Regularized 4D Gauss-Bonnet Gravity

• The framework introduces a dynamical Gauss-Bonnet term in 4D via dimensional reduction, conformal counterterms, and Kaluza-Klein methods.
• Modified black hole solutions and cosmological dynamics reveal deviations from Einstein gravity due to higher-curvature corrections.
• Rigorous perturbation analysis exposes strong coupling and gradient instabilities, constraining the physical viability of these models.

The Regularized 4-Dimensional Gauss-Bonnet Framework is a class of modified gravity models that extend Einstein-Hilbert theory by non-trivial incorporation of the Gauss-Bonnet (GB) invariant in four dimensions. In conventional four-dimensional general relativity, the GB term is topological and does not alter the classical field equations due to Lovelock’s theorem. The regularized framework achieves dynamical relevance of GB corrections by dimensional continuation, Kaluza-Klein reduction, or conformal counterterms, resulting in well-defined field equations with higher-curvature interactions, typically cast in the Horndeski or generalized Proca class. This approach fundamentally modifies black hole solutions, stellar structure, cosmological evolution, and gravitational wave phenomenology, and is subject to rigorous theoretical and observational viability constraints. Below, we systematically detail the geometric construction, action regularization procedure, field equations, canonical solutions, and stability criteria.

1. Dimensional Reduction and Regularization Schemes

The prototypical construction begins with the D-dimensional Einstein-Gauss-Bonnet action: $$S_D = \frac{1}{16\pi G_D}\int d^D x\,\sqrt{-g}\;\Bigl[R + \hat\alpha\,\mathcal{R}_{\rm GB}^{(D)}\Bigr]$$ with the Lanczos-Gauss-Bonnet combination

$$\mathcal{R}_{\rm GB}^{(D)} = R^2 - 4\,R_{AB}R^{AB} + R_{ABCD}R^{ABCD}$$

which is dynamically trivial in $D=4$ (Euler characteristic). To derive an effective 4D dynamics, one implements a singular rescaling $\hat\alpha \to \alpha/(D-4)$ and then takes $D \to 4$ after variation. The Kaluza-Klein procedure further involves compactifying on a $(D-4)$-dimensional maximally symmetric internal space with metric-size scalar $\phi$, integrating out internal coordinates, and subtracting topological counterterms (Tsujikawa, 2022). Conformal regularization yields an equivalent (up to field redefinitions) scalar-tensor action after subtracting the conformal image of the GB term (Easson et al., 2020, Fernandes et al., 2020). Variants also exist in Riemann-Cartan (torsionful) geometry (Qiu et al., 30 Oct 2025), Weyl geometry (vector-tensor) (Charmousis et al., 17 Apr 2025), and with non-Riemannian volume-forms (Guendelman et al., 2018).

2. Regularized 4D Action and Field Equations

The limiting action in a spatially flat internal space is

$$S_{\rm 4D} = \int d^4x\,\sqrt{-g}\left\{R + \alpha\left[4\,G^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi - \phi\,R_{\rm GB}^2 - 8 X\Box\phi + 8 X^2 \right]\right\}$$

where $X = -\tfrac12\nabla^\mu\phi\nabla_\mu\phi$, and $G^{\mu\nu}$ is the Einstein tensor in 4D. In Horndeski language, this is encoded as: $$G_2 = \alpha(8X^2),\quad G_3=8\alpha X,\quad G_4=1+4\alpha X,\quad G_5=4\alpha \ln|X|$$ (Tsujikawa, 2022, Easson et al., 2020). Upon variation, the modified Einstein equations are: $$G_{\mu\nu} + \alpha\,{\cal H}_{\mu\nu} = T^{(\phi)}_{\mu\nu}$$ where ${\cal H}_{\mu\nu}$ combines all higher-derivative GB interactions, and $T^{(\phi)}_{\mu\nu}$ stems from the scalar sector. The scalar field $\phi$ obeys a nonlinear PDE derived from this Horndeski structure.

3. Canonical Solutions: Black Holes and Cosmology

For static, spherically symmetric backgrounds: $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{h(r)} + r^2\,d\Omega^2,\quad \phi=\phi(r)$$ Hairy black hole solutions exist for spatially flat internal space ($\lambda=0$): $$f(r) = h(r) = 1 + \frac{r^2}{2\alpha}\bigg[1 - \sqrt{1 + \frac{8\alpha M}{r^3}}\bigg],\quad \phi'(r) = \frac{1}{r}\left[\frac{1}{\sqrt{h}-1}\right]$$ which asymptote to Schwarzschild for $\alpha \to 0$ (Tsujikawa, 2022).

In cosmology, this framework produces Friedmann equations with additional $H^4$ corrections (from the GB scalar), modifying both early-universe expansion and late-time dark energy behavior (Feng et al., 2020, Bayarsaikhan et al., 2022). The energy density and pressure associated with the GB sector are

$$\rho_{\rm GB} = -3\hat\alpha H^4,\quad p_{\rm GB} = \hat\alpha H^2(3H^2+4\dot H)$$

and the observable tensor-mode gravitational wave speed is shifted: $$c_T^2 = 1 + \frac{4\hat\alpha\dot H}{1+2\hat\alpha H^2}$$

4. Viability: Strong Coupling and (In-)Stability

Linear perturbation analysis in the regularized theory reveals critical pathologies:

• Strong coupling problem: The kinetic term for certain even-parity perturbation modes vanishes everywhere, i.e. the coefficient $\mathcal{K}(r)\equiv 0$ for all $r$, leading to the breakdown of perturbative stability (Tsujikawa, 2022).
• Gradient instability: The effective squared speeds for angular propagation of even-parity perturbations become negative or divergent near the event horizon and at spatial infinity, e.g., $c_\Omega^2\to-\infty$ near $r_+$ and $c_{r,even}^2/B_2\approx-2$ at large $r$, signaling unresolvable instability (Tsujikawa, 2022).
• Eikonal and cosmological-constant instabilities: In asymptotically de Sitter/anti-de Sitter backgrounds, one finds additional parameter-dependent unstable modes (Cuyubamba, 2020). Negative or extreme values of the GB coupling $\alpha$ can cause high-$\ell$ or low multipole instabilities.

These issues persist for any finite GB coupling after regularization and cast doubt on the physical viability of the simplest hairy black hole branch and associated cosmological backgrounds.

5. Phenomenological Implications and Observational Constraints

Nevertheless, the regularized 4D EGB framework produces explicit predictions for gravitational wave speed, black hole thermodynamics, and compact object structure (Feng et al., 2020, Övgün, 2021, Panotopoulos et al., 31 Dec 2025). Phenomenologically:

• The corrections to stellar structure (modified TOV equations) can increase maximum mass and alter compactness for neutron stars or exotic bosonic dark-matter stars (Mazhari, 17 Oct 2025, Panotopoulos et al., 31 Dec 2025).
• The existence of confining charge modifies black hole shadow and quasinormal ringdown properties, offering testable consequences via astronomical observations (EHT, LIGO/Virgo) (Övgün, 2021).
• Astrophysical and multimessenger constraints, such as from GW170817/GRB170817A, demand the dimensionless GB parameter $|\tilde\alpha|\lesssim 10^{-15}$, ruling out significant deviations from GR in the late universe unless tuned to extremely small coupling (Feng et al., 2020).

6. Generalizations and Branch Structure

Variants of the regularized framework allow non-minimal scalar coupling, vector-tensor (Proca-type) theories via Weyl geometry, explicit non-Riemannian volume forms, and torsionful Cartan extensions (Charmousis et al., 17 Apr 2025, Qiu et al., 30 Oct 2025, Bayarsaikhan et al., 2022, Guendelman et al., 2018). These generalizations escape some constraints of the scalar-tensor branch and admit additional “hair” parameters, effective cosmological constant generation via disformal transformations, and long-range torsion fields. However, the underlying regularization principle remains the singular dimensional limit (or conformal subtraction/counterterm constructions), and their relation to the original purely metric theory is formal rather than strict. Key solution families include black holes with primary vector or scalar hair, modified cosmological attractors, and domain wall spacetimes.

7. Action Principle, Uniqueness, and Theoretical Ambiguities

Canonical variational completion studies demonstrate that purely metric dimensional regularizations (without extra scalar or vector fields) yield field equations with no consistent finite action in $D=4$; instead, the limiting Lagrangian diverges and does not admit a well-defined variational principle (Hohmann et al., 2020, Mahapatra, 2020). Construction via conformal counterterm or dimensional reduction generically introduces auxiliary scalar or vector degrees of freedom, yielding Horndeski-class or generalized Proca actions. Scheme dependence of the boundary counterterms and nonuniqueness of the bulk 4D action remain unsolved issues (Mahapatra, 2020).

Summary Table: Regularized 4D Gauss-Bonnet Construction

Scheme	Key Features	Instabilities/Issues
Dimensional rescaling ($D\to4$)	Scalar-tensor action in Horndeski class	Strong coupling, gradient instability (Tsujikawa, 2022)
Kaluza-Klein reduction	Scalar dilaton from flat internal space	Equivalent to Horndeski, perturbed instability
Conformal counterterm	Uses subtraction of conformal GB term, scalar emerges	Well-posed action, extra degree of freedom
Weyl/Proca generalization	Vector-tensor, primary hair, disformal cosmology	No canonical metric-only action
Non-Riemannian volume-form	Integration constant GB term, non-analytic electrodynamics	Transcends standard metric theory
Riemann-Cartan (torsionful)	Intrinsic torsion hair sourced by regularized GB term	Novel black hole and cosmology phenomenology

The regularized 4D Einstein-Gauss-Bonnet paradigm provides a mathematically rigorous vehicle for higher-curvature corrections beyond GR in four-dimensional spacetimes. While the action is closed and second order in the Horndeski or generalized Proca class, physical viability is limited by unavoidable strong coupling and perturbative instabilities in the simplest branches, and significant observational constraints restrict deviations from GR to be minute in cosmology and astrophysics. Generalizations (vector, torsion, non-minimal coupling) remain an active area of research for extending the theory’s applicability and phenomenological signatures.