Vigneron, Barzegar, and Read (arXiv:2506.03936) recently asked what constraints arise when a single affine connection is required to be compatible with both a Galilean and a Carrollian spacetime structure at once, and showed that in the torsion-free, non-orthogonal case such bi-compatible connections exist nontrivially, with residual freedom given exactly by a static spatial curvature. Their analysis restricted itself to torsion-free connections in the non-orthogonal regime τμvμ≠0\tau_{\mu}v^{\mu}\neq 0 and explicitly set aside the orthogonal case τμvμ=0\tau_{\mu}v^{\mu}=0; the classification of such connections with torsion allowed, the treatment of the orthogonal regime, and the identification of the gravitational theories these geometries can support remain untreated in the literature and are not among the source's own listed open questions. This problem targets that untreated extension: solving it would complete the connection-classification program for non-Lorentzian (Galilean and Carrollian) metric-affine geometry and would supply kinematic input to the program the source itself proposed as a concluding open question — the construction of Galilean and Carrollian metric-affine gravitation on the unified scaffolding the source authors constructed.

Background

In metric-affine gravity one describes spacetime with a general affine connection ∇\nabla whose torsion Tαμν=2Γα[μν]T^{\alpha}{}_{\mu\nu}=2\Gamma^{\alpha}{}_{[\mu\nu]} (antisymmetric part of the connection) and non-metricity Qαμν=∇αgμνQ_{\alpha\mu\nu}=\nabla_{\alpha}g_{\mu\nu} (failure of the covariant derivative of the metric to vanish) are not constrained to zero, generalizing the Levi-Civita connection of general relativity (GR). In parallel, one can take the c→∞c\to\infty (Galilean) or c→0c\to 0 (Carrollian) limits of Lorentzian geometry: a Galilean structure is a pair (τμ,hμν)(\tau_{\mu},h^{\mu\nu}) with τ\tau a nowhere-vanishing clock 1-form and hμνh^{\mu\nu} a positive semidefinite contravariant spatial metric with dim⁡ker⁡h=1\dim\ker h=1, τ∈ker⁡h\tau\in\ker h; a Carrollian structure is a pair (vμ,γμν)(v^{\mu},\gamma_{\mu\nu}) with vv a nowhere-vanishing vector field and γ\gamma a positive semidefinite spatial metric with v∈ker⁡γv\in\ker\gamma. Neither structure identifies vectors with covectors, so each supports several inequivalent 'non-metricities' (e.g. Q^μν=∇μτν\hat Q_{\mu\nu}=\nabla_{\mu}\tau_{\nu} and Q^αμν=∇αhμν\hat Q_{\alpha}{}^{\mu\nu}=\nabla_{\alpha}h^{\mu\nu} for the Galilean case; Qˇμν=∇μvν\check Q_{\mu}{}^{\nu}=\nabla_{\mu}v^{\nu} and Qˇαμν=∇αγμν\check Q_{\alpha\mu\nu}=\nabla_{\alpha}\gamma_{\mu\nu} for the Carrollian case), and torsion and non-metricity cease to be independent quantities, being linked by algebraic identities such as ταTαμν=−2Q^[μν]+2ωμν\tau_{\alpha}T^{\alpha}{}_{\mu\nu}=-2\hat Q_{[\mu\nu]}+2\omega_{\mu\nu} with ω=dτ\omega=\mathrm{d}\tau the Coriolis-related field. Schwartz (arXiv:2403.15460) gave a full classification of general affine connections on Galilei manifolds in terms of independently specifiable tensor fields, and Vigneron–Barzegar–Read (arXiv:2506.03936) supplied the dual classification of general Carrollian connections with both torsion and non-metricity, showed that both families arise as limits of the most general Lorentzian structures (the causal regularity condition of global hyperbolicity forcing τ=Ndt\tau=N\mathrm{d}t globally in the Galilean case), constructed for the first time an ultra-relativistic geometric trinity of gravitational theories (the c→0c\to 0 analogue of the 'geometric trinity' in which GR's curvature degrees of freedom are traded for torsion, as in the teleparallel equivalent of GR (TEGR), or for non-metricity, as in the symmetric teleparallel equivalent (STEGR), and of the non-relativistic trinity of Wolf–Read–Vigneron, arXiv:2308.07100), and then asked what happens when the compatibility requirements are imposed on both a Galilean and a Carrollian structure simultaneously. In the torsion-free case with τμvμ≠0\tau_{\mu}v^{\mu}\neq 0 (normalizable to 1) they proved: the compatibility conditions Q^μν=0\hat Q_{\mu\nu}=0, Q^αμν=0\hat Q_{\alpha}{}^{\mu\nu}=0, Qˇμν=0\check Q_{\mu}{}^{\nu}=0, Qˇαμν=0\check Q_{\alpha\mu\nu}=0, T=0T=0 force vv to be a collineation (a vector whose flow preserves the tensor) of all the structure tensors and of the Riemann tensor, force dτ=0\mathrm{d}\tau=0, make the two induced spatial connections coincide so there is a unique spatial Ricci curvature, and leave as the only residual freedom a spatial Riemann tensor that is static (Ricci of the form diag(0,Rij)\mathrm{diag}(0,\mathcal{R}_{ij}) in an adapted basis); moreover such a connection is also Levi-Civita-compatible with a metric g=diag(±1,hij)g=\mathrm{diag}(\pm 1,h_{ij}), so it is simultaneously Galilean-, Carrollian-, and metric-compatible, and it coincides with the 'reference connection' introduced by Vigneron (arXiv:2204.13980) to control non-Euclidean spatial topologies in the non-relativistic limit. Two cases of the simultaneous-compatibility question were, however, excluded from the source's scope and remain untreated in the literature: the general case with torsion allowed, and the 'orthogonal' case τμvμ=0\tau_{\mu}v^{\mu}=0, which the source explicitly set aside (noting it would make the flow of vv tangent, and hence periodic, along a closed spatial foliation). On the physical side, the source's own concluding open questions — recovering a Newtonian-type potential in the Carrollian trinity, identifying the common structure of the ultra-relativistic trinity (analogous to the 'Maxwell gravitation' common core of the non-relativistic trinity found by March–Wolf–Read, arXiv:2309.06889), a gauge/fibre-bundle perspective on general Carroll connections, and above all the construction of general Galilean and Carrollian metric-affine theories of gravitation on this scaffolding — presuppose detailed kinematics of general Galilean and Carrollian connections, of which the simultaneously compatible ones form a special class.

Problem Statement

On a smooth 4-dimensional manifold carrying both a Galilean structure (τμ,hμν)(\tau_{\mu},h^{\mu\nu}) and a Carrollian structure (vμ,γμν)(v^{\mu},\gamma_{\mu\nu}), classify all affine connections ∇\nabla — with torsion allowed — that are simultaneously compatible with both structures, i.e. satisfying ∇μτν=0\nabla_{\mu}\tau_{\nu}=0, ∇αhμν=0\nabla_{\alpha}h^{\mu\nu}=0, ∇μvν=0\nabla_{\mu}v^{\nu}=0, and ∇αγμν=0\nabla_{\alpha}\gamma_{\mu\nu}=0, in the two cases the source excluded from its scope and did not treat: (a) the general torsionful case Tαμν≠0T^{\alpha}{}_{\mu\nu}\neq 0 in the non-orthogonal regime τμvμ≠0\tau_{\mu}v^{\mu}\neq 0 (normalizable to 1), together with the orthogonal regime τμvμ=0\tau_{\mu}v^{\mu}=0 that the source set aside, in which torsion is likewise unrestricted so that the torsion-free subclass is included; and (b) the associated global questions of existence and constrainedness (foliation, completeness, and periodicity properties of vv and τ\tau). The requested classification comprises a decomposition of the connection into independently specifiable tensorial data, the complete set of algebraic and differential identities relating torsion, the four non-metricities, ω=dτ\omega=\mathrm{d}\tau, and the expansion tensor Θ=12Lvγ\Theta=\tfrac{1}{2}\mathcal{L}_{v}\gamma (with Lv\mathcal{L}_{v} the Lie derivative along vv), the induced curvature structure, and a characterization of the residual freedom — thereby completing, in the bi-compatible setting, the program that Schwartz carried out for Galilean connections and Vigneron–Barzegar–Read for Carrollian connections individually. As the physical half of the same objective, determine what gravitational content such bi-compatible geometries can carry: identify which sectors of general Galilean and Carrollian metric-affine gravitation can be formulated on simultaneously compatible backgrounds — a question related to, but distinct from, the source's concluding Question 4, which calls for constructing general Galilean and Carrollian metric-affine theories of gravitation on the full unified scaffolding without restriction to bi-compatible backgrounds — and in particular whether any nontrivial dynamics beyond the static spatial curvature of the torsion-free case is possible on them. A sector of Galilean or Carrollian metric-affine gravitation here means a choice of which of the connection's geometric quantities — curvature, torsion, and the non-metricities — are permitted to be nonzero and dynamical, in the manner of the geometric trinity of gravitational theories (the curvature, torsion, and non-metricity sectors) as classified for each non-Lorentzian structure in the cited sources; such a sector is said to be formulable on a simultaneously compatible background when its action and field equations are well defined on the bi-compatible geometry and admit a simultaneously compatible connection as a solution, so that the compatibility conditions do not force the connection out of that class. A complete resolution requires all of the following parts, each of which is a self-contained sub-question: (1) the general torsionful classification in the non-orthogonal regime τμvμ≠0\tau_{\mu}v^{\mu}\neq 0; (2) the classification of the orthogonal regime τμvμ=0\tau_{\mu}v^{\mu}=0, which includes the torsion-free subclass; (3) the global existence, foliation, completeness, and periodicity questions for vv and τ\tau; and (4) the gravitational-content question, i.e. the identification of the sectors formulable on bi-compatible backgrounds and whether any nontrivial dynamics beyond the static spatial curvature of the torsion-free case is possible. Resolving the core classification (1) alone, or any proper subset of (1)–(4), is a partial resolution rather than a full one.

References

  1. Quentin Vigneron, Hamed Barzegar, James Read, 'Affine connections for Galilean and Carrollian structures: a unified perspective', Classical and Quantum Gravity 42, 175018 (2025), arXiv:2506.03936, doi:10.1088/1361-6382/adfc1e, https://arxiv.org/abs/2506.03936 (primary source)
  2. Philip K. Schwartz, 'The classification of general affine connections in Newton–Cartan geometry: Towards metric-affine Newton–Cartan gravity', Classical and Quantum Gravity 42, 015010 (2025), arXiv:2403.15460, https://arxiv.org/abs/2403.15460
  3. William J. Wolf, James Read, Quentin Vigneron, 'The Non-Relativistic Geometric Trinity of Gravity', General Relativity and Gravitation 56, 126 (2024), arXiv:2308.07100, https://arxiv.org/abs/2308.07100
  4. Quentin Vigneron, 'Non-relativistic regime and topology: topological term in the Einstein equation', Foundations of Physics 54, article 15 (2024), arXiv:2204.13980, doi:10.1007/s10701-023-00749-z, https://arxiv.org/abs/2204.13980
  5. Eleanor March, William J. Wolf, James Read, 'On the geometric trinity of gravity, non-relativistic limits, and Maxwell gravitation', Philosophy of Physics 2(1), article 15 (2024), arXiv:2309.06889, https://arxiv.org/abs/2309.06889
  6. Philip K. Schwartz, James Read, Quentin Vigneron, 'Weyl-type theorems in Galilei and Carroll geometry', arXiv:2606.00799 (2026), https://arxiv.org/abs/2606.00799
  7. Luca Ciambelli, Puttarak Jai-akson, 'Foundations of Carrollian Geometry', Physics Reports (2026), arXiv:2510.21651 (v1 2025, v4 2026), doi:10.1016/j.physrep.2026.05.005, https://arxiv.org/abs/2510.21651
  8. Andrew James Bruce, 'Carrollian Lie Algebroids: Taming Singular Carrollian Geometries', Classical and Quantum Grav. 43, 025001 (2026), arXiv:2510.03877, doi:10.1088/1361-6382/ae3426, https://arxiv.org/abs/2510.03877
  9. Eleanor March, James Read, 'Non-relativistic Twistor Theory: Newtonian Limits and Gravitationally Induced Collapse', Foundations of Physics (2026), doi:10.1007/s10701-026-00940-y, https://doi.org/10.1007/s10701-026-00940-y
  10. Arkachur Bhattacharya, Pushkar Soni, 'A unified expansion of Einstein's gravity', arXiv:2607.16459 (2026), https://arxiv.org/abs/2607.16459
  11. F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne'eman, 'Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance', Physics Reports 258, 1–171 (1995), arXiv:gr-qc/9402012, doi:10.1016/0370-1573(94)00111-F, https://arxiv.org/abs/gr-qc/9402012