Hořava–Lifshitz-type quantum gravity breaks local Lorentz symmetry by foliating spacetime into spatial slices with a preferred time direction, and its viability hinges on whether Lorentz invariance can re-emerge at macroscopic scales. Knorr (2019) constructed a covariant functional renormalization group flow on foliated spacetimes that preserves background diffeomorphism symmetry, and showed at leading order — background-field approximation, linear order in the foliation-breaking couplings and in Newton's and the cosmological constant — that Lorentz-symmetry breaking is a relevant perturbation: a small breaking seeded near the Planck scale is magnified by factors of order unity toward low energies, so observational bounds on Lorentz violation apply essentially scale-independently. The open problem is whether this conclusion survives beyond these approximations: computing the flow at higher order in the breaking couplings (where foliation-gauge dependence is expected) and beyond the background-field approximation, including matter fields together with matter-sector Lorentz-breaking operators and their feedback on the breaking, and replacing the Gaussian-separatrix estimate by a self-consistent multi-scale computation based on the asymptotically safe ultraviolet fixed point, together with the associated Wick-rotation/CDT connection and parameterisation-robustness questions. Substantial adjacent progress exists — fluctuation-level foliated flows, minimally coupled matter, Wick-rotation conditions, and matter-sector percolation of Lorentz violation — but no work redoes the relevance/magnification analysis beyond linear order or with matter-sector Lorentz-breaking operators, so the problem remains open.

Background

Local Lorentz symmetry — the invariance of physics under boosts linking space and time — is extremely well tested in the matter sector, and quantum-gravity models that break it are tightly constrained. One prominent class, Hořava–Lifshitz (HL) gravity, foliates spacetime into spatial slices labeled by a preferred time parameter and keeps only foliation-preserving diffeomorphisms (transformations that map slices to slices) rather than the full spacetime diffeomorphism group; this permits anisotropic scaling between space and time with higher spatial derivatives, improving ultraviolet behavior. A long-standing open issue in HL gravity is whether Lorentz invariance can be restored, at least to high precision, at large length scales: if quantum fluctuations diminish the breaking, HL gravity is viable, while if they enhance it, the severe low-energy bounds on Lorentz violation must already be met at the Planck scale.

The main non-perturbative tool here is the functional renormalization group (FRG): the Wetterich equation gives the scale dependence of the effective average action, a scale-dependent quantum effective action interpolating between the classical action and the full quantum action. A coupling is called relevant if it grows as the coarse-graining scale is lowered toward the infrared (IR). In the background-field approximation one splits the metric into a fixed background plus fluctuations and evaluates the flow on the background; this is less reliable than the fluctuation (vertex) approach, which computes correlation functions of the fluctuating fields. Within the gravitational asymptotic-safety program, the Reuter non-Gaussian fixed point (NGFP) — an interacting ultraviolet (UV) fixed point of the RG flow with finitely many relevant directions — is the proposed UV completion of gravity; a trajectory emanating from it (a separatrix near the Gaussian fixed point is another common proxy) supplies the running of Newton's constant gg used in scale integrations.

Earlier FRG work on foliated gravity includes the demonstration of asymptotic freedom in the matter-coupled HL setup (D'Odorico, Saueressig, Schutten 2014) and the ADM-based analyses of Biemans, Platania and Saueressig (2017): decomposing the metric by the Arnowitt–Deser–Misner (ADM) split into lapse, shift and spatial metric equips spacetime with a foliation structure, and non-Gaussian fixed points of foliated (gravity-)systems were identified. Knorr (2019) introduced a complementary, covariant implementation: an RG flow on foliated spacetimes with a foliation-preserving gauge fixing that retains full background diffeomorphism invariance, so the Lorentz-breaking couplings can be treated as small perturbations of the Lorentz-invariant theory. Linearizing the flow in the three breaking couplings (and in gg and the dimensionless cosmological constant λ\lambda), he found that the flow is generically not attracted toward the diffeomorphism-invariant subspace — General Relativity does not emerge from the RG running of foliated gravity — and that breaking couplings seeded at the Planck scale are magnified by order-unity factors (about 1.1–1.9 in his diagonalized/original couplings) when running to the infrared, with the natural consequence that observational constraints on Lorentz violation are essentially scale-independent. The quantitative estimate integrated the flow along the separatrix between the Gaussian and the non-Gaussian fixed point only up to a Planck-scale cutoff, using the semiclassical running g(k)≈k2/MPl2g(k) \approx k^{2}/M_{\mathrm{Pl}}^{2}, because within the linearized approximation the limit ΛUV→∞\Lambda_{\mathrm{UV}} \to \infty is not well defined.

The paper identifies the directions relevant to establishing the robustness of this conclusion. Its explicit outlook names the addition of matter fields with their own Lorentz-breaking operators, to test whether matter interactions change the relevance of the breaking and to disentangle the different limiting propagation speeds ('speeds of light') of the sectors, and the link to discrete Monte-Carlo approaches such as Causal Dynamical Triangulations (CDT, a lattice-style path-integral quantization built on a causal foliation, in which an anisotropy parameter plays the role of the breaking couplings) together with the associated Wick rotation (analytic continuation between Euclidean and Lorentzian signature, which on foliatable manifolds may be well defined via the lapse function). The body of the paper further acknowledges as limitations of the leading-order computation that terms of higher order in the breaking couplings are expected to be foliation-gauge dependent, that the background-field approximation was used for technical simplicity, that other choices of the solution of the foliation constraints (and the dependence on the chosen linearised parameterisation) will be considered elsewhere, and that ΛUV→∞\Lambda_{\mathrm{UV}} \to \infty cannot sensibly be taken within the linearised approximation — motivating a self-consistent multi-scale computation incorporating the nontrivial NGFP within an asymptotically safe scenario.

Problem Statement

Determine whether the foliation-based breaking of Lorentz symmetry in quantum gravity is a relevant perturbation robustly — that is, whether Knorr's leading-order conclusion, that a small Lorentz-breaking seeded near the Planck scale is magnified (by factors of order unity) under the renormalization group flow toward low energies, so that observational bounds on Lorentz violation are essentially scale-independent and must already be met at the Planck scale, survives beyond the approximations in which it was derived. Concretely, the directions bearing on this question comprise: (1) the flow of the foliation-breaking couplings beyond linear order in the breaking and beyond the background-field approximation, including whether the higher-order terms introduce foliation-gauge dependence and how any such gauge dependence is controlled or removed; (2) the inclusion of matter fields together with matter-sector Lorentz-breaking operators, to determine whether matter interactions change the relevance of the gravitational symmetry breaking, and to disentangle the different limiting propagation speeds of the gravitational and matter sectors; (3) a self-consistent multi-scale computation of the infrared magnification along trajectories of the asymptotically safe non-Gaussian fixed point, so that the ultraviolet cutoff can be taken to infinity rather than being fixed near the Planck scale; together with (4) the connection of the continuum foliated flow to discrete foliation-based approaches such as Causal Dynamical Triangulations via a well-defined Wick rotation for foliatable manifolds, and (5) the dependence of the conclusion on the linearised field parameterisation and on the chosen solution of the foliation constraints.

References

  1. Benjamin Knorr, "Lorentz symmetry is relevant", Physics Letters B 792 (2019) 142–148, DOI 10.1016/j.physletb.2019.01.070, arXiv:1810.07971, https://arxiv.org/abs/1810.07971
  2. Astrid Eichhorn, Alessia Platania, Marc Schiffer, "Lorentz invariance violations in the interplay of quantum gravity with matter", Physical Review D 102 (2020) 026007, DOI 10.1103/PhysRevD.102.026007, arXiv:1911.10066, https://arxiv.org/abs/1911.10066
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