Infinite-Curvature Corrections in Gravity

• Infinite-curvature corrections are modifications to gravitational frameworks that incorporate an infinite series of higher-order curvature invariants to achieve UV-completion and regularize singular geometries.
• They impact black hole thermodynamics, quasinormal mode spectra, and near-horizon symmetries, leading to novel phenomenology distinct from general relativity.
• These corrections enable nonperturbative regular solutions in black holes and cosmological models while also introducing challenges like dynamic instabilities and analytic convergence concerns.

Infinite-curvature corrections refer to modifications of gravitational and geometric frameworks involving an infinite series of curvature-dependent terms—whether in the gravitational action, symmetry algebra, or matter couplings. These corrections commonly appear in attempts to formulate UV-complete theories of gravity, in the study of singularity resolution, in the algebraic description of near-horizon symmetries, and in the systematics of black hole solutions and perturbations. The dominant motif is the summation or resummation of higher-order curvature invariants, which may generate regular geometries, nontrivial thermodynamic properties, or intricate symmetry enhancements beyond those accessible with a finite number of curvature terms.

1. Infinite-Curvature Corrections in Gravitational Actions

Infinite-curvature corrections arise naturally when promoting the Einstein–Hilbert action to include arbitrary powers of curvature, particularly in dimensions $D>4$ via Lovelock or quasi-topological densities: $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$ where each $\mathcal{Z}_n$ is an $n$th-order (quasi-)topological invariant constructed from contractions of Riemann tensors, and $\{\alpha_n\}$ are coupling constants (Konoplya et al., 2024, Arbelaez, 29 Sep 2025, Konoplya et al., 2024).

For four-dimensional theories, dimensional regularization and conformal rescaling enable higher-order Lovelock invariants to contribute nontrivially, producing scalar–tensor actions in the Horndeski class (Fernandes, 10 Apr 2025, Tsujikawa, 26 May 2025, Felice et al., 15 Jul 2025). The infinite sum can be organized so that the field content and equations of motion remain second order, avoiding Ostrogradsky ghosts under generic circumstances.

A related class consists of nonlocal, infinite-derivative (IDG) gravities, where analytic form factors in the d'Alembertian $\Box$ generate infinite towers of terms ($R\Box^n R$, $C\Box^n C$, etc.), e.g.,

$$L = \frac12\left[ \kappa^{-1} R - \frac13 R\, \mathcal{F}_0(\Box)\, R + C^{abcd} \mathcal{F}_2(\Box) C_{abcd} \right]$$

with $$\mathcal{F}_i(\Box)=\sum_{n=0}^{\infty} f_{i,n}\Box^n$$ (Kolář et al., 2023).

The convergence of such series or the necessity for analytic continuation/regularization (e.g., zeta-function, Pauli–Villars) is a central concern; in string-theoretic or other UV-complete settings, exponential suppression of the $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$0 is expected at large $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$1 (Chernicoff et al., 26 Jun 2025).

2. Algebraic and Symmetry Structures under Infinite-Curvature Corrections

Infinite-curvature corrections are reflected in the enhancement and structure of gravitational symmetry algebras. Near-horizon symmetry analyses, especially in black hole contexts, show that the inclusion of all curvature orders organizes the space of conserved supertranslation charges into an infinite-dimensional module, explicitly tied to the BMS-like symmetry acting on horizon cross-sections (Chernicoff et al., 26 Jun 2025).

Algebraically, systematic dressing of flat-space isometry algebras by curvature corrections produces infinite-dimensional Lie algebras such as the free Lie or "Poincaré$$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$2" algebra (Gomis et al., 2020). In this construction, additional generators at each "level" correspond to noncommuting generalizations of translations and boosts, encoding the curvature corrections as a formally infinite series with each generator scaling as $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$3 for curvature radius $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$4 and level $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$5. These algebraic structures admit nonlinear realizations and encode, order by order, the $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$6-curvature corrections to geodesic equations in a fully recursive fashion.

3. Regular Black Holes and Singularity Resolution

Infinite-curvature corrections enable the construction of regular black hole solutions—spacetimes free of curvature singularities at $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$7. In $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$8-dimensional theories with a generating function $$S = \int d^D x\,\sqrt{-g} \left[ R + \sum_{n=2}^{\infty} \alpha_n \mathcal{Z}_n \right],$$9, static, spherically symmetric metrics

$\mathcal{Z}_n$0

are governed by the algebraic master equation $\mathcal{Z}_n$1 (Konoplya et al., 2024, Konoplya et al., 2024, Arbelaez, 29 Sep 2025). For suitable choices of $\mathcal{Z}_n$2, this function $\mathcal{Z}_n$3 resums the infinite series to produce nonperturbative solutions (e.g., the Dymnikova black hole, with $\mathcal{Z}_n$4 given via a Lambert $\mathcal{Z}_n$5-function (Konoplya et al., 2024)) that are nonsingular and approach a de Sitter core as $\mathcal{Z}_n$6.

Nonlocal infinite-derivative gravity (IDG) models produce similar regularization mechanisms, with the nonlocality scale $\mathcal{Z}_n$7 "smearing out" classically singular sources and rendering curvature invariants finite in pp-wave solutions—contrasting sharply with general relativity where certain curvature components diverge (Kolář et al., 2023).

Table: Core Features of Infinite-Curvature-corrected Regular BHs

Theory/Class	Regularization Mechanism	Key Black Hole Property
Lovelock/Quasi-top.	Algebraic resummation of $\mathcal{Z}_n$8	$\mathcal{Z}_n$9 near $n$0
Nonlocal/IDG	Nonlocal form-factors $n$1	All curvature invariants finite

These solutions are fundamentally nonperturbative: no finite truncation in $n$2 achieves the same singularity resolution, but for physical observables (e.g., QNM frequencies) low-order truncations suffice to approximate the full infinite-tower predictions within $n$3–$n$4 accuracy when the couplings are in the phenomenologically viable range (Konoplya et al., 2024, Konoplya et al., 2024).

4. Cosmological Models and Inflationary Singularities

Dimensional regularization of Lovelock invariants and infinite curvature sums in Horndeski-type scalar–tensor theories enable the replacement of the Big Bang singularity with a past-eternal de Sitter inflationary phase ('geometric inflation') (Fernandes, 10 Apr 2025): $n$5 with $n$6 setting both the inflationary Hubble scale and the size of the regularized black hole interior.

However, recent analyses have revealed severe pathologies: for background solutions $n$7, linear scalar perturbations are infinitely strongly coupled (vanishing kinetic term) at all times, and nonlinear perturbations grow uncontrollably. Even with off-attractor initial conditions, the kinetic coefficients for perturbations diverge or become ghostly as $n$8. The tensor sector is generically Laplacian-unstable during inflation. These features render the homogeneous, singularity-free inflationary background physically illegitimate (Tsujikawa, 26 May 2025).

5. Linear Stability and Thermodynamics: Pathologies and Degeneracies

Infinite-curvature corrections often regularize metric components and curvature invariants, yet may simultaneously induce severe dynamical instabilities:

• Regular black hole backgrounds with $n$9 in 4D exhibit vanishing kinetic terms for even-parity scalar perturbations (strong coupling), and suffer ghost and Laplacian instabilities in the odd-parity (tensor) sector near $\{\alpha_n\}$0, independent of the choice of $\{\alpha_n\}$1 (Felice et al., 15 Jul 2025). Thus, while the background is smooth, the solution is dynamically ruled out.
• For planar-horizon black holes in four-dimensional regularized curvature-tower theories, both the regular scalarized and trivial GR branches have identical thermodynamic quantities (mass, entropy, temperature, and free energy) at fixed temperature: the infinite tower of corrections is 'thermodynamically invisible' in the planar case (Cisterna et al., 29 May 2025). For spherical horizons, this degeneracy is partially lifted only when the Gauss–Bonnet (quadratic) coupling is included, shifting the free energy and potentially favoring one branch over the other depending on its sign.

6. Effect on Quasinormal Modes and Gravitational-Wave Phenomenology

The ringdown phase of regular black holes in infinite-curvature-corrected theories exhibits characteristic shifts:

• As the curvature-coupling scales $\{\alpha_n\}$2 increase, the real part of the fundamental QNM frequency $\{\alpha_n\}$3 decreases and the damping rate $\{\alpha_n\}$4 is suppressed—leading to a lower oscillation frequency and longer-lived modes relative to GR (Konoplya et al., 2024, Arbelaez, 29 Sep 2025).
• These trends persist across all spin sectors, multipoles, and dimensions $\{\alpha_n\}$5 (Arbelaez, 29 Sep 2025); overtones exhibit non-GR-like phenomenology, including unconventional modes with vanishing frequency.
• The QNM spectrum converges rapidly with only the first $\{\alpha_n\}$6 curvature corrections, further justifying the use of low-order truncations in phenomenological studies.
• Gravitational-wave observatories could, in principle, detect these deviations as percent-level shifts in the observed ringdown signal, providing indirect evidence for singularity-resolution mechanisms involving infinite curvature corrections.

7. Infinite-Series Issues: Convergence, Analytic Structure, and Physical Constraints

Infinite-curvature sums introduce nontrivial analytic and physical challenges:

• The convergence of the series is non-automatic; for regular black holes, convergence of $\{\alpha_n\}$7 must be guaranteed for $\{\alpha_n\}$8 in the relevant domain (Arbelaez, 29 Sep 2025). In effective field theory, exponential suppression of $\{\alpha_n\}$9 at high $\Box$0 is physically motivated, particularly from string theory (Chernicoff et al., 26 Jun 2025).
• Divergent or non-analytic generating functions (as in the Dymnikova solution) signify genuinely nonperturbative phenomena that are inaccessible to any finite order of curvature expansion (Konoplya et al., 2024).
• Thermodynamic stability and the validity of the generalized Wald entropy formula constrain the signs and magnitudes of $\Box$1, as negative or ghostly couplings can violate the second law (Chernicoff et al., 26 Jun 2025).
• From the symmetry side, infinite curvature corrections "dress" existing BMS-like near-horizon symmetry algebras, but do not introduce new central extensions; the semi-direct sum $\Box$2 structure persists (Chernicoff et al., 26 Jun 2025).

In summary, infinite-curvature corrections are a foundational ingredient in modern studies of gravity beyond General Relativity, enabling the construction of regular geometries, encoding symmetry enhancements, and providing rich phenomenology—while simultaneously introducing subtle analytic and dynamical constraints that are under active investigation.