Open problems in the torsional Newton–Cartan large-$c$ expansion of general relativity: non-twistless torsion, torsion-coupled motion, inequivalent dynamics, and physical regimes

Van den Bleeken (2017) showed that performing the manifestly covariant large-$c$ (inverse speed of light) expansion of general relativity without the customary no-pole assumption on the Levi-Civita connection yields an effective non-relativistic gravitational theory on twistless-torsional Newton–Cartan (TTNC) geometry, describing a strong-gravity, strong-time-dilation regime beyond the post-Newtonian expansion, and listed open directions for future work. Later work has resolved two of them: an action principle for the GR-derived TTNC equations now exists via the $1/c^2$-expanded Einstein–Hilbert action (Hansen–Hartong–Obers), and the higher-order expansions that fix the subleading metric fields have been computed in a 3+1 formulation (Elbistan–Hamamci–Van den Bleeken–Zorba; Elbistan). This statement isolates the surviving open core: (1) the expansion without the assumption that the leading stress-tensor coefficient $\mathcal{T}^{(-4)}_{\mu\nu}$ vanishes — the effective theory with non-twistless torsion, its consistency, geometry, and action; (2) a physical interpretation of the expanded geodesic equations whose velocity-coefficient fields are forced non-zero by torsion; (3) the classification of inequivalent TTNC equations of motion and of the relativistic parents that produce them; (4) the identification of realistic regimes (e.g. neutron-star interiors, early compact-binary inspiral) where the strong-time-dilation/low-velocity expansion applies, including the posed challenge of an analytic two-body problem in this expansion.

Background

The post-Newtonian (PN) expansion approximates general relativity (GR) for slowly moving bodies in weak fields; it is simultaneously an expansion in the inverse square of the speed of light, $1/c^2$, and in Newton's constant. A manifestly coordinate-covariant version of the $1/c$ expansion was developed in the classical literature on the Newtonian limit of GR: one expands the metric as $g_{\mu\nu}=c^2\tau_\mu\tau_\nu+\dots$, where the leading term defines the clock one-form $\tau_\mu$ of Newton–Cartan geometry. Newton–Cartan (NC) geometry is the non-relativistic analogue of Lorentzian geometry: instead of a single non-degenerate metric it carries a degenerate inverse spatial metric $h^{\mu\nu}$ together with the clock one-form $\tau_\mu$ obeying $h^{\mu\nu}\tau_\nu=0$; the connection compatible with both generically has torsion built from $d\tau$. When $\tau\wedge d\tau=0$ (twistless torsion) the clock one-form is hypersurface-orthogonal, spacetime is still foliated by hypersurfaces of absolute simultaneity, and the geometry is called twistless-torsional Newton–Cartan (TTNC); when $d\tau=0$ time is absolute and torsion vanishes. Van den Bleeken (2017) redid this expansion while allowing the Levi-Civita connection to have a $c^{-2}$ pole. Under the assumption that the leading coefficient of the expanded stress tensor, $\mathcal{T}^{(-4)}_{\mu\nu}$ (the coefficient of $c^4$), vanishes, the leading-order Einstein equations force the torsion to be twistless, and the next-to-leading and next-to-next-to-leading orders give a closed set of TTNC equations for the fields $\tau_\mu$, $h_{\mu\nu}$ and the strong time-dilation potential. The effective theory extends Newtonian gravity into a regime of strong gravitational time dilation: for example, re-expanding the Schwarzschild metric with the mass scaled as $M\sim c^2$ gives a solution in which the Newtonian potential vanishes entirely and all gravity resides in torsion plus a curved spatial metric. Two of the paper's open directions have since been settled. Hansen, Hartong and Obers constructed the $1/c^2$-expanded Einstein–Hilbert action with Lagrange multipliers, first for Newtonian gravity (2019) and then for the full leading and subleading theory coupled to matter (2020); its equations of motion contain the GR-derived TTNC equations. In their scheme the twistless condition is imposed off shell, and matter whose stress tensor rises above order $c^4$ is excluded precisely because it would source non-twistless torsion and hence, in their words, 'acausal' NC spacetimes — so the general-torsion case is deliberately deferred, not solved. Ergen, Hamamci and Van den Bleeken (2020) included odd powers of $c$, identified the expansion as an expansion around the stationary sector of GR, and gave a leading-order variational principle. Elbistan, Hamamci, Van den Bleeken and Zorba (2023) recast the expansion in a 3+1 form (a decomposition of the metric, dual to the Arnowitt–Deser–Misner (ADM) split of GR in that the roles of tangent and co-tangent spaces are interchanged, and non-degenerate as $c\to\infty$) and computed effective Lagrangians to higher order than before — up to $c^{-2}$ including odd powers and up to $c^{-4}$ in the even ($1/c^2$) sector — confirming the $c^{-2}$ Lagrangian reproduces the 2017 TTNC equations; Elbistan (2026) pushed this to $c^{-3}$. In these formulations the subleading fields are dynamical, so the 2017 question of fixing the inert fields $\hat B_\mu$ and $\hat\gamma^{\mu\nu}$ at higher orders is answered in reorganized variables. TTNC geometry is also the covariant framework for non-projectable Hořava–Lifshitz gravity — a gravity theory with anisotropic scaling between space and time proposed to be power-counting renormalizable, 'non-projectable' meaning its lapse function is allowed to depend on the spatial coordinates — whose dictionary to dynamical NC geometry was built by Hartong and Obers (2015), and 'type I'/'type II' variants of TTNC differing by gauge transformations have since been distinguished (von Blanckenburg and Schwartz 2025). What no later work has done is to drop the assumption excluding non-twistless torsion, to give the torsion-deformed particle motion a physical interpretation, to classify the inequivalent TTNC dynamics, or to demonstrate a realistic physical regime for the theory.

Problem Statement

The problem is the surviving open core of the future-work questions posed in Van den Bleeken (2017). Its shared objective is to complete the mathematical and physical characterization of the torsional (TTNC) large-$c$ limit of general relativity: to determine how general the non-relativistic limits of GR are, what the resulting theory says about motion in strong time dilation, how its dynamics compare with other non-relativistic gravity theories, and where it applies physically. The four directions are independent components of this single shared objective: none of them is a prerequisite for the others, and a self-contained answer to any one of them stands on its own. Within direction (1), the derivation of the expanded equations and geometry, the consistency question and the action-principle question belong to that single direction. The items already resolved by later work (the action principle for the twistless equations and the higher-order expansions fixing the subleading fields $\hat B_\mu$ and $\hat\gamma^{\mu\nu}$) are excluded and documented in the Background above. The open questions are: (1) General torsion: perform the large-$c$ expansion of the Einstein equations without assumption (27) of the source paper, i.e. with $\mathcal{T}^{(-4)}_{\mu\nu}\neq 0$ — allowing stress-tensor components of sufficiently leading order in $c$ that the twistless condition is no longer enforced — and derive the resulting effective non-relativistic equations and geometry with non-twistless torsion; determine whether this expansion is consistent (Hansen–Hartong–Obers suggest such Newton–Cartan spacetimes are 'acausal' — is the non-twistless limit of GR viable, pathological, or viable only in a restricted sense?), and whether it admits an action principle. (2) Torsion-coupled motion: physically interpret the expanded geodesic equations in a torsional background. In this expansion the particle velocity is itself expanded in powers of $1/c$, with coefficient fields $V^{(i)\mu}$, and torsion forces the leading coefficient $V^{(-1)\mu}$ to be non-zero; the observable content of the resulting trajectories in the strong-time-dilation regime remains unexplained. (3) Inequivalent dynamics: classify the inequivalent systems of TTNC gravitational equations compatible with the same symmetries, determine which relativistic parents (which expansions, gauge choices, or gauged algebras) produce them, and establish whether the GR-derived equations are distinguished among them. (4) Physical regimes: identify physically realistic situations in which the strong-time-dilation/low-velocity regime of the large-$c$ expansion applies, with a quantitative domain of validity; the general challenge is posed in Van den Bleeken (2017), and candidate examples identified in follow-up work (Ergen–Hamamci–Van den Bleeken; Hansen–Hartong–Obers) are neutron-star interiors and the early inspiral of compact binaries, including the concrete challenge — posed by Ergen–Hamamci–Van den Bleeken — of whether the two-body problem can be addressed analytically in this expansion.

References

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