Abstract
Hermann Weyl’s work is difficult to classify as physics, mathematics, philosophy or history of science. Perhaps because of his wide audience, perhaps also because of his aesthetic preferences, Weyl likes to use analogies and metaphors in order to provide insights about the most difficult and abstract problems of the twentieth-century science.
Die einfache Tatsache, daß ich eine Plastelinkugel in meiner Hand zu einer beliebigen Mißgestalt zerdrücken kann, die ganz anders aussieht als eine Kugel, scheint den Riemannschen Standpunkt ad absurdum zu führen.
The simple fact that I can squeeze a ball of plasticine with my hands into any irregular shape totally different from a sphere would seem to reduce Riemann’s view to an absurdity.
H. Weyl, Space-Time-Matter, first edition, p. 90.
This article has been translated from French to English by Pascale Pelletier, in collaboration with the author who thanks her for her patient and precise work.
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Notes
- 1.
As in the following, we use Bose’s translation, revised if necessary.
- 2.
See Sect. 11.1.3.
- 3.
- 4.
Barbour and Pfister (1995, p. 530).
- 5.
- 6.
One part of this section is identical with a passage of the general introduction of the current volume.
- 7.
Einstein (1916).
- 8.
See Norton (1993, pp. 808–sq.) for a good overview of the foundational debates on general relativity, during the first 80 years of existence of this theory.
- 9.
Amongst the bibliographical references given by Weyl for his chapter IV, we find Kretschmann’s article “Über den physikalischen Sinn der Relativitätspostulate”. Therefore, Weyl had probably been influenced by Kretschmann’s famous argument, according to which the general covariance principle had no physical meaning by itself, since every physical theory can be expressed in a covariant form by a tensorial reinterpretation. See also Norton (1993).
- 10.
Weyl (1918b, p. 181), Weyl (1919a, §26, p. 192), and Weyl (2010, p. 226):
A new physical factor appears only when it is assumed that the metrical structure of the world is not given a priori, but that the above quadratic form is related to matter by generally invariant laws. Only this fact justifies us in assigning the name “general theory of relativity” to our reasoning; we are not simply giving it to a theory which has merely borrowed the mathematical form of relativity.
- 11.
For example, in 1924, see Weyl (1924, p. 197).
- 12.
The expression appeared first in Einstein (1918, p. 197). Cf. Barbour and Pfister (1995, p. 10). Weyl does not explicitly refer to Mach within paragraph 12 of Raum-Zeit-Materie. Mach appears however in the bibliographical references of chapter IV, in Weyl (1921, p. 291, bibliographical note 2). Weyl in (1924, p. 198) acknowledged that Mach was the father of the principle of determination of inertia by cosmic matter.
- 13.
In particular the famous text: Riemann (1919).
- 14.
See Weyl’s texts quoted in Michel (2006, p. 198).
- 15.
See the bibliography of Bernard (2015b, p. 198) for the list of Weyl’s works on the problem of space – in its technical meaning – and a historical discussion. Secondary reading on this subject is abundant, see Coleman and Korté (2001, p. 198), Scholz (2004, p. 198), Laugwitz (1958, p. 198), Bernard (2015a, p. 198), Bernard (2018, p. 198), and Weyl (2015, vol. 2).
- 16.
Weyl (1918b, p. 90).
- 17.
Weyl (1919a, p. 90)].
- 18.
Weyl (1921, p. 90).
- 19.
Weyl (1924, p. 198).
- 20.
Weyl (1949, 86–87;105).
- 21.
Weyl (1934, p. 129).
- 22.
Weyl (1923, pp. 44–45). We find there the problem of the tension between the homogeneity of space, as a form of appearances, and the heterogeneity of the metric ; and the solution consisting in moving the metric simultaneously with matter.
- 23.
I will specify in this article the most important changes in Raum-Zeit-Materie that I personally noticed. Concerning Philosophy of Mathematics and Natural Science, I have received the information from Carlos Lobo.
- 24.
- 25.
See the end of Sect. 11.1.2.
- 26.
The notion of matter which is present in the first four editions of Raum-Zeit-Materie does not form a discrete set of particles but a field. Weyl was then taking up the programme of Gustav Mie which consisted in bringing out the notion of matter from the notion of field.
- 27.
See for example Einstein (1920).
- 28.
- 29.
- 30.
One part of this section is identical with a passage of the general introduction of the current volume. There are more details there.
- 31.
- 32.
For the shift from the notion of homogeneity to the notion of congruence then to the notion of group of congruences, see Weyl (2010, 5–6;11–15) or Weyl (1923, 44–49). For the use of the theory of groups to found the notion of metric in a context of differential geometry, see Weyl (2010, §18), Eckes (2011, §18) or the texts in relation to the problem of space, in its technical meaning (cf. footnote 15).
- 33.
Weyl (2010, p. 11):
Space is a form of appearances 〈Form der Erscheinungen〉, and, by being so, is necessarily homogeneous. It would appear from this that out of the rich abundance of possible geometries included in Riemann’s conception, only the three special cases mentioned come into consideration from the outset, and that all the others must be rejected without further examination as being of no account: parturiunt montes, nascetur ridiculus mus! Riemann held a different opinion, as is evidenced by the concluding remarks of his essay [⋯ ] Only now that Einstein has removed the scales from our eyes by the magic light of his theory of gravitation do we see what these words actually mean.
- 34.
- 35.
In the specific case of general relativity, we have (Weyl 2015, p. 44):
According to Einstein, the metric structure of the universe is not homogeneous. How is this possible, given that space and time are forms of appearances?
- 36.
See Weyl (1918b, pp. 88–90) and the corresponding parts in the three editions of Raum-Zeit-Materie that follow.
- 37.
In Raum-Zeit-Materie, Weyl does not refer to material qualities. He only refers to “the material” 〈das Materiale〉. However in other texts of the same time, such as Weyl (1923, introduction), Weyl calls the material content “qualitative” and describes the homogeneity of space by the fact of being able to move these qualities towards any point.
- 38.
Weyl (1918b, p. 88). The fact that Weyl choses as an example the density of electric charge 〈Elektrizitätsdichte〉 is significant. Perhaps he has already in mind his own theory (to be published in 1918, see Weyl (1918a)) in which the metric field is the carrier of the gravitational and electromagnetic interactions, simultaneously. So, if something like a principle of metric determination by matter is to be considered in such a conceptual framework, it cannot take the form “mass determines the metric” any longer but, instead, “mass and electric charge determine the metric”.
- 39.
The “pull-forward” (“pull-back”) terminology is posterior to Weyl. See Iftime and Stachel (2006, p. 1243). Besides Weyl does not mention here the fact that a region S cannot be moved towards any region S′. Instead, as shown by the process used, one must take a region S′ diffeomorphic to the first one. For instance a simply connected region cannot be transformed into an annular region. Weyl is explicit about it further on in the text (Weyl 2010, p. 98).
- 40.
Weyl insists on the fact that, in order to determine the visual shape of a portion of matter, one must not only know the metric coefficients for the portion S of space-time where the matter is but also for all the space-time points through which the light rays which, emitted from S, will reach the observer. The body of the latter is represented by a point-eye set on a point outside S. The necessity to take into account the metric on the intermediate trajectory is clear as soon as we think of phenomena such as light rays deflection by gravity or, in an anachronistic way, the gravitational lenses phenomena. Weyl will come back to the necessity to take into account the intermediate metric field to differentiate the rotation of the stellar compass from the rotation of the stars themselves in Weyl (1924, p. 198, left hand column).
For Weyl, having the visual observer intervene in order to define the shape of a material object is an important step, in view of his attachment to the Husserlian phenomenology during the years which we are considering; Weyl, here, uses significantly the term “experiences of consciousness” 〈Bewußtseinserlebnissen〉 (Weyl 1918b, p. 89). Concerning this point-eye idea or Ego-centre 〈Ich-Zentrum〉, and its phenomenological function, see in particular analyses in Ryckman (2009, p. 286), Mancosu and Ryckman (2005, p. 89), and Bernard (2013, 241–sq.). See also Kerszberg’s article in the present volume.
However concerning the specific issue with which we are dealing, we do no need to discuss it further. The “pulling forward” of the metric in the space-time region which separates the observer from the element of matter that is observed is technically expressed in the same way as the pulling forward of the region S itself. The point-eye representing the observer is also pulled forward.
- 41.
- 42.
Iftime and Stachel (2006, Appendix: “Active and Passive Covariance”).
- 43.
Added in the fourth edition.
- 44.
On this issue, see also the analogy of flexible sheet metal in Weyl (2015, p. 44).
- 45.
- 46.
A question will remain whether matter could even totally emerge from matter itself. It is the question raised by Mach’s principle below.
- 47.
- 48.
Any n–dimensional manifold that is (arcwise) connected and of class \(\mathbb {C}^p\) for p = 0, 1, ⋯ , ∞ is not only homogeneous but even maximally isotropic. Given any two points P, P′ of M, and given {v 1, ⋯ , v n−1}, \(\{v^{\prime }_1, \cdots , v^{\prime }_{n-1}\}\) two families of linearly independent vectors, respectively, taken in T P(M) and in \(T_{P'}(M)\). Then there is a diffeomorphism of class \(\mathbb {C}^p\) which sends P to P′, sends the infinitesimal straight line < v 1 > to \(<v^{\prime }_1>\), sends the infinitesimal plane < v 1, v 2 > to \(<v^{\prime }_1, v^{\prime }_2>\), ⋯ and finally sends the infinitesimal hyperplane < v 1, ⋯ , v n−1 > to \(<v^{\prime }_1, \cdots , v^{\prime }_{n-1}>\).
- 49.
On that question, see Bernard (2013, Chap. III, 3.).
- 50.
Weyl (2015, p. 44).
- 51.
See our Footnote 12.
- 52.
This point partly follows from the structure of Raum-Zeit-Materie. The text which we have explained is taken from §12, therefore from chapter II, while general relativity is mentioned only in chapter IV. See what we have said p. 297 about Weyl’s standpoint towards the generalised covariance principle.
- 53.
See our comments in Bernard (2018, p. 3).
- 54.
I refer to the paragraphs numbering in the fourth edition.
- 55.
- 56.
Aristotle, Physics, book I, Chap. 2.
- 57.
Diogenes Laercius, Lives, Doctrines and Sentences of Famous Philosophers, VI, Chap. 2 [Diogenes].
- 58.
After the passage evoked previously, Aristotle eventually admitted that even if the Eleatic opinion of the immobility and unity of the world is obviously false, dedicating efforts to refute Parmenides and Melissus may be physically instructive. It is of course also the case for Zeno.
- 59.
T μν is the energy-momentum tensor and G μν a tensor only dependant of the metric field and its derivatives, which would still need to be determined. See Norton (1987, pp. 162–sq.).
- 60.
Weyl uses himself this approximation in Weyl (2010, p. 205).
- 61.
Weyl (2010, p. 205; 262–263).
- 62.
Weyl (2010, 229).
- 63.
- 64.
Here, we leave aside the issues arising at the boundary, when joining the two solutions.
- 65.
Weyl (1918b, p. 90).
- 66.
Giving some sense to such a moving of matter would require setting the problem in a really dynamic framework, without only considering the initial state and the final equilibrium state. In Massenträgheit und Kosmos, Weyl will be able to give sense to such a global movement with his Boats-Lake Analogy. See further Sect. 11.4.7.
- 67.
- 68.
In particular, Cassirer, in the line of the Marbourgh school, criticises Kant’s philosophy of space, in so far as it is too strongly connected to perception data, leading to an exclusive focus on Euclidean geometry. See Cassirer (1910, chapitre III, particularly p 106) and Cassirer (1923, chap. V). According to Cassirer, the a priori notion of space which must be incorporated to science needs rather to be based on the driving forces of mathematical analysis and numerical symbols.
- 69.
Weyl (1918b, p. 88): “Genaueres hierüber in Kap. IV.”.
- 70.
Weyl (1924, p. 198): 〈Plastelinmasse〉.
- 71.
The notion of homogeneity appears punctually in Massenträgheit und Kosmos; however it is not the homogeneity as an a priori property required from space “per se”, but, instead, the (local or global) homogeneity of some configurations of matter, considered as particular cases, or the metric homogeneity of some specific solutions to the Einstein equations, particularly the de Sitter’s one. Homogeneity has become the exception rather than the a priori rule.
- 72.
Weyl (1924, p. 197). There is an implicit reference to the famous biblical sentence “And I tell you that you are Peter (Céphas = Rock), and on this rock I will build my church”.
- 73.
Saul of Tarsus was the Jewish name of the man later known as the Apostle Paul or Saint Paul in the New Testament. It is said that he was initially a Pharisee, violent towards Christians, before converting and joining Jesus Christ. He changed his name from Saul to Paul to mark this conversion. There is a German phrase “change from Saul to Paul” 〈sich vom Saulus zu Paulus wandeln〉, used to describe a radical change of personality or behaviour.
- 74.
About Einstein’s abandonment of Mach’s principle, see letter of 02.02.1954 to F. Pirani, the extract of which is reported and translated into English in Renn (2007, p. 61). In Norton (1987, pp. 180–sq.), it explains that this abandonment by Einstein of Mach’s principle takes the form of a shift from an overt antirealism towards space (then identified with the naked manifold) to a realism towards space (then identified with the metric field, called “ether”). Other references about Einstein’s position concerning Mach’s principle in: Barbour and Pfister (1995, P. 10; 67–90), Norton (1993, pp. 808–sq.) and Torretti (1983, section 6.2). Before abandoning Mach’s principle, Einstein gave it very variable forms. According to Norton and Torretti (1983, p. 201), Einstein’s change of mind about Mach’s principle began in the years 1918–1919.
- 75.
The question whether Weyl had real persons in mind behind his characters is minor. What is important is to underscore the fact that Paul’s intellectual evolution is close to Einstein’s and Weyl’s. Paul and Peter, in the dialogue, say that they first met in the United States in 1915. Einstein and Weyl met as early as 1913 at the E.T.H. of Zurich. Weyl arrived at the institute when Einstein was there, working with his friend Grossman at elaborating general relativity. In the dialogue, Paul tells Peter that the latter should well know the axial symmetry solutions of the theory of general relativity, since he raised the problem of their existence. Weyl (with Lense and Thirring) is amongst the first scientists who published such solutions (see Weyl 2010, §32; and bibliography note 22 of chapter IV; Weyl 1919b).
- 76.
Weyl (1924, p. 187, right hand column). See our Footnote 10 above.
- 77.
Thus, since at the time one consensually believed in the static nature of the cosmos, Newton’s bucket experiment was explained by Mach by the fact that the bucket is in motion (rotating) relatively to the frame of reference defined by the static whole of cosmic masses.
- 78.
However, there is no consensus on Mach’s real purpose. See Barbour and Pfister (1995, 9–65;90–sq.).
- 79.
Weyl (2015, p. 1).
- 80.
When a metric is attributed to space-time, we shall expect the world-lines to be time-like.
- 81.
Kerszberg (1986, p. 1).
- 82.
This case contrasts with the tensorial density \(\mathbb {T}^{\mu \nu }\) which always depends on the metric (or at least its determinant) and with tensors of more complex matter (with pressure, etc.) which we shall consider later.
- 83.
- 84.
See the formulation of the principle (C) above, in which the properties considered in order to characterise matter were motion, charge and mass. If, according to the fpba, motion disappeared, there would only be mass and charge left, construed as simple scalars, as in §12 of Raum-Zeit-Materie.
- 85.
Section 11.3.1.
- 86.
The reader is referred to the literature on the evolution of Weyl’s belief in his theory of unification of gravitation with electromagnetism (1918). See Afriat (2009, p. 197, right hand column).
- 87.
Weyl (1924, p. 199, right hand column).
- 88.
Weyl (1924, p. 199).
- 89.
See our Footnote 74 about the similar rejection found in Einstein’s thought.
- 90.
Weyl (1924, p. 199, left hand column).
- 91.
This late arrival of cosmology in the debate had already been noticed in Raum-Zeit-Materie. See our Footnote 43.
- 92.
Weyl (1924, pp. 199–200).
- 93.
It is a classic in relativistic literature. See Barbour and Pfister (1995, “bucket experiment” p. 531).
- 94.
- 95.
Typically, along the lines of Choquet-Bruhat and Geroch, the first derivatives are not given but, instead, a second-order tensor giving the external curvature of the initial hypersurface within the manifold which is to be generated.
- 96.
The difficult problem of the existence and unicity (up to a diffeomorphism) of the solutions to the Einstein equations, with several types of initial conditions and regularity hypotheses, has been subject since the 1950s to major progress, thanks to Choquet’s work. See Choquet-Bruhat (1952, 1969, 67–sq.). She has shown the existence and unicity (up to a diffeomorphism) of a local solution to the Einstein equations, within the neighbourhood of a space-like hypersurface on which the Cauchy boundary conditions were given. The existence and unicity results are valid for the Einstein equations without sources but also with sources like perfect fluids or electromagnetic fields. See Choquet-Bruhat and Geroch (1969, p. 331).
To obtain global results, we must add hypotheses such as the global hyperbolic character of the manifold. See Choquet-Bruhat and Geroch (1969, p. 331). The problem becomes more complicated, sometimes with no solution or no unicity, if the correct regularity conditions are not posed, if the initial conditions are ill-defined or if hypotheses similar to global hyperbolicity are not available (which has everything to do with Weyl’s principle in cosmology).
In the 1920s, we do not know whether results, at least partial, similar to the ones obtained by Choquet were already available. The most ancient reference given by Choquet is Darmois (1927, p. 331). Einstein’s and Weyl’s convictions on the possibility to correctly pose the Cauchy problem for the Einstein equations could be based on the similarity of these equations with the Laplace and Poisson equations, for which the results of existence and unicity were well known, and on some successful attempts to the univocal determining of a metric in a few specific cases (in the first place those considered by Schwarzschild). For a presentation by Weyl of the theorem of existence and unicity of the solution to a system of partial differential equations, see Weyl (2015, Appendix 3, 2nd part).
- 97.
In Barbour and Pfister (1995, p. 93), Ehlers notes that the energy-momentum tensor, until it is coupled with a metric, does not properly describe the field of matter, and to this day, no physical theory can describe the field of matter before a metric is given. Thus he agrees that Mach’s principle, if it stipulated that “matter in itself [i.e. prior to any metric consideration] determines the metric” would be neither true nor false but even pure nonsense. He mentions that Einstein eventually admitted it in his letter to Pirani of 02.02.1954: “the T μν which must represent “matter”, always presupposes g μν”, quoted from Einstein’s letter to Pirani of 02.02.1954 in Torretti (1983, p. 202). Einstein then suggests to avoid from now on speaking of Mach’s principle in regard to general relativity.
- 98.
See p. 322 above.
- 99.
Stachel (1969) showed that, when we limit ourselves to a field of matter with restricted properties, then we can find dynamic variables describing sources, independently from any metric data. In these particular cases, the tensor T μν has the properties of a simple tensorial field that can be defined on the naked manifold. Stachel illustrates it with three cases:
-
a scalar field without a mass,
-
an incoherent matter ([dust]),
-
an incoherent radiation.
We can then calculate the T μν outside the initial Cauchy surface (by solving the conservation equations), before solving the Einstein equations to have g μν. Imposing from the start, the conservation equations enable then to obtain the conditions to the integration of the Einstein equations, whatever the metric ultimately retained. In that sense, we can calculate the dynamic of the sources before knowing the space-time geometry. In these particular cases, the metric only appears on one side of the Einstein equations, contrary to the general case. However, this does not invalidate the fact that a given field T μν, even of one of these very simple types, shall take totally different physical significations according to the specific metric to which it is correlated. Moreover, Stachel shows that this early calculation of T μν on the whole manifold does not generate any extra restriction on g μν, which still fully depends on the initial conditions that can be freely chosen.
-
- 100.
Afriat and Caccese in Afriat (2010, pp. 16–17) argue that we can sometimes attain a notion of matter without using any metric. After having considered various types of metric tensor, they conclude:
Generally, then, the reliance of matter on the metric seems to depend on the kind of matter; in particular on how rich, structured and complicated it is. The simplest matter –absent matter– can do without the metric; the more frills it acquires, the more it will need the metric.
Of course, these affirmations do not raise any problem if we replace everywhere the word “matter” by “tensor T μν”. However what is precisely debatable is the possibility that the tensor T μν alone, before being coupled with a metric, represents a well-determined state of matter. So, for example, even if the tensor T μν = ρ.u μ u ν (“dust”) does not depend on g μν, it will represent a very different state of the matter, depending on the metric to which it is correlated on a considered hypersurface.
- 101.
Let us remember that the Schwarzchild radius of a massive body is proportional to its mass, while the geometrical radius of the object grows much more slowly based on the mass (if geometry was Euclidean, this radius would of course grow as \(\sqrt [3]{m}\)). Therefore, the initial density being fixed, a ball of matter will become a black hole as soon as its radius is large enough. The hypothesis of a constant density then loses its coherence.
- 102.
This is in particular the case when we solve the Einstein equations in the peculiar case of a stationary solution, with specific symmetries, as in the calculus made by Schwarzschild for his “interior metric”. In this kind of simple situation, the application of “Mach’s principle” takes a form that is reminiscent of Weyl’s formulation of the
principle in Raum-Zeit-Materie, that is: g = F(ρ⋯ ).
- 103.
Of course, for any relativistic calculation of the metric on a void region, the problem does not arise since the energy-momentum tensor is simply null, therefore, independent per se from any metric. This particularly includes the approximate derivation, by Einstein (1915), of the metric which surrounds a point mass, the exact solution suggested by Schwarzschild (1916a), or again the metric interior to Thirring’s hollow sphere (Thirring 1918, 1921) or the metric near a Lense-Thirring rotating massive body (Lense and Thirring 1918). However in the case of the calculation by Schwarzschild of the metric interior to a spheric mass of perfect fluid, incompressible and at rest (Schwarzschild 1916b, pp. 16–17), the problem arises since T μν should appear under the general form:
$$\displaystyle \begin{aligned} T^{\mu\nu}=\left(\rho+p\right)u^{\alpha{}}u^\beta-p.g^{\alpha\beta} \end{aligned}$$which explicitly depends on the metric. In Schwarzschild’s text, it is however clear that the T μν is not a starting data of the problem. In a significant manner, Schwarzschild starts with the mixed tensor \(T^1_1=T^2_2=T^3_3=-p\) and \(T^4_4=\rho _0\) (ρ 0 is a constant, since the fluid is incompressible, p depends on the radial coordinate as per a function which will be determined by the stability hypothesis). The presence of symmetry hypotheses indeed enables Schwarzschild to specify the general form of the metric, before calculating. But it is clear that the metric (therefore the T μν) is only perfectly determined after the field equations have been solved.
- 104.
See in particular Barbour and Pfister (1995, pp. 188–sq.).
- 105.
About Mach’s principle construed as a as a rule imposed on the boundary conditions to select some cosmological models 333, see Reinhardt (1973, pp. 531–534) and Barbour and Pfister (1995, pp. 39;77;79–83;95;97;148;190–195;228;238–239;443). About the idea of realising Mach’s ideas within the limits of initial relational data between elements of matter, in a context wider than general relativity, see Barbour and Pfister (1995, 107;111–112;204–207;218–222;443–444).
- 106.
In Weyl (1924, p. 201), Weyl remarks that the hypothesis of a static universe, like in Einstein (1917, p. 201), is equivalent to determining the state of the metric in the past:
The difficulty that arises from the spatial horizon is evidently resolved by [the choice of a] closed space ; but it remains nevertheless, since it is located everywhere in the universe continuum which can deform [⋯] in the same manner as a mollusc. The restriction to static conditions is indeed opaque and debatable.
Weyl then develops an analogy with electromagnetism and asks how Coulomb’s equations, in the static case, derive from Maxwell’s equations. Then he concludes:
The formation of this field F inevitably results from the variable electromagnetic field laws, if we add the hypothesis that space was deprived of a field at the beginning of the sequence. If so, it is not because the field is fixed on the infinitely distant spatial horizon, but, instead the link comes from the world boundary of the past which goes back to an infinitely distant [time].
This argument is also developed in Weyl (1949, §23 C). Besides the hypothesis of a static nature, on which Weyl insists, it seems to me that the homogeneity and symmetry hypotheses on which Einstein and cosmologists usually rely in their derivations have also a metric significance.
- 107.
- 108.
Weyl (1924, p. 202, right hand column).
- 109.
Weyl (1924, p. 202, left hand column).
- 110.
Weyl (1924, p. 202, between both columns).
- 111.
Weyl (1924, p. 203), my translation.
- 112.
Weyl (1924, p. 202, right hand column).
- 113.
In Massenträgheit und Kosmos, it is de Sitter’s. In Philosophy of Mathematics and Natural Science, Weyl will return to the same argument using Minkowski metric as the rest metric of the ether. This changes nothing to the argument that follows.
- 114.
Weyl explicitly refers to Einstein’s article from 1914.
- 115.
Weyl (1924, p. 202, right hand column): “though [de Sitter’s metric] is, per se, qualitatively totally determined, there are however an infinity of possible ways for this state to be realised in the continuum of the world”.
- 116.
Weyl (1924, p. 203, left hand column).
- 117.
Here we use Weyl’s terminology. He very often uses the word “orientation” to refer to the different manners in which a same geometrical object may be expressed in coordinates, that is the different manners in which it can unfold on the manifold. See, for instance, Weyl (2015, pp. 44–45) or Weyl (1921, p. 126).
The orientation of a geometrical object, in that sense, may have a purely subjective status, resulting from an arbitrary choice, as when we consider the orientation of the Riemannian metric at a singular point of the manifold or, on the contrary, have an objective invariant sense, as when we consider the variation of the orientation of the metric throughout an open domain.
In the same manner, here Weyl says that the absolute orientation of Minkowski metric in one region considered in isolation has no physical significance. But the relative change of the orientation of the metric in passing from a region to another one makes sense. Weyl generally illustrates this type of behaviour by referring to the discussions on the differentiation between right hand and left hand in Kant’s and Leibniz’s works.
- 118.
Weyl (1949, pp. 104–107).
- 119.
In Philosophy of Mathematics and Natural Science, contrary to Massenträgheit und Kosmos, Weyl uses Minkowski’s metric (and not de Sitter’s) for the ether, when he develops the analogy. The cosmological preference for de Sitter’s metric will nevertheless be reasserted (and justified in the same manner) by Weyl a few pages further.
- 120.
It is therefore posterior to the German edition of Philosophy of Mathematics and Natural Science, but not to the English edition.
- 121.
Weyl (1934, p. 125). See above p. 336.
- 122.
Weyl (1934, p. 128).
- 123.
This is what the Einsteinian literature has called the “point-coincidence argument” since Stachel’s suggestion, see Norton (1999, p. 128).
- 124.
See Weyl (2015, pp. 17–19) in which the inertial and causal structures respectively correspond to the projective and conformal structures.
- 125.
Weyl (2010, p. 125). See above p. 306
- 126.
They are in great number in comparison with the text discussed previously .
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Bernard, J. (2019). The Plasticine Ball Argument. In: Bernard, J., Lobo, C. (eds) Weyl and the Problem of Space. Studies in History and Philosophy of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-11527-2_11
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