Abstract
Einstein and Hilbert both struggled to reconcile general covariance and causality in their early work on general relativity. In Einstein’s case, this first led to his infamous “hole argument”, a stumbling block that persuaded him early on that generally covariant field equations for gravitation could never be found. After his breakthrough to general covariance in the fall of 1915, the resolution came in form of the “point-coincidence argument.” Hilbert from the beginning took a different view of the “causality problem,” though he shifted his position somewhat in the light of Einstein’s breakthrough in November 1915. Nevertheless, his aim was to establish initial conditions that would lead to a well-defined Cauchy problem in general relativity. Hilbert consistently advocated the use of coordinate conditions in order to obtain solutions of the field equations that would maintain the causal ordering of events. Einstein’s “causality problem” thus differs from that of Hilbert, and the latter was never a victim of Einstein’s “hole argument.”
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Notes
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For more on Einstein’s (mis)appropriation of Mach’s principle, see Barbour (2005).
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The “December Proofs” (Hilbert 1915a) were recently discovered (see Corry, L., J. Renn, and J. Stachel (1997) and contain significant differences from the published version (Hilbert 1915b). For discussion of these differences, and differing opinions on their significance, see Renn and Stachel (1999), and Sauer (1999, 2005).
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Some commentators have mistakenly asserted that Hilbert’s equations are not generally covariant. On this issue we wholeheartedly support Ohanian’s recent statement when he writes: “The fact is that Hilbert’s variational equations are covariant, but he supplements them, correctly, by extra, noncovariant, coordinate conditions that are needed to make the solution unique, as is well known to anybody who has ever tried to construct a solution of the Einstein equations.” (Ohanian 2008, p. 355 (n. 56 to p. 221)).
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See, once again, Ohanian (2008, n. 56), cited above.
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Indeed, to go further, Hilbert’s epistemological analysis of the differing status that should be accorded to general covariance versus causality might perhaps be suggestive to those working on the interpretation of General Relativity as a gauge theory, and the associated “problem of time” in quantum gravity.
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Stachel (1992) writes that while Einstein was “always a bit vague about just what he meant by causality” in his hole argument, Hilbert on the other hand “gave a quite precise meaning to the concept”, formulating it in the context of whether the field equations can be expressed in Cauchy normal form. Surely it is right that Hilbert was led to think about causality in the context of general covariance by Einstein’s concerns in the summer of 1915, and Stachel is of course exactly right that Hilbert’s version of the problem is stated in the precise mathematical language of the Cauchy problem. What we wish to emphasize is that the problem Hilbert thus arrives at is importantly different from that with which Einstein wrestled in his hole argument.
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See section on Hilbert’s causality problem, above.
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References
Barbour, J. B. (2005). Absolute or relative motion? Vol. 2: The deep structure of general relativity. Oxford: Oxford University Press.
Brading, K. A., & Ryckman, T. A. (2008). Hilbert’s ‘Foundations of Physics’: Gravitation and electromagnetism within the axiomatic method. Studies in History and Philosophy of Modern Physics, 39, 102–153.
Brading, K. A., & Ryckman, T. A. (2009). Hilbert’s Axiomatic Method and his “Foundations of Physics”: Reconciling causality with the axiom of general invariance. In (Lehner et al. 2012, 175–199).
Corry, L., Renn, J., & Stachel, J. (1997). Belated decision in the Hilbert-Einstein priority dispute. Science, 278, 1270–1273.
Earman, J., Janssen, M., & Norton, J. (Eds.). (1993). The attraction of gravitation: New studies in the history of general relativity (Einstein studies vol. 5, pp. 30–62). Boston: Birkhäuser.
Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 49, 769–822. Reprinted in (Einstein 1996, 284–337). Translated by W. Parrett and G. Jeffrey as “The Foundation of the General Theory of Relativity”, in H. Lorentz et al. (1923), 109–65.
Einstein, A. (1995). The collected papers of Albert Einstein. Vol. 4: The swiss years, writings, 1912–1914, M. J. Klein, A. J. Kox, & R. Schulmann (Eds.). Princeton: Princeton University Press.
Einstein, A. (1996). The collected papers of Albert Einstein. Vol. 6: The Berlin years, writings, 1914–1917, A. J. Kox, M. J. Klein, & R. Schulmann (Eds.). Princeton: Princeton University Press.
Einstein, A., & Grossmann, M. (1914). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation. Zeitschrift für Mathematik und Physik, 62, 225–259. Reprinted in (Einstein 1995, 302–344).
Eisenstaedt, J., & Kox, A. J. (Eds.). (1992). Studies in the history of general relativity (Einstein studies, vol. 3). Boston: Birkhäuser.
Hilbert, D. (1915a). Die Grundlagen der Physik (Erste Mitteilung). Annotated “Erste Korrektur meiner erste Note”, printer’s stamp date “6 Dez. 1915”. 13 pages with omissions. Published in (Sauer and Majer 2009, 317–330). Translation as “The foundations of physics (Proofs of first communication)”, in (Renn and Schemmel 2007, 989–1001).
Hilbert, D. (1915b). Die Grundlagen der Physik (Erste Mitteilung). In Nachrichten. Königliche Gesellschaft der Wissenschaften zu Göttingen. Mathematische-Physikalische Klasse (pp. 395–407). Reprinted in (Sauer and Majer 2009, 28–46). Translation as “The foundations of physics (first communication)”, in (Renn and Schemmel 2007, 1003–1015).
Hilbert, D. (1916). Das Kausalitätsprinzip in der Physik. Typescript of lectures, dated 21 and 28 November 1916. Bibliothek des Mathematisches Institut, Universität Göttingen. 17 pages. Published in (Sauer and Majer 2009, 335–346).
Hilbert, D. (1917). Die Grundlagen der Physik (Zweite Mitteilung). Nachrichten. Königliche Gesellschaft der Wissenschaften zu Göttingen. Mathematische-Phyikalische Klasse (pp. 53–76). Reprinted in (Sauer and Majer 2009, 47–72). Translation as “The foundations of physics (second communication)”, in (Renn and Schemmel 2007, 1017–1038).
Hilbert, D. (1919/1992). Natur und mathematisches Erkennen. Vorlesung gehalten 1919-1920 in Göttingen. D. E. Rowe, Hrsg. Basel: Birkhäuser.
Howard, D., & Norton, J. (1993). Out of the labyrinth? Einstein, Hertz, and the Göttingen answer to the hole argument. In (Earman, Janssen and Norton 1993, 30–62).
Howard, D., & Stachel, J. (Eds.). (1989). Einstein and the history of general relativity (Einstein studies, vol. 1). Basel: Birkhäuser.
Janssen, M., & Renn, J. (2007). Untying the knot: How Einstein found his way back to field equations discarded in the Zurich notebook. In M. Janssen, J. D. Norton, J. Renn, T. Sauer, & J. Stachel (Eds.), The genesis of general relativity (vol. 2, pp. 839–925). Einstein’s Zurich notebook: Commentary and essays. Dordrecht: Springer.
Lehner, C., Renn, J., & Schemmel, M. (Eds.). (2012). Einstein and the changing worldviews of physics (Einstein studies, vol. 12). New York: Springer.
Majer, U. (1995). Geometry, intuition and experience: From Kant to Husserl. Erkenntnis, 42, 261–285.
Norton, J. D. (1984). How Einstein found his field equations: 1912-1915. Historical Studies in the Physical Sciences, 14, 253–316. Reprinted in (Howard and Stachel 1989, 101–159)
Norton, J. D. (1993). General covariance and the foundations of general relativity: Eight decades of dispute. Reports on Progress in Physics, 56, 791–858.
Ohanian, H. C. (2008). Einstein’s mistakes. New York: Norton.
Renn, J., & Schemmel, M. (2007). The genesis of general relativity. Gravitation in the twilight of classical physics: The promise of mathematics. Dordrecht: Springer.
Renn, J., & Stachel, J. (1999). Hilbert’s foundation of physics: From a theory of everything to a constituent of general relativity. Berlin: Max-Planck-Institut für Wissenschaftsgeschichte, Preprint 118. Reprinted in (Renn and Schemmel 2007, 858–973).
Ryckman, T. A. (2005). The reign of relativity. Oxford: Oxford University Press.
Sauer, T. (1999). The relativity of discovery: Hilbert’s first note on the foundations of physics. Archive for History of Exact Sciences, 53, 529–575.
Sauer, T. (2005). Einstein equations and Hilbert action: What is missing on page 8 of the proofs for Hilbert’s first communication on the foundations of physics? Archive for History of Exact Sciences, 59, 577–590.
Sauer, T., & U. Majer (Eds.). (2009). David Hilbert’s Lectures on the Foundations of Physics, 1915–1927. Berlin: Springer.
Stachel, J. (1989). Einstein’s search for general covariance, 1912–1915. In (Howard and Stachel 1989, 63–100). Based on a paper circulating privately since 1980.
Stachel, J. (1992). The Cauchy problem in general relativity - The early years. In (Eisenstaedt and Kox 1992, 407–418).
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Brading, K., Ryckman, T. (2018). Hilbert on General Covariance and Causality. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_3
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