Einstein-Gauss-Bonnet Theory

• Einstein-Gauss-Bonnet theory is a higher-curvature gravity framework that incorporates the Gauss-Bonnet term to maintain second-order field equations in dimensions greater than four.
• It utilizes both perturbative and numerical methods to construct higher-dimensional black objects, analyzing phenomena like the Gregory-Laflamme instability and correlated thermodynamic stability.
• The Gauss-Bonnet correction enriches the phase diagram by creating a stability window in d=6 that smooths nonuniformity and alters key properties such as mass, tension, and entropy.

Einstein-Gauss-Bonnet (EGB) theory is a higher-curvature generalization of Einstein gravity that adds a specific quadratic curvature invariant—the Gauss-Bonnet (GB) term—to the gravitational action. Originating from Lovelock’s theorem, the EGB theory provides the unique extension of general relativity that maintains second-order field equations in dimensions $d > 4$, making it a central framework for probing gravitational phenomena in higher dimensions, as well as in string-inspired and phenomenological models of gravity.

1. Structure of the Einstein-Gauss-Bonnet Action

The EGB action in $d$ dimensions is given by

$$S = \frac{1}{16\pi G} \int d^d x \sqrt{-g} \left[ R + \alpha \left( R^2 - 4 R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \right) \right],$$

where $R$ is the Ricci scalar, $R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ are the Ricci and Riemann tensors, $\alpha$ is the Gauss–Bonnet coupling constant (with dimensions of $[\text{length}]^2$), and $G$ is the gravitational constant. The bracketed quadratic expression is the Gauss–Bonnet invariant, which is dynamically trivial in $d=4$ but contributes nontrivially for $d\geq 5$.

This higher-curvature correction is motivated both by the consistency requirements of Lovelock’s theorem and by low-energy limits of string theory, where it emerges as the leading correction to general relativity. The contribution of the GB term guarantees that the field equations remain second order in the metric, thereby avoiding the Ostrogradsky instability associated with higher derivatives.

2. Black String Solutions and Higher-Dimensional Phenomena

A principal application of EGB theory is the construction and analysis of higher-dimensional black objects, such as black strings and their stability properties.

Construction of Black Strings

Uniform black string (UBS) solutions in EGB theory are described by the metric ansatz

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

where $z$ parametrizes the additional “compact” direction. In the presence of the Gauss–Bonnet term, $a(r)$ cannot be taken as trivial and instead must vary with the radial coordinate.

Two main techniques are employed for constructing solutions:

• Perturbative Approach: The GB term is treated as a small perturbation, and the metric functions are expanded in powers of $\alpha$. For instance, setting $a(r)=1+\alpha\,a_1(r)+...$ and similarly for $b(r)$ and $f(r)$, the field equations reduce order by order to ODEs for the correction functions $a_1(r),b_1(r),f_1(r),...$. Explicit analytic first-order solutions are derived for $d=5,6$ (see (Brihaye et al., 2010)).
• Numerical Construction: For fully nonlinear regimes, the ODE system is solved numerically with boundary conditions at the horizon ($r=r_h$) and at spatial infinity, where asymptotic flatness is imposed. Near the horizon, the expansions are $a(r) = a_0 + a_1(r-r_h) + ...$ with proper regularity of $b(r),f(r)$.

Gregory-Laflamme Instability and Stability Analysis

The dynamical stability of UBSs is probed using linearized perturbations. Perturbations of the form $X(r,z) = X_0(r) + \epsilon X_1(r)\cos(k z) + ...$ are substituted into the coupled EGB field equations, yielding an eigenvalue problem for $k^2$, the square of the critical wavenumber. If $k^2>0$, the mode is unstable, leading to the well-known Gregory–Laflamme instability of black strings.

A noteworthy outcome is that, while in pure Einstein gravity all black strings ($d\ge5$) are unstable (i.e., $k^2>0$), in EGB theory a qualitatively distinct behavior emerges for $d=6$. For a sufficiently large value of $\beta = \alpha / r_h^2$, $k^2$ becomes negative and the string is dynamically stable. For $d=5$ and $d>6$, instability persists across admissible parameters.

3. Thermodynamic Properties and Correlated Stability

EGB gravity has significant effects on the thermodynamics of higher-dimensional black objects, altering quantities such as mass, tension, temperature, and entropy.

• Mass and Tension: Asymptotic expansions yield

$$g_{tt} \sim -1 + \frac{c_t}{r^{d-4}}, \quad g_{zz} \sim 1 + \frac{c_z}{r^{d-4}},$$

with mass $M$ and tension $\mathcal{T}$ depending on $c_t$ and $c_z$. The GB corrections modify the relationships between these quantities.

• Entropy: From Wald's formula, for the horizon $\Sigma_h$,

$$S = \frac{1}{4G} \int_{\Sigma_h} d^{d-2}x\,\sqrt{h} \left(1 + \frac{\alpha}{2}\widetilde{R}\right),$$

where $\widetilde{R}$ is the intrinsic scalar curvature of the horizon cross-section.

• Temperature: The Hawking temperature is

$T_H = \frac{\sqrt{b'(r_h)f'(r_h)}}{4\pi}.$

• Smarr Relation: A generalized Smarr formula links asymptotic and horizon data:

$M(1-n) = T_H S,$

with $n$ the relative string tension.

• Specific Heat and Thermodynamic Stability: For $d=6$, a branch of solutions with positive specific heat ($C_p = T_H (\partial S/\partial T_H) > 0$) exists at sufficiently large $\beta$, indicating thermodynamic stability, in contrast with $d=5$ or $d>6$, where $C_p<0$ everywhere.
• Gubser–Mitra (Correlated Stability) Conjecture: The correspondence between dynamical ($k^2$) and thermodynamic ($C_p$) stability is realized explicitly. In $d=6$, the transition to thermodynamic stability (positive $C_p$) matches the onset of dynamical stability (negative $k^2$), in precise accord with the conjecture.

4. Nonuniform Black Strings and Phase Structure

The existence and properties of nonuniform black strings (NUBS) are also investigated, focusing on $d=6$. Using an ansatz with explicit $z$-dependence for the metric components,

$$ds^2 = -e^{2A(r,z)}\frac{r^2}{r^2+r_0^2}dt^2 + e^{2B(r,z)}(dr^2 + dz^2) + e^{2C(r,z)}(r^2+r_0^2)d\Omega_3^2,$$

the equations reduce to a complex system of nonlinear PDEs, which are solved numerically by discretization and Newton–Raphson methods.

A nonuniformity parameter,

$$\lambda = \frac{1}{2}\left(\frac{R_{\text{max}}}{R_{\text{min}}} - 1\right),$$

measures the horizon’s deformation; $\lambda=0$ corresponds to the uniform black string. As in Einstein gravity, the NUBS branch bifurcates from the UBS branch at the threshold of (in)stability. The Gauss–Bonnet term "smooths" the deformation: increasing $\alpha$ for fixed temperature reduces $\lambda$, rendering the string more uniform and shifting the phase boundaries within the space of solutions.

5. Summary of New Phenomena and Implications

EGB theory reveals new qualitative features beyond those of pure Einstein gravity in higher dimensions:

• Modification of Charges and Horizons: Gauss–Bonnet contributions alter the mass, tension, area, and entropy, as well as horizon properties, compared to their purely Einstein counterparts.
• Dimension-Selective Stabilization: While dynamical and thermodynamic instability persists in most dimensions, $d=6$ exhibits a stable phase for sufficiently large GB coupling, aligned with a positive specific heat regime.
• Phase Diagram Richness: The presence and smoothing of nonuniform phases, alongside the existence of (correlated) thermodynamically and dynamically stable branches, signal a richer landscape of possible phases and transitions, potentially including transitions between UBS, NUBS, and localized black holes.

A concise table summarizing these key features is provided below:

Dimension (d)	UB String Stability	Thermodynamic Stability	Nonuniform Branches (NUBS)	GB Effects
5	Unstable	$C_p<0$ everywhere	Present	Quantitative charge corrections
6	Stable for large $\beta$	$C_p>0$ for large $\beta$	Present; less nonuniform for larger $\alpha$	Qualitative change: stability window
$>6$	Unstable	$C_p<0$ everywhere	Present	Similar to $d=5$; no stable regime

The presence of a correlated stability window in $d=6$ and the associated extension of the phase diagram underscore the importance of higher-curvature corrections for the phase structure of higher-dimensional objects and could have consequences for the endpoint of instabilities, the nature of compactification, and cosmological applications in string-inspired models.