Higher Curvature Corrections

• Higher curvature corrections are modifications to Einstein gravity that add higher-order contractions of curvature tensors to address UV effects and resolve classical singularities.
• They enable nonperturbative regular black hole solutions with finite curvature invariants and a de Sitter core, offering a mechanism to overcome singularities.
• The corrections impact gravitational observables such as quasinormal spectra and serve as effective models capturing string α′ effects and holographic dualities.

Higher curvature corrections are modifications to gravitational theories obtained by augmenting the Einstein–Hilbert action with terms containing higher-order contractions of the curvature tensor. These corrections are central to modern research in high-energy, gravitational, string, and holographic field theory, playing a critical role in attempts to regularize classical singularities, encode string theoretic α′ effects, explore black hole microphysics, and probe the structure of quantum gravity. Higher curvature terms arise naturally both as ultraviolet (UV) corrections in low energy effective actions from string theory, and as leading corrections in expansions of quantum gravity path integrals.

1. Mathematical Structure and Theories

Higher curvature corrections are incorporated by generalizing the gravitational action: $$S = \int d^D x\,\sqrt{-g}\,\left[R + \sum_{n=2}^\infty \ell^{2(n-1)} \alpha_n \mathcal{L}_n \right]$$ where $\mathcal{L}_n$ denote scalar densities constructed from the Riemann tensor (and its contractions) at order $n$, $\ell$ is the cutoff length scale (e.g., string length), and $\alpha_n$ are dimensionless couplings. The prototypical examples include:

• Gauss–Bonnet term ($n=2$): $$R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$
• Quasi-topological cubic and higher powers for $n\geq3$, e.g., $c_1R^3 + c_2 R R_{\mu\nu}R^{\mu\nu} + \cdots$

In spherical symmetry, special combinations (e.g., Lovelock, quasi-topological densities) can be constructed so that the equations of motion remain second order in derivatives, circumventing Ostrogradsky instability (Konoplya et al., 12 Mar 2024).

2. Nonperturbative Regular Black Hole Solutions

The inclusion of an infinite tower of higher curvature terms enables the construction of static, spherically symmetric black hole metrics that are everywhere regular and possess a de Sitter core. For instance, the D-dimensional extension of the Dymnikova solution has the form: $$f(r) = 1 - \mu r^{3-D} \left[1 - e^{-\alpha \mu/r^{D-1}}\right]$$ where $\mu$ is proportional to the ADM mass, and $\alpha>0$ determines the strength of the tower. Near $r=0$, the solution approaches a de Sitter core, with all curvature invariants finite, effectively resolving the classical singularity (Konoplya et al., 13 Apr 2024). These metrics cannot be reproduced by any finite truncation of curvature invariants; the action must depend nonanalytically on an auxiliary field, indicating an intrinsically nonperturbative modification of general relativity. The exact integration of the field equations requires a generating function with essential singularities (e.g., $h(\psi) = \psi e^{1/(\alpha \psi)}$), precluding a purely polynomial expansion in curvature invariants.

3. Linear Stability and Quasinormal Spectra

The perturbative analysis around these higher-curvature-corrected backgrounds leads to wave equations of the form: $$\frac{d^2\Psi}{dr_*^2} + [\omega^2 - V_\ell(r)]\Psi = 0, \quad dr_*/dr = 1/f(r)$$ with effective potentials that depend on the detailed form of $f(r)$ and the number of spacetime dimensions $D$. Quasinormal frequencies $\omega$ can be computed numerically using spectral (e.g., Bernstein polynomials with Chebyshev collocation) and high-order WKB-Padé methods, both showing rapid convergence and agreement for low-lying modes.

Empirically, the fundamental (least damped) mode shifts only mildly for moderate $\alpha$, while higher overtones show increased sensitivity, with the possibility of purely damped, non-oscillatory modes at critical coupling. This spectrum is very well-approximated by truncating the correction series to a handful of terms when $\alpha \ll 1$ (Konoplya et al., 12 Mar 2024, Konoplya et al., 13 Apr 2024).

4. Physical and Theoretical Implications

The main physical consequences are as follows:

• Singularity Resolution: Higher curvature corrections can generate black holes with finite curvature invariants, replacing the singular core with a de Sitter patch. The Kretschmann scalar and related invariants are bounded, with a de Sitter–like scaling determined by the series coefficients.
• Nonperturbative Structure: There does not exist a finite-order, polynomial-in-curvature Lagrangian yielding exact regular solutions of this kind; the requisite action must be defined via a nonanalytic generating function. This is reminiscent of nonlocal ghost-free modifications (e.g., infinite derivative theories) where propagators are exponentially suppressed in the UV.
• Gravitational Wave Observables: As the quasinormal spectrum is modified, even small higher-curvature couplings can, in principle, leave imprints on gravitational wave ringdown signals, providing observational access to UV modifications of gravity.
• Astrophysical and Holographic Applications: Regular black holes offer a toy model of singularity-free collapse, relevant for information paradox considerations and horizon microphysics. Extensions to AdS or rotating configurations open avenues in holographic dualities, where higher curvature corrections are linked to $1/N$ or $\alpha'$ effects in the boundary theory (Konoplya et al., 12 Mar 2024, Konoplya et al., 13 Apr 2024).

5. Truncation vs. Infinite Series and Convergence

An essential point is that while the infinite tower is required for the exact regular solution, physical observables such as the horizon position $r_H$, temperature $T_H$, and fundamental quasinormal frequency $\omega_0$ converge rapidly when computed with only a small number of terms in the curvature expansion (e.g., keeping up to fifth order). Deviations between truncated and infinite towers become nonperturbatively small for reasonable coupling strengths (i.e., $O(e^{-1/\alpha})$ corrections), justifying the widespread use of Lovelock and Gauss–Bonnet truncations as phenomenological models (Konoplya et al., 12 Mar 2024, Konoplya et al., 13 Apr 2024).

Truncation Order	Observable Accuracy	Physical Regime
2–5 terms (e.g., Lovelock)	$\ll 1\%$ for $\omega_0$	$\alpha \ll 1$ (weak regime)
Infinite tower	Essential for exact nonperturbative solution	$\alpha$ finite; regular black holes

6. Interpretive Significance and Outlook

The existence of regular, nonperturbative black hole solutions arising from higher curvature towers provides robust evidence that the classical singularity predicted by general relativity is not generic once consistent UV corrections are incorporated. The necessity of summing infinitely many higher-derivative terms for certain properties, such as metric regularity, points to fundamental nonlocality in the gravitational sector at short distances. Such nonperturbative actions may furnish a phenomenological bridge between local gravitational EFTs and more exotic UV completions, such as string theory or nonlocal ghost-free gravities.

Furthermore, the fact that truncated series provide excellent approximations for many physical observables in realistic regimes substantiates the practical utility of Lovelock-type models in gravitational phenomenology while clarifying their intrinsic limitations for singularity resolution (Konoplya et al., 12 Mar 2024, Konoplya et al., 13 Apr 2024).

• Dymnikova regular black hole and infinite tower construction: (Konoplya et al., 13 Apr 2024)
• Quasinormal mode analysis, truncation convergence: (Konoplya et al., 12 Mar 2024)
• Nonperturbative non-analyticity of curvature generating functions: (Konoplya et al., 13 Apr 2024)