Modified dispersion relations (MDRs) are a standard tool of quantum gravity phenomenology, encoding Planck-scale corrections to the relativistic energy-momentum relation. Hohmann, Pfeifer and Wagner (Phys. Rev. D 110, 104030, 2024) proved that generic one-particle MDRs violate the weak equivalence principle (WEP): the spatial gravitational acceleration of a test particle depends on its mass unless the Hamiltonian is two-homogeneous (homogeneous of degree two) in the four-momenta, and they quantified the violation through a general Eötvös-factor formula, bounding the kappa-Poincaré deformation with MICROSCOPE data. Their analysis deliberately assumes that the one-particle dispersion relation, fed into the Hamilton equations of motion, fully determines the dynamics. The open problem is whether this WEP-violation conclusion survives in a complete deformed-special-relativity (DSR) framework, in which modified energy-momentum addition laws, deformed local Lorentz transformations, a multi-particle sector, and a consistent coupling to gravity on generic curved spacetimes are present; what the corresponding field-theoretic Eötvös factor for realistic composite test bodies (atoms and nuclei) is; and how the analysis extends to arbitrary post-Newtonian order. A 2025 follow-up has taken a first step in the deformed Galilean (Newtonian) sector, but the general question remains open.

Background

A modified dispersion relation (MDR) replaces the standard relativistic relation H(x,p)=m2H(x,p)=m^2 between a particle's energy and momentum by a deformed function, typically parameterized by a Planck-scale correction; MDRs are the common phenomenological currency of several quantum-gravity-motivated frameworks. Deformed (or doubly) special relativity (DSR) goes further and implements a deformed relativity principle: the Planck energy is made an observer-independent scale, which entails not only an MDR but also deformed Lorentz transformations, modified energy-momentum addition laws (in the best-studied example, the kappa-Poincaré Hopf algebra — a deformation of the Poincaré spacetime-symmetry algebra in which the coproduct, the algebraic rule that dictates how a symmetry acts on a multi-particle state, is nontrivial and thereby fixes the modified addition law), and a nontrivial multi-particle sector. A long-standing issue in DSR is the soccer-ball problem, namely how composite macroscopic bodies avoid accumulating Planck-size deformation effects; Amelino-Camelia argued that the total momentum of a collection of particles can be left undeformed, which underlies the multi-particle sector used in the source paper. The weak equivalence principle (WEP), equivalently the universality of free fall, states that the gravitational acceleration of a test body is independent of its mass and composition; it is quantified by the Eötvös factor, the normalized difference of the accelerations of two test bodies, which the MICROSCOPE satellite mission constrained to about 10−1510^{-15} (P. Touboul et al., Phys. Rev. Lett. 129, 121102, 2022). Hohmann, Pfeifer and Wagner (2024) performed a 3+13+1 decomposition (a split of spacetime into spatial slices plus time) and slow-velocity (1/c1/c) expansion of the parametrization-invariant (invariant under re-choice of the worldline evolution parameter) relativistic point-particle action induced by a generic MDR on a curved background, and proved that the spatial acceleration depends on the particle's mass unless the Hamiltonian is two-homogeneous (homogeneous of degree two) in the four-momenta (or the Lagrangian differs from the homogeneous case only by a total derivative). They derived a general Eötvös-factor formula for polynomial MDRs, applied it to the kappa-Poincaré dispersion relation in the bicrossproduct basis (one standard parametrization of that deformation) on curved spacetime, and used MICROSCOPE data to bound the deformation parameter as Ξ^−1≥1015 GeV/c2\hat{\Xi}^{-1}\geq10^{15}\,{\rm GeV}/c^2. For macroscopic test bodies they modeled the body as a rigid collection of NN constituents and found that the deformation is suppressed by 1/N1/N, so the Eötvös factor depends only on constituent masses. Crucially, this whole analysis assumes that in the one-particle sector the dispersion relation plus Hamilton equations fully determine the motion: the multi-particle sector, deformed Lorentz transformations, and modified addition laws of complete DSR frameworks are not included, and on generic (non-maximally-symmetric) curved spacetimes the construction of those DSR features is itself incomplete in the literature. The one-particle MDR analysis is therefore best interpreted as constraining Lorentz-violating scenarios rather than fully developed DSR models. The same technical apparatus (the 3+13+1 decomposition and slow-velocity expansion of the parametrization-invariant action) is the entry point to Finsler gravity, a generalization of Riemannian geometry in which the length functional is homogeneous but not quadratic in the tangent vector, and the authors identify the post-Newtonian (pN, an expansion in powers of 1/c1/c) analysis of Finsler gravity with dynamical field equations as a further avenue.

Problem Statement

Determine whether the weak equivalence principle (universality of free fall) survives Planck-scale dispersion modifications once the full deformed-special-relativity structure is included, replacing the one-particle modified-dispersion-relation (MDR) assumption of Hohmann, Pfeifer and Wagner (2024), and derive the experimentally comparable Eötvös factor. Concretely, following the source's discussion, the problem comprises these related objectives, presented by the source as distinct future research avenues with objective (1) as its core question: (1) Within a complete DSR framework — one that includes modified energy-momentum addition laws, deformed local Lorentz transformations, and a multi-particle sector, implemented on generic curved spacetimes and coupled consistently to gravity — derive the test-body velocity, spatial gravitational acceleration, and Eötvös factor, and decide whether the generic mass-dependent acceleration found for one-particle MDRs persists, is amplified, or can be removed by the full DSR structure (for instance by a modified identification of the particle's velocity or a modified coupling to the gravitational field). (2) Derive, quantum-field-theoretically, the composition-dependent Eötvös factor for realistic macroscopic test bodies such as atoms and nuclei, whose mass is dominated by binding energies and whose number of effective constituents is ambiguous, going beyond the rigid-body model that yields a 1/N1/N suppression with the Eötvös factor depending only on constituent masses. (3) Extend the 1/c1/c expansion so that the Eötvös factor can be quantified and constrained at arbitrary post-Newtonian order. (The post-Newtonian analysis of Finsler gravity with dynamical field equations is listed by the source as a separate additional application of the same 3+13+1/slow-velocity apparatus, not as part of this objective.) A complete answer must state, for a complete DSR framework meeting the requirements of objective (1), constructed or extended to generic curved spacetimes with a consistent gravitational coupling, whether the weak equivalence principle holds, with the explicit Eötvös factor η=2∣aA−aB∣/(aA+aB)\eta=2|a_A-a_B|/(a_A+a_B) — the normalized difference of the accelerations aAa_A and aBa_B of two test bodies AA and BB (or its vanishing) — and its comparison with equivalence-principle experiments such as MICROSCOPE; objective (1) is the required core of a complete answer, and objectives (2) and (3) are optional extensions along the avenues indicated by the source. Until such an analysis exists, the one-particle MDR bounds are to be understood as constraining Lorentz-violating scenarios rather than fully developed deformed-relativity models.

References

  1. Manuel Hohmann, Christian Pfeifer, Fabian Wagner, 'Weak equivalence principle and nonrelativistic limit of general dispersion relations', Physical Review D 110, 104030 (2024), DOI 10.1103/PhysRevD.110.104030, arXiv:2404.18811, https://arxiv.org/abs/2404.18811
  2. Giuseppe Fabiano, Domenico Frattulillo, Christian Pfeifer, Fabian Wagner, 'Universality of free fall in Planck-scale deformed Newtonian gravity', Physical Review D 112, 124008 (2025), DOI 10.1103/whyz-n93x, arXiv:2510.02046, https://arxiv.org/abs/2510.02046
  3. Pierre Touboul et al. (MICROSCOPE Collaboration), 'MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle', Physical Review Letters 129, 121102 (2022), DOI 10.1103/PhysRevLett.129.121102, https://doi.org/10.1103/PhysRevLett.129.121102
  4. Pasquale Bosso, Fabrizio Illuminati, Luciano Petruzziello, Fabian Wagner, 'Spin couplings as witnesses of Planck scale phenomenology', European Physical Journal C 85, 269 (2025), DOI 10.1140/epjc/s10052-025-13941-4, arXiv:2403.19389, https://arxiv.org/abs/2403.19389
  5. Joseph Balsells, Martin Bojowald, 'Geometry and proper time of a relativistic quantum clock', Physical Review D 110, 124061 (2024), DOI 10.1103/PhysRevD.110.124061, arXiv:2410.08156, https://arxiv.org/abs/2410.08156
  6. Arshid Shabir, Mir Faizal, 'Mass-Dependent Minimal Length and Deformed Quantum Mechanics from Heterotic String Theory', Foundations of Physics 56, 17 (2026), DOI 10.1007/s10701-026-00917-x, https://doi.org/10.1007/s10701-026-00917-x
  7. Giovanni Amelino-Camelia, 'Planck-Scale Soccer-Ball Problem: A Case of Mistaken Identity', Entropy 19(8), 400 (2017), DOI 10.3390/e19080400, https://doi.org/10.3390/e19080400