Inertial Motion, Explanation, and the Foundations of Classical Spacetime Theories

Part of the Einstein Studies book series (EINSTEIN, volume 13)


I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to “explain” inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of “explain” is rather different from what one might have expected. This sense of explanation is connected with a view of theories—I call it the “puzzleball view”—on which the foundations of a physical theory are best understood as a network of mutually interdependent principles and assumptions.


General relativity Newtonian gravitation Explanation Inertial motion 



Versions of this work have been presented at CEA-Saclay (Paris), University of Wuppertal, University of Pittsburgh, University of Chicago, University of Texas at Austin, Columbia University, Brown University, New York University, University of Western Ontario (twice!), University of California–Berkeley, University of California–Irvine, Yale University, University of Southern California, Notre Dame University, Ludwig-Maximilians University, and at the ‘New Directions’ conference in Washington, DC. I am grateful to very helpful comments and discussion from all of these audiences, and particularly to (in no special order) John Manchak, Giovanni Valente, Craig Callender, Alexei Grinbaum, Harvey Brown, David Wallace, Chris Smeenk, Wayne Myrvold, Erik Curiel, Ryan Samaroo, John Norton, John Earman, Howard Stein, Mike Tamir, Bryan Roberts, Shelly Kagan, Sahotra Sarkar, Josh Dever, Bob Geroch, Bill Wimsatt, David Albert, Tim Maudlin, Sherri Roush, Josh Schechter, Chris Hill, Don Howard, Katherine Brading, Eleanor Knox, and Radin Dardashti. Special thanks are due to David Malament, Jeff Barrett, Kyle Stanford, and Pen Maddy for many helpful discussions and comments on previous versions of this work. Thank you, too, to Dennis Lehmkuhl for organizing the 2010 workshop on which this volume is based, and for editing the volume.


  1. 1.
    Batterman, R., 2002. The Devil in the Details. Oxford University Press, New York.zbMATHGoogle Scholar
  2. 2.
    Blanchet, L., 2000. Post-Newtonian gravitational radiation. In: Schmidt, B. (Ed.), Einstein’s Field Equations and Their Physical Implications. Springer, Berlin, pp. 225–271.Google Scholar
  3. 3.
    Bromberger, S., 1966. Why-questions. In: Brody, B. A. (Ed.), Readings in the Philosophy of Science. Prentice Hall, Inc., Englewood Cliffs, pp. 66–84.Google Scholar
  4. 4.
    Brown, H., 2005. Physical Relativity. Oxford University Press, New York.CrossRefzbMATHGoogle Scholar
  5. 5.
    Cartan, E., 1923. Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie). Annales scientifiques de l’École Normale Supérieure 40, 325–412.zbMATHGoogle Scholar
  6. 6.
    Cartan, E., 1924. Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (suite). Annales scientifiques de l’École Normale Supérieure 41, 1–25.zbMATHGoogle Scholar
  7. 7.
    Christian, J., 1997. Exactly soluble sector of quantum gravity. Physical Review D 56 (8), 4844 –4877.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Curiel, E., 2012. On tensorial concomitants and the non-existence of a gravitational stress-energy tensor, available at:
  9. 9.
    Curiel, E., 2017. Towards a theory of spacetime theories. Birkauser, Boston, Ch. A Primer on Energy Conditions.Google Scholar
  10. 10.
    Damour, T., 1989. The problem of motion in Newtonian and Einsteinian gravity. In: Hawking, S. W., Israel, W. (Eds.), Three Hundred Years of Gravitation. Cambridge University Press, New York, pp. 128–198.Google Scholar
  11. 11.
    Dixon, W. G., 1964. A covariant multipole formalism for extended test bodies in general relativity. Il Nuovo Cimento 34 (2), 317–339.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dixon, W. G., 1975. On the uniqueness of the Newtonian theory as a geometric theory of gravitation. Communications in Mathematical Physics 45, 167–182.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Duval, C., Künzle, H. P., 1978. Dynamics of continua and particles from general covariance of Newtonian gravitation theory. Reports on Mathematical Physics 13 (3).Google Scholar
  14. 14.
    Earman, J., Friedman, M., 1973. The meaning and status of Newton’s law of inertia and the nature of gravitational forces. Philosophy of Science 40, 329.CrossRefGoogle Scholar
  15. 15.
    Earman, J., Glymour, C., 1978. Einstein and Hilbert: Two months in the history of general relativity. Archive for History of Exact Sciences 19 (3), 291–308.Google Scholar
  16. 16.
    Earman, J., Glymour, C., 1978. Lost in the tensors: Einstein’s struggles with covariance principles 1912–1916. Studies in the History and Philosophy of Science 9 (4), 251–278.Google Scholar
  17. 17.
    Eddington, A. S., 1924. The Mathematical Theory of Relativity. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  18. 18.
    Ehlers, J., Geroch, R., 2004. Equation of motion of small bodies in relativity. Annals of Physics 309, 232–236.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Einstein, A., Grommer, J., 1927. Allgemeine Relativitätstheorie und Bewegungsgesetz. Verlag der Akademie der Wissenschaften, Berlin.zbMATHGoogle Scholar
  20. 20.
    Einstein, A., Infeld, L., Hoffman, B., 1938. The gravitational equations and the problem of motion. Annals of Mathematics 39 (1), 65–100.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Feynman, R. C., 1967. The Character of Physical Law. Cornell University Press, Ithaca, NY.Google Scholar
  22. 22.
    Friedrichs, K. O., 1927. Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und der Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz. Mathematische Annalen 98, 566–575.CrossRefzbMATHGoogle Scholar
  23. 23.
    Geroch, R., Jang, P. S., 1975. Motion of a body in general relativity. Journal of Mathematical Physics 16 (1), 65.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Geroch, R., Weatherall, J. O., 2017. Equations of motion. Unpublished manuscript.Google Scholar
  25. 25.
    Harper, W. L., 2012. Isaac Newton’s Scientific Method: Turning Data into Evidence about Gravity and Cosmology. Oxford University Press, New York.zbMATHGoogle Scholar
  26. 26.
    Havas, P., 1989. The early history of the ‘problem of motion’ in general relativity. In: Howard, D., Stachel, J. (Eds.), Einstein and the History of General Relativity. Vol. 11 of Einstein Studies. Birkhäuser, Boston, pp. 234–276.Google Scholar
  27. 27.
    Hawking, S. W., Ellis, G. F. R., 1973. The Large Scale Structure of Space-time. Cambridge University Press, New York.CrossRefzbMATHGoogle Scholar
  28. 28.
    Kennefick, D., 2005. Einstein and the problem of motion: A small clue. In: Kox, A. J., Eisenstaedt, J. (Eds.), The Universe of General Relativity. Vol. 11 of Einstein Studies. Birkhäuser, Boson, pp. 109–124.CrossRefGoogle Scholar
  29. 29.
    Kvanvig, J., 2007. Coherentist theories of epistemic justification. In: Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy. Stanford University, Stanford, CA, available at:
  30. 30.
    Lovelock, D., 1971. The Einstein tensor and its generalizations. Journal of Mathematical Physics 12 (3), 498–501.CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Lovelock, D., 1972. The four-dimensionality of space and the Einstein tensor. Journal of Mathematical Physics 13 (6), 874–876.CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Malament, D., 2012. A remark about the “geodesic principle” in general relativity. In: Frappier, M., Brown, D. H., DiSalle, R. (Eds.), Analysis and Interpretation in the Exact Sciences: Essays in Honour of William Demopoulos. Springer, New York, pp. 245–252.Google Scholar
  33. 33.
    Malament, D. B., 2012. Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. University of Chicago Press, Chicago.Google Scholar
  34. 34.
    Mathisson, M., 1931. Bewegungsproblem der feldphysik und elektronenkonstanten. Zeitschrift für Physik 69, 389Â408.Google Scholar
  35. 35.
    Mathisson, M., 1931. Die mechanik des materieteilchens in der allgemeinen relativitätstheorie. Zeitschrift für Physik 67, 826–844.Google Scholar
  36. 36.
    Misner, C. W., Thorne, K. S., Wheeler, J. A., 1973. Gravitation. W. H. Freeman.Google Scholar
  37. 37.
    Navarro, J., Sancho, J., 2008. On the naturalness of Einstein’s equation. Journal of Geometry and Physics 58, 1007–1014.Google Scholar
  38. 38.
    Newman, E. T., Posadas, R., 1969. Motion and structure of singularities in general relativity. Physical Review 187 (5), 1784–1791.CrossRefzbMATHGoogle Scholar
  39. 39.
    Newman, E. T., Posadas, R., 1971. Motion and structure of singularities in general relativity, ii. Journal of Mathematical Physics 12 (11), 2319–2327.CrossRefGoogle Scholar
  40. 40.
    Quine, W. V. O., 1951. Two dogmas of empiricism. The Philosophical Review 60, 20–43.CrossRefzbMATHGoogle Scholar
  41. 41.
    Quine, W. V. O., 1960. Carnap and logical truth. Synthese 12 (4), 350–374.CrossRefzbMATHGoogle Scholar
  42. 42.
    Reyes, G. E., 2009. A derivation of Einstein’s vacuum field equations, available at:
  43. 43.
    Sachs, R. K., Wu, H., 1973. General Relativity for Mathematicians. Springer-Verlag, New York.zbMATHGoogle Scholar
  44. 44.
    Sauer, T., Trautman, A., 2008. Myron Matthison: What little we know of his life,
  45. 45.
    Souriau, J.-M., 1974. Modèle de particule à spin dans le champ électromagnétique et gravitationnel. Annales de l’Institut Henri Poincaré Sec. A 20, 315.Google Scholar
  46. 46.
    Stein, H., 1967. Newtonian space-time. The Texas Quarterly 10, 174–200.Google Scholar
  47. 47.
    Sternberg, S., 2003. Semi-riemannian geometry and general relativity, available at:
  48. 48.
    Sus, A., 2011. On the explanation of inertia, unpublished.Google Scholar
  49. 49.
    Tamir, M., 2011. Proving the principle: Taking geodesic dynamics too seriously in Einstein’s theory, available online at
  50. 50.
    Taub, A. H., 1962. On Thomas’ result concerning the geodesic hypothesis. Proceedings of the National Academy of the USA 48 (9), 1570–1571.CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Thomas, T. Y., 1962. On the geodesic hypothesis in the theory of gravitation. Proceedings of the National Academy of the USA 48 (9), 1567–1569.CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Trautman, A., 1965. Foundations and current problem of general relativity. In: Deser, S., Ford, K. W. (Eds.), Lectures on General Relativity. Prentice-Hall, Englewood Cliffs, NJ, pp. 1–248.Google Scholar
  53. 53.
    van Fraassen, B., 1980. The Scientific Image. Clarendon Press, Oxford.CrossRefGoogle Scholar
  54. 54.
    Wald, R. M., 1984. General Relativity. University of Chicago Press, Chicago.CrossRefzbMATHGoogle Scholar
  55. 55.
    Weatherall, J. O., 2011. The motion of a body in Newtonian theories. Journal of Mathematical Physics 52 (3), 032502.Google Scholar
  56. 56.
    Weatherall, J. O., 2011. On the status of the geodesic principle in Newtonian and relativistic physics. Studies in the History and Philosophy of Modern Physics 42 (4), 276–281.Google Scholar
  57. 57.
    Weatherall, J. O., 2012. A brief remark on energy conditions and the Geroch-Jang theorem. Foundations of Physics 42 (2), 209–214.CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Wimsatt, W. C., 1981. Robustness, reliability, and overdetermination. In: Brewer, M., Collins, B. (Eds.), Scientific Inquiry in the Social Sciences. Jossey-Brass, San Francisco, pp. 123–162.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Logic and Philosophy of ScienceUniversity of CaliforniaIrvineUSA

Personalised recommendations