Uniqueness of Galilean and Carrollian limits of gravitational theories and application to higher derivative gravity

TL;DR

This work proves the equivalence of multiple non-Lorentzian limit constructions (PNR, PUL, ZS, IS) and their gauging counterparts (GAG, CAG) for Galilean and Carrollian gravity. It then introduces a general curvature-expansion framework and an algorithm that converts nth-order Galilean and Carrollian expansions of arbitrary finite-order or higher-derivative gravity theories into constrained optimization problems. The authors apply this framework to $f(R)$, $f(g_{\mu\nu},R_{\mu\nu\rho\sigma})$, and the most general $f(g_{\mu\nu},R_{\mu\nu\rho\sigma},\nabla_{\mu})$ HDG theories, deriving LO/NLO limits for Carrollian and Galilean cases and establishing conditions for GR modifications. A key finding is that polynomial HDG terms cannot modify GR in both limits simultaneously, underscoring a form of uniqueness for the non-Lorentzian limits and guiding future searches for GR-violating HDG theories with consistent double-limit behavior.

Abstract

We show that the seemingly different methods used to derive non-Lorentzian (Galilean and Carrollian) gravitational theories from Lorentzian ones are equivalent. Specifically, the pre-nonrelativistic and the pre-ultralocal parametrizations can be constructed from the gauging of the Galilei and Carroll algebras, respectively. Also, the pre-ultralocal approach of taking the Carrollian limit is equivalent to performing the ADM decomposition and then setting the signature of the Lorentzian manifold to zero. We use this uniqueness to write a generic expansion for the curvature tensors and construct Galilean and Carrollian limits of all metric theories of gravity of finite order ranging from the $f(R)$ gravity to a completely generic higher derivative theory, the $f(g_{μν},R_{μνσρ},\nabla_μ)$ gravity. We present an algorithm for calculation of the $n$-th order of the Galilean and Carrollian expansions that transforms this problem into a constrained optimization problem. We also derive the condition under which a gravitational theory becomes a modification of general relativity in both limits simultaneously.