Abstract
This paper contains the introduction of the conformal curvature tensor now known as the Weyl tensor, which is widely used in geometry and gravity theory. Its classification by Petrov type has played a major role in, for example, studies of exact solutions and of gravitational radiation. The paper also introduces what is now called Weyl geometry, a generalization of Riemannian geometry which has been used in gravity theories alternative to general relativity, and contains ideas leading on to Weyl’s concept of projective curvature.
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Notes
I refer to the presentation in my book Raum, Zeit, Materie, Springer 1918 (in the sequel cited as RZM), and the literature cited there.
A first communication about this appeared under the title “Gravitation und Elektrizität” in Sitzungsber. d. K. Preuß. Akad. d. Wissenschaften 1918, p. 465.
Naturally, traditional geometry leaves the path of this, its principal task, and immediately takes on the less specific one by not making space itself anymore the object of its investigation, but the structures possible in space, special classes and their properties they are endowed with on the basis of the space-metric.
I am bold enough to believe that the totality of physical phenomena can be derived from a single universal world law of greatest mathematical simplicity.
See also H. Weyl, Das Kontinuum (Leipzig 1918), specifically pp. 77 ff.
RZM, §13.
RZM, §6.
In the following we will use Einstein’s convention that summation is always to be carried out over indices which occur twice in a formula without our finding it necessary to always place a summation sign in front of it.
I thus differentiate between “coordinate system” and “frame of reference.”
See also Hessenberg, “Vektorielle Begründung der Differentialgeometrie,” Math. Ann. vol. 78 (1917), pp. 187–217, especially p. 208.
With this comment, I would like to correct a mistake on page 183 of my book Raum, Zeit, Materie.
In this one could follow the approach I have taken in RZM, §14.
Translator: Längenkrümmung (longitudinal or length) curvature. I would assume that this should refer to length since it is about the length measurement not being integrable. But I do not know of a suitable way to express this in English.
Translator: Wirbel (curl or vortex).
Translator: also known as the Jacobian determinant.
The relation between scalar and scalar density corresponds completely to that between function and Abelian differential in the theory of algebraic functions.
Quantitätsgrößen: quantities of extent.
Translator: Situsgeometrie: This refers to the structure available on a bare differentiable manifold.
Translator: Stromstärke (current) refers to the integral of the current density over a cross section. It has units of Amperes. It would be more appropriate here to say current density.
Translator: Wirkungsgröße: action (quantity)
Translator: index added for consistency
Weyl, Ann. d. Phys vol. 54 (1917), p. 117 (§*2); F. Klein, Nachr. d. K. Gesellsch. d. Wissensch. zu Göttingen, math.-physik. Kl., 25. Jan. 1918.
Cp. G. Mie, Annalen der Physik, Vol. 37, 39, 40 (1912/13), or the presentation of Mie’s theory in RZM §25; D. Hilbert, Die Grundlagen der Physik (1. Mitteilung), Nachr. d. K. Gesellsch. d. Wissensch. zu Göttingen, Sitzung vom 20. Nov. 1915.
Translator: Elektrizität (electricity).
Translator: Eliminante (eliminant).
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An editorial note to this paper can be found in this issue at https://doi.org/10.1007/s10714-022-02930-7.
Original paper: “Reine Infinitesimalgeometrie”, Mathematische Zeitschrift 2, 1918, pp. 384–411.
Translated, in two parts, by Jürgen Renn and Jörg Frauendiener. Renn’s translation of the first part of the paper appeared as “Purely Infinitesimal Geometry (Excerpt)” in: Janssen, M., Norton, J.D., Renn, J., Sauer, T., Stachel, J. (eds) The Genesis of General Relativity. Boston Studies in the Philosophy of Science, vol 250, pp. 2013-2029. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-4000-9_49.
Editorial responsibility: Malcolm A. H. MacCallum, m.a.h.maccallum@qmul.ac.uk.
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Weyl, H. Republication of: Purely infinitesimal geometry by Hermann Weyl. Gen Relativ Gravit 54, 51 (2022). https://doi.org/10.1007/s10714-022-02931-6
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DOI: https://doi.org/10.1007/s10714-022-02931-6