {Machine review of arXiv:2507.23758}
}
read the original abstract

This paper aims to highlight Pascual Jordan's axiomatic definition of the covariant derivative, as set out in his 1952 textbook "Schwerkraft und Weltall". Developed in light of his \emph{Erweiterte Gravitationstheorie} - a projective reformulation of relativity theory that incorporates a variable gravitational constant - Jordan's definition resembles those in contemporary usage. The paper contextualises Jordan's work within the broader historical frameworks of differential geometry and projective relativity, with a particular focus on the Princeton relativity group led by Oswald Veblen and Luther Pfahl Eisenhart. It also provides a summary of Jordan's formalism, focusing particularly on his definition of the covariant derivative, as well as a brief history of the origin and development of the covariant derivative.

Discussion (0). Continue with ORCID to comment.

Reference graph

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