Skip to main content
SpringerLink
Log in
Book cover

Between Rationalism and Empiricism pp 453–512Cite as

Spacetime, Invariance, Covariance

  • Chapter

  • 284 Accesses

Abstract

The papers collected in this chapter concern the physical structure of space, time and spacetime. Problems connected with invariance properties of physical theories based on invariance properties of their corresponding spacetime theories are well-known, e.g. the Galileo invariance of Newton’s dynamics, the Lorentz invariance of classical electrodynamics and the general covariance of Einstein’s field equations. In [31], [33] and [34] problems of the formulation, the meaning and the physical content of invariance and covariance statements are dealt with. Less well-known is the question of characterizing the structure and theory of spacetime by assumptions as plausible as possible from the physical point of view. The classical case is Helmholtz’ characterization of the riemannian spaces of constant curvature by means of their group of free mobility of rigid bodies. Since today we believe in Einstein’s general theory of relativity, a more appropriate characterization should refer to the Lorentz manifolds that are the infinitesimal version of Minkowski spacetime. No such characterization seems to be known. But Hermann Weyl succeeded in 1923 in characterizing a slightly more general class of manifolds which he named after Pythagoras ([32]).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. “...the nature of the metric field...is essentially one and therefore absolutely determined...in it the aprioristic essence of the space-time structure is expressed” (Weyl 1923b, p.45)

    Google Scholar 

  2. Scheibe 1999, VIII.3 (Newton/Cartan theory of gravitation)

    Google Scholar 

  3. Scheibe 1997b, IV. 1–3

    Google Scholar 

  4. First published as Scheibe 1982c. The completion of this paper was made possible by a Visiting Fellowship at the Center for Philosophy of Science at the University of Pittsburgh. The author wants to express his gratitude to the chairman and the director of the Center, Professor A. Grünbaum and Professor L. Laudan, for the kind invitation and generous hospitality.

    Google Scholar 

  5. Bourbaki 1968, Ch. IV

    Google Scholar 

  6. Shoenfield 1967, Ch. 9

    Google Scholar 

  7. Bourbaki 1968, Ch. IV

    Google Scholar 

  8. ibid.

    Google Scholar 

  9. Trautman 1972, p. 85

    Google Scholar 

  10. Ludwig 1978

    Google Scholar 

  11. Stegmüller 1979

    Google Scholar 

  12. Scheibe 1979 (this vol. ch. III.11)

    Google Scholar 

  13. Trautman 1972, p. 85

    Google Scholar 

  14. Ehlers 1973

    Google Scholar 

  15. Rosen 1972

    Google Scholar 

  16. ibid.

    Google Scholar 

  17. Bourbaki 1968, Ch. IV

    Google Scholar 

  18. Iyanaga/ Kawada 1977, 92 D

    Google Scholar 

  19. ibid., 108 Z; Kobayashi/Nomizu 1963, p. 1

    Google Scholar 

  20. Ludwig 1978

    Google Scholar 

  21. ibid.

    Google Scholar 

  22. Havas 1964, Trautman 1967, Künzle 1972, Misner et al. 1973, Ch. 12

    Google Scholar 

  23. First published as Scheibe 1988e.

    Google Scholar 

  24. Weyl 1919, p. 25 ff; 1922 a; 1922 b; 1923 a; 1923 b, 7. und 8. Vorlesung; 1923 c, §19.

    Google Scholar 

  25. Weyl 1923 b, p. 45; 1949, p. 134.

    Google Scholar 

  26. Weyl 1923 b, p. 61.

    Google Scholar 

  27. Weyl 1923 b, p. 46.

    Google Scholar 

  28. Weyl 1923b, p. 46.

    Google Scholar 

  29. Weyl 1919, p. 27.

    Google Scholar 

  30. Laugwitz 1958. In this paper Laugwitz extends “Weyl’s first space problem” to the pseudo-riemannian case and prepares the ground for a similar characterization also in this case. However, he did not succeed then and has recently informed me that nobody seems to have continued his work.

    Google Scholar 

  31. Scheibe 1957, p. 172, 176.

    Google Scholar 

  32. Kobayashi and Nomizu 1963, p. 288.

    Google Scholar 

  33. Scheibe 1957, p. 192.

    Google Scholar 

  34. Klingenberg 1959, p. 301.

    Google Scholar 

  35. Cartan 1923, esp. pp. 171–174.

    Google Scholar 

  36. Kobayashi and Nomizu 1963, p. 288 f; Kobayashi and Nagano 1965, Theorem 2, p. 86; Freudenthal 1960, p. 107; 1964, p. 157 ff; 1967, p. 326.

    Google Scholar 

  37. For a modern treatment of Weyl’s geometries see Folland 1970. Our version is slightly more general than Folland’s: the conformal viewpoint is applied only locally.

    Google Scholar 

  38. More accurately stated, we have obtained the concept of a group field of a given nature Q3 where Q3 is a class of conjugate Lie-subgroups of the general linear group, see Scheibe 1957, §2. No group within Q3 is distinguished by our concept. By contrast, the concept of a G-structure requires such a distinction. However, the generalization of Weyl’s metric connection, to be given presently, also requires it.

    Google Scholar 

  39. Scheibe 1957, p. 175 Satz 1. With respect to the “historical myth” I want to destroy it should also be emphasized that, whereas an Ok-structure uniquely determines a torsionfree linear connection compatible with it, an Ok-field does not do this.

    Google Scholar 

  40. Weyl 1922 a, §1; 1922 b, V; 1923 b, 7. und 8. Vorlesung; 1923 c, §§ 17 und 19.

    Google Scholar 

  41. Weyl 1922b, p. 221 (p.344 of the Ges. Abh. II).

    Google Scholar 

  42. Weyl 1923b, p. 47, my italics.

    Google Scholar 

  43. It should be noticed that the otherwise somewhat clumsy concept of a class of linear connections related to a group field in the manner indicated has the advantage that it avoids the explicit dependency on the group G: It thus makes the whole concept of a metric dependent only on a class of conjugate groups representing the nature of the metric. This is very much in the spirit of Weyl! The following lemma (A) is in Weyl (1923 b), p. 49 f.

    Google Scholar 

  44. A second postulate used in this argument — the socalled postulate of the freedom of the metric — can be proven, cf. Weyl 1923 b, p. 49, Scheibe 1957, p. 196, Hilfssatz 1.

    Google Scholar 

  45. Cartan 1923; Freudenthal 1960.

    Google Scholar 

  46. Weyl 1923b, p. 46.

    Google Scholar 

  47. Wigner 1979, p. 16 f.

    Google Scholar 

  48. Scheibe 1982c (this vol. sect. VII.31).

    Google Scholar 

  49. Kobayashi and Nagano 1965, p. 86, theor. 1.

    Google Scholar 

  50. Weyl 1923 b, p. 45.

    Google Scholar 

  51. Weyl 1923 b, p. 49.

    Google Scholar 

  52. Weyl 1949, p. 137.

    Google Scholar 

  53. Coffa 1979, p. 277.

    Google Scholar 

  54. Weyl 1949, p. 137.

    Google Scholar 

  55. Weyl 1949, p. 134.

    Google Scholar 

  56. Künzle 1972. It is interesting that although Newton’s theory in a sense is a limiting case of Einstein’s, Weyl’s postulate does not survive the limiting process.

    Google Scholar 

  57. Weyl (1923 c), § 29

    Google Scholar 

  58. First published as Scheibe 1991f.

    Google Scholar 

  59. Einstein 1916b,sect. 3

    Google Scholar 

  60. ibid. sect. 2

    Google Scholar 

  61. ibid. sect.3

    Google Scholar 

  62. Kretschmann 1917, p. 576

    Google Scholar 

  63. Einstein 1918, p. 242

    Google Scholar 

  64. Einstein/ Infeld 1938, p.212

    Google Scholar 

  65. Einstein 1916b, sect. 2

    Google Scholar 

  66. ibid. sect. 3

    Google Scholar 

  67. Cartan 1923/4

    Google Scholar 

  68. Bourbaki 1968, Ch. IV

    Google Scholar 

  69. For the following see Scheibe 1982c (this vol. VII.31)

    Google Scholar 

  70. For details see Iyanaga/ Kawada 1977. 92 D and (a narrower concept) 108 Z.

    Google Scholar 

  71. A recent exception is Dixon 1978, pp. 42ff

    Google Scholar 

  72. Klein 1925

    Google Scholar 

  73. The Standard monograph is Schouten 1954

    Google Scholar 

  74. See Misner/ Thorne/ Wheeler 1973. What I am emphasizing is that, although the definition of, say, the concept of a vector field need not refer to coordinate Systems, the definition is based on the concept of a differentiable manifold and this concept usually is defined by using coordinate Systems.

    Google Scholar 

  75. A different analysis of the principle of general covariance can be found in Weinberg 1972, pp. 91ff

    Google Scholar 

  76. For some further thoughts on the matter see Scheibe 1982c (this vol. VII.31)

    Google Scholar 

  77. Anderson 1967 and 1971

    Google Scholar 

  78. Friedman 1973

    Google Scholar 

  79. First published as Scheibe 1994b.

    Google Scholar 

  80. Wigner 1979, 30 f.

    Google Scholar 

  81. Loc. cit., 16 f.

    Google Scholar 

  82. Dodd 1984, 41.

    Google Scholar 

  83. See, for instance, Genz und Decker 1991; Mayer-Kuckuk 1989.

    Google Scholar 

  84. For details see Bourbaki 1968, Ch. IV.

    Google Scholar 

  85. The first to have applied species of structures to physies in a systematic fashion is Ludwig in 1978, 21990.

    Google Scholar 

  86. See Scheibe 1982c (this vol. VII.31).

    Google Scholar 

  87. This is due to the von Neumann-Stone theorem, see Emch 1984, Ch. 8.3 f

    Google Scholar 

  88. See, for instance, Bethge und Schröder 1986, Ch. 5.

    Google Scholar 

  89. Einstein’s favorite view of the matter is discussed in Scheibe 1991f (this vol. VII.35).

    Google Scholar 

  90. Kant, Critique of Pure Reason, B 39.

    Google Scholar 

  91. Sommerfeld 1948.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moorbirkenkamp 2A, 22391, Hamburg, Germany

    Prof. Dr. Erhard Scheibe

Authors
  1. Prof. Dr. Erhard Scheibe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Philosophie, Universität Dortmund, Emil-Figge-Strasse 50, 44227, Dortmund, Germany

    Brigitte Falkenburg

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media New York

About this chapter

Cite this chapter

Scheibe, E. (2001). Spacetime, Invariance, Covariance. In: Falkenburg, B. (eds) Between Rationalism and Empiricism. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0183-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0183-7_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6555-9

  • Online ISBN: 978-1-4613-0183-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation