Quasi-Topological Black Holes in Higher Dimensions

Updated 18 August 2025

Quasi-topological black holes are solutions in higher-derivative gravity, reducing complex field equations to algebraic polynomials for static, symmetric metrics.
They incorporate specific cubic and higher-order curvature invariants that modify horizon structures and thermodynamic properties, including entropy and phase behavior.
These models offer rich holographic insights by tuning dual CFT central charges and revealing van der Waals–like and multi-critical phase transitions.

Quasi-topological black holes are solutions to higher-derivative gravitational theories—specifically quasi-topological gravity—that admit analytic or algebraic black hole metrics in more than four dimensions with higher curvature corrections beyond the Einstein-Hilbert and Gauss–Bonnet terms. Distinguished by the presence of specific cubic and higher-order curvature invariants, these theories are engineered so that, despite their generically fourth- and higher-order equations of motion, highly symmetric (static, spherically symmetric or AdS-planar) ansätze reduce the field equations to algebraic polynomial constraints for the metric function. This property maintains the calculability of black hole solutions and their thermodynamics, and expands the class of dual conformal field theory (CFT) models accessible via holography.

1. Formulation of Quasi-topological Gravity and Black Hole Ansatz

The foundational quasi-topological gravity action in $D \geq 5$ involves the Einstein–Hilbert term, Gauss–Bonnet (GB) term (with coupling $\lambda$ ), and a specifically constructed cubic curvature invariant (with coupling $\mu$ ) (Myers et al., 2010). The action is schematically

$S = \int d^Dx \sqrt{-g} \left[ \frac{1}{16\pi G_D}\left(R - 2\Lambda + \lambda\, \mathcal{L}_2 + \mu\, \mathcal{L}_3\right) + \ldots \right]$

where $\mathcal{L}_2$ is the GB term and $\mathcal{L}_3$ denotes the quasi-topological cubic invariant, carefully fixed to ensure the reduction of the full equations of motion to an algebraic polynomial for metrics of high symmetry.

For example, in $D=5$ , imposing the planar AdS black hole ansatz,

$ds^2 = \frac{r^2}{L^2}\left[-N(r)^2 f(r)\,dt^2 + dx^2 + dy^2 + dz^2\right] + \frac{L^2}{r^2 f(r)} dr^2,$

leads to a single algebraic constraint on $f(r)$ : $1 - f(r) + \lambda f(r)^2 + \mu f(r)^3 = \frac{\omega^4}{r^4}$ where $\omega$ is a mass parameter and $L$ gives the AdS scale (Myers et al., 2010). For hyperbolic or spherical horizons, $f(r)$ at the horizon is shifted, and in higher dimensions the right-hand side is generalized to $\omega^{D-1}/r^{D-1}$ .

This prescription generalizes to higher-order (quartic and quintic) quasi-topological gravities, where further scalar combinations of the Riemann tensor can contribute but are chosen so that the field equations for spherically (or planar) symmetric spacetimes remain algebraic polynomials of degree equal to the highest order of the curvature invariants present (Olamaei et al., 2023, Ghanaatian et al., 2013, Ghanaatian et al., 2014).

2. Structure and Characteristics of Black Hole Solutions

Polynomial Determination: For each instance (cubic, quartic, quintic, ...) the black hole solution is determined by the corresponding algebraic polynomial, e.g.,

$\hat\mu_5 \Psi^5 + \hat\mu_4 \Psi^4 + \hat\mu_3 \Psi^3 + \hat\mu_2 \Psi^2 - \Psi + \kappa = 0$

with $\Psi$ an auxiliary function related to the metric (typically involving $f(r)$ ), $\kappa$ containing the mass, charge, and cosmological constant, and $\hat\mu_k$ the coefficients for higher-curvature interactions (Olamaei et al., 2023).

Vacua and Branch Selection: The roots of this polynomial not only determine the black hole horizon structure, but also specify the possible maximally symmetric vacua (for fixed $r$ , constant $f$ ). Only one of the roots typically provides a suitable AdS asymptotic and is ghost-free. The sign of the derivative of the polynomial at the vacuum, $-\partial_f P(f_\infty)$ , governs the absence of ghost-like graviton modes.

Horizon Structure: Unlike Einstein or GB gravity, quasi-topological corrections can support more intricate horizon structures. For instance, in the presence of negative cubic/quartic couplings, a double-horizon geometry may arise even for uncharged, spherically symmetric black holes (Brenna et al., 2012, Ghanaatian et al., 2018). For hyperbolic topology, even three horizon solutions can appear in quintic theories (Olamaei et al., 2023).

Special Cases:

For Einstein-Gauss-Bonnet-quasi-topological models in $D=5$ , the cubic equation for the metric function can be solved analytically, although only one root is physically acceptable for ghost-free/causal propagation.
For quartic and higher-order theories, the polynomial equations typically require numerical or perturbative solution, though analytic structures emerge in decoupling or extremal limits (Ghanaatian et al., 2013, Ghanaatian et al., 2014, Olamaei et al., 2023).
For example, pure quintic truncation admits exact expressions for $f(r)$ , illuminating the isolated effect of the highest-order term (Olamaei et al., 2023).

3. Thermodynamics and Holographic Properties

Entropy and Wald Corrections: The black hole entropy receives higher-derivative corrections calculable via the Wald formula,

$S = \frac{A}{4} \left[ 1 + 2k\hat{\mu}_2 \frac{(n-1)L^2}{(n-3)r_+^2} - 3k^2\hat{\mu}_3 \frac{(n-1)L^4}{(n-5)r_+^4} + 4k^3\hat{\mu}_4 \frac{(n-1)L^6}{(n-7)r_+^6} - 5k^4\hat{\mu}_5 \frac{(n-1)L^8}{(n-9)r_+^8} \right]$

where $r_+$ denotes the location of the event horizon and $k$ is the horizon curvature (Olamaei et al., 2023, Ghanaatian et al., 2013). The area law is violated generically in presence of higher-curvature terms.

First Law and Mass: The first law $dM = T dS + \Phi dQ$ extends as usual. The temperature is obtained from the surface gravity, which, after fixing $f(r_+)=0$ , must account for the nontrivial dependence (often only accessible numerically) of the solutions on $r_+$ . The mass is defined via background subtraction or thermodynamic conjugacy, maintaining consistency with the extended first law.

Thermal Stability: The Hessian of the mass with respect to $S$ and $Q$ is used to paper global thermodynamic stability, with AdS backgrounds generally supporting stable large black holes for admissible coupling ranges, while flat and dS backgrounds fail to provide overlapping regions of positive temperature and positive-definite Hessian (Ghanaatian et al., 2018, Olamaei et al., 2023).

Holography and CFT Duals: Quasi-topological gravity with three independent gravitational couplings offers the capacity to dial dual CFT central charges (not just the central charge but also the parameter controlling the three-point function of the stress tensor in $d=4$ ), extending the universality class accessible by AdS/CFT correspondence beyond Einstein and GB gravity (Myers et al., 2010). The fact that linearized graviton fluctuations about AdS backgrounds reduce to second-order equations ensures a simple dual CFT stress-tensor structure at the two-point (but not three- or higher-point) level (Myers et al., 2010).

4. Generalizations, Matter Couplings, and Rotating Black Holes

Lifshitz and Non-AdS Geometries: By coupling massive gauge fields or matter sectors, quasi-topological black holes can be constructed for Lifshitz-invariant spacetimes (metrics with anisotropic scaling). The higher-curvature terms act as effective matter, sometimes allowing Lifshitz geometries even in the absence of explicit matter fields (Brenna et al., 2011, Ghanaatian et al., 2014).

Nonlinear and Non-Abelian Matter: Quartic quasi-topological gravity admits black holes charged under nonlinear electrodynamics (e.g., power–Maxwell or power–Yang–Mills). The field equations and horizon structure remain controlled by quartic (or higher-degree) polynomials, while thermodynamic stability persists in allowed parameter domains (Ali et al., 2022, Ghanaatian et al., 2013).

Scalarization and Hair: When coupled to scalars, especially with non-minimal or exponential couplings, quasi-topological black holes exhibit spontaneous scalarization. A tachyonic instability controlled by coupling parameters leads to scalarized branches labeled by the number of radial nodes; the $n=0$ fundamental black holes are dynamically preferred and stable, while excited ( $n\geq1$ ) states are unstable (Myung et al., 2020).

Rotating Solutions: Rotational generalizations are technically constrained. For slow rotation (small rotation parameters), perturbative solutions exist, where the equation governing the angular sector is second-order due to the inherited structure from the quasi-topological and Lovelock densities (Fierro et al., 2020). Finite rotation solutions are generally inaccessible with the Kerr–Schild ansatz, suggesting a breakdown of that mechanism in non-Einsteinian higher-curvature theories.

5. Phase Structure, Microstructure, and Critical Phenomena

Multi-criticality: Generalized quasi-topological gravities (GQTG) can be tuned to support multi-critical points (N-tuple points) in the black hole phase space (Lu et al., 2023). The Maxwell equal area law is reformulated via the “K-rule” in terms of a compact integral constructing a geometric criterion for the emergence of multiple coalescing first-order transitions (e.g., quadruple or quintuple critical points requiring several independent higher-curvature densities).

Phase Transitions and Black Hole Chemistry: Quasi-topological black holes exhibit van der Waals–like first-order phase transitions, swallowtail free energy diagrams, isolated and multiple critical points, triple points, and $\lambda$ –lines (continuous lines of second-order phase transitions), paralleling phenomena in condensed-matter systems (Dykaar et al., 2017, Mir et al., 2019). The extended thermodynamic phase space (treating $\Lambda$ as pressure) reveals rich critical behavior not present in pure Einstein gravity; for instance, one finds non-mean-field critical exponents at isolated critical points and reentrant transitions.

Thermodynamic Geometry and Microstructure: Thermodynamic geometry, specifically the Ruppeiner scalar curvature, has been used as an empirical probe of black hole microstructure in four-dimensional quasi-topological solutions (Tiwari et al., 14 Aug 2025). The sign of the scalar curvature encodes dominant repulsive (positive $\mathcal{R}$ ) or attractive (negative $\mathcal{R}$ ) inter-microstate interactions, with zero crossings and divergences marking second-order phase transitions. A single zero crossing typically signifies a transition between repulsive and attractive regimes, directly correlating to criticality in the system.

6. Regular Black Holes and Singularities

Resolution of Singularities via Quasi-topological Dressing: Recent results demonstrate that by promoting the Lagrangian to a function $h$ of a complete set of curvature invariants (beyond just $R$ , $R_{\mu\nu}R^{\mu\nu}$ , etc.), with suitable coefficients, quasi-topological gravity can yield static, spherically symmetric black holes that are completely regular—i.e., curvature invariants remain bounded everywhere (Frolov et al., 25 Nov 2024). The corresponding reduced action leads to a master equation for a basic invariant $p$ , which for a special class becomes a linear second-order ODE, solvable in terms of special (Lerch transcendent) functions. At the core ( $r\to0$ ), the geometry becomes de Sitter-like (regular), while at asymptotic infinity the spacetime approaches Schwarzschild-Tangherlini. There exists a mass gap and a universal scaling property for these regular black holes.

Super-extremal Black Holes and Quartic Degeneracy: In theories with quasitopological electromagnetic corrections, super-extremal dyonic black holes have been identified where the horizon is quartically degenerate—i.e., the metric function and its first three derivatives vanish at the horizon. These configurations are characterized by a universal (parameter-independent) value for the mass-to-radius ratio, fixed by a hypergeometric parameter unique to the theory (Hod, 4 Apr 2024).

7. Summary Table: Key Theoretical Attributes

Property/Feature	Description / Effect	Reference(s)
Field equations for symmetric metrics	Reduce to algebraic polynomial (cubic, quartic, etc.)	(Myers et al., 2010, Olamaei et al., 2023)
Entropy formula	Wald entropy, area law + higher-curvature corrections	(Ghanaatian et al., 2013, Olamaei et al., 2023)
Horizon structure	Multiple horizons, extremal & super-extremal cases with novel features	(Brenna et al., 2012, Hod, 4 Apr 2024)
Thermodynamic phase structure	Van der Waals transitions, $\lambda$ -lines, multi-criticality	(Dykaar et al., 2017, Lu et al., 2023)
Linearized graviton equations	Second-order on AdS backgrounds, no ghost propagation	(Myers et al., 2010)
Holographic interpretation	Gravity side with tunable CFT central charges (three parameters)	(Myers et al., 2010)
Regular black holes	Achieved with nonlinear curvature invariants, de Sitter-like cores	(Frolov et al., 25 Nov 2024)

These properties collectively enable quasi-topological black holes to serve as versatile models for exploring the impact of higher-curvature corrections on black hole mechanics, their thermodynamic and phase structure, the holographic landscape of dual CFTs, and the possible regularization of classical singularities. This framework continues to be extended to wider classes of matter couplings, rotating and nontrivial topology solutions, and provides a fertile testing ground for probing beyond-Einstein gravity in both high-energy and mathematical physics contexts.