Carrollian limit of quadratic gravity

Technical Summary: Carrollian Limit of Quadratic Gravity

Problem Statement
The paper investigates the Carrollian limit (the ultralocal limit where the speed of light $c \to 0$) of general quadratic gravity in four dimensions. Quadratic gravity, an effective field theory derived from string theory or proposed as a renormalizable gravity theory, includes terms quadratic in curvature ($R^2$ and $R_{\mu\nu}R^{\mu\nu}$) alongside the standard Einstein-Hilbert term. While the Carrollian limit of General Relativity (GR) is well-understood, its application to quadratic gravity presents a challenge: the naive limit of the action leads to a theory that is drastically different from the Carrollian limit of GR, potentially losing the Ricci scalar term entirely due to the different scaling orders of the curvature terms. Furthermore, quadratic gravity is generally plagued by unphysical ghost and tachyonic degrees of freedom. The authors aim to determine under what conditions the Carrollian limit of quadratic gravity remains a modification of the Carrollian limit of GR and to classify these theories while imposing conditions to remove tachyons.

Methodology
The authors employ the Pre-Ultralocal (PUL) parametrization of the metric, a framework well-suited for Carrollian expansions where the speed of light $c$ is treated explicitly.

1. Parametrization: The spacetime metric is decomposed into vertical (timelike) and horizontal (spacelike) sub-bundles using fields such as the temporal one-form $T_\mu$, the spatial metric $\Pi_{\mu\nu}$, and their expansions in powers of $c^2$.
2. Curvature Expansion: The Riemann tensor, Ricci tensor, and Ricci scalar are expanded in powers of $c$. The authors derive the PUL versions of these tensors, identifying that the Ricci scalar $R$ scales as $c^{-2}$ (dominated by extrinsic curvature terms $K_{\mu\nu}$), while quadratic terms like $R_{\mu\nu}R^{\mu\nu}$ and $R^2$ scale as $c^{-4}$.
3. Action Analysis: The general quadratic gravity action, $S = \int (R - \alpha R_{\mu\nu}R^{\mu\nu} + \beta R^2)\sqrt{-g}d^4x$, is expanded. The authors analyze the behavior of the parameters $\alpha$ and $\beta$ as functions of $c$. They posit that for the resulting theory to be a modification of the Carrollian limit of GR, $\alpha$ and $\beta$ must depend on $c$ in specific ways (e.g., $\alpha \sim c^n, \beta \sim c^m$).
4. Classification and Constraints: Based on the powers of $c$ in $\alpha$ and $\beta$, the authors classify the resulting theories. They then impose conditions derived from the linearized theory to eliminate tachyonic modes, restricting the parameter space.
5. Magnetic Limit: For the ghost-free and tachyon-free theories, the authors derive the "magnetic limit" (a specific truncation of the next-to-leading order, NLO, action) to study the dynamics and symmetries.

Key Contributions and Results
The paper establishes that the Carrollian limit of quadratic gravity is not unique; it depends critically on how the coupling constants $\alpha$ and $\beta$ scale with $c$.

• Scaling Requirements: If $\alpha$ and $\beta$ are independent of $c$, the $c^{-4}$ quadratic terms dominate, and the $c^{-2}$ Ricci scalar term vanishes, resulting in a theory unrelated to GR. To recover a modification of GR, $\alpha$ and $\beta$ must scale with positive powers of $c$.
• Classification of Theories: By analyzing the leading order (LO) and next-to-leading order (NLO) expansions, the authors identify four nonequivalent Carrollian theories that modify GR:
  • (2,2): $\alpha \sim c^2, \beta \sim c^2$. Modifies GR at LO with both $R_{\mu\nu}R^{\mu\nu}$ and $R^2$ terms.
  • (2,4): $\alpha \sim c^2, \beta \sim c^4$. Modifies GR at LO with $R_{\mu\nu}R^{\mu\nu}$ terms; $R^2$ terms appear at NLO.
  • (4,2): $\alpha \sim c^4, \beta \sim c^2$. Modifies GR at LO with $R^2$ terms; $R_{\mu\nu}R^{\mu\nu}$ terms appear at NLO.
  • (4,4): $\alpha \sim c^4, \beta \sim c^4$. The LO is identical to GR; modifications appear at NLO.
• Tachyon Removal: The authors apply the conditions $\alpha \leq 0$ and $\alpha - 3\beta \geq 0$ (derived from the linearized theory) to eliminate tachyons. This restricts the viable theories:
  • Theories (2,4) and (4,2) are identified as the primary candidates for tachyon-free modifications.
  • For (2,4), the LO is equivalent to GR, while the NLO introduces $R^2$ corrections.
  • For (4,2), the LO is equivalent to an $R + R^2$ theory.
• Magnetic Limit and Dynamics:
  • For the (2,4) theory, the magnetic limit action modifies the GR magnetic limit with quartic terms of the extrinsic curvature and imposes an additional constraint.
  • For the (4,2) theory, the analysis reveals that the LO equations can be interpreted as GR with an emergent cosmological constant (value $-1/(10\beta')$) arising from the $R^2$ term, even without an explicit cosmological constant in the Lagrangian. The magnetic limit action for this case includes this cosmological term and spatial derivatives, breaking the strict ultralocality found in the electric limit.

Significance
The paper provides a systematic classification of Carrollian limits for quadratic gravity, demonstrating that a consistent limit (one that modifies rather than replaces GR) requires a non-trivial dependence of coupling constants on the speed of light. The work bridges the gap between high-energy effective theories (like those from string theory) and the ultralocal physics relevant to black hole horizons. By identifying tachyon-free sectors, the authors isolate specific theories (particularly those equivalent to $R + \beta R^2$) that are renormalizable and free of ghosts. The discovery of an emergent cosmological constant in the (4,2) case suggests new mechanisms for generating vacuum energy in Carrollian regimes. The authors conclude that these results lay the groundwork for studying particle dynamics near black hole horizons and potentially identifying Carrollian analogs of Schwarzschild-Bach black hole solutions.