The Type II non-relativistic geometric trinity of gravity: action-level teleparallel and symmetric-teleparallel Newton–Cartan theories from the 1/c expansion

The non-relativistic geometric trinity of gravity (Wolf, Read, and Vigneron, 2024) established three equivalent geometrical formulations of non-relativistic gravity — Newton–Cartan theory (NC, based on curvature), its teleparallel equivalent (TENC, based on torsion), and its symmetric teleparallel equivalent (STENC, based on non-metricity) — but only in the 'Type I' regime (closed clock form) and only at the level of the equations of motion. Because action principles for the Type I theories provably do not exist, the equivalence cannot be lifted to the level of the action within Type I, and the associated Noether analysis of gravitational stress-energy and conserved quantities is unavailable. The open problem, stated explicitly in the source's conclusion, is to construct the 'Type II' analogues of TENC and STENC that arise from the systematic 1/c expansion of the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR), and thereby to establish — or rigorously rule out — a full action-level non-relativistic geometric trinity: well-posed Lagrangians for all three Type II vertices, a demonstration of their equivalence at the level of the action, and the associated Noether currents, including gravitational stress-energy in the torsional and non-metric non-relativistic settings. No construction of Type II TENC or STENC and no action-level non-relativistic trinity is known in the literature to date; the problem remains open.

Background

In general relativity (GR), the 'geometric trinity of gravity' comprises three equivalent formulations of one and the same dynamics: standard GR, in which gravity is encoded in the curvature of the Levi-Civita connection; the teleparallel equivalent of general relativity (TEGR), in which the connection is flat but has torsion (the antisymmetric part of the connection coefficients); and the symmetric teleparallel equivalent of general relativity (STEGR), in which the connection is flat and torsion-free but fails to preserve the metric (non-metricity, $Q_{\mu\nu\rho}=\nabla_\mu g_{\nu\rho}$ , carries the gravitational degrees of freedom). The three formulations agree at the level of the field equations and, unlike the case of many modified-gravity theories, their equivalence can be exhibited at the level of the action, differing by boundary terms.

Each vertex also admits a non-relativistic (large speed of light) analogue. Newton–Cartan (NC) geometry replaces the Lorentzian metric by a spatial metric $h^{\mu\nu}$ and a clock one-form $\tau_\mu$ defining absolute time, together with a compatible Galilean connection $\hat{\Gamma}$ . Read and Teh (2018) and Schwartz (2023) constructed the teleparallel equivalent of Newton–Cartan gravity (TENC), in which the gravitational potential is carried by the torsion of a flat Weitzenböck-type connection (a connection of zero curvature but non-zero torsion, which vanishes in a preferred inertial frame) and which arises as the $1/c^2$ limit of TEGR; Wolf, Read, and Vigneron (2024) completed the trinity by constructing the symmetric teleparallel equivalent of Newton–Cartan gravity (STENC) as the $1/c^2$ limit of STEGR, showing equivalence with NC and TENC at the level of the equations of motion.

This non-relativistic trinity is, however, restricted to the 'Type I' regime, in which the clock form is required to be closed ( $d\tau=0$ ). Hansen, Hartong, and Obers showed that a more systematic expansion of GR in inverse powers of the speed of light (the 1/c expansion), keeping subleading fields and relaxing the clock-form condition to $\tau\wedge d\tau=0$ , yields 'Type II' Newton–Cartan theories. Type II theories reproduce many effects previously believed to require relativity and much of GR's solution space, and — crucially — provably admit action principles, whereas Type I ones provably do not (Hansen–Hartong–Obers 2019, 2020; an action for Type II NC is explicitly constructed in their 2020 paper). It is known in the relativistic case that different vertices of the trinity are more or less apt for different physical questions (for example, black-hole boundaries), and the relativistic trinity has been extended to modified-gravity theories $f(R)$ , $f(T,B)$ , and $f(Q,\tilde B)$ of the curvature, torsion, and non-metricity scalars (with boundary terms $B$ , $\tilde B$ ), as in Erdmenger, Heß, Meyer, and Matthaiakakis (2024). Whether analogous structures exist on the non-relativistic side is unexplored, and the interpretive questions about gravitational stress-energy $t^{\mu\nu}$ and conserved quantities — tensorial in TEGR and STEGR but dependent on connection gauge choices, and currently out of reach non-relativistically precisely because no suitable action exists — can, in the source authors' view, only be addressed within the Type II setting.

Problem Statement

Construct the 'Type II' analogues of TENC and STENC — the torsion-based and non-metricity-based non-relativistic gravitational theories that arise from the systematic 1/c expansion of TEGR and STEGR respectively, in parallel with the way Type II Newton–Cartan theory arises from the 1/c expansion of GR — and determine whether a full action-level non-relativistic geometric trinity can be established. Addressing this requires (1) exhibiting well-posed action principles (Lagrangians) for the Type II torsional and Type II non-metric theories, together with their relation to the known action for Type II Newton–Cartan gravity; (2) demonstrating the equivalence of the three Type II vertices at the level of the action — including the boundary terms relating the Lagrangians — or else proving a rigorous obstruction to such an action-level trinity under precisely stated assumptions; (3) carrying out the associated Noether analysis (derivation of the conserved currents following from the variational symmetries of the action), in particular deriving the gravitational stress-energy and conserved quantities for the Type II theories, and clarifying the status of the conservation laws of the Type I TENC and STENC field equations; and (4) specifying how the Type I theories (with $d\tau=0$ ) are recovered from the Type II ones under the appropriate restriction. Both outcomes are resolutions of the question: establishing the action-level trinity through (1)–(4), or rigorously ruling it out through the obstruction of (2), which states exactly which action-level property fails and under which assumptions. The problem may be posed in the geometric sector or with matter coupling, at the same generality of the 1/c expansion scheme at which Type II Newton–Cartan theory is defined.

References

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James Read, Nicholas Teh, "The Teleparallel Equivalent of Newton-Cartan Gravity", Classical and Quantum Gravity 35 (2018) 18LT01, DOI 10.1088/1361-6382/aad70d, arXiv:1807.11779, https://arxiv.org/abs/1807.11779
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