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  • Open Access

Violation of energy conditions and the gravitational radius of the proton

Adrian Dumitru1,2,* and Jorge Noronha3,†

  • *Contact author: adrian.dumitru@baruch.cuny.edu
  • †Contact author: jn0508@illinois.edu
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Phys. Rev. D 112, 094057 – Published 26 November, 2025

DOI: https://doi.org/10.1103/91br-d218

Abstract

The energy-momentum tensor (EMT) of the proton encodes fundamental information about its mass, pressure, and shear distributions. Using recent lattice QCD data for the gravitational form factors, we show that the Breit-frame Wigner EMT may be of Hawking-Ellis type IV in the proton’s core. Such EMT violates all pointwise energy conditions and lacks a causal rest frame so that the usual mechanical picture fails at short distances. We define the gravitational radius—a new hadronic observable—marking the scale where the EMT becomes ordinary (type I) and the classical interpretation is restored. We also derive from the averaged null energy condition nonperturbative, model-independent quantum field theory constraints on gravitational form factors.

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References (56)

  1. X.-D. Ji, A QCD analysis of the mass structure of the nucleon, Phys. Rev. Lett. 74, 1071 (1995).
  2. D. E. Kharzeev, Mass radius of the proton, Phys. Rev. D 104, 054015 (2021).
  3. X.-D. Ji, Gauge-invariant decomposition of nucleon spin, Phys. Rev. Lett. 78, 610 (1997).
  4. M. V. Polyakov, Generalized parton distributions and strong forces inside nucleons and nuclei, Phys. Lett. B 555, 57 (2003).
  5. M. V. Polyakov and P. Schweitzer, Forces inside hadrons: Pressure, surface tension, mechanical radius, and all that, Int. J. Mod. Phys. A 33, 1830025 (2018).
  6. V. D. Burkert, L. Elouadrhiri, and F. X. Girod, The pressure distribution inside the proton, Nature (London) 557, 396 (2018).
  7. P. E. Shanahan and W. Detmold, Pressure distribution and shear forces inside the proton, Phys. Rev. Lett. 122, 072003 (2019).
  8. K. Kumerički, Measurability of pressure inside the proton, Nature (London) 570, E1 (2019).
  9. C. Lorcé, H. Moutarde, and A. P. Trawiński, Revisiting the mechanical properties of the nucleon, Eur. Phys. J. C 79, 89 (2019).
  10. V. D. Burkert, L. Elouadrhiri, and F. X. Girod, Determination of shear forces inside the proton, arXiv:2104.02031.
  11. Z.-E. Meziani, Gluonic energy momentum tensor form factors of the proton, in Proceedings of the 16th Conference on Quark Confinement and the Hadron Spectrum (2025), arXiv:2505.05671.
  12. V. D. Burkert, L. Elouadrhiri, F. X. Girod, C. Lorcé, P. Schweitzer, and P. E. Shanahan, Colloquium: Gravitational form factors of the proton, Rev. Mod. Phys. 95, 041002 (2023).
  13. D. C. Hackett, D. A. Pefkou, and P. E. Shanahan, Gravitational form factors of the proton from lattice QCD, Phys. Rev. Lett. 132, 251904 (2024).
  14. I. Y. Kobzarev and L. B. Okun, Gravitational interaction of fermions, Zh. Eksp. Teor. Fiz. 43, 1904 (1962).
  15. H. Pagels, Energy-momentum structure form factors of particles, Phys. Rev. 144, 1250 (1966).
  16. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2023).
  17. M. Visser, Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, Melville, NY, 1996).
  18. R. M. Wald, General Relativity (Chicago University Press, Chicago, USA, 1984).
  19. T. A. Roman, Quantum stress energy tensors and the weak energy condition, Phys. Rev. D 33, 3526 (1986).
  20. P. Martin-Moruno and M. Visser, Classical and semi-classical energy conditions, Fundam. Theor. Phys. 189, 193 (2017).
  21. S. Abdolrahimi, D. N. Page, and C. Tzounis, Ingoing Eddington-Finkelstein metric of an evaporating black hole, Phys. Rev. D 100, 124038 (2019).
  22. P. Martin-Moruno and M. Visser, Hawking-Ellis classification of stress-energy tensors: Test fields versus backreaction, Phys. Rev. D 103, 124003 (2021).
  23. F. J. Tipler, Energy conditions and spacetime singularities, Phys. Rev. D 17, 2521 (1978).
  24. T. A. Roman, On the “averaged weak energy condition” and penrose’s singularity theorem, Phys. Rev. D 37, 546 (1988).
  25. A. Borde, Geodesic focusing, energy conditions and singularities, Classical Quantum Gravity 4, 343 (1987).
  26. M. S. Morris, K. S. Thorne, and U. Yurtsever, Wormholes, time machines, and the weak energy condition, Phys. Rev. Lett. 61, 1446 (1988).
  27. J. L. Friedman, K. Schleich, and D. M. Witt, Topological censorship, Phys. Rev. Lett. 71, 1486 (1993); 75, 1872(E) (1995).
  28. N. Graham and K. D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76, 064001 (2007).
  29. E. Witten, Light rays, singularities, and all that, Rev. Mod. Phys. 92, 045004 (2020).
  30. T. Faulkner, R. G. Leigh, O. Parrikar, and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, J. High Energy Phys. 09 (2016) 038.
  31. T. Hartman, S. Kundu, and A. Tajdini, Averaged null energy condition from causality, J. High Energy Phys. 07 (2017) 066.
  32. R. L. Jaffe, Ambiguities in the definition of local spatial densities in light hadrons, Phys. Rev. D 103, 016017 (2021).
  33. G. A. Miller, Defining the proton radius: A unified treatment, Phys. Rev. C 99, 035202 (2019).
  34. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, USA, 1995).
  35. X.-B. Tong, J.-P. Ma, and F. Yuan, Perturbative calculations of gravitational form factors at large momentum transfer, J. High Energy Phys. 10 (2022) 046.
  36. K. A. Mamo and I. Zahed, Diffractive photoproduction of J/ψ and ϒ using holographic QCD: Gravitational form factors and GPD of gluons in the proton, Phys. Rev. D 101, 086003 (2020).
  37. K. A. Mamo and I. Zahed, Nucleon mass radii and distribution: Holographic QCD, lattice QCD and GlueX data, Phys. Rev. D 103, 094010 (2021).
  38. K. A. Mamo and I. Zahed, J/ψ near threshold in holographic QCD: A and D gravitational form factors, Phys. Rev. D 106, 086004 (2022).
  39. M. Fujita, Y. Hatta, S. Sugimoto, and T. Ueda, Nucleon D-term in holographic quantum chromodynamics, Prog. Theor. Exp. Phys. 2022, 093B06 (2022).
  40. H.-T. Ding, X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, R. Zhang, and Y. Zhao, QCD predictions for meson electromagnetic form factors at high momenta: Testing factorization in exclusive processes, Phys. Rev. Lett. 133, 181902 (2024).
  41. H. Maeda and C. Martinez, Energy conditions in arbitrary dimensions, Prog. Theor. Exp. Phys. 2020, 043E02 (2020).
  42. P. Martin-Moruno and M. Visser, Essential core of the Hawking–Ellis types, Classical Quantum Gravity 35, 125003 (2018).
  43. P. Martin-Moruno and M. Visser, Semiclassical energy conditions for quantum vacuum states, J. High Energy Phys. 09 (2013) 050.
  44. H. Maeda, Nonstaticity with type II, III, or IV matter field in f(Rμνρσ,gμν) gravity, Gen. Relativ. Gravit. 53, 90 (2021).
  45. H. Maeda and T. Harada, Criteria for energy conditions, Classical Quantum Gravity 39, 195002 (2022).
  46. A. C. Wall, Proving the achronal averaged null energy condition from the generalized second law, Phys. Rev. D 81, 024038 (2010).
  47. K. Tanaka, Operator relations for gravitational form factors of a spin-0 hadron, Phys. Rev. D 98, 034009 (2018).
  48. X.-B. Tong, J.-P. Ma, and F. Yuan, Gluon gravitational form factors at large momentum transfer, Phys. Lett. B 823, 136751 (2021).
  49. M. Goharipour, F. Irani, M. H. Amiri, H. Fatehi, B. Falahi, A. Moradi, and K. Azizi (MMGPDs Collaboration), Can we determine the exact size of the nucleon?: A comprehensive study of different radii, Nucl. Phys. B1017, 116962 (2025).
  50. R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall, Quantum focusing conjecture, Phys. Rev. D 93, 064044 (2016).
  51. R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer, and A. C. Wall, Proof of the quantum null energy condition, Phys. Rev. D 93, 024017 (2016).
  52. A. C. Wall, Lower bound on the energy density in classical and quantum field theories, Phys. Rev. Lett. 118, 151601 (2017).
  53. S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang, A general proof of the quantum null energy condition, J. High Energy Phys. 09 (2019) 020.
  54. C. Lorcé, New explicit expressions for Dirac bilinears, Phys. Rev. D 97, 016005 (2018).
  55. D. C. Hackett, P. R. Oare, D. A. Pefkou, and P. E. Shanahan, Gravitational form factors of the pion from lattice QCD, Phys. Rev. D 108, 114504 (2023).
  56. X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Pion distribution amplitude at the physical point using the leading-twist expansion of the quasi-distribution-amplitude matrix element, Phys. Rev. D 106, 074505 (2022).

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Phys. Rev. D 112, 094057– Published 26 November, 2025
  • Received 5 June 2025
  • Revised 23 October 2025
  • Accepted 3 November 2025
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