Lovelock Gravity: Concepts and Applications

• Lovelock gravity is a diffeomorphism-invariant theory in D>4 that uses polynomial curvature invariants to yield second-order field equations and avoid Ostrogradsky ghosts.
• It features a rich vacuum structure, including multiple maximally symmetric and product-space solutions, which are key to understanding spontaneous compactification and symmetry breaking.
• Extensions to scalar-tensor and conformal frameworks enable the theory to model generalized black holes, cosmological dynamics, and non-perturbative phenomena in higher dimensions.

Lovelock gravity is the unique, most general diffeomorphism-invariant metric theory in $D>4$ dimensions whose field equations are of second order in the metric and free of Ostrogradsky ghost instabilities. It is constructed as a polynomial in the curvature, with each term being the dimensionally continued Euler density. The theory encompasses the Einstein-Hilbert action as its first-order term and, in higher dimensions, admits further nontrivial curvature invariants such as the Gauss-Bonnet ($k=2$) and higher-order Lovelock terms. Its algebraic structure underlies a diverse set of gravitational phenomena, including multi-vacuum structure, symmetry breaking, spontaneous compactification, generalized black holes, scalar-tensor couplings, and unique cosmological behaviors.

1. Lovelock Action and Second-Order Field Equations

Lovelock gravity is defined through the action

$$S = \frac{1}{16\pi G_D} \int d^D x\,\sqrt{-g}\,\sum_{k=0}^{K_\text{max}} \alpha_k\,\mathcal{L}_k,$$

where each Lovelock Lagrangian density for order $k$ is given by

$$\mathcal{L}_k = \frac{1}{2^k}\,\delta^{\mu_1\nu_1\cdots\mu_k\nu_k}_{\alpha_1\beta_1\cdots\alpha_k\beta_k} \prod_{r=1}^k R^{\alpha_r\beta_r}{}_{\mu_r\nu_r},$$

with the generalized Kronecker delta enforcing antisymmetry, and $\alpha_k$ are dimensionful couplings. In $D$ dimensions, $K_\text{max} = \lfloor(D-1)/2\rfloor$ is dictated by the vanishing of Euler densities beyond this order.

Varying the action yields field equations of the schematic form: $$\sum_{k=0}^{K_\text{max}} \alpha_k G^{(k)}_{\mu\nu} = T_{\mu\nu},$$ where each $G^{(k)}_{\mu\nu}$ is a generalized "Einstein tensor" involving contractions of $k$ Riemann tensors but only two derivatives of the metric. The Lovelock property ($\nabla_a X^{(k)abcd} = 0$ for the relevant momenta) guarantees unitarity, and the linearized equations remain second order, forbidding higher-derivative ghosts (Brustein et al., 2012).

2. Vacuum Structure, Symmetry Breaking, and Product-Space Solutions

A maximally symmetric vacuum in Lovelock gravity satisfies

$$R_{\mu\nu\rho\sigma} = \frac{2\Lambda_\mathrm{eff}}{{(D-1)(D-2)}} \left(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}\right),$$

and yields, upon substitution, a characteristic polynomial for $\Lambda_\mathrm{eff}$,

$\sum_{k=0}^K c_k\,\Lambda_\mathrm{eff}^k = 0,$

with $c_k$ determined by $D$ and the $\alpha_k$ (Kastor et al., 2015). This polynomial admits up to $K$ real roots—corresponding to distinct maximally symmetric vacua (Minkowski, dS, AdS).

Critical surfaces in coupling space (where the discriminant vanishes) separate regions with differing numbers of real vacua. In those symmetry-breaking sectors where no fully symmetric vacuum survives, Lovelock gravity admits reduced symmetry, product-space vacua (manifolds of the form $\mathcal{M}_p \times \mathcal{N}_{D-p}$, each factor maximally symmetric with curvatures $K_1$, $K_2$).

Explicit studies in Gauss-Bonnet gravity ($K=2$):

• In $D = 5$, the symmetry-breaking regime is everywhere covered by product vacua of $AdS_3 \times S^2$ or $S^3 \times AdS_2$ type.
• In $D = 6$, there exists a strip in $(\alpha_2, \Lambda_0)$-space where neither maximally symmetric nor simple product vacua exist, indicating the presence of more intricate vacuum structure, including higher-factor or warped products.

Vacuum phase transitions occur when couplings cross critical values. These phenomena have key applications in spontaneous compactification scenarios, and the stability analysis of each vacuum is crucial for physical relevance (Kastor et al., 2015, Nishida, 2013).

3. Spontaneous Dimensional Reduction and Emergent Lower-Dimensional Gravity

If only a single Lovelock term of rank $p$ is used, the theory admits as vacuum a direct product $M^{m} \times S^{n}$, with $D = m + n$ and $S^n$ a sphere of arbitrary radius $r$ (Nishida, 2013). Specifically, in $m = 4$, all Lovelock densities beyond $k=1$ vanish.

An algebraic condition $n+1 \le 2p < n+3$ ensures that only the Einstein–Hilbert term survives after reduction. The resultant effective action is: $$S_\mathrm{eff} = \frac{1}{2\kappa_4^2} \int d^4x\,\sqrt{-g_{(4)}}\,R^{(4)},$$ with Newton's constant determined by the $D$-dimensional coupling, $p$, $n$, and $r$. The 4D cosmological constant vanishes identically. Hence, spontaneous compactification in Lovelock gravity can yield conventional vacuum general relativity in four dimensions absent a bare cosmological constant (Nishida, 2013).

4. Black Hole Solutions, Regularity, and Non-Perturbative Features

The pure Lovelock vacuum equations support spherically symmetric black hole metrics of the generalized form: $$ds^2 = -\mu(r) dt^2 + \frac{dr^2}{\mu(r)} + r^2 d\Sigma^2,$$ where $$\mu(r) = \gamma \pm \left(2 m(r)/r^{d-2n-1}\right)^{1/n}$$ and $d\Sigma^2$ represents the transverse geometry, which may be spherical or hyperbolic (Estrada et al., 2024). When regularized matter profiles are introduced, singularities of Lovelock black holes can be smoothed, yielding de Sitter (dS) or anti-de Sitter (AdS) cores and extremal remnants.

For odd Lovelock order $n$, the transverse geometry is always spherical; for even $n$, it can be hyperbolic. Critical spacetime dimension $d = 2n + 1$ entails special features—trivial vacuum solutions that are Lovelock-flat but generally not Riemann-flat, akin in the $n=1$ case to the global monopole solution (Dadhich et al., 2012). The addition of a cosmological constant yields higher-dimensional analogues of the BTZ black hole, with horizon structure and thermodynamics governed by the Lovelock order and couplings.

In the regularized large-order ($t\to\infty$) Lovelock theory, the black hole singularity is provably eliminated, with all curvature invariants remaining finite at $r=0$, provided unique vacuum conditions are enforced (Casalino et al., 2020, Estrada et al., 2024).

5. Scalar-Tensor and Horndeski–Lovelock Extensions

Generalizations of Lovelock gravity include the coupling to scalar fields in ghost-free, second-order form:

• Lovelock scalar-tensor gravity (e.g., Lovelock–Brans–Dicke) features a scalar parameter $\phi$ multiplying curvature invariants (e.g., Ricci scalar, Gauss–Bonnet, Chern–Pontryagin), with equations reducing to GR for large Brans–Dicke parameter unless a topological balance condition is met (Tian et al., 2015).
• Coupling of scalars kinetically to Lovelock tensors generalizes the Horndeski framework; the equations of motion remain strictly second-order, and novel cosmological solutions arise in higher-dimensional backgrounds (Gao, 2018).

The Lovelock–Horndeski model in seven dimensions yields cyclic, eternally expanding, and multiple de Sitter cosmologies, with the rich phenomenology set by the scalar coupling to the Lovelock sector.

6. Cosmological Dynamics and Universal Solution Structure

Lovelock gravity modifies Friedmann equations by introducing higher-order terms in the Hubble parameter: $$\sum_{p=0}^n \widetilde{\alpha}_p \left(H^2 + \frac{k}{a^2}\right)^p = F(\rho; d),$$ with $\widetilde\alpha_p$ absorbing dimensional and coupling factors (Nikolaev, 2024, Sheykhi et al., 2011). The presence of multiple roots yields both universal (Type II) solutions identical to GR and unique (Type I) Lovelock-driven branches, including vacuum exponential expansions (Lovelock de Sitter solutions) and new pressure-free open-universe solutions with scale factor $a(t) \propto 1/\sinh(C_1 t)$, unattainable in classical GR.

Non-perturbative regularized Lovelock gravity gives rise to attractor and bounce cosmologies, with late-time de Sitter fixed points robust against linear perturbations (Casalino et al., 2020, Nikolaev, 2024).

7. Mathematical Structure, Conformal Extensions, and Well-Posedness

Lovelock gravity admits higher-curvature generalizations of familiar conformal tensors—Weyl, Schouten, Cotton, Bach—each constructed from antisymmetrized products of Riemann tensors and traced to ensure conformal invariance in appropriate dimensions ($D=4k$) (Kastor, 2013). Conformal Lovelock theories leverage these tensors to build conformally invariant actions and study renormalization and holographic properties in $AdS/CFT$ contexts.

The Cauchy problem is locally well-posed when the harmonic-gauge-reduced Lovelock equations are cast as a quasilinear, symmetric hyperbolic system, ensuring existence, uniqueness, and continuous dependence on data at the same level of rigor as GR (Willison, 2014).

• Symmetry Breaking and Vacua: (Kastor et al., 2015)
• Spontaneous Dimensional Reduction: (Nishida, 2013)
• Regular Black Holes and Critical Dimensions: (Estrada et al., 2024, Dadhich et al., 2012)
• Scalar-Tensor and Horndeski–Lovelock: (Tian et al., 2015, Gao, 2018)
• f(Lovelock) Generalization: (Bueno et al., 2016)
• Regularized Lovelock and Non-perturbative Limit: (Casalino et al., 2020)
• Cosmology and Universal Solutions: (Nikolaev, 2024)
• Entropic Force and Thermodynamics: (Sheykhi et al., 2011)
• Conformal Lovelock Gravity: (Kastor, 2013)
• Hamiltonian Structure: (Kunstatter et al., 2012)
• Local Well-posedness: (Willison, 2014)