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Komar superpotentials in extended theories of gravity

H. Arthur Weldon*

  • *Contact author: hweldon@mail.wvu.edu
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Phys. Rev. D 114, 084021 – Published 7 October, 2026

DOI: https://doi.org/10.1103/wzzq-1xfz

Abstract

This paper investigates Lagrangians with arbitrary dependence on the Riemann tensor and any number of derivatives. Noether’s second theorem is applied separately to LG and LM, where the matter Lagrangian depends on a scalar field. For LG diffeomorphism invariance requires that the generalized Einstein tensor satisfy ∇αEαβ=0 and that there is a current satisfying 0=∂α(gJG2α). The main result is an explicit formula for the superpotential Φ[αμ] that solves ∂μΦ[αμ]=gJG2α. Some applications to f(R) gravity and various quadratic gravity theories are discussed. There are analogous results, including superpotentials, for the matter sector whenever the couplings to the matter fields contain derivatives of the metric.

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References (43)

  1. E. Noether, Invariante Variationsprobleme, Gott. Nachr. 1918, 235 (1918), arXiv:physics/0503066.
  2. R. Utiyama, New energy and energy-tensor of a gravitational field, Prog. Theor. Phys. 72, 83 (1984).
  3. S. DeHaro, Noether’s theorems and energy in general relativity, in The Philosophy and Physics of Noether’s Theorems, edited by J. Read and N. Teh (Cambridge University Press, Cambridge, England, 2022), p. 197-256.
  4. R. J. McLeod, A brief review of Noether’s theorems and their applications to general relativity, arXiv:2106.04393.
  5. R. Ilin and S. Paston, Energy-momentum pseudotensor and superpotential for generally covariant theories of gravity of general form, Universe 6, 173 (2020).
  6. R. V. Ilin, Noether currents in theories with higher derivatives and theories with differential field transformation in action (DFTA), Eur. Phys. J. Plus 137, 1144 (2022).
  7. S. Aoki and T. Onogi, Conserved non-Noether charge in general relativity: Physical definition versus Noether’s second theorem, Int. J. Mod. Phys. A 37, 2250129 (2022).
  8. N. B. e Sá, M. A. S. Pinto, and T. Trindale, Covariant conservation laws, local invariance and Noether’s second theorem, Eur. Phys. J. C 86, 612 (2026).
  9. N. B. e Sá, Improper currents in theories with local invariance, Ann. Phys. (Amsterdam) 485, 170331 (2026).
  10. A. Freese, Reflections on Noether’s second theorem and the energy-momentum tensor, Phys. Rev. D 113, 016011 (2026).
  11. A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Ann. Phys. (Berlin) 49, 769 (1916) English translation in H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity (Dover, Toronto, Canada, 1952), p. 149.
  12. C. Møller, On the localization of the energy of a physical system in the general theory of relativity, Ann. Phys. (N.Y.) 4, 347 (1958).
  13. P. A. M. Dirac, General Theory of Relativity (Princeton University Press, Princeton, NJ, 1996), pp. §31—§32.
  14. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1975), p. §96.
  15. A. Komar, Covariant Conservation Laws in General Relativity, Phys. Rev. 113, 934 (1959).
  16. A. Komar, Asymptotic covariant conservation laws for gravitational radiation, Phys. Rev. 127, 1411 (1962).
  17. A. Komar, Positive-definite energy density and global consequences for general relativity, Phys. Rev. 129, 1873 (1962).
  18. P. K. Townsend, Black holes, arXiv:gr-qc/9707012.
  19. S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rep. 509, 167 (2011).
  20. T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Modified gravity and cosmology, Phys. Rep. 513, 1 (2012).
  21. S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models, Phys. Rep. 505, 59 (2011).
  22. S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rep. 692, 1 (2017).
  23. S. Capozziello, V. F. Cardone, and V. Salzano, Cosmography of f(R) gravity, Phys. Rev. D 78, 064504 (2008).
  24. A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Relativity 13, 3 (2010).
  25. T. P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451 (2010).
  26. S. Kumar, R. C. Nunes, S. Pan, and P. Yadav, New late-time constraints on f(R) gravity, Phys. Dark Universe, 42, 101281 (2023).
  27. G. Montani, L. A. Escamilla, N. Carlevaro, and E. Di Vanetino, Decay of f(R) quintessence into dark matter: Mitigating the Hubble tension?, Phys. Rev. D 113, 023507 (2026).
  28. L. Alvarez-Gaume, A. Kehagias, C. Dounnas, D. Lüst, and A. Riotto, Aspects of quadratic gravity, Fortschr. Phys. 64, 176. (2016).
  29. B. Holdom and J. Ren, Quadratic gravity: From weak to strong, Int. J. Mod. Phys. D 25, 1643004 (2016).
  30. A. Salvio, Quadratic gravity, Front. Phys. 6, 77 (2018).
  31. J. Daas, K. Kuijpers, F. Saueressig, M. F. Wondrak, and H. Falcke, Probing quadratic gravity with the Event Horizon Telescope, Astron. Astrophys. 673, A53 (2023).
  32. J. F. Donoghue and G. Menezes, On quadratic gravity, Il Nuovo Cimento 45C, 26 (2022).
  33. G. P. de Brito, Quadratic gravity in analogy to quantum chromodynamics: Light fermions in its landscape, Phys. Rev. D 109, 086005. (2024).
  34. D. Buccio, J. F. Donoghue, G. Menezes, and R. Percacci, Physical running of couplings in quadratic gravity, Phys. Rev. Lett. 133, 021604 (2024).
  35. J. Kuntz, Unitarity through PT symmetry in quantum quadratic gravity, Classical Quantum Gravity 42, 175003. (2025).
  36. A. De Felice and S. Tsujikawa, Solar system constraints on f(G) models, Phys. Rev. D 80, 063516 (2009).
  37. D. Glavan and C. Lin, Einstein-Gauss-Bonnet gravity in four-dimensional spacetime, Phys. Rev. Lett. 124, 081301 (2020).
  38. P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne, The 4D Einstein-Gauss-Bonnet theory of gravity: A review, Classical Quantum Gravity 39, 063001 (2022).
  39. T. Ortín, Komar integrals for theories of higher order in the Riemann curvature and black-hole chemistry, J. High Energy Phys. 08 (2021) 023.
  40. J. N. Goldberg, Conservation laws in general relativity, Phys. Rev. 111, 315 (1958).
  41. P. G. Bergmann, Conservation laws in general relativity as the generators of coordinate transformations, Phys. Rev. 112, 287 (1958).
  42. V. Iyer and R. M. Wald, Comparison of the Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52, 4430 (1995).
  43. T. Biswas, E. Gerwick, T. Koivisto, and A. Maxumdar, Towards singularity-and ghost-free theories of gravity, Phys. Rev. Lett. 108, 031101 (2012).

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Phys. Rev. D 114, 084021– Published 7 October, 2026
  • Received 7 August 2026
  • Accepted 9 September 2026
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