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

TL;DR

This work constructs and analyzes f(𝑅)-dGB gravity, a non-Horndeski theory that combines $f(\mathcal{R})$ gravity with EdGB terms and can be reformulated as a two-scalar (bi-scalar) theory. By focusing on static, asymptotically flat BH solutions, the authors reveal that $f(\mathcal{R})$ corrections modify BH geometry yet preserve EdGB’s nonperturbative features such as a minimum mass and multiple branches; they also identify a mechanism that suppresses the Ricci scalar divergence inside the horizon, while the overall singularity structure and elliptic regions remain akin to EdGB. A hyperbolicity analysis shows that the theory’s dynamical system becomes strongly degenerate inside BHs, with an interior elliptic region persisting despite higher-curvature terms, indicating that ill-posedness is not resolved nonperturbatively by simply adding higher-curvature operators. The results underscore both the potential and the limitations of nonperturbative UV extensions of gravity and motivate nonlinear evolution studies of BH formation and evaporation in this framework, as well as the exploration of more complete UV completions beyond finite higher-curvature towers.

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.

Figures (8)

• Figure 1: Exterior profile of the scalar field $\chi$ for a BH solution in the quadratic case ($n = 2$). The orange solid and the blue dotted lines denote the profile before and after the application of the symmetry transformation \ref{['eq:symmetrylambda']}, respectively. The green dashed line denotes the asymptotic behavior in Eq. \ref{['eq:chiExpansionInfQuadratic']}. While the application of the symmetry generally leaves the profile of $\chi$ unaltered, as expected, large discrepancies appear close to the outer boundary, probably due to the high sensitivity of these solutions to numerical errors.
• Figure 2: Exterior profiles of the scalar fields for a BH solution in the quartic case ($n = 4$). Upper panel: behavior of $\chi$. Colors and conventions are the same as in the upper panel of Fig. \ref{['fig:SolutionProfilesQuadratic']}, with the asymptotic behavior being now given by Eq. \ref{['eq:chiExpansionInfQuartic']}. The application of the symmetry transformation \ref{['eq:symmetrylambda']} introduces errors close to the outer boundary, as in the quadratic case. Lower panel: behavior of $\phi$ after applying the transformation \ref{['eq:symmetrylambda']} (orange solid line), compared against the asymptotic behavior in Eq. \ref{['eq:phiExpansionInf']} (blue dashed line).
• Figure 3: Event horizon radius as a function of BH mass, both normalized by the square root of the dilaton coupling constant $\lambda$, for different theories. The dashed gray line corresponds to EdGB theory with the exponential coupling \ref{['eq:DilatonicCoupling']}. The other curves correspond, instead, to the $f(\mathcal{R})$-dGB theory, with $f(\mathcal{R}) = \mathcal{R} + \kappa \mathcal{R}^n$. Solid lines correspond to the quadratic case ($n = 2$) for different values of $\kappa/\widetilde{M}^2=\left(1,10,100,1000\right)$, corresponding to $\ell/\widetilde{M} \approx \left(1, 3.16, 10, 31.6\right)$ in terms of the reference length scale $\ell$ (violet, green, orange and blue, respectively). The dashed red line corresponds to the quartic case ($n = 4$) with $\kappa = 10^{15} \, \widetilde{M}^6$, corresponding to $\ell \approx 316 \, \widetilde{M}$.
• Figure 4: Kretschmann scalar (solid curves) as a function of the distance from $r_\text{S}$ for representative BH solutions of $f(\mathcal{R})$-dGB gravity, with particular values of $\kappa$ and different values of the dilaton coupling constant $\lambda$. The top (bottom) panel shows the quadratic (quartic) $f(\mathcal{R})$–dGB case. In both figures, the vertical dashed lines mark the position of the horizons of each solution and are color–coded consistently with the corresponding solid curves.
• Figure 5: Behavior of the curvature invariants as a function of $r-r_\text{S}$ in EdGB, quadratic and quartic $f(\mathcal{R})$-dGB theories (top, middle and bottom panels, respectively), shown for representative values of the coupling constants $\kappa$ and $\lambda$. The gray dotted line marks a reference scaling $\sim (r-r_\text{S})^{-1}$ for the Kretschmann scalar in the interior region. In all panels, the vertical dot-dashed line indicates the location of the event horizon.