Generic Theory of Geometrodynamics from Noether’s Theorem for the \(\mathrm {Diff}(M)\) Symmetry Group

  • Jürgen StruckmeierEmail author
  • David Vasak
  • Johannes Kirsch
Part of the FIAS Interdisciplinary Science Series book series (FIAS)


We work out the most general theory for the interaction of spacetime geometry and matter fields—commonly referred to as geometrodynamics—for spin-0 and spin-1 particles. Actually, we present a Hamilton–Lagrange–Noether formulation of the gauge theory of gravitation. It is based on the minimum set of postulates to be introduced, namely (i) the action principle and (ii) the form-invariance of the action under the (local) diffeomorphism group. The second postulate thus implements the Principle of General Relativity, also referred to as the Principle of General Covariance. According to Noether’s theorem, this physical symmetry gives rise to a conserved Noether current, from which the complete set of theories compatible with both postulates can be deduced. This finally results in a new generic Einstein-type equation, which can be interpreted as an energy-momentum balance equation emerging from the Lagrangian \(\mathscr {L}_{R}\) for the source-free dynamics of gravitation and the energy-momentum tensor of the source system \(\mathscr {L}_{0}\). Provided that the system has no other symmetries—such as SU(N)—the canonical energy-momentum tensor turns out to be the correct source term of gravitation. For the case of massive spin-1 particles, this entails an increased weighting of the kinetic energy over the mass as the source of gravity, compared to the metric energy momentum tensor, which constitutes the source of gravity in Einstein’s General Relativity. We furthermore confirm that a massive vector field necessarily acts as a source for torsion of spacetime. From the viewpoint of our generic Einstein-type equation, Einstein’s General Relativity constitutes the particular case for scalar and massless vector particle fields, and the Hilbert Lagrangian \(\mathscr {L}_{R,\mathrm {H}}\) as the model for the source-free dynamics of gravitation.



First of all, we want to remember our revered academic teacher Walter Greiner, whose charisma and passion for physics inspired us to stay engaged in physics for all of our lives.

The authors thank Patrick Liebrich and Julia Lienert (Goethe University Frankfurt am Main and FIAS), and Horst Stoecker (FIAS, GSI, and Goethe University Frankfurt am Main) for valuable discussions. D.V. and J.K. thank the Fueck Foundation for its support.


  1. 1.
    W. Greiner, Classical Mechanics, 2nd edn. (Springer, Berlin, 2010)CrossRefGoogle Scholar
  2. 2.
    J. Struckmeier, J. Muench, D. Vasak, J. Kirsch, M. Hanauske, H. Stoecker, Phys. Rev. D 95, 124048 (2017). Scholar
  3. 3.
    A. Einstein, Private letter to Hermann Weyl (ETH Zürich Library, Archives and Estates, 1918)Google Scholar
  4. 4.
    D. Kehm, J. Kirsch, J. Struckmeier, D. Vasak, M. Hanauske, Astron. Nachr./AN 338(9–10), 1015 (2017).
  5. 5.
    F.W. Hehl, P. von der Heyde, G.D. Kerlick, J.M. Nester, Rev. Mod. Phys. 48(3), 393 (1976). Scholar
  6. 6.
    E. Noether, Nachrichten der Königlichen Gesesellschaft der Wissenschaften Göttingen. Mathematisch-Physikalische Klasse 57, 235 (1918)Google Scholar
  7. 7.
    J. Struckmeier, A. Redelbach, Int. J. Mod. Phys. E 17, 435 (2008). Scholar
  8. 8.
    M. Gaul, C. Rovelli, Lect. Notes Phys. 541, 277 (2000)ADSCrossRefGoogle Scholar
  9. 9.
    P. Matteucci, Rep. Math. Phys. 52, 115 (2003)., arXiv:gr-qc/0201079
  10. 10.
    M. Godina, P. Matteucci, Int. J. Geom. Methods Mod. Phys. 2, 159 (2005)., arXiv:math/0504366
  11. 11.
    J. Schouten, Ricci-Calculus (Springer. Berlin (1954). Scholar
  12. 12.
    F.W. Hehl, Rep. Math. Phys. 9(3), 55 (1976)ADSCrossRefGoogle Scholar
  13. 13.
    F.W. Hehl, in Proceedings, 15th Workshop on High Energy Spin Physics (DSPIN-13), Dubna, Russia, 8–12 October 2013 (2014)Google Scholar
  14. 14.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W. H. Freeman and Company, New York, 1973)Google Scholar
  15. 15.
    P. Jordan, Ann. der Phys. 428(1), 64 (1939)ADSCrossRefGoogle Scholar
  16. 16.
    D. Sciama, Mon. Not. R. Astron. Soc. 113, 34 (1953)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Feynman, W. Morinigo, W. Wagner, Feynman Lectures On Gravitation, Frontiers in Physics (Westview Press, Boulder, 2002)Google Scholar
  18. 18.
    S. Hawking, The Theory of Everything (New Millenium Press, 2003)Google Scholar
  19. 19.
    S. Carroll, Spacetime and Geometry (Prentice Hall, Englewood Cliffs, 2013)Google Scholar
  20. 20.
    T.W.B. Kibble, J. Math. Phys. 2, 212 (1961)ADSCrossRefGoogle Scholar
  21. 21.
    D.W. Sciama, in Recent Developments in General Relativity (Pergamon Press, Oxford; PWN, Warsaw, 1962), pp. 415–439. Festschrift for InfeldGoogle Scholar
  22. 22.
    J. Plebanski, A. Krasinski, An Introduction to General Relativity and Cosmology (Cambridge University Press, Cambridge, 2006)CrossRefGoogle Scholar
  23. 23.
    J. Struckmeier, P. Liebrich, J. Muench, M. Hanauske, J. Kirsch, D. Vasak, L. Satarov, H. Stoecker, Int. J. Mod. Phys. E 28(1), 1950007 (2019)., arXiv:1711.10333
  24. 24.
    K. Hayashi, T. Shirafuji, Prog. Theor. Phys. 64(3), 866, 883, 1435, 2222 (1980).

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Jürgen Struckmeier
    • 1
    • 2
    • 3
    Email author
  • David Vasak
    • 1
  • Johannes Kirsch
    • 1
  1. 1.Frankfurt Institute for Advanced Studies (FIAS)Frankfurt am MainGermany
  2. 2.Goethe-UniversitätFrankfurt am MainGermany
  3. 3.GSI Helmholtzzentrum für Schwerionenforschung GmbHDarmstadtGermany

Personalised recommendations