Generalized Quasi-Topological Gravity

• Generalized quasi-topological gravity is a higher-curvature theory that extends Lovelock gravities while preserving second-order linearized equations and analytic tractability.
• It produces unique, single-function black hole solutions with modified thermodynamic properties and a robust embedding in the AdS/CFT framework.
• The theory supports consistent holographic duals and cosmological models, ensuring ghost-free dynamics and controlled corrections from higher-curvature terms.

Generalized quasi-topological gravity (GQTG) refers to a class of higher-curvature gravitational theories that generalize the properties of Lovelock and quasi-topological gravities. These extensions are distinguished by their ability to produce modified gravitational dynamics—especially in dimensions D ≥ 4 or 5—while preserving several desirable structural features: second-order linearized equations on maximally symmetric backgrounds, non-hairy generalizations of Schwarzschild-like black holes (i.e., with a single metric function f(r) and g_{tt}g_{rr} = –1), and a uniquely tractable coupling to the AdS/CFT correspondence, black hole thermodynamics, and cosmological evolution. GQTGs, including both proper (differential) and quasi-topological (algebraic) branches, are now regarded as universal representatives for higher-curvature gravity due to their recurrence relations, explicit covariant construction to arbitrary order, and their thermodynamic and holographic consistency.

1. Structural Foundations and Defining Properties

GQTGs arise from a systematic extension of the gravitational Lagrangian by higher-order contractions of the Riemann tensor, aiming to retain tractable field equations and physical properties:

• Action and Ansätze: The typical action is:

$$S = \frac{1}{16\pi G} \int d^Dx\,\sqrt{-g}\left[-2\Lambda + R + \sum_n \sum_i \mu^{(n)}_i\,\mathcal{R}_i^{(n)}\right]$$

where each $\mathcal{R}_i^{(n)}$ is a basis invariant of order $n$ constructed from the Riemann tensor and the metric.

• Single-function Black Holes: When the action is evaluated on static, spherically symmetric metrics

$$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_{D-2}^2,$$

the equations reduce to a single ODE (up to second order in $f(r)$), or to a purely algebraic equation in the quasi-topological subclass. The unique "non-hairy" character (no extra functions) is crucial for analytic tractability and black hole thermodynamics.

• Second-order Linearized Equations: For small perturbations $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$ around a maximally symmetric background ($\bar{g}_{\mu\nu}$, such as AdS or Minkowski), the linearized field equations remain second order, propagating only the massless, transverse, traceless graviton and avoiding ghosts (Bueno et al., 2019, Bueno et al., 2022).
• Proper vs. Quasi-topological: In $D \geq 5$, at a given curvature order $n$, there exist a unique quasi-topological density (algebraic equation for $f$) and $(n-2)$ proper GQTG densities (second-order ODE for $f$) (Bueno et al., 2022, Bueno et al., 2019, Moreno et al., 2023). In $D = 4$, only a single (proper) GQTG exists at each order—never purely algebraic.

2. Explicit Construction and Classification

The classification and construction of GQTGs are achieved via recursive relations and by "uplifting" reduced ansatz expressions to fully covariant invariants:

• Recursive Construction: Recursive relations between lower and higher order densities allow for systematic generation at all $n$ (Bueno et al., 2019):

$$\mathcal{S}_{n+5} = -\frac{3(n+3)}{4(D-1)(n+1)} \mathcal{S}_1 \mathcal{S}_{n+4} + \frac{3(n+4)}{4(D-1)n} \mathcal{S}_2 \mathcal{S}_{n+3} - \frac{(n+3)(n+4)}{4(D-1)n(n+1)} \mathcal{S}_3 \mathcal{S}_{n+2}$$

with similar (dimension-dependent) recursions for the quasi-topological (algebraic) branch.

• Covariant Dictionary: Each density constructed on the SSS ansatz (tensors A, B, ψ from second derivatives and curvature components) is mapped via a dictionary to unique off-shell covariant invariants (Moreno et al., 2023), allowing general application beyond SSS metrics.
• Dimensional Dependence: For $D \geq 5$, there are $(n-1)$ inequivalent GQTG densities at curvature order $n$: one quasi-topological plus $(n-2)$ proper GQTGs. In $D=4$, due to trace identities and vanishing of higher Lovelock terms, only one proper GQTG exists at each order, realizing for the first time an explicit covariant representative at every $n$ (Moreno et al., 2023, Bueno et al., 2022).

3. Black Hole Solutions and Thermodynamics

• Black Hole Equations: For SSS metrics, GQTG field equations for $f(r)$ reduce to a second-order ODE (or algebraic equation in the quasi-topological case). The most general GQTG yields equations of the form:

$\mathcal{F}[f, f', f'', r] = C$

where $C$ is related to the ADM mass.

• Thermodynamics: The mass $M$, Hawking temperature $T$, and Wald entropy $S$ are computed using standard techniques but generalized for higher-curvature corrections:
  • Mass from the integration constant $C$.
  • Temperature from $f'(r_+)$ at the event horizon $r_+$.
  • Wald entropy,

  $$S = -2\pi \int_{\mathcal{H}} d^{D-2}x \sqrt{h}\, P^{abcd} \epsilon_{ab}\epsilon_{cd},$$

  where $$P^{abcd} = \partial \mathcal{L} / \partial R_{abcd}$$ and $\epsilon_{ab}$ is the binormal.

• First Law and Free Energy: The first law $dM = TdS$ is verified for black hole solutions in all GQTGs without further conditions on the couplings (Bueno et al., 2022).

• Embedding Function: Both the maximally symmetric vacuum structure and black hole thermodynamics are encapsulated by the "embedding function" $h(x)$ evaluated on the background curvature. Black hole quantities (mass and entropy) can often be entirely written in terms of $h$ and its derivatives (Bueno et al., 2022):

$$h(x) = \frac{16\pi G L^2}{(D-1)(D-2)}\left[\mathcal{L}(x) - \frac{2x}{D}\mathcal{L}'(x)\right].$$

4. Physical Consistency, Holography, and Hydrodynamics

• Ghost and Causality Constraints: Maintaining second-order linearized equations ensures freedom from massive ghosts and higher-derivative instabilities (Bueno et al., 2019, Chernicoff et al., 2016). Additional bounds on couplings arise by requiring causality (e.g., no superluminal propagation) and unitarity in the dual CFT (Myers et al., 2010).

• AdS/CFT Dictionary: The GQTG couplings map onto dual CFT data including central charges ($a$, $c$) and three-point function coefficients ($t_2$, $t_4$) (Myers et al., 2010). Curvature-cubed and higher terms allow for independent tuning of these parameters, which is impossible in Lovelock gravity.

• Shear Viscosity and $\eta/s$ Bound: The ratio of shear viscosity to entropy density is generically modified by GQT contributions:

$$\frac{\eta}{s} = \frac{1}{4\pi}\left[1 - 4\lambda - 36\mu(\cdots)\right]$$

where the terms in parentheses depend on the coupling constants and AdS background. The lower bound can be driven below $1/(4\pi)$—violating the original KSS bound—yet remains finite and bounded by the region of physical couplings (Myers et al., 2010, Peng et al., 2018, Mir et al., 2019).

• Holographic Hydrodynamics: In black brane backgrounds, the inclusion of GQTG terms leads to new features:
  • Existence of multiple butterfly velocities for chaos diagnostics (Peng et al., 2018).
  • Structure of thermoelectric DC conductivities and their invariance under certain classes of higher-curvature terms (Peng et al., 2018, Li et al., 2017).

5. Cosmological and Thermodynamic Implications

• Generalized Friedmann Equations: When applied to FLRW cosmology, GQTGs yield modified Friedmann equations involving higher powers of $H^2 + k/a^2$ but preserve second-order evolution, ensuring independence from ghosts and maintenance of well-posed cosmological dynamics (Dehghani et al., 2013).
• Thermodynamics of Apparent Horizons: The same modified entropy–area relations used for black hole horizons yield, upon replacing the horizon radius by the apparent horizon in cosmology, a precise coincidence between the gravitational field equations obtained from a variational principle and those derived from the first law of thermodynamics (Dehghani et al., 2013, Sheykhi et al., 2014).
• Generalized Second Law: The generalized second law of thermodynamics (increase of total horizon plus matter entropy) can be established in GQTG cosmology under commonly assumed conditions (Dehghani et al., 2013).

6. Extensions to Arbitrary and Infinite Order, Classification, and Universality

• Arbitrary-order Densities: GQTG and quasi-topological (algebraic) gravities have been constructed to all orders in curvature using recursive relations and explicit covariant uplift procedures (Bueno et al., 2019, Moreno et al., 2023). For $D \geq 5$, there exists one unique quasi-topological (algebraic) gravity per order $n$, with proper GQTGs spanning the remainder.
• Universality: It has been proven that via suitable (invertible) metric redefinitions, any higher-curvature gravity action built from Riemann contractions (and, with certain caveats, even including derivatives of curvature) is equivalent, at the level of static spherically symmetric solutions and their thermodynamics, to a GQTG—a universality result (Bueno et al., 2019, Bueno et al., 2019).
• Covariant Representatives and Dictionaries: Recent works have produced explicit, all-order formulas for covariant representatives of each GQTG class, completing the algebraic dictionary between SSS ansatz terms and off-shell invariants (Moreno et al., 2023), thus giving a practical method for enumerating and constructing higher-order terms in any dimension.

7. Open Problems and Future Directions

• Rotating and Non-static Solutions: Slowly rotating black holes in both cubic and quartic quasi-topological gravities have been constructed; however, the search for fully rotating solutions is ongoing and appears to require methods beyond the standard Kerr–Schild ansatz (Fierro et al., 2020).
• Generalizations and Applications: EGQT theories in $2+1$ dimensions, SQT (EOM-trivial) gravities with vanishing contributions on a broad class of backgrounds (Bueno et al., 2022, Chen, 2022), and extensions employing non-linear functions of multiple curvature invariants for singularity resolution and regular black holes (Frolov et al., 25 Nov 2024) are active areas of development.
• Holography and Field Theory Duals: GQTGs serve as toy models for exploring the landscape of holographically dual non-supersymmetric CFTs with tunable three-point function parameters, as well as for refining hydrodynamic and quantum information bounds.
• Classification and Systematics: The explicit quadratic growth (for $D\geq5$) or linear uniqueness (for $D=4$) of GQTG densities at each order, as established via generating functions and algebraic identities, provides a solid foundation for further mathematical classification and exploration of allowed physical theories.

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Generalized quasi-topological gravities form a robust, systematized, and mathematically tractable sector of higher-curvature gravity, distinguished by their unique combination of analytic solvability, extended parameter space, universality (under field redefinitions), and consistent thermodynamic and holographic properties. They provide a template for analyzing quantum gravity effects, black hole microphysics, and holographically dual field theories in an effective action framework (Myers et al., 2010, Hennigar et al., 2017, Ahmed et al., 2017, Bueno et al., 2019, Bueno et al., 2019, Bueno et al., 2022, Moreno et al., 2023).