Gauss-Bonnet Black Holes

• Gauss–Bonnet black holes are solutions in extended gravity theories that incorporate a quadratic curvature term, modifying horizon and thermodynamic properties across various dimensions.
• They exhibit rich structures including modified horizon topologies, charge dynamics, rotation effects, and non-Einstein branches which influence stability and emission characteristics.
• Their extended thermodynamics reveals deviations from the pure area law with critical behavior and van der Waals–like phase transitions in higher-dimensional AdS scenarios.

Gauss–Bonnet black holes are black-hole solutions of gravitational theories in which the Einstein–Hilbert action is supplemented by the quadratic Gauss–Bonnet invariant

$$\mathcal{L}_{GB}=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^2.$$

In $D>4$ this term is the first nontrivial and dominant higher-curvature correction, motivated in particular by string theory and extra-dimensional scenarios, whereas four- and three-dimensional constructions discussed in the literature use regularized scalar-tensor formulations or related lower-dimensional limits. Across these settings, the Gauss–Bonnet sector often changes the metric only softly while substantially modifying horizon structure, entropy, critical behavior, stability, radiation, and rotation (Konoplya et al., 2010, Marks et al., 2021, Hennigar et al., 2020).

1. Higher-curvature framework

In standard Einstein–Gauss–Bonnet gravity, the action is written as

$$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$

or equivalently in closely related conventions with $\alpha\mathcal{L}_{GB}$ multiplying the curvature-squared sector. The corresponding field equations are second order in the metric despite the higher-curvature terms, which is the defining Lovelock property of the Gauss–Bonnet combination (Bhamidipati et al., 2017, Peng, 2021).

For static solutions, the basic ansatz remains of Schwarzschild type,

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

or, more generally,

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$

with the horizon geometry encoded in $h_{ij}$. In five-dimensional charged AdS examples, the metric function takes the square-root form characteristic of Gauss–Bonnet theory,

$$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$

and similar square-root branches occur in many other dimensions and matter sectors (Das et al., 2019).

A central structural distinction concerns dimensionality. In the five-dimensional dyonic analysis, the Gauss–Bonnet term is stated to be topological in four or fewer dimensions but dynamically relevant in five dimensions (Panahiyan et al., 2018). By contrast, four-dimensional Gauss–Bonnet-de Sitter black holes are formulated through a regularized scalar-tensor Horndeski-type action, and the three-dimensional rotating Gauss–Bonnet BTZ solutions arise in a lower-dimensional scalar-tensor formulation with an auxiliary scalar $\phi=\ln(r/l)$ (Marks et al., 2021, Hennigar et al., 2020). This distinction is essential: “Gauss–Bonnet black hole” denotes a family of theories rather than a single universal action.

2. Static geometries and horizon topology

The standard higher-dimensional Einstein–Gauss–Bonnet black hole already exhibits the basic pattern that recurs throughout the subject. For nonrotating black holes in $D>4$, the metric can be written as

$D>4$0

with a horizon mass relation and a Hawking temperature

$D>4$1

A key result is that turning on $D>4$2 makes the black hole colder even though the geometry is modified only “softly” at the level of the metric (Konoplya et al., 2010).

A major generalization replaces constant-curvature horizons by Einstein spaces with nontrivial Weyl curvature. In $D>4$3 dimensions, the Dotti–Gleiser condition

$D>4$4

introduces a new parameter $D>4$5 into the metric function and quasi-local mass. In this sector, the Weyl tensor contributes explicitly to the field equations, branch structure, and thermodynamics, and the Dotti–Gleiser solution is the unique vacuum solution when the warp factor is non-constant (Maeda, 2010). The closely related six-dimensional analysis reaches the same conclusion from a different angle: the horizon’s full internal curvature enters the bulk equations, and the black-hole potential acquires a new “charge-like” parameter $D>4$6 tied to the horizon Weyl curvature and Euler characteristic. In that setting, $D>4$7 can add horizons or generate branch singularities, and exotic horizons such as $D>4$8 and Bergman space are admitted under the Gauss–Bonnet constraints (0906.4953).

Five-dimensional topological solutions exhibit further departures from the spherical paradigm. One vacuum static AdS solution has horizon topology $D>4$9 with

$$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$0

while another uses a Sol-manifold horizon. For the Sol-manifold solution, the Gauss–Bonnet coupling is fixed by

$$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$1

and both the total energy and entropy vanish, $$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$2 and $$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$3 (Peng, 2021). These examples make clear that Gauss–Bonnet black holes are not confined to maximally symmetric horizon geometries.

An additional exact class appears in anisotropic-scaling spacetimes inspired by Lifshitz geometry. There the metric asymptotically obeys

$$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$4

with consistency requiring $$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$5. Gauss–Bonnet gravity admits an exact vacuum solution, independent of $$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$6, provided

$$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$7

and the branch structure distinguishes pseudo-hyperbolic black holes from pseudo-spherical naked singularities (Mahmoudi et al., 2022).

3. Charge, rotation, and non-Einstein branches

Gauge fields enrich the Gauss–Bonnet sector without changing its characteristic square-root structure. In five-dimensional dyonic solutions, the electric and magnetic fields are independent and enter the metric only through the combination $$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$8, so the geometry enjoys electric-magnetic duality under interchange of the labels $$I=\frac{1}{16\pi}\int d^Dx\,\sqrt{-g}\left[R-2\Lambda+\alpha_{\rm GB}\left(R_{\gamma\delta\mu\nu}R^{\gamma\delta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}+R^2\right)\right],$$9 and $\alpha\mathcal{L}_{GB}$0. The exact metric function has the form

$\alpha\mathcal{L}_{GB}$1

and can support up to three horizons depending on $\alpha\mathcal{L}_{GB}$2, $\alpha\mathcal{L}_{GB}$3, $\alpha\mathcal{L}_{GB}$4, and $\alpha\mathcal{L}_{GB}$5 (Panahiyan et al., 2018).

In the large-$\alpha\mathcal{L}_{GB}$6 limit, static Einstein–Maxwell–Gauss–Bonnet black holes with cosmological constant admit an effective membrane description in terms of mass, charge, and momentum densities $\alpha\mathcal{L}_{GB}$7, $\alpha\mathcal{L}_{GB}$8, and $\alpha\mathcal{L}_{GB}$9. The exact static charged metric is retained, but the large-$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$0 expansion turns the nonlinear problem into effective equations on the horizon. This framework yields analytic quasinormal frequencies and a sharp stability criterion: for positive Gauss–Bonnet coupling, asymptotically flat and AdS black holes are always stable, whereas sufficiently positive cosmological constant can trigger instability; for negative Gauss–Bonnet coupling, instability can occur even in flat or AdS space (Chen et al., 2017).

Rotation introduces qualitatively new structure. In five-dimensional asymptotically flat Einstein–Gauss–Bonnet gravity, black holes with two equal-magnitude angular momenta are described by a cohomogeneity-1 ansatz with enhanced $ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$1 symmetry. These solutions possess a regular $ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$2 horizon, an ergoregion, and an extremal endpoint where the Hawking temperature vanishes and the angular momenta attain their extremal values. Their entropy splits into Einstein and Gauss–Bonnet pieces,

$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$3

and the first law takes the form

$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$4

Most global properties remain Myers–Perry-like, but the Gauss–Bonnet term deforms the geometry and modifies the domain of existence and thermodynamic response (Brihaye et al., 2010).

Lower-dimensional rotating solutions exhibit a different novelty. The rotating three-dimensional Gauss–Bonnet BTZ black hole is an exact solution of the regularized theory,

$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$5

with effective AdS scale

$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$6

These black holes possess an outer horizon and an ergoregion but do not have an inner horizon. For $ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$7, the spacetime ends at a branch singularity before any inner horizon forms; for $ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$8, the origin is a curvature singularity (Hennigar et al., 2020).

4. Thermodynamics, entropy, and criticality

A defining feature of Gauss–Bonnet black holes is that entropy is generically not given by the pure area law. In $ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,$9-dimensional Dotti–Gleiser black holes, the Wald entropy is

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$0

and the first law reads

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$1

The same pattern appears in many variants: for five-dimensional Einstein–Gauss–Bonnet black holes in a string cloud background,

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$2

while for Gauss–Bonnet gravity’s rainbow the entropy is modified by both the Wald term and the rainbow factor,

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$3

(Maeda, 2010, Herscovich et al., 2010, Hendi et al., 2015).

Extended phase-space thermodynamics is especially rich in AdS. For neutral five-dimensional Gauss–Bonnet black holes, the cosmological constant is treated as pressure, $$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$4, the mass is enthalpy, and there is a van der Waals–type critical point with

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$5

Because $$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$6, rectangular cycles in the $$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$7-$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$8 plane are natural, and the resulting critical heat engines satisfy

$$ds^2=-f(r)\,dt^2+f^{-1}(r)\,dr^2+r^2 h_{ij}dx^i dx^j,$$9

In this sense, the Gauss–Bonnet coupling $h_{ij}$0 plays a role analogous to charge in Reissner–Nordström–AdS criticality, while preserving a distinct critical scaling (Bhamidipati et al., 2017).

A more recent holographic reformulation varies both $h_{ij}$1 and $h_{ij}$2 and treats the dual CFT central charges $h_{ij}$3 and $h_{ij}$4 as thermodynamic variables. The extended first law becomes

$h_{ij}$5

and the criticality conditions in either the $h_{ij}$6 or $h_{ij}$7 ensemble yield a universal critical Gauss–Bonnet coupling

$h_{ij}$8

The corresponding first-order small/large black-hole transitions occur for $h_{ij}$9, not $$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$0, and are bounded above by causality: $$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$1 for A-charge criticality and $$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$2 for C-charge criticality. The critical exponents are mean-field,

$$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$3

(Cui et al., 2024).

De Sitter thermodynamics requires a different ensemble because of the cosmological horizon. In both the $$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$4 cavity formulation and the four-dimensional regularized theory, the Euclidean action is evaluated with thermodynamic data fixed at a finite-radius cavity, and the locally measured temperature is the redshifted Hawking temperature at the cavity wall. This setting produces Hawking–Page-like transitions to empty de Sitter space, first-order small/large black-hole transitions, triple points, zeroth-order transitions, reentrant behavior, and, in the charged case, a swallowtube in $$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$5-$$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$6-$$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$7 space. The charged canonical ensemble excludes evaporation to empty space, so the relevant competition is between charged black-hole branches rather than between black holes and thermal radiation (2002.01567, Marks et al., 2021).

Not all Gauss–Bonnet black holes are thermodynamically stable. The five-dimensional $$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$8 vacuum solution has

$$f(r)=1+\frac{r^2}{4\gamma}\left(1-\sqrt{1-\frac{8\gamma}{l^2}+\frac{64\gamma M}{3\pi r^4}-\frac{2\gamma Q^2}{3\pi r^6}}\right),$$9

and is therefore thermodynamically unstable, whereas the Sol-manifold solution has vanishing entropy and energy (Peng, 2021). For Dotti–Gleiser black holes with $\phi=\ln(r/l)$0, $\phi=\ln(r/l)$1, $\phi=\ln(r/l)$2, and $\phi=\ln(r/l)$3, the heat capacity is negative, and for $\phi=\ln(r/l)$4 the free energy is positive, so these black holes are not globally stable in de Sitter space (Maeda, 2010).

5. Radiation, perturbations, and observational probes

A recurring misconception is that small metric deformations imply small physical effects. Gauss–Bonnet Hawking radiation provides a direct counterexample. For higher-dimensional nonrotating Einstein–Gauss–Bonnet black holes, the wave equations for tensor, vector, and scalar gravitational perturbations all reduce to

$\phi=\ln(r/l)$5

and the Hawking emission rate is

$\phi=\ln(r/l)$6

The key result is that graviton emission is suppressed by many orders when Gauss–Bonnet corrections are present. The suppression is the product of quick cooling and the exponential suppression of the tensor-graviton grey-body factor, whose WKB form is

$\phi=\ln(r/l)$7

For $\phi=\ln(r/l)$8 and $\phi=\ln(r/l)$9, the total energy-emission rate can be about $D>4$0 times smaller than in the Schwarzschild case, so the black-hole lifetime is much longer than standard Schwarzschild–Tangherlini estimates would suggest (Konoplya et al., 2010).

Perturbative stability depends sensitively on the sign of the Gauss–Bonnet coupling. In the large-$D>4$1 effective theory, the charge perturbation has frequency

$D>4$2

so it is always damped, but the scalar-type gravitational sector can become unstable. For positive Gauss–Bonnet coupling, asymptotically flat and AdS black holes remain stable, whereas sufficiently positive cosmological constant in de Sitter can trigger a $D>4$3-instability. For negative Gauss–Bonnet coupling, the instability criterion can be satisfied even in asymptotically flat space, and at the onset of instability a nontrivial static zero mode appears, indicating a new nonspherical solution branch (Chen et al., 2017).

Matter sectors further diversify the stability problem. In five-dimensional Einstein–Gauss–Bonnet gravity with a string cloud source,

$D>4$4

the Hawking temperature is

$D>4$5

the free energy can exhibit a Hawking–Page transition, and the evaporation law

$D>4$6

implies that evaporation is obstructed for $D>4$7, infinite for $D>4$8, and finite for $D>4$9 with $D>4$00 (Herscovich et al., 2010).

Observational diagnostics include shadows and high-frequency emission. For five-dimensional charged Gauss–Bonnet black holes, the shadow radius $D>4$01 is determined by the photon sphere and the celestial coordinates. The principal qualitative result is asymptotics-dependent: increasing the Gauss–Bonnet parameter $D>4$02 increases the shadow in AdS but decreases it in asymptotically flat spacetime, while increasing $D>4$03 decreases the shadow in both cases. A plasma with refractive index

$D>4$04

further shrinks the shadow. Since the limiting absorption cross section scales as

$D>4$05

the energy emission rate decreases when $D>4$06 decreases; the paper concludes that increasing $D>4$07 decreases the emission rate and that plasma suppresses it further (Das et al., 2019).

6. Broader variants and conceptual developments

Energy-dependent deformations show that the Gauss–Bonnet sector can be combined with modified dispersion relations. In Gauss–Bonnet gravity’s rainbow, the spacetime depends on probe energy through rainbow functions $D>4$08 and $D>4$09, with line element

$D>4$10

The Hawking temperature, entropy, charge, and mass are all modified by $D>4$11 and $D>4$12, but the thermodynamic structure survives: $D>4$13 In the canonical ensemble, the heat capacity can have a positive root, one or two divergences, and corresponding second-order phase transitions, with the critical radii shifted by the rainbow parameters (Hendi et al., 2015).

Scalar–Gauss–Bonnet theories add another layer by coupling a scalar directly to the Gauss–Bonnet invariant. In extended scalar-tensor-Gauss–Bonnet gravity with a massive scalar field,

$D>4$14

the scalar decays as

$D>4$15

and the scalar mass suppresses the hair, moves the solutions closer to Schwarzschild, enlarges the domain of existence for linear and exponential couplings, and shifts the bifurcation point of spontaneously scalarized branches to smaller black-hole masses (Doneva et al., 2019).

Recent rotating scalar–Gauss–Bonnet constructions add a quadratic Gauss–Bonnet correction through an auxiliary field $D>4$16, so the action effectively contains a $D>4$17 term. With quadratic-exponential coupling

$D>4$18

Kerr remains an exact solution because $D>4$19, but a tachyonic effective mass

$D>4$20

can trigger spontaneous scalarization. The numerical solutions show that both rotation and the quadratic Gauss–Bonnet term shrink the scalarized domain, reduce the scalar charge, and often leave the scalarized black holes with higher entropy than Kerr black holes of the same mass and spin (Liu et al., 17 Mar 2025).

Lower-dimensional regularized theories also sharpen the conceptual scope of the subject. In three dimensions, exact charged Gauss–Bonnet black holes can be constructed with a zero-point length regularization

$D>4$21

which makes the gravitational and electromagnetic potentials finite at the origin while leaving scalar curvature invariants singular there. The first law acquires an extra work term conjugate to $D>4$22,

$D>4$23

and the entropy is reduced relative to the classical $D>4$24 result by the stringy correction (Jusufi et al., 2023). This suggests that “Gauss–Bonnet black hole” is best understood as a broad higher-curvature category spanning Lovelock black holes, regularized lower-dimensional models, scalarized branches, and holographically reinterpreted thermodynamic systems rather than a single canonical geometry.