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Cartan rediscovered in general relativity

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Abstract

Élie Cartan’s invariant integral formalism is extended to gauge field theory, including general relativity. This constitutes an alternative procedure, as shown in several examples, that is equivalent when no second class constraints are present to the Rosenfeld, Bergmann, Dirac algorithm. In addition, a Hamilton–Jacobi formalism is developed for constructing explicit phase space functions in general relativity that are invariant under the full four-dimensional diffeomorphism group. These identify equivalence classes of classical solutions of Einstein’s equations. Each member is dependent on intrinsic spatial coordinates and also undergoes non-trivial evolution in intrinsic time. Furthermore, the construction yields series expansion solutions of the field equations for all of the components of the metric tensor, including lapse and shift, in the intrinsic temporal and spatial coordinates. The intrinsic coordinates are determined by the spacetime geometry in terms of Weyl scalars. The implications of this analysis for an eventual quantum theory of gravity are profound.

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Notes

  1. This will in the following be referred to as S22.

  2. A translation into English by Delphenich [10] is available.

  3. See S22 for more historical details.

  4. [9], p. 15.

  5. [9], p. 6–7.

  6. We have interchanged the meanings of Cartan’s original d and \(\delta \) notation.

  7. See S22.

  8. We employ the conventions of [21].

  9. These calculations are carried out explicitly in [37], and also in [40] where the projectability requirement is extended to Einstein–Yang–Mills theory. In the latter case, and indeed whenever additional gauge symmetries are present, it turns out that the full diffeomorphism-related symmetry is not present. Rather, it must be combined with additional gauge transformations [35, 38, 39]. The Cartan invariant integral and the derivation of the the gauge generator from the vanishing Noether charge is now also extended to Einstein–Yang–Mills theory [49] where a Yang–Mills gauge transformation must accompany the diffeomorphism-induced transformations in order to attain Legendre projectability.

  10. See [24] p. 119 and equation (1.1).

  11. See [15] for details.

  12. See [1], p. 192.

  13. See S22 for a detailed historical account of Cartan’s work, and its extension by Paul Weiss to field theory.

  14. See [47] for Bergmann’s use of Noether’s notation. Note also that this is a special case of the variations that were represented by \(\delta _0\) in S22.

  15. See for a translation into English of this article [44] and [52] for a careful analysis of the article and its relation to later work.

  16. See S22.

  17. See [47] for an account of this history.

  18. See [47], pp. 246–247 for further discussion.

  19. See S22.

  20. As noted in [37], no local Gribov gauge fixing ambiguities arise with the use of scalar phase space functions, and an example is given of their appearance for a non-scalar geometric object.

  21. See [41], p. 084015–7.

  22. On the other hand this distinction, between active and passive is fictional—as has been noted by [55], p. 439.

  23. ([3], p. 279.

  24. [6], p. 243

  25. [5], p. 17.

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  1. Austin College, 900 North Grand Ave, Sherman, TX, 75090, USA

    Donald Salisbury

  2. Max Planck Institute for the History of Science, Boltzmannstrasse 22, 14195, Berlin, Germany

    Donald Salisbury, Jürgen Renn & Kurt Sundermeyer

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  1. Donald Salisbury
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Salisbury, D., Renn, J. & Sundermeyer, K. Cartan rediscovered in general relativity. Gen Relativ Gravit 54, 116 (2022). https://doi.org/10.1007/s10714-022-03003-5

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