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Special Relativity from Measuring Rods

  • John Stachel
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 76)

Abstract

The mathematical structures associated with a space-time theory, such as the special theory of relativity (SRT) — or the general theory (GRT) for that matter — are numerous and interrelated in complex ways.1 One may start their analysis with the concept of a point set, the elements of which are identified with events in space-time.2 Imposing a continuity structure on this set leads to the concept of space-time as a four-dimensional topological manifold. Restriction to a differentiable structure then leads to the concept of space-time as a differentiable manifold. Various additional mathematical structures may now be introduced on this manifold: projective, affine, conformal and pseudo-metrical (a metrical structure with Minkowski signature). Each of these mathematical structures is closely associated with the behavior of some idealized physical entity in space-time. The projective structure is associated with the trajectories of structureless free test particles. If each particle carries some intrinsic measure of duration along its trajectory, it reflects the affine structure. The conformal structure is associated with the wave fronts of massless fields, such as the electromagnetic. A pseudo-metrical structure with Minkowski signature implies the existence of two fundamentally distinct types of interval which cannot be transformed into one another by any operation of the symmetry group defining the geometry (the inhomo- geneous Lorentz group for SRT).3 These two distinct types of interval are called spacelike and timelike, and physically quite distinct entities — measuring rods and clocks — are associated with their respective measurement.

Keywords

Special Relativity Inertial Frame Lorentz Transformation Conformal Structure Transformation Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Berzi, V. and Gorini, V. 1969. ‘Reciprocity Principle and the Lorentz Transformations,’ J. Math Phys. 10, 1518–1524.CrossRefGoogle Scholar
  2. Carter, B. and Quintana, H. 1972. ‘Foundations of General Relativistic High-Pressure Elasticity Theory,’Proc. Roy. Soc. Lond. A 331, 57–83.CrossRefGoogle Scholar
  3. Dixon, W. G. 1978. Special Relativity: The Foundation of Macroscopic Physics. Cambridge: Cambridge University Press.Google Scholar
  4. Dorling, J. 1976. ‘Special Relativity Out of Euclidean Geometry,’ unpublished.Google Scholar
  5. Ehlers, J. 1973a. ‘The Nature and Structure of Spacetime.’ In J. Mehra (ed.), The Physicist’s Conception of Nature, pp. 71–91. Dordrecht and Boston: D. Reidel.Google Scholar
  6. Ehlers, J. 1973b. ‘Survey of General Relativity Theory.’ In W. Israel (ed.), Relativity, Astrophysics and Cosmology, pp. 1–125. Dordrecht: D. Reidel.Google Scholar
  7. Ehlers, J., Pirani, F. A. E., and Schild, A. 1972. The Geometry of Free Fall and Light Propagation.’ In L. O’Raifeartaigh (ed.), General Relativity. Oxford: Clarendon Press.Google Scholar
  8. Einstein, A. 1905. Zur Elektrodynamik bewegter Körper,Ann. Phys. 17, 891–921.CrossRefGoogle Scholar
  9. Einstein, A. 1911. ‘Zum Ehrenfestchen Paradoxon,’Physik. Zeitschr.12, 509–510.Google Scholar
  10. Einstein, A. 1954. ‘What Is the Theory of Relativity?’ In his Ideas and Opinions, pp. 227–232. New York: Crown.Google Scholar
  11. Einstein, A. 1979. Autobiographical Notes: A Centennial Edition. LaSalle/Chicago: Open Court.Google Scholar
  12. Feenberg, E. 1979. ‘Distant Synchrony and the One-Way Velocity of Light,’ Found, Phys. 9, 329–337.CrossRefGoogle Scholar
  13. Frank, P. and Rothe, H. 1911. Ueber die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte System. Ann. der Phys. 34, 825–855.CrossRefGoogle Scholar
  14. Herglotz, G. 1911. Ueber die Mechanik des deformierbaren Korpers vom Standpunkte der Relativitatstheorie,Ann. Phys. 36, 493–533.CrossRefGoogle Scholar
  15. Ignatowsky, W. 1910. ‘Einige allgemeine Bemerkungen zum Relativitatsprinzip,’ Physik, Zeitschr. 10, 972–975.Google Scholar
  16. Jammer, M. 1979. ‘Some Foundational Problems in the Special Theory of Relativity.’ In G. Toraldo di Francia (ed.), Problems in the Foundations of Physics, pp. 202–236. Amsterdam/New York/Oxford: North-Holland.Google Scholar
  17. Lee, A. R. and Kalotas, T. M. 1975. ‘Lorentz Transformations from the First Postulate,’ Am. J. Phys. 43, 434–437.CrossRefGoogle Scholar
  18. Lévy-Leblond, J. M. 1976. ‘One More Derivation of the Lorentz Transformation,’ Am. J. Phys. 44, 271–277.CrossRefGoogle Scholar
  19. Minkowski, H. 1909. Raum und Zeit. Leipzig/Berlin: B. G. Teubner.Google Scholar
  20. Pauli, W. 1958. Theory of Relativity. New York: Pergamon.Google Scholar
  21. Pfarr, J. (ed.). 1981. Protophysik und Relativitatstheorie. Mannheim/Vienna/Zurich: B. I. Wissenschaftsverlag.Google Scholar
  22. Poincaré, H. 1887. ‘Sur les hypothèses fondamentales de la géometric,’ Bull. Soc. Math. France 15, 203–216.Google Scholar
  23. Poincaré, H. 1952. ‘Non-Euclidean Geometries.’ In his Science and Hypothesis, pp. 35–50. New York: Dover.Google Scholar
  24. Robb, A. A. 1914. A Theory of Time and Space. Cambridge: Cambridge University Press.Google Scholar
  25. Robb, A. A. 1921. The Absolute Relations of Time and Space. Cambridge: Cambridge University Press.Google Scholar
  26. Robb, A. A. 1936. The Geometry of Space and Time. Cambridge: Cambridge University Press.Google Scholar
  27. Schwartz, H. M. 1962. ‘Axiomatic Deduction of the General Lorentz Transformations,’ Am. J. Phys. 30, 697–707.CrossRefGoogle Scholar
  28. Soper, D. E. 1976. Classical Field Theory. New York: Wiley-Interscience.Google Scholar
  29. Süssmann, G. 1969. ‘Begründung der Lorentz-Gruppe mit Symmetrie-und Relativitats-Annahmen,’ Zeitschr. Naturf. 24a, 495–498.Google Scholar
  30. Synge, J. L. 1960. Relativity: The General Theory. Amsterdam: North-Holland.Google Scholar
  31. Synge, J. L. 1964. Relativity: The Special Theory. 2nd ed. Amsterdam: North-Holland.Google Scholar
  32. Weyl, H. 1923. Mathematische Analyse des Raumproblems. Berlin: Springer.Google Scholar

Copyright information

© D. Reidel Publishing Company 1983

Authors and Affiliations

  • John Stachel
    • 1
  1. 1.Boston UniversityUSA

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