The general affine connection compatible with a Carrollian structure — allowing both torsion and non-metricity — was classified by Vigneron, Barzegar and Read (2025), who used it to construct the first ultra-relativistic geometric trinity of gravitational theories and posed, as their first explicitly acknowledged open question, under what conditions such a connection can be gauge-fixed or otherwise constrained so as to recover a potential-based ultra-relativistic gravitational theory analogous to potential-based Newtonian gravity. In the Galilean (non-relativistic) case this recovery proceeds through the antisymmetric Coriolis form; its Carrollian dual, the tensor χμν=∇(μτν)\chi_{\mu\nu}=\nabla_{(\mu}\tau_{\nu)}, is symmetric, so the analogous mechanism is unavailable and a different strategy is required.

Background

In Lorentzian general relativity, gravity is encoded in the curvature of the Levi-Civita connection (the unique torsion-free, metric-compatible connection of the spacetime metric). The 'geometric trinity of gravity' generalizes this picture: the teleparallel equivalent of general relativity (TEGR) encodes gravity in the torsion of a flat, metric connection, where torsion is the antisymmetric part Tαμν=2Γα[μν]T^{\alpha}{}_{\mu\nu}=2\Gamma^{\alpha}{}_{[\mu\nu]} of the connection (square brackets denoting antisymmetrization); the symmetric teleparallel equivalent (STEGR) encodes it in the non-metricity, i.e. the covariant derivative ∇αgμν\nabla_{\alpha}g_{\mu\nu} of the metric, of a flat, torsion-free connection. All three theories are dynamically equivalent up to boundary terms.

These non-Lorentzian analogues are organized by degenerate metric structures. A Galilean structure is a pair (τμ,hμν)(\tau_{\mu},h^{\mu\nu}) of a nowhere-vanishing 'clock' 1-form τ\boldsymbol{\tau} and a positive-semidefinite contravariant 'spatial' metric h\boldsymbol{h} whose kernel is spanned by τ\boldsymbol{\tau}; it underlies Newton-Cartan theory, the geometric reformulation of Newtonian gravity. A Carrollian structure is the ultra-relativistic counterpart: a pair (vμ,γμν)(v^{\mu},\gamma_{\mu\nu}) of a nowhere-vanishing vector field v\boldsymbol{v} lying in the kernel of a positive-semidefinite metric γ\boldsymbol{\gamma}; such structures arise naturally on null hypersurfaces such as black-hole horizons and null infinity. Taking Galilean limits of TEGR and STEGR, Wolf, Read and Vigneron (2024) completed a non-relativistic geometric trinity, and March, Wolf and Read (2024) showed that its dynamical common core is 'Maxwell gravitation', a theory possessing a standard of absolute rotation but no standard of non-rotational acceleration. In the Galilean case it is known that suitable gauge fixing — essentially the vanishing of spatial torsion — suffices to recover standard, potential-based Newtonian gravity from the trinity members (Read and Teh 2018; Schwartz 2023).

Compatible connections on such degenerate structures are not unique, and their most general ('metric-affine') form carries both torsion and non-metricity, subject to identities that make the two non-independent. Schwartz (2025) classified general Galilean connections: relative to a natural reference connection, the difference is built from the non-metricities, the torsion, and a Coriolis 2-form κμν=∇[μvαγν]α\kappa_{\mu\nu}=\nabla_{[\mu}v^{\alpha}\gamma_{\nu]\alpha}, which encodes the acceleration and rotation of a chosen set of observers. Vigneron, Barzegar and Read (2025) derived the dual classification for Carrollian structures: the general Carrollian connection is fixed by its non-metricities, its torsion, and the symmetric tensor χμν=∇(μτν)\chi_{\mu\nu}=\nabla_{(\mu}\tau_{\nu)} (round brackets denoting symmetrization; τ\boldsymbol{\tau} is a choice of Ehresmann connection, i.e. a 1-form dual to v\boldsymbol{v}), the dual of the Coriolis field, together with the expansion tensor Θμν=12Lvγμν\Theta_{\mu\nu}=\tfrac{1}{2}\mathcal{L}_{\boldsymbol{v}}\gamma_{\mu\nu}, where Lv\mathcal{L}_{\boldsymbol{v}} denotes the Lie derivative along v\boldsymbol{v}. They then constructed, for the first time, an ultra-relativistic geometric trinity of gravitational theories.

The open issue studied here is the first question the authors pose in their conclusion. In the Galilean case, the recovery of a potential-based theory (Newtonian gravity with its scalar gravitational potential satisfying a Poisson-type equation) relies crucially on the antisymmetry of the Coriolis form; the Carrollian dual χμν\chi_{\mu\nu} is symmetric, and Vigneron, Barzegar and Read state that this 'presents roadblocks to a straightforward understanding of the circumstances under which one has, for example, a potential-based Carrollian spacetime theory'. (Bekaert and Morand call the expansion Θμν\Theta_{\mu\nu} a 'Carrollian potential', but Vigneron, Barzegar and Read explicitly note that this is not obviously the analogue of the Newtonian gravitational potential that is wanted.) A different strategy, with its own physical justification, is required.

Problem Statement

Under what conditions does the general Carrollian metric-affine framework admit a gauge fixing or constraint on the connection — imposed at fixed Carrollian structure (vμ,γμν)(v^{\mu},\gamma_{\mu\nu}) and possibly involving the choice of the Ehresmann 1-form τ\boldsymbol{\tau} dual to v\boldsymbol{v}, in analogy with the gauge fixings of the connection in the Galilean precedent — which may, but need not, be expressed on the classification data of Vigneron, Barzegar and Read (the reduced torsion and reduced non-metricities, i.e. the parts of the torsion and non-metricity that remain freely specifiable once the algebraic identities linking Carrollian torsion and non-metricity are imposed, and which, as Vigneron, Barzegar and Read note, are not uniquely fixed by those identities alone, except for the reduced torsion of a metric Carrollian connection; the symmetric tensor χμν=∇(μτν)\chi_{\mu\nu}=\nabla_{(\mu}\tau_{\nu)}, where τ\boldsymbol{\tau} is a 1-form dual to the Carrollian vector field v\boldsymbol{v} and round brackets denote symmetrization; and the expansion tensor Θμν=12Lvγμν\Theta_{\mu\nu}=\tfrac{1}{2}\mathcal{L}_{\boldsymbol{v}}\gamma_{\mu\nu}, where Lv\mathcal{L}_{\boldsymbol{v}} is the Lie derivative along v\boldsymbol{v} and γμν\gamma_{\mu\nu} is the degenerate Carrollian metric, Θμν\Theta_{\mu\nu} being an intrinsic object of the Carrollian structure, requiring no connection) — under which it reduces to a potential-based ultra-relativistic (Carrollian) gravitational theory, playing the structural role that potential-based Newtonian gravity plays for the non-relativistic geometric trinity?

The question asks for three connected things. First, what counts as 'potential-based' in the Carrollian setting is itself to be made precise, guided by analogy with the Newtonian case, in which the gravitational interaction is mediated by a scalar potential satisfying a Poisson-type equation sourced by matter, though the Carrollian realization need not take the same form; the expansion tensor Θμν\Theta_{\mu\nu}, called a 'Carrollian potential' by Bekaert and Morand, is explicitly flagged by Vigneron, Barzegar and Read as not obviously the desired analogue. The notion so made precise must be non-trivial — not a stipulation under which the reduction holds vacuously, nor one satisfied by the expansion tensor Θμν\Theta_{\mu\nu} alone — and must be tied to the Newtonian case by a stated limit, notwithstanding that the Carrollian realization itself need not take the same form. Second, the conditions themselves are sought: a gauge-fixing rule for the general Carrollian metric-affine framework, in analogy with the uniform rule (essentially vanishing spatial torsion) of the Galilean case, rather than a construction tailored to one special Carrollian structure, or a proof that none exist within the general framework, identifying the obstruction. Third, the physical justification for the strategy adopted is to be given, since the recovery of the Newtonian potential in the Galilean case relies crucially on the antisymmetry of the Coriolis form κμν\kappa_{\mu\nu}, whose Carrollian dual χμν\chi_{\mu\nu} is symmetric. Whether such a gauge-fixing rule (or constraint) exists, or whether none exists, is the question.

References

  1. Q. Vigneron, H. Barzegar, J. Read, 'Affine connections for Galilean and Carrollian structures: a unified perspective', Class. Quantum Grav. 42, 175018 (2025), arXiv:2506.03936, DOI: 10.1088/1361-6382/adfc1e, https://arxiv.org/abs/2506.03936
  2. P. K. Schwartz, 'The classification of general affine connections in Newton-Cartan geometry: Towards metric-affine Newton-Cartan gravity', Class. Quantum Grav. 42, 015010 (2025), arXiv:2403.15460, DOI: 10.1088/1361-6382/ad922f, https://arxiv.org/abs/2403.15460
  3. P. K. Schwartz, 'Teleparallel Newton-Cartan gravity', Class. Quantum Grav. 40, 105008 (2023), arXiv:2211.11796, DOI: 10.1088/1361-6382/accc02, https://arxiv.org/abs/2211.11796
  4. J. Read, N. Teh, 'The Teleparallel Equivalent of Newton-Cartan Gravity', Class. Quantum Grav. 35, 18LT01 (2018), arXiv:1807.11779, DOI: 10.1088/1361-6382/aad70d, https://arxiv.org/abs/1807.11779
  5. W. J. Wolf, J. Read, Q. Vigneron, 'The Non-Relativistic Geometric Trinity of Gravity', Gen. Rel. Grav. 56, 126 (2024), arXiv:2308.07100, DOI: 10.1007/s10714-024-03308-7, https://arxiv.org/abs/2308.07100
  6. E. March, W. J. Wolf, J. Read, 'On the geometric trinity of gravity, non-relativistic limits, and Maxwell gravitation', Phil. Phys. 2, 15 (2024), arXiv:2309.06889, DOI: 10.31389/pop.80, https://arxiv.org/abs/2309.06889
  7. X. Bekaert, K. Morand, 'Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective', J. Math. Phys. 59, 072503 (2018), arXiv:1505.03739, DOI: 10.1063/1.5030328, https://arxiv.org/abs/1505.03739
  8. A. J. Bruce, 'Carrollian Lie Algebroids: Taming Singular Carrollian Geometries', Class. Quantum Grav. 43, 025001 (2026), arXiv:2510.03877, DOI: 10.1088/1361-6382/ae3426, https://arxiv.org/abs/2510.03877
  9. P. K. Schwartz, J. Read, Q. Vigneron, 'Weyl-type theorems in Galilei and Carroll geometry' (2026), arXiv:2606.00799, https://arxiv.org/abs/2606.00799
  10. S. Blitz, G. Herczeg, D. McNutt, 'Potential Carroll Structures and Special Carrollian Manifolds' (2026), arXiv:2601.20068, https://arxiv.org/abs/2601.20068
  11. E. March, J. Read, 'Non-relativistic Twistor Theory: Newtonian Limits and Gravitationally Induced Collapse', Found. Phys. 56, 37 (2026), DOI: 10.1007/s10701-026-00940-y, https://doi.org/10.1007/s10701-026-00940-y