Gravity with higher-curvature terms and second-order field equations: $f(\mathcal{R})$ meets Gauss-Bonnet

Abstract

General Relativity is expected to break down in the high-curvature regime. Beyond an effective field theory treatment with higher-order operators, it is important to identify consistent theories with higher-curvature terms at the nonperturbative level. Two well-studied examples are $f(\mathcal{R})$ gravity and Einstein-dilaton-Gauss-Bonnet (EdGB) gravity. The former shares the same vacuum solutions as General Relativity, including black holes, while the latter suffers from well-posedness issues due to quadratic curvature terms in the strong-coupling regime. We show that combining these two theories leads to genuinely new phenomena beyond their simple superposition. The resulting framework falls outside Horndeski's class, as it can be recast as a gravitational theory involving two nonminimally coupled scalar fields with nontrivial mutual interactions. This construction naturally extends EdGB gravity to include arbitrary higher-curvature terms, providing a versatile setting to address fundamental questions. Focusing on quadratic and quartic corrections, we find that: (i) black holes are modified by $f(\mathcal{R})$ terms, unlike the case without Gauss-Bonnet interactions; (ii) the resulting solutions retain the qualitative nonperturbative features of EdGB black holes with certain couplings, such as a minimum mass and multiple branches; (iii) a nontrivial mechanism suppresses the divergence of the Ricci scalar in the black-hole interior; (iv) still, even with quartic corrections, the singularity structure and elliptic regions inside the horizon remain similar to those of pure EdGB gravity. This suggests that, at the nonperturbative level, the theory's ill-posedness cannot be resolved by adding individual higher-order terms. This conjecture could be tested by studying the nonlinear dynamics, which remains governed by second-order field equations.

14 pages, 8 figures, ancillary Mathematica notebook provided as supplemental material

References in corpus (33)

• f(R) Theories Of Gravity
• Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models
• Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution
• From k-essence to generalised Galileons
• Black hole hair in generalized scalar-tensor gravity
• Black holes in scalar-tensor gravity
• Black hole hair in generalized scalar-tensor gravity: An explicit example
• Dilatonic Black Holes with Gauss-Bonnet Term
• Distinguishing Boson Stars from Black Holes and Neutron Stars from Tidal Interactions in Inspiraling Binary Systems
• Dilatonic Black Holes in Higher-Curvature String Gravity II: Linear Stability
• Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations
• Dynamics of spontaneous black hole scalarization and mergers in Einstein-scalar-Gauss-Bonnet gravity
• Evolution of Einstein-scalar-Gauss-Bonnet gravity using a modified harmonic formulation
• Well-posed formulation of Lovelock and Horndeski theories
• Black hole solutions with dilatonic hair in higher curvature gravity
• Scalarized Black Hole dynamics in Einstein dilaton Gauss-Bonnet Gravity
• Robust approach to f(R) gravity
• Gravitational Collapse in Einstein dilaton Gauss-Bonnet Gravity
• Nonlinear studies of binary black hole mergers in Einstein-scalar-Gauss-Bonnet gravity
• Hyperbolicity in Spherical Gravitational Collapse in a Horndeski Theory
• Dynamics of a Z2 symmetric EdGB gravity in spherical symmetry
• Black Hole Relics in String Gravity: Last Stages of Hawking Evaporation
• Black hole scalarization with Gauss-Bonnet and Ricci scalar couplings
• Towards numerical relativity in scalar Gauss-Bonnet gravity: 3+1 decomposition beyond the small-coupling limit
• Particle-like ultracompact objects in Einstein-scalar-Gauss-Bonnet theories
• Causality in gravitational theories with second order equations of motion
• Black Hole Scalarization in Gauss-Bonnet Extended Starobinsky Gravity
• Late-Time Cosmology of Scalar-Coupled $f(R,\mathcal{G})$ Gravity
• Properties of ultra-compact particle-like solutions in Einstein-scalar-Gauss-Bonnet theories
• Nonperturbative gedanken experiments in Einstein-dilaton-Gauss-Bonnet gravity: nonlinear transitions and tests of the cosmic censorship beyond General Relativity
• What is the fate of Hawking evaporation in gravity theories with higher curvature terms?
• Breaking black-hole uniqueness at supermassive scales
• Stable non-linear evolution in regularised higher derivative effective field theories