Skip to main content
Log in

Einstein’s Principle of Equivalence and the Heuristic Significance of General Covariance

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

The philosophy of physics literature contains conflicting claims on the heuristic significance of general covariance. Some authors maintain that Einstein's general relativity distinguishes itself from other theories in that it must be generally covariant, for example, while others argue that general covariance is a physically vacuous and trivial requirement applicable to virtually any theory. Moreover, when general covariance is invested with heuristic significance, that significance as a rule is assigned to so-called “active” general covariance (or diffeomorphism equivalence), underwritten by the principle of background independence, rather than to general covriance as Einstein understood it (in the sense of coordinate transformations). While agreeing with the latter group of commentators that general covariance indeed carries heuristic significance, I argue that a background independent theory need not be generally covariant and that instead the Principle of Equivalence as Einstein understood it (i.e., not the infinitesimal version of the principle) provides the key to understanding the heuristic power of general covariance as a “mathematical sieve” for determining the gravitational field law. However, heuristic significance accrues to general covariance only indirectly, by its encompassing the relativity of frames underwritten by Einstein's principle of equivalence.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Einstein’s \(G_{\mu \nu }\) is our \(R_{\mu \nu }\) or the Ricci tensor.

  2. Norton [22]. Compare, for instance, Friedman [15, p. 212; Barbour [2, pp. 203, 206; Earman [4, p. 445].

  3. See, for instance, [34].

  4. Rynasiewicz [32, pp. 444–445] points out that the point-coincidence argument in Einstein’s private correspondence of around the same time period (1915–1916) differs from the published version in that the former argues not for a requirement of general covariance but rather for the physical equivalence of (in modern terminology) diffeomorphic models.

  5. See, for example, Einstein’s Princeton lectures: “… absolutum means not only ‘physically real’, but also ‘independent in its physical properties, having a physical effect, but not itself influenced by physical conditions’” [7, p. 55].

  6. A “finite Galilean region” is, for Einstein, a region of space and time, neither infinitely small nor infinitely large, within which the special theory of relativity holds. The Principle of Equivalence as Einstein understands it applies strictly to such finite Galilean regions. See Sect. 6 below.

  7. For example, Einstein to Besso, January 3, 1916: “… [coordinate] system \(K\) obviously is not a physical reality” [11, p. 171].

  8. Principle of Relativity. Nature’s laws are merely statements about temporal-spatial coincidences; therefore, they find their only natural expression in generally covariant equations” [12, p. 33]

  9. The same could be said of a unimodular theory that imposes a coordinate restriction. For in such a case we have a non-generally covariant field law that still proscribes absolute background spacetime structure.

  10. Similarly Friedman [15, p. 212]: “What is new about general relativity is only the necessity for a generally covariant formulation, a necessity due to the use of nonflat space-times in which no inertial coordinate systems exist.”

  11. In a 1916 letter to Lorentz [11, p. 170], Einstein mentions three considerations influencing his eventual rejection of the “Entwurf” theory. First, the field equations failed to yield the correct perihelion motion of Mercury; second, they failed for rotating coordinates; and third, the maximum degree of covariance he was able to achieve did not uniquely determine the field equations.

  12. After his 1918 response to Kretschmann Einstein seems to have dropped the point-coincidence argument from his published writings. The 1922 Princeton Lectures, for instance, mention only the rotating disk argument for general covariance.

  13. Which is not to say that all else is in fact equal. One reason for restricting the covariance of special relativity to Lorentz transformations, for instance, is that those transformations correspond to the theory’s relativity principle.

  14. On Kretschmann’s paper, see, for instance, Norton [25, Sect. 5, pp. 817–819] and Rynasiewicz [32].

  15. On the difference between Einstein’s employment and the modern employment of the term “coordinate system,” see Norton [23].

  16. It is useless, that is, not for the reason Einstein gives—that it retains a “three-dimensional” way of thinking, but rather because the motion of other bodies cannot be uniquely determined relative to such a mollusk, thus defeating the purpose of a reference body.

  17. Most influential on this point has been Anderson [1].

  18. See, for instance, Stachel [34, p. 66]. While Stachel suggests that the requirement for a second-rank tensor derives from the fact that the stress-energy tensor is a second-rank tensor, the vacuum field equation (or the equation of the “pure gravitational field,” as Einstein called it) itself requires a second-rank tensor.

  19. For a more in-depth account of active transformations and active general covariance see, for example, Friedman [35, pp. 46–61] or Norton [23, pp. 1225–1230].

  20. Strictly speaking, a lesser degree of covariance than general covariance can still give rise to hole indeterminism, such as unimodular formulations of general relativity.

  21. See also Norton [25, pp. 826–828].

  22. Dieks suggests that general relativity achieves a “local” equivalence of frames [3, p. 185], but such a proposal proves untenable for reasons given below in Sect. 5. In short, the proposal amounts to reverting to an infinitesimal principle of equivalence which cannot discriminate between geodesic and non-geodesic motion.

  23. The letter is item 16-148 of the Einstein Archive. I have quoted from Norton [22, p. 39].

  24. See John Norton’s definitive treatment of the Einstein’s own version of the Principle of Equivalence [22]. Most of my analysis here is based on Norton’s paper.

  25. See Einstein to Moritz Schlick, March 21, 1917 [11, pp. 305–306]: “[Your] derivation of the motion of a point… assumes that, seen from the local coordinate system, the point moves in a straight line. Nothing can be derived from this, however. The local coordinate system is generally of importance only at the infinitesimal level, and at the infinitesimal level every uninterrupted line is straight.” Einstein proceeds to explain the true principle of equivalence for “finite (matter-free) portions of the world” in which the special theory of relativity holds. For a detailed analysis of Einstein’s response to Schlick, see Norton [22, pp. 34–39].

  26. On Einstein’s rotating disk thought experiment see the definitive paper by John Stachel [33].

  27. See, for instance, Einstein and Infeld’s account of the famous elevator experiment: “Our new c.s. [coordinate system] rigidly connected with the freely falling lift differs from the inertial c.s. in only one respect. In an inertial c.s., a moving body on which no forces are acting will move uniformly forever. The inertial c.s. as represented in classical physics is neither limited in space nor time. The case of the observer in our lift is, however, different. The inertial character of his c.s. is limited in space and time. Sooner or later the uniformly moving body will collide with the wall of the lift, destroying the uniform motion. Sooner or later the whole lift will collide with the earth, destroying the observers and their experiments. The c.s. is only a “pocket edition” of a real inertial c.s. This local character of the c.s. is quite essential. If our imaginary lift were to reach from the North Pole to the Equator, with the handkerchief placed over the North Pole and the watch over the Equator, then, for the outside observer, the two bodies would not have the same acceleration; they would not be at rest relative to each other. Our whole argument would fail! The dimensions of the lift must be limited so that the equality of acceleration of all bodies relative to the outside observer may be assumed” [14, p. 228].

  28. By “at rest” Einstein evidently means in contact with the disk. Renn and Sauer note that key to Einstein’s understanding of the Principle of Equivalence is the interpretation of inertial forces as gravitational forces, so that Einstein interprets such inertial forces (e.g., centrifugal force on the rotating disk) by analogy to inertial forces in play with Newton’s rotating bucket. [30, pp. 144–145].

  29. I am perplexed by Janssen’s assertion, in an otherwise insightful paper, that a body off the disk (“person next to the disk”) experiences a centripetal Coriolis field that compensates for the centrifugal force of its orbital motion around the disk [19, p. 65]. Such an “off-disk” body would not be affected by the gravitational field of the disk at all, being outside the finite Galilean region within which the principle of equivalence holds. Furthermore, the Coriolis component of the disk’s gravitational field is not centripetal.

  30. Strictly speaking, Mach’s Principle, as defined by Einstein in his 1918 response to Kretschmann, is not a principle of relative motion per se but rather the somewhat narrower principle that “[t]he G-field is completely determined by the masses of the bodies” [12, p. 33]. On Einstein’s grapplings with Mach’s ideas see Norton [25, Sects. 3.7 and 3.8, pp. 806–809].

  31. In terms of Anderson’s [1] approach to symmetry, we proceed the opposite way as Kretschmann, seeking not the smallest sub-group of transformations consistent with the theory’s relativity principle, but rather the largest group leaving the “absolute objects” of the theory invariant. In the case of the special theory of relativity, this procedure once again singles out the Lorentz group (the absolute object of interest being the Minkowski metric) and, since general relativity countenances no absolute objects at all (that is, it allows for no absolute background spacetime structure), Anderson concludes that its symmetry group coincides with the general group—the “manifold mapping group” or group of all automorphisms (i.e., active general covariance).

    Of the two accounts of symmetry and its relation to relativity principles, Kretschmann’s seems better conceived than Anderson’s. For if, in the case of the special theory of relativity, Anderson’s symmetry group (the Lorentz group) corresponds just to the relativity principle of interest, why should not the symmetry group (general group) of the general theory of relativity likewise single out its relativity principle, which it does not? Indeed, it is somewhat misleading for Anderson to even call the (active) general covariance group a “symmetry group” of the general theory of relativity, since a symmetry group is supposed to pick out a theory’s absolute objects, which remain invariant under that group, whereas in general relativity the so-called symmetry group (general group) picks out instead the dynamical objects which remain invariant under the general group of transformations (on this point see Pooley [26, p. 199].

    Renn and Sauer [30, pp. 145 and 302] suggest that Einstein erroneously focused on his covariance requirement because he lacked the modern understanding of relativity principles in terms of spacetime symmetry. But this is to eviscerate the Principle of Equivalence, since symmetry does not distinguish PE fields from inertial frames in which the Riemann tensor also vanishes. That is the point of Einstein’s observation to Laue, in the 1950 letter quoted above, that it is the non-vanishing of the Christoffel symbols that registers the presence of a real gravitational field, not the non-vanishing of the Riemann tensor.

  32. Renn and Sauer [30] add a fifth heuristic criterion: the main differential operator for the metric tensor must be of second order, by analogy to the Poisson equation for the gravitational potential in classical gravitational field theory. However, this criterion can be folded into (3) (recovery of the Newtonian limit for weak fields).

  33. Renn subsequently combines criteria (1) and (2) into the single criterion of a generalized relativity principle [28, p. 46].

  34. For a detailed account of the Entwurf theory’s failure on rotating frames, including Einstein’s eventual recognition thereof, see Janssen [18]. The Entwurf field equation yields for rotating coordinates \(g_{44} = 1 - (3/4)\omega^{2} \left( {x^{2} + y^{2} } \right)\) instead of the correct \(g_{44} = 1 - \omega^{2} \left( {x^{2} + y^{2} } \right).\)

  35. See Norton [24, Sect. 6, pp. 132–141] on Einstein’s use of variational techniques to determine that the Entwurf field equations had the maximum degree of covariance permitted by the hole argument.

References

  1. Anderson, J.: Principles of Relativity Physics. Academic Press, New York (1967)

    Book  Google Scholar 

  2. Barbour, J.: General covariance and best matching. In: Callender, C., Huggett, N. (eds.) Physics Meets Philosophy at the Planck Scale, pp. 199–212. Cambridge University, Cambridge (2001)

    Chapter  Google Scholar 

  3. Dieks, D.: Another look at general covariance and the equivalence of reference frames. Stud. Hist. Philos. Sci. Part B 37(1), 174–191 (2006)

    Article  MathSciNet  Google Scholar 

  4. Earman, J.: Two challenges to the requirement of substantive general covariance. Synthese 148(2), 443–468 (2006)

    Article  MathSciNet  Google Scholar 

  5. Earman, J., Norton, J.: What price spacetime substantivalism: the hole story. Br. J. Philos. Sci. 38, 515–525 (1987)

    Article  MathSciNet  Google Scholar 

  6. Einstein, A.: The foundation of the general theory of relativity. In: Lorentz et al. (ed.). The Principle of Relativity, translated by W. Perrett and G.B. Jeffery, 109–164. Dover, New York. First published in 1916 (1952)

  7. Einstein, A.: The Meaning of Relativity, fifth edition. Princeton University, Princeton. First published in 1922 (1956)  

  8. Einstein, A.: Relativity: The Special and General Theory. Fifteenth edition. Trans. R.W. Lawson. Random House, New York. First published in 1916 (1961)

  9. Einstein, A.: Autobiographical Notes, translated by P.A. Schilpp. Open Court, LaSalle, IL. First published in 1949 (1979)

  10. Einstein, A.: On Friedrich Kottler’s Paper: ‘On Einstein’s Equivalence Hypothesis and Gravitation’. In The Collected Papers of Albert Einstein, vol. 6 (English Translation Supplement), translated by A. Engel. Princeton University Press, Princeton, pp. 237–239. First published in 1916 (1997)

  11. Einstein, A.: The Collected Papers of Albert Einstein, vol. 8: The Berlin Years: Correspondence, 1914–1918 (English Translation Supplement), translated by A.M. Hentschel. Princeton University, Princeton (1998)

  12. Einstein, A.: On the foundations of the general theory of relativity. In The Collected Papers of Albert Einstein, vol. 7 (English Translation Supplement), translated by A. Engel. Princeton University Press, Princeton, pp. 33–35. First published in 1918 (2002)

  13. Einstein, A., Grossmann, M.: Entwurf einer verallgemeinerten Relativitätstherie und einer Theorie der Gravitation. Teubner, Leipzig (1913)

    MATH  Google Scholar 

  14. Einstein, A., Infeld, L.: The Evolution of Physics: From Early Concepts to Relativity and Quanta. Simon & Schuster, New York. First published in 1938 (2008)

  15. Friedman, M.: Foundations of Spacetime Theories. Princeton University, Princeton (1983)

    Book  Google Scholar 

  16. Goenner, H., Renn, J., Ritter, J., Sauer, T. (eds.): The Expanding Worlds of General Relativity. Einstein Studies, vol. 7. Birkhäuser, Boston (1999)

    Book  Google Scholar 

  17. Howard, D., Stachel, J. (eds.): Einstein and the History of General Relativity. Birkhäuser, Boston (1989)

    Google Scholar 

  18. Janssen, M.: Rotation as the Nemesis of Einstein’s Entwurf Theory. In [16], pp. 127–157

  19. Janssen, M.: Of pots and holes: Einstein’s bumpy road to general relativity. Ann. Phys. 14(Supplement), 58–85 (2005)

    Article  MathSciNet  Google Scholar 

  20. Kretschmann, E.: “Über den physicalischen Sinn der Relativitätspostulate, A. Einsteins neue und seine ursprügliche Relativitätstheorie. Ann. Phys. 53, 575–614 (1917)

    MATH  Google Scholar 

  21. Norton, J.: Einstein, the hole argument and the reality of space. In: Forge, J. (ed.) Measurement, Realism and Objectivity, pp. 153–188. D. Reidel, Dordrecht (1987)

    Chapter  Google Scholar 

  22. Norton, J.: What was Einstein’s principle of equivalence? In [17], pp. 5–47

  23. Norton, J.: Coordinates and covariance: Einstein’s view and the modern view. Found. Phys. 19(10), 1215–1263 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  24. Norton, J.: How Einstein found his field equations, 1912–1915. In [17], pp. 101–159

  25. Norton, J.: General covariance and the foundations of general relativity: eight decades of dispute. Rep. Progress Phys. 56, 791–858 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  26. Pooley, O.: Substantive general covariance: another decade of dispute. In: Suárez, M., Dorato, M., Rédei, M. (eds.) EPSA Philosophical Issues in the Sciences, pp. 197–209. Springer, New York (2010)

    Chapter  Google Scholar 

  27. Pooley, O.: Background independence, diffeomorphism invariance and the meaning of coordinates. In: Lehmkuhl, D., Schiemann, G., Scholz, E. (eds.) Towards a Theory of Spacetime Theories, pp. 105–143. Birkhäuser, Boston (2017)

    Chapter  Google Scholar 

  28. Renn, J.: Standing on the shoulders of a dwarf: general relativity—a triumph of Einstein and Grossmann’s erroneous Entwurf theory. In: Cox, A.J., Eisenstaedt, J. (eds.) The Universe of General Relativity, pp. 39–51. Birkhäuser, Boston (2005)

    Chapter  Google Scholar 

  29. Renn, J., Sauer, T.: Heuristics and mathematical representation in Einstein’s search for a gravitational field equation. In [16], pp. 87–125

  30. Renn, J., Sauer, T.: Pathways out of classical physics. In: Janssen, M., Norton, J.D., Renn, J., Sauer, T., Stachel, J. (eds.) The Genesis of General Relativity, Boston Studies in the Philosophy of Science, vol. 250, pp. 113–312. Springer, Dordrecht (2007)

    Google Scholar 

  31. Ryckman, T.: The Reign of Relativity: Philosophy and Physics 1915–1925. Oxford University Press, Oxford (2005)

    Book  Google Scholar 

  32. Rynasiewicz, R.: Kretschmann’s analysis of covariance and relativity principles. In [16], pp. 431–462

  33. Stachel, J.: The rigidly rotating disk as the ‘missing link’ in the history of general relativity. In [17], 48–62

    ADS  MathSciNet  Google Scholar 

  34. Stachel, J.: “Einstein’s search for general covariance, 1912–1915. In [17], pp. 63–100

    ADS  MathSciNet  Google Scholar 

  35. Zahar, E.: Einstein’s Revolution: A Study in Heuristic. Open Court, LaSalle (1989)

    Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Joseph K. Cosgrove.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cosgrove, J.K. Einstein’s Principle of Equivalence and the Heuristic Significance of General Covariance. Found Phys 51, 27 (2021). https://doi.org/10.1007/s10701-021-00434-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-021-00434-z

Keywords

Navigation