Einstein-Gauss-Bonnet Gravity

• Einstein-Gauss-Bonnet Gravity is a higher-curvature theory that augments the Einstein-Hilbert action with a quadratic Gauss-Bonnet term, producing second-order field equations in dimensions greater than four.
• It underpins rich phenomenology including modified black hole thermodynamics, early universe inflation or bouncing scenarios, and novel scalar-tensor formulations in four dimensions.
• The theory leverages dimensional reduction and regularization techniques to incorporate string-theoretic and quantum gravity corrections while avoiding Ostrogradsky instabilities.

Einstein-Gauss-Bonnet Gravity is a class of higher-curvature modifications of general relativity that incorporates the Gauss-Bonnet (GB) term—a particular combination of quadratic curvature invariants—into the gravitational action. Distinctive for generating second-order field equations without introducing ghosts in dimensions greater than four, it is central to both fundamental theoretical investigations (including string-theoretic corrections, braneworld scenarios, and quantum gravity) and in the construction of novel black hole and cosmological solutions. In four dimensions, implementation of Gauss-Bonnet corrections requires special limiting prescriptions and typically leads to scalar-tensor (Horndeski) extensions.

1. Foundations and Action Principle

The D-dimensional Einstein-Gauss-Bonnet action generalizes Einstein-Hilbert by the inclusion of the GB density $\mathcal{G}_{D}$: $$S = \int d^D x \sqrt{-g}\ \Big[ R - 2\Lambda + \alpha\, \mathcal{G}_D \Big] + S_{\text{matter}}$$ where

$$\mathcal{G}_D = R^2 - 4 R_{ab} R^{ab} + R_{abcd}R^{abcd}$$

and $\alpha$ is the dimensionful Gauss-Bonnet coupling. For $D>4$, $\mathcal{G}_D$ is no longer topological and yields nontrivial contributions to the equations of motion: $G_{ab} + \alpha\, H_{ab} = T_{ab}$ with $H_{ab}$ built from contractions of two Riemann tensors as in Lovelock's construction. The theory thus evades Ostrogradsky instabilities and preserves only the massless graviton as its propagating degree of freedom (Fernandes et al., 2022, Glavan et al., 2019).

In $D=4$, $\mathcal{G}_4$ is a total derivative and does not contribute to the classical field equations if added directly. Modifications to induce dynamics require either:

• Rescaling: $\alpha \to \alpha/(D-4)$ with subsequent $D\to4$ limit at the level of the equations of motion or, equivalently, compactification/Kaluza-Klein reductions (Glavan et al., 2019, Fernandes et al., 2022, Easson et al., 2020, Bruck et al., 2018).
• Scalar-tensor generalization: Coupling $\mathcal{G}_4$ to a dynamical scalar $\phi$ such that nontrivial variations survive in four dimensions. The resulting theory falls within the Horndeski class and maintains second-order field equations (Fernandes et al., 2022, Easson et al., 2020).

2. Black Hole Solutions and Thermodynamics

In higher dimensions ($D\geq5$), static black hole solutions in Einstein-Gauss-Bonnet gravity include the Boulware-Deser solution. For example, in $D=5$: $$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega_3^2,$$

$$f(r) = 1 + \frac{r^2}{4\alpha} \left(1 \mp \sqrt{1 + \frac{8\alpha \mu}{r^4}}\right)$$

where the minus branch recovers general relativity as $\alpha \to 0$ (Kleihaus et al., 2012). Black hole entropy deviates from the Bekenstein-Hawking area law: $$S_H = \frac{1}{4G}\int_{\mathcal{H}} d^{D-2}x\, \sqrt{h} (1 + \alpha \widetilde{R})$$

In four dimensions, black hole solutions require the regularized GB limit and yield corrections to thermodynamic quantities: $$S = \frac{A}{4} + 2\pi\alpha\ln\frac{A}{A_0},\quad A=4\pi r_+^2$$ with the metric function (GR-branch, asymptotically flat) (Fernandes et al., 2022): $$f(r)=1+\frac{r^2}{2\alpha}\left[1-\sqrt{1+\frac{8\alpha M}{r^3}}\right]$$ For spherically symmetric gravitational collapse, the GB term generically increases the time to singularity formation, moderates the strength of the curvature singularity, and can impact the visibility of singularities as in $5D$ collapse, sometimes leading to cosmic censorship violation (Ghosh et al., 2010).

The topological nature of $\mathcal{G}_4$ introduces an additive, Euler-characteristic-dependent contribution to entropy: $$S = \frac{A}{4G} + \frac{\alpha\pi}{G}\chi_\mathcal{H}$$ This term does not cause second law violation in physically meaningful regimes ($|\alpha|/G \ll 1$) for mergers in both asymptotically flat and AdS spacetimes (Chatterjee et al., 2013).

3. Cosmological Dynamics and Early Universe Phenomenology

Einstein-Gauss-Bonnet gravity modifies the Friedmann equation in both $4D$ and higher-dimensional FLRW cosmologies. In $D=4$: $H^2 + \alpha H^4 = \frac{8\pi G}{3}\rho$ and, in $D=5$: $$H^2 + k/a^2 + \alpha(H^2 + k/a^2)^2 = \frac{\kappa^2_5}{6}\rho$$ At high curvature (early universe), the $H^4$ term can drive inflationary or bouncing scenarios even without a potential (Gomez et al., 2022, Sberna, 2017). Acceleration, nonsingular bounces, and late-time de Sitter self-acceleration can emerge, with the GB coupling dictating the regimes.

In scalar-Gauss-Bonnet (sGB) cosmologies, the action is: $$S = \int d^4x\sqrt{-g}\left[\frac{1}{2}R - \frac{1}{2}(\nabla\phi)^2 - V(\phi) + \frac{1}{8}f(\phi)\mathcal{G}\right]$$ Analytic families of exact (power-law, exponential) and reconstructed solutions exist for various $f(\phi)$ and $V(\phi)$. A key finding is that, in the absence of a potential ($V=0$), all nonsingular or bouncing branches are generically unstable to tensor (gravitational wave) perturbations unless a sufficiently large potential term is introduced (Sberna, 2017, Chakraborty et al., 2018, Fomin et al., 2017). Viable inflationary models satisfying Planck and GW170817 constraints are possible, with non-minimal couplings enabling blue-tilted tensor spectra and potentially detectable primordial gravitational waves (Oikonomou et al., 22 Mar 2025).

4. Dimensional Reduction, Kaluza-Klein Origins, and Effective Theories

From a higher-dimensional Lovelock gravity perspective, dimensional reduction (over extra dimensions with a scale factor/radion $b(x)$) yields $4D$ scalar-tensor (Horndeski) theories where the scalar degree of freedom encodes the size of the compact space. For the Gauss-Bonnet combination, the reduction leads uniquely to second-order (ghost-free) dynamics: $$S_{4D} = \int d^4x\sqrt{-g}\left[G_4(b, X)R + P(b, X) + \cdots\right]$$ with $X=-\frac{1}{2}(\partial b)^2$ and $G_4$, $P$ as specified functions of $b$ and the higher-dimensional geometry (Bruck et al., 2018).

Consistent reduction to 4D Einstein gravity with arbitrary effective gravitational and cosmological constants can be achieved by flux-dressing the internal space with $p$-form fields and tuning the Lovelock couplings (Canfora et al., 2021). The physical 4D theory thereby inherits novel corrections (additional entropy term, shifted mass formulae) and new classes of black brane/string solutions.

5. Boundary Terms, Asymptotics, and Gravitational Radiation

Einstein-Gauss-Bonnet gravity requires precise boundary term constructions for a well-posed variational principle. The Gauss-Bonnet-Katz boundary term, obtained via continuation of the Chern-Weil transgression, provides such a term that recovers the standard Gibbons-Hawking(-Myers) boundary term in appropriate product backgrounds. This ensures the correct mass formulae (Boulware-Deser, etc.) and regularizes infrared divergences (Deruelle et al., 2017).

In asymptotic analysis, scalar-tensor reductions of Gauss-Bonnet gravity (Horndeski-type) present nontrivial but non-radiative scalar behavior: the news function associated to the scalar field vanishes, and thus no new radiative degrees arise at null infinity. The Gauss-Bonnet modifications enter only at subleading (post-Newtonian) order in waveforms and have no leading-order effect on Bondi mass loss (Lu et al., 2020).

6. Stability, Emergent Universes, and Dynamical Dimensional Reduction

In $4D$ Einstein-Gauss-Bonnet cosmology, static universe solutions exist for both closed and open spatial curvature and are stable against both homogeneous and inhomogeneous scalar perturbations for suitable equations of state ($w$ ranges) (Li et al., 2020). This supports "emergent universe" scenarios in which a static phase naturally precedes inflation.

In certain classes of coupling parameters, $5D$ Einstein-Gauss-Bonnet gravity admits no maximally symmetric vacuum, and the physically realized vacua are warped-product spacetimes that dynamically reduce to $4D$ on large scales. This provides a compelling mechanism for the spontaneous emergence of four macroscopic dimensions without ad hoc compactification (Izaurieta et al., 2012).

7. Open Issues, Constraints, and Phenomenological Implications

While Einstein-Gauss-Bonnet gravity is free from Ostrogradsky ghosts and admits a broad class of analytically tractable solutions, several issues remain:

• The ambiguity and physical viability of the $D\to4$ limit: Scalar-tensor (Horndeski) completions are consistent; naive rescaled approaches are generally ill-posed (Easson et al., 2020, Fernandes et al., 2022).
• Strong coupling and extra degrees of freedom in time-dependent/higher-order backgrounds.
• Constraints from observations (e.g., GW170817) force the GB coupling to be small: $|\alpha| \lesssim 10^9$–$10^{17}\ \mathrm{m}^2$ depending on the probe (Fernandes et al., 2022).
• Nontrivial Gauss-Bonnet corrections generically produce higher-curvature modifications detectable only in high-energy/early-universe or strong-gravity regimes.
• Open directions include construction of rotating solutions in $4D$ scalar-Gauss-Bonnet gravity, full nonlinear simulations of mergers, and the statistical foundation for GB-induced entropy terms.

Einstein-Gauss-Bonnet gravity thus provides a rigorous, constrained, and phenomenologically rich framework for exploring higher-curvature corrections to general relativity in both higher and, via appropriate limits, four dimensions (Fernandes et al., 2022, Gomez et al., 2022, Ghosh et al., 2010, Deruelle et al., 2017).