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Axiomatic Thinking in Physics—Essence or Useless Ornament?

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Axiomatic Thinking II

Abstract

In the first part of this contribution, I will present aspects and attitudes towards “axiomatic thinking” in various branches of theoretical physics. In the second and more technical part, which is approximately of the same size, I will focus on mathematical results that are relevant for axiomatic schemes of space-time in connection with attempts to axiomatize Special and General Relativity.

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Notes

  1. 1.

    The German originals of all translated quotations will be given in the appendix.

  2. 2.

    A very readable biography of Hertz is by Fölsing [24].

  3. 3.

    In his book on mechanics, [72], which form the first volume of his famous six-volume lecture series on theoretical physics (“Vorlesungen über Theoretische Physik”) Arnold Sommerfeld mentions Heinrich Hertz’s book on Mechanics in connection with the idea to eliminate the notion of “force” with the following words: “Heinrich Hertz hat dieses Programm mit meisterhafter Konsequenz durchgeführt. Aber zu fruchtbaren Folgerungen ist seine Methode kaum gelangt; insbesondere für den Anfänger ist sie völlig ungeeignet”.

  4. 4.

    A biography of Carathéodory is that of Georgiadou [26].

  5. 5.

    More precisely, the assumptions are as follows: The set M of equilibrium states of a thermodynamical system is assumed to be a smooth manifold. The inner energy of the system is represented by a real-valued function U on M; it is a function of state. “Heat” is also a real-valued quantity, but in contrast to energy it is not associated to states, i.e. points in M, but rather to “quasi-static processes”, i.e. piecewise \(C^1\) curves in M. These are called “quasi-static” because the process is assumed to remain within M, i.e. to only proceed in a succession of equilibrium states, which means that the real-time process must be “sufficiently slow”. Hence, heat is represented by a one-form that we denote by \(\omega \). Another one-form is that of “reversible work”, which we denote by \(\alpha \). Typically one has \(\alpha =-p\,dV+\cdots \), where p stands for pressure and V for volume, both of which are functions of state. The first law of thermodynamics then says that \(dU=\omega +\alpha \). Now, a quasi-static process \(\gamma \) (piecewise \(C^1\) a curve in M) is called “adiabatic” if \(\omega (\dot{\gamma })=0\), i.e. if the heat associated to this process vanishes. Carathéodory’s principle of adiabatic inaccessibility then says that for any \(p\in M\) and any neighbourhood \(V_p\subset M\) of p there is a point \(q\in V_p\) such that no adiabatic process exists connecting p and q. If \(T_pM\) denotes the tangent space to M at \(p\in M\) and \(TM=\cup _{p\in M}T_pM\) its tangent bundle, we define the “kernel distribution” as the subbundle \(K_\omega :=\{v\in T(M): \omega (v)=0\}\subset T(M)\). It is clear that the principle of adiabatic inaccessibility holds if \(K_\omega \) is integrable: just choose for q a point in \(V_p\) that is not on the same integral submanifold (leaf) as p. The no-trivial mathematical result of Carathéodory [14] is the proof of the converse, that is, adiabatic inaccessibility is not sufficient but also necessary for \(K_\omega \) to be integrable. Integrability is equivalent to \(d\omega \vert _{K_\omega }=0\) or \(\omega \wedge d\omega =0\), as has been known from Frobenius [25] long before [14]. This implies the existence of a function T on M such that \(d(\omega /T)=0\) (T is a so-called integrating denominator). Hence, (locally) there is a function S on M such that \(\omega /T=dS\); this is how temperature T and entropy S emerge from the principle of adiabatic inaccessibility.

  6. 6.

    This Weyl extended in the third and fifth editions of 1921 and 1923.

  7. 7.

    Robb’s system in its final form is complex and takes more than 400 pages for its presentation and discussion Robb [70]. It contains 21 axioms which are based on the relations of “before” and “after” which can be reduced to that of lightlike connectability. It was later shown by Mundy [59] that this system is amenable to considerable simplification.

  8. 8.

    This concept was only spelled out 40 years after Carathéodory by Marzke and Wheeler [52] based on Marzke [51].

  9. 9.

    Weyl’s “Axiomatik” lecture has 40 paragraphs grouped into five chapters: chapter I (paragraphs 1–9) on “Geometrie”; chapter II (paragraphs 10–15) on “Die Raum-Zeit-Lehre der Speziellen Relativitätstheorie”; chapter III (paragraphs 16–25) on “Raum und Zahl”; chapter IV (paragraphs 26–36) on “Algebra”; chapter V (paragraphs 37–40) on “Topologie”.

  10. 10.

    See [16, 18] for the formal definition of a path structure and [28, 64] for the discussion of the law if inertia and its impact on space-time structure.

  11. 11.

    We recall that an affine space inherits a natural topology from its associated vector space. This is also the natural manifold topology it receives from the atlas of affine charts, i.e. it is the coarsest (in the sense of “fewest” open sets) topology in which all chart maps are continuous.

  12. 12.

    The set of timelike curves considered here includes nowhere differentiable ones; the notion of being timelike cannot in this case be defined by the tangent vector and has to be generalized.

  13. 13.

    [35] is a combination of Hilbert [33, 34] with various changes taking into account suggestions and criticism by Klein [39] and results of Noether [61], which also led to additional axioms.

  14. 14.

    Here we use the following terminology: An “autoparallel” is a curve \(\lambda \mapsto x(\lambda )\) that satisfies the differential equation \({\ddot{x}}^a+\Gamma ^a_{bc}{\dot{x}}^b{\dot{x}}^c=f(\lambda ){\dot{x}}^a\), where f remains unspecified. A “geodesic” is a curve that satisfies this differential equation for \(f\equiv 0\). Note that \(\Gamma ^a_{bc}\) are the components of the connection (the Christoffel symbols in Riemannian and semi-Riemannian geometry). Any autoparallel can be turned into a geodesic by reparametrization. This fixes the parameter \(\lambda \) up to affine transformations: \(\lambda \mapsto \lambda ':=a\lambda +b\), where \(a\in \mathbb {R}-\{0\}\) and \(b\in \mathbb {R}\). Sometimes people speak of “unparametrized geodesics” instead of “autoparallels”, but we shall not adopt this terminology here.

  15. 15.

    In General Relativity, a “clock” is usually defined as any device that is capable of measuring the proper length (or any preferred parameter affinely equivalent to that) of a timelike curve. This may be its own worldline (if it defines any) or that of another object. How this can be done—in principle—with light rays alone, thereby giving operational meaning to the concept of a “Lichtuhr” already mentioned by Hilbert [34, p. 54], has been explained by [52] based on [51].

  16. 16.

    This difficult-to-access paper was republished as “Golden Oldie” by Ehlers et al. [20].

  17. 17.

    A “Weyl structure” is equivalent to a “Weyl geometry” in Matveev and Scholz [53]’s terminology. The latter is defined by an equivalence class of pairs \((g,\varphi )\) with equivalence relation \((g,\varphi )\sim (g',\varphi ')\Leftrightarrow \) \(g'=\exp (\Omega )g\) and \(\varphi '=\varphi +d\Omega \). It is easy to see by the obvious generalization of the standard Koszul formula that for any given pair \((g,\varphi _g)\) there is a unique torsion-free \(\nabla \) satisfying (10.4).

  18. 18.

    The “first clock-effect” just means that initially synchronized clocks generally show different readings when connecting two timelike separated events by different worldlines. Geometrically this just refers to the trivial fact that the lengths of paths connecting two given points in space-time depend on the paths. The “second clock-effect” refers to the (mathematical) possibility that the ticking rates of the two clocks may also differ upon being brought together, depending on their pre-history. A general, non-integrable Weyl geometry allows for such second clock-effects.

  19. 19.

    Note that a Weyl structure\((M,\mathscr {C},\nabla )\) only includes a conformal equivalence class \(\mathscr {C}\) of semi-Riemannian metrics. This suffices to define timelike curves as those curves \(\gamma \) where \(g(\dot{\gamma },\dot{\gamma })<0\) for one—and hence any—representative \(g\in \mathscr {C}\), which also does not depend of the parametrization. But lengths of curves are not determined by \(\mathscr {C}\) so that the notion of a “clock” cannot be defined as before in footnote 15, namely, as a device measuring the proper lengths of timelike curves (or any of the affinely equivalent parameters). Instead, in Weylian space-times, a “clock” is defined, according to Perlick [63], by a device that allows to measure any of the preferred parameters within the affine equivalence class of curve parameters with respect to which the acceleration \(\ddot{\gamma }=\nabla _{\dot{\gamma }}\dot{\gamma }\) is \(\mathscr {C}\)-perpendicular to the direction of the curve, i.e. \(g(\ddot{\gamma },\dot{\gamma })=0\) for one—and hence any—\(g\in \mathscr {C}\).

  20. 20.

    There are various approaches, not all of which allow the notion of lightlike geodesics. A scheme in which this is possible has been introduced and used in [42]. See [41] for a review and references [58] for a discussion of conditions on Finsler metrics that lead to Lorentzian-like two-component lightcones.

  21. 21.

    The idea is simple: Take a Lagrange function (square of the “Finsler function”) \(L(x,\dot{x})=\exp \bigl (2\sigma (x,\dot{x})\bigr )\,g_{ab}(x){\dot{x}}^a{\dot{x}}^b\) with \(\sigma :TM\rightarrow \mathbb {R}\) any smooth function. This obviously leads to the same conformal structure. As shown by Matveev and Trautman [75], it has the same projective structure if \(\sigma \) satisfies \(\partial \sigma /\partial x^b-(\partial \sigma /\partial {\dot{x}}^a)\Gamma ^a_{bc}{\dot{x}}^c=0\), where \(\Gamma ^a_{bc}\) are the components of the Levi-Civita connection (Christoffel symbols) for g.

  22. 22.

    To prove that the \(C^2\) requirement is, in fact, sufficient, to deduce the Lorentzian nature of the geometry is one of the crucial steps in [19] which is not proven in detail. It relies on the classification of quadrics in projective 3-space (not stated by the authors) from which the further axiom \(L_2\) then picks out those containing the required two components of the set of lightlike directions.

  23. 23.

    This was just the central idea behind Paul Finsler’s generalization of Riemannian geometry that he developed in his 1918 thesis under the supervision of Carathéodory: to generalize the geodesic variational principle as much as possible while maintaining existence and uniqueness for the solutions of the resulting Euler-Lagrange equations [23].

  24. 24.

    In fact, after this paper was written, my attention was drawn to the very recent work of Bernal et al. [8] in which a far more detailed analysis of the compatibility of Finsler geometries with the EPS scheme was made. In particular, the authors also discuss the justification of the \(C^2\) assumption in axiom \(L_1\) in more detail as here and as by Lämmerzahl and Perlick [41] and also come to the conclusion that it may well be relaxed. I thank Christian Pfeifer for pointing out this reference to me.

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Acknowledgements

I sincerely thank the organizers for inviting me to the conference Axiomatic Thinking: One hundred years since Hilbert’s address in Zürich, held at Zürich University on 14–15 September 2017.

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  1. Leibniz University of Hannover, Institute for Theoretical Physics, Appelstraße 2, 30167, Hannover, Germany

    Domenico Giulini

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  1. Domenico Giulini
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  1. Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal

    Fernando Ferreira

  2. Carl Friedrich von Weizsäcker-Zentrum, Universität Tübingen, Tübingen, Germany

    Reinhard Kahle

  3. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Giovanni Sommaruga

Appendix: German Originals of Quotations

Appendix: German Originals of Quotations

Reichenbach

From “Axiomatik der relativistischen Raum-Zeit Lehre”

[65, p. 1–2], quoted on page 231:

Es ist der Wert einer axiomatischen Darstellung, dass sie den Inhalt einer wissenschaftlichen Theorie in wenigen Sätzen zusammenfasst; jedes Urteil über die Aussagen der Theorie darf sich dann auf ein Urteil über die Axiome beschränken, denn in ihnen ist jeder Satz der Theorie schon implizite enthalten. ... die Frage der mathematischen Axiome ist geklärt durch die Entdeckung, dass die mathematischen Axiome Definitionen sind, d.h. willkürliche Fest-setzungen, über die es kein wahr oder falsch gibt, und dass nur die logischen Eigenschaften des Systems, Widerspruchsfreiheit, Unabhängigkeit, Eindeutigkeit, Vollständigkeit Gegenstand der Kritik sein können.

Die Physik unterscheidet sich jedoch von der Mathematik in einer wesentlichen Beziehung. Ihre Sätze wollen mehr sein als konsequente Folgen willkürlicher Setzungen; sie wollen für die Wirklichkeit Geltung besitzen.

Das Urteil ‘wahr’ oder ‘falsch’ bedeutet deshalb in der Physik etwas wesentlich anderes als in der Mathematik; es ist eine außerlogische Beziehung, es besagt das Zutreffen oder Nichtzutreffen eines Wahrnehmungserlebnisses. Und die Frage nach der Wahrheit erscheint dem Physiker als das eigentlich Interessante; denn wenn sie bejaht wird, darf er seine Theorie in einem gewissen Sinne als eine Beschreibung der Wirklichkeit bezeichnen.

Die axiomatische Darstellung einer physikalischen Theorie ist zunächst den gleichen Gese-tzen unterworfen wie in der Mathematik [...]. Aber gerade weil die physikalischen Axiome ebenfalls die ganze Theorie schon implizite enthalten, überträgt sich der Geltungsanspruch auch auf sie; die physikalischen Axiome dürfen nicht willkürlich, sie müssen wahr sein. Wahr bedeutet hier wieder ein Tatsachenurteil, welches letzten Endes die Wahrnehmung fällt.

Einstein

From “Geometrie und Erfahrung”

[37, Doc. 52, p. 385–6], quoted on page 233:

Insofern sich sie Sätze der Mathematik auf die Wirklichkeit beziehen sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.

From discussions at Bad-Nauheim (1920)

[37, Doc. 46, p. 353], quoted on page 236:

Es ist eine logische Schwäche der Relativitätstheorie in ihrem heutigen Zustande, dass sie Maßstäbe und Uhren gesondert einführen muss, statt sie als Lösungen von Differentialgleichungen konstruieren zu können. Was aber die Zuverlässigkeit der Konsequenzen hinsichtlich der Beziehung auf das empirische Fundament der Theorie anbelangt, so sind die Konsequenzen, welche das Verhalten der starren Körper und Uhren betreffen, die am besten gesicherten.

From “Geometrie und Erfahrung”

[37, Doc. 52, p. 386], quoted on page 257:

Der von der Axiomatik erzielte Fortschritt besteht nämlich darin, dass durch sie das Logisch-Formale vom sachlichen oder anschaulichen Gehalt sauber getrennt wurde; nur das Logisch-Formale bildet gemäß der Axiomatik den Gegenstand der Mathematik, nicht aber der mit dem Logisch-Formalen verknüpfte anschauliche oder sonstige Inhalt.

Hertz

From the introduction of “Prinzipien der Mechanik”

[32, p. 1–3], quoted on page 236:

Wir machen uns innere Scheinbilder oder Symbole der äußeren Gegenstände, und zwar machen wir sie von solcher Art, dass die denknotwendigen Folgen der Bilder stets wieder die Bilder seien von den naturnotwendigen Folgen der abgebildeten Gegenstände.

Die Bilder, von welchen wir reden, sind unsere Vorstellungen von den Dingen; sie haben mit den Dingen die eine wesentliche Übereinstimmung, welche in der Erfüllung der genann-ten Forderung liegt, aber es ist für ihren Zweck nicht nötig, dass die irgend eine weitere Übereinstimmung mit den Dingen haben.

Eindeutig sind die Bilder, welche wir uns von den Dingen machen wollen, noch nicht be-stimmt durch die Forderung, dass die Folgen der Bilder wieder die Bilder der Folgen seien.

Von zwei Bildern desselben Gegenstandes wird dasjenige das zweckmäßigere sein, welches mehr wesentliche Beziehungen des Gegenstandes wiederspiegelt als das andere, welches, wie wir sagen wollen, das deutlichere ist. Bei gleicher Deutlichkeit wird von zwei Bildern dasjenige zweckmäßiger, welches neben den wesentlichen Zügen die geringere Zahl überflüssiger oder leerer Beziehungen enthält, welches also das einfachere ist.

Pauli

From [62, p. 775], quoted on page 250:

Wie immer man sich im Einzelnen zu diesen Argumenten stellen mag, so viel scheint sicher zu sein, dass zu den Grundlagen der bisher aufgestellten Theorien erst neue, der Kontinu-umsauffassung des Feldes fremde Elemente hinzukommen müssen, damit man zu einer befriedigenden Lösung des Problems der Materie gelangt.

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Giulini, D. (2022). Axiomatic Thinking in Physics—Essence or Useless Ornament?. In: Ferreira, F., Kahle, R., Sommaruga, G. (eds) Axiomatic Thinking II. Springer, Cham. https://doi.org/10.1007/978-3-030-77799-9_10

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