Open problems in the covariant $1/c^2$ expansion of general relativity: radiative modes, matching to full GR, and quantization of non-relativistic gravity

The essay 'Gravity between Newton and Einstein' (Hansen, Hartong and Obers, Int. J. Mod. Phys. D 28 (2019) 1944010) showed that truncating the covariant off-shell $1/c^2$ expansion of the Einstein-Hilbert action at next-to-leading order yields a non-relativistic gravity theory on torsional Newton-Cartan geometry that passes the three classical tests of general relativity through strong gravitational time dilation alone, yet admits no gravitational-wave solutions and has instantaneous dynamics on a preferred time foliation. The essay, together with the subsequent program review (Hartong, Obers and Oling, Front. Phys. 11 (2023) 1116888), identifies four open directions for this framework: which realistic matter models source torsion ($d\tau\neq 0$) and the strong time-dilation solutions; at which order and by what mechanism higher orders of the expansion restore propagating gravitational-wave degrees of freedom; whether the quantum version of non-relativistic gravity is more attainable than its notoriously difficult relativistic parent; and what the limitations of the truncation are for dynamical, nonstatic, strongly radiating systems and how it matches to full general relativity. The problem is open.

Background

Newtonian gravity can be recovered from general relativity geometrically through Newton-Cartan (NC) geometry (introduced by Cartan in 1923-24): a manifold equipped with a clock one-form $\tau_\mu$ (whose kernel defines the instantaneous spatial slices) and a degenerate inverse spatial metric $h^{\mu\nu}$. Classically the clock form is closed, $d\tau=0$, expressing absolute time; the Newtonian gravitational potential then enters the connection. The traditional route from general relativity to Newtonian gravity is the post-Newtonian (PN) expansion of the Einstein equations in powers of $1/c^2$, where 1PN denotes the order-$c^{-2}$ corrections used, for example, in gravitational-wave source modeling, and 2.5PN the order-$c^{-5}$ terms at which radiation-reaction effects first enter. A modern refinement, initiated by Van den Bleeken (2017) and developed by Hansen, Hartong and Obers, expands the Einstein-Hilbert action itself covariantly and off shell in powers of $1/c^2$ without assuming weak fields. Truncating this expansion at next-to-leading order defines an action for 'non-relativistic gravity' on a torsional NC geometry: the connection acquires torsion $\propto v^\lambda\partial_{[\mu}\tau_{\nu]}$ when $d\tau\neq 0$, and in the twistless-torsional case ($\tau\wedge d\tau=0$, so a time foliation still exists) the non-relativistic lapse function multiplying $dt$ becomes physical and encodes gravitational time dilation — a genuinely strong-field effect absent from Newtonian gravity. In this 'type II' torsional NC theory (the program's label for the twistless-torsional case $d\tau\neq 0$, as opposed to 'type I' with $d\tau=0$) the $1/c^2$ expansion of the Schwarzschild solution terminates after one order, so Schwarzschild is an exact solution with the same lapse $\sqrt{1-r_s/r}$ as in general relativity; the essay 'Gravity between Newton and Einstein' showed that the theory thereby passes all three classical tests — perihelion precession, light deflection, and gravitational redshift — which are pure time-dilation effects. However, the equations of motion of the truncated theory admit no gravitational-wave solutions: propagating gravitational degrees of freedom are 'true relativistic phenomena', and the theory has instantaneous interactions on a preferred foliation of time. The essay therefore leaves open the four directions stated above, which the program review of Hartong, Obers and Oling (2023) subsequently surveys. Their context: (i) in the expansion of general relativity coupled to matter, the matter content controls whether $d\tau=0$ or $d\tau\neq0$; (ii) classic work on the PN expansion already shows obstructions to restoring radiation — Rendall (1992) proved that PN expansions are not compatible with asymptotic flatness, because the far zone of any radiating source is dominated by gravitational waves and the $1/c$ series has a finite radius of validity, with tail terms breaking the naive Taylor expansion (retardation effects in fact first enter through odd powers of $1/c$, analyzed by Ergen, Hamamci and Van den Bleeken); the standard Blanchet-Damour and Will-Wiseman schemes match a near-zone $1/c$ expansion to a far-zone multipolar expansion in the coupling $G$ in harmonic gauge; (iii) the quantum question is motivated by the Bronstein cube — the cube of limits in the $(c,G,\hbar)$ parameter space whose corners connect Newtonian gravity, special-relativistic field theory, non-relativistic quantum mechanics, and full quantum gravity — with the suggestion that quantizing the non-relativistic (preferred-foliation, instantaneous) theory might be a 'third road' toward quantum gravity, echoing how Hořava-Lifshitz gravity (Hořava 2009), a gravity theory with a preferred foliation and anisotropic scaling between space and time, achieves power-counting renormalizability; and (iv) matching the truncation to full general relativity is required whenever both descriptions overlap, as in radiating astrophysical systems.

Problem Statement

Following the open directions identified in the essay of Hansen, Hartong and Obers (2019) and surveyed in the program review of Hartong, Obers and Oling (2023), the task is to complete the non-relativistic $1/c^2$ (large-$c$) expansion of general relativity into a framework that fills the gap the truncated theory leaves between Newton and Einstein. The single scientific objective is that non-relativistic gravity — the next-to-leading-order truncation of the covariant off-shell $1/c^2$ expansion of the Einstein-Hilbert action on torsional Newton-Cartan geometry — be embedded in a framework that recovers, within the expansion, the genuinely relativistic phenomena it omits, and admits a quantum version. The linked questions are: (1) Radiative sector: carry the $1/c^2$ expansion to higher orders and determine at which order, and by what mechanism, propagating gravitational-wave degrees of freedom and other radiative effects reappear — the truncated equations admit none. (2) Quantum theory: construct the quantum theory corresponding to the non-relativistic action, and determine whether quantization on the preferred-foliation, instantaneous-interaction background is more attainable than its notoriously difficult relativistic parent — the 'third road' suggested in the essay. (3) Matching and validity: determine the limitations of the non-relativistic truncation for dynamical, nonstatic, or strongly radiating astrophysical systems, and construct the matching of its solutions to full general relativity in regimes where both descriptions overlap. Radiation restoration and matching are distinct aspects of the same higher-order expansion: determining the order and mechanism by which propagating modes reappear does not by itself construct the matching to full general relativity, and conversely; the quantum question is posed about that same non-relativistic action. Each question is an independent aspect of the single objective.

References

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