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

Technical Summary: Uniqueness of Galilean and Carrollian Limits and Application to Higher Derivative Gravity

Problem Statement
The paper addresses the fragmentation in methods used to derive non-Lorentzian gravitational theories (Galilean and Carrollian limits) from Lorentzian ones. While various techniques exist—such as pre-nonrelativistic (PNR) and pre-ultralocal (PUL) parametrizations, infinite scaling (IS), zero signature (ZS) approaches, and the gauging of Galilei and Carroll algebras (GAG and CAG)—their equivalence has not been rigorously established across all contexts. Furthermore, there is a lack of a unified, algorithmic framework to compute the $n$-th order expansions for generic higher derivative gravity (HDG) theories, ranging from $f(R)$ gravity to the most general $f(g_{\mu\nu}, R_{\mu\nu\sigma\rho}, \nabla_\mu)$ theories.

Methodology
The authors employ a comparative analysis of existing derivation methods followed by the construction of a generalized expansion algorithm.

1. Equivalence Proofs: The paper demonstrates that distinct methods yield identical metric structures and connection definitions:

   • Galilean Limit: The PNR parametrization is shown to be constructible from the gauging of the Galilei algebra (GAG). Similarly, the PNR approach is equivalent to the Infinite Scaling (IS) method (taking the norm of the normal vector to infinity in an ADM decomposition).
   • Carrollian Limit: The PUL parametrization is constructed from the gauging of the Carroll algebra (CAG). The PUL approach is proven equivalent to the Zero Signature (ZS) approach (setting the metric signature to zero in an ADM decomposition).
   • Unified Framework: By introducing a real parameter $\epsilon$ in the ADM decomposition (where $\epsilon$ represents the signature, taking values $1, -1, 0$), the authors unify the derivation of these limits. The limit $\epsilon \to 0$ corresponds to the Carrollian limit, while $\epsilon \to -\infty$ corresponds to the Galilean limit.

2. Algorithmic Expansion: Leveraging the established uniqueness, the authors derive a generic expansion for curvature tensors. They transform the problem of calculating the $n$-th order of Galilean and Carrollian expansions for arbitrary HDG theories into constrained optimization problems.

   • The Lagrangian is expanded in powers of the speed of light ($c$ or $c^{-1}$).
   • The coefficients of these expansions are determined by solving optimization problems (specifically, finding the minimum or maximum power of $c$) subject to constraints derived from the tensor structure of the theory.
   • The Simplex method is identified as the computational tool to solve these optimization problems.

Key Contributions and Results

• Unification of Methods: The paper provides a rigorous proof that the seemingly disparate methods (parametrization vs. algebra gauging vs. signature limits) are mathematically equivalent. This establishes a "unique" limit for any given Lorentzian theory.
• General Algorithm for HDG: A systematic algorithm is presented to compute the Leading Order (LO) and Next-to-Leading Order (NLO) for any metric theory of gravity of finite order.
  • For $f(R)$ gravity, the authors derive explicit formulas for the electric (LO) and magnetic (NLO) Carrollian limits and the Newton-Cartan (LO) and Torsional Newton-Cartan (NLO) Galilean limits.
  • For $f(g_{\mu\nu}, R_{\mu\nu\sigma\rho})$ gravity, the expansion is formulated as a constrained optimization problem where the order of the limit depends on the powers of the Riemann tensor and metric in the Lagrangian.
  • For the most general HDG ($f(g_{\mu\nu}, R_{\mu\nu\sigma\rho}, \nabla_\mu)$), the algorithm incorporates covariant derivatives, expanding them into specific orders of $c$ and formulating a multi-variable optimization problem to determine the dominant terms.
• Conditions for GR Modification: The paper derives specific conditions under which a gravitational theory acts as a modification of General Relativity (GR) in these limits.
  • For $f(R)$ theories, a modification of GR in the Galilean limit requires specific constraints on the power of $c$ in the coupling constants (e.g., $N \leq -2$), while the Carrollian limit requires $N \geq 2$.
  • Mutual Exclusivity: A significant result is the demonstration that no polynomial $f(R)$ or $f(g_{\mu\nu}, R_{\mu\nu\sigma\rho})$ theory can simultaneously be a modification of GR in both the Galilean and Carrollian limits. The conditions required for one limit are mutually exclusive with those for the other.
  • The authors note that while theories of the form (4.26) and quadratic gravity fail to satisfy the conditions for simultaneous modification, the general algorithm allows for the search of such theories in the broader class of $f(g_{\mu\nu}, R_{\mu\nu\sigma\rho}, \nabla_\mu)$, though no such theory is explicitly constructed in this work.

Significance and Claims
The paper claims that establishing the equivalence of these derivation methods removes ambiguity in the definition of non-Lorentzian gravity. By converting the expansion of complex curvature tensors into constrained optimization problems, the authors provide a computationally efficient tool for studying these limits.

• Galilean Context: Higher-order expansions are noted as useful for improving the accuracy of dynamical systems in the post-Newtonian approximation.
• Carrollian Context: While higher-order Carrollian expansions currently lack immediate physical application, the authors suggest that with the growing interest in Carrollian physics (e.g., black hole horizons, holography), an algorithmic approach to compute these orders will become beneficial.
• Future Work: The paper modestly concludes that the search for a theory that modifies GR in both limits simultaneously remains an open problem for future research, and the analysis of specific dynamical systems using the proposed algorithm is left for subsequent work.

The authors do not claim to have discovered new physical phenomena but rather to have provided a rigorous mathematical framework and a computational algorithm that unifies existing approaches and facilitates the systematic study of higher-derivative corrections in non-Lorentzian regimes.