---
title: Generalized Quasi-Topological Gravity
url: https://www.emergentmind.com/topics/generalized-quasi-topological-gravity
type: topic
---

# Generalized Quasi-Topological Gravity

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 [1909.07983, 2203.05589].

- **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 $f$) [2203.05589, 1909.07983, 2304.08510]. 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 $n$ [1909.07983]:
  $$
  \mathcal{S}_{n+5} = -\frac{3(n+3)}{4(D-1)(n+1)} \mathcal{S}_1 \mathcal{S}_{n+4} + \frac{3(n+4)}{4(D-1)n} \mathcal{S}_2 \mathcal{S}_{n+3} - \frac{(n+3)(n+4)}{4(D-1)n(n+1)} \mathcal{S}_3 \mathcal{S}_{n+2}
  $$
  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 [2304.08510], 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 $n$ [2304.08510, 2203.05589].

## 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 [2203.05589].

- **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 [2203.05589]:
  $$
  h(x) = \frac{16\pi G L^2}{(D-1)(D-2)}\left[\mathcal{L}(x) - \frac{2x}{D}\mathcal{L}'(x)\right].
  $$

## 4. Physical Consistency, Holography, and Hydrodynamics

- **Ghost and Causality Constraints:** Maintaining second-order linearized equations ensures freedom from massive ghosts and higher-derivative instabilities [1909.07983, 1612.00389]. Additional bounds on couplings arise by requiring causality (e.g., no superluminal propagation) and unitarity in the dual CFT [1004.2055].

- **AdS/CFT Dictionary:** The GQTG couplings map onto dual CFT data including central charges ($a$, $c$) and three-point function coefficients ($t_2$, $t_4$) [1004.2055]. 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 [1004.2055, 1802.00697, 1902.10906].

- **Holographic Hydrodynamics:** In black brane backgrounds, the inclusion of GQTG terms leads to new features:
  - Existence of multiple butterfly velocities for chaos diagnostics [1802.00697].
  - Structure of thermoelectric DC conductivities and their invariance under certain classes of higher-curvature terms [1802.00697, 1708.07198].

## 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 [1306.4510].

- **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 [1306.4510, 1404.0260].

- **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 [1306.4510].

## 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 [1909.07983, 2304.08510]. 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 [1906.00987, 1909.07983].

- **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 [2304.08510], 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 [2012.06618].

- **Generalizations and Applications:** EGQT theories in $2+1$ dimensions, SQT (EOM-trivial) gravities with vanishing contributions on a broad class of backgrounds [2212.00637, 2301.00235], and extensions employing non-linear functions of multiple curvature invariants for singularity resolution and regular black holes [2411.16050] 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.

---

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 [1004.2055, 1703.01631, 1703.11007, 1906.00987, 1909.07983, 2203.05589, 2304.08510].

Source: https://www.emergentmind.com/topics/generalized-quasi-topological-gravity