Einstein-Maxwell-Scalar-Gauss-Bonnet Overview

• EMSGB theory is a four-dimensional gravitational framework in which the metric, Maxwell field, and scalar field interact through a non-minimal Gauss-Bonnet term, enabling phenomena such as spontaneous scalarization.
• The theory features diverse coupling formulations—quadratic, linear, and Gaussian—that lead to a range of solutions including charged hairy black holes, traversable wormholes, and NUT-charged geometries, with models solved via symmetry reduction and numerical methods.
• Analyses of EMSGB reveal rich phenomenology, from the testing of strong cosmic censorship and horizon regularity in de Sitter backgrounds to holographic superconductivity and complex thermodynamic phase structures in AdS spaces.

Searching arXiv for recent and foundational papers on Einstein-Maxwell-Scalar-Gauss-Bonnet theory and closely related variants. arXiv Search Query: all:"Einstein-Maxwell-Scalar-Gauss-Bonnet theory" OR all:"Einstein-Maxwell-scalar-Gauss-Bonnet" OR all:"Einstein-scalar-Gauss-Bonnet Maxwell" Einstein-Maxwell-Scalar-Gauss-Bonnet theory denotes a class of four-dimensional gravitational models in which the metric $g_{\mu\nu}$, a Maxwell field $A_\mu$, and a scalar field interact through the Einstein-Hilbert sector together with a non-minimal scalar coupling to the Gauss-Bonnet invariant,

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$

Across its main formulations, the theory is used to study curvature-induced scalarization, charged hairy black holes, Cauchy-horizon regularity, holographic condensates in anti-de Sitter space, NUT-charged geometries, and exact traversable wormholes (Sang et al., 2022, Hunter et al., 2020, Butler et al., 2023, Guo et al., 2024, Cañate et al., 2019, Erices et al., 28 Apr 2026).

1. Action-level formulations

There is no single universally normalized EMSGB action in the literature. Instead, the subject comprises a closely related family of models sharing the same structural ingredients: gravity, electromagnetism, a scalar sector, and a scalar-dependent Gauss-Bonnet interaction. A representative de Sitter form is

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$

with quadratic coupling $f(\phi)=\tfrac12\alpha\phi^2$. Other papers adopt a linear scalar-Gauss-Bonnet term together with a linear scalar-Maxwell interaction, or a Gaussian-type coupling $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$. In AdS holography, the scalar is charged and the gauge interaction is contained in $|D_\mu\psi|^2$, with $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$. One exact wormhole construction replaces standard Maxwell electrodynamics by the power-Maxwell Lagrangian $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$ (Sang et al., 2022, Hunter et al., 2020, Guo et al., 2024, Cañate et al., 2019, Erices et al., 28 Apr 2026).

Formulation	Coupling choice	Typical setting
Quadratic scalar-GB	$f(\phi)=\tfrac12\alpha\phi^2$	RNdS perturbations and SCC
Linear scalar-GB plus scalar-Maxwell	$A_\mu$0	Asymptotically flat hairy black holes; NUT solutions
Gaussian scalar-GB	$A_\mu$1	Canonical black-hole thermodynamics
Charged scalar in AdS	$A_\mu$2	Holographic superconductors and scalarization
EsGB with power-Maxwell NLED	$A_\mu$3	Exact traversable wormholes

A recurrent source of notation ambiguity is that $A_\mu$4, $A_\mu$5, and $A_\mu$6 label different couplings in different subliteratures. In the SCC analysis, $A_\mu$7 is instead the spectral ratio $A_\mu$8, not a Lagrangian parameter (Sang et al., 2022).

2. Field equations and symmetry reductions

Variation with respect to $A_\mu$9, $$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$0, and the scalar yields Einstein, Maxwell, and scalar equations modified by the Gauss-Bonnet coupling. In the quadratic-coupling de Sitter model these take the form

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$1

In the linear-coupling asymptotically flat model, the scalar equation becomes

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$2

and the Maxwell equation is modified to

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$3

In the AdS charged-scalar model one has

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$4

together with a Maxwell equation containing the charged-scalar current (Sang et al., 2022, Hunter et al., 2020, Guo et al., 2024).

Most concrete analyses proceed by symmetry reduction. For static, spherically symmetric black holes the metric is commonly written either as

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$5

or as

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$6

with a purely electric potential $$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$7. In the NUT sector the ansatz is

$$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$8

with $$R_{\rm GB}^2 = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} -4R_{\mu\nu}R^{\mu\nu} +R^2.$$9 and $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$0. In AdS holography the standard planar ansatz is

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$1

These reductions turn the full field equations into coupled ODE systems, which are then solved by shooting, asymptotic matching, pseudospectral discretization, or direct numerical integration depending on the problem (Butler et al., 2023, Hunter et al., 2020, Guo et al., 2024).

3. Scalarization and charged hairy black holes

The central mechanism in EMSGB theory is spontaneous scalarization. In the quadratic-coupling model, small fluctuations around $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$2 acquire an effective mass-squared contribution $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$3. Near a sufficiently compact horizon, where $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$4 is large, a tachyonic instability develops once $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$5 passes a threshold, and the nonlinear evolution yields a new static and spherical black-hole branch with $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$6 (Sang et al., 2022).

A closely related instability appears in the Gaussian-coupling formulation of charged asymptotically flat black holes. There, linearization about Reissner-Nordström gives $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$7, and the scalarized branch bifurcates from the RN branch above a critical coupling $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$8. The RN solution remains

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\, \Bigl[ R-2\Lambda -F_{\mu\nu}F^{\mu\nu} -\tfrac12\,\nabla_\mu\phi\,\nabla^\mu\phi +f(\phi)\,R_{\rm GB}^2 \Bigr],$$9

while the scalarized branch has $f(\phi)=\tfrac12\alpha\phi^2$0 and satisfies a nontrivial horizon regularity condition (Erices et al., 28 Apr 2026).

In the linear scalar-GB plus scalar-Maxwell model, the resulting charged hairy black holes are described as scalarised black hole solutions with secondary scalar hair, parametrised in terms of the black hole's mass and charge. For the asymptotically flat static ansatz, the electric field obeys

$f(\phi)=\tfrac12\alpha\phi^2$1

and the far-field behavior is

$f(\phi)=\tfrac12\alpha\phi^2$2

Here $f(\phi)=\tfrac12\alpha\phi^2$3, $f(\phi)=\tfrac12\alpha\phi^2$4, and $f(\phi)=\tfrac12\alpha\phi^2$5 are the ADM mass, electric charge, and scalar charge. In perturbation theory, $f(\phi)=\tfrac12\alpha\phi^2$6, so the scalar hair is secondary in the specific sense adopted there, although the full numerical solution still depends on the horizon data and the choice of branch (Hunter et al., 2020).

Regularity at the horizon imposes nontrivial algebraic constraints on $f(\phi)=\tfrac12\alpha\phi^2$7, and the corresponding discriminant yields parameter bounds for physically acceptable solutions. This leads to a standard but important clarification: EMSGB scalarization is not the unrestricted appearance of arbitrary scalar hair, but the emergence of branches constrained by horizon regularity, asymptotic flatness, and coupling-dependent reality conditions (Hunter et al., 2020).

4. Cauchy horizons, quasinormal modes, and strong cosmic censorship

In the de Sitter sector, EMSGB theory has been studied as a test case for strong cosmic censorship in Reissner-Nordström-de Sitter backgrounds. The relevant spacetime has three Killing horizons at radii $f(\phi)=\tfrac12\alpha\phi^2$8. Christodoulou’s formulation of SCC requires that the maximal analytic extension fail at the inner horizon $f(\phi)=\tfrac12\alpha\phi^2$9, and the key regularity question is whether perturbations are sufficiently mild for a weak extension across the Cauchy horizon to exist (Sang et al., 2022).

The criterion used in the EMSGB analysis is Sobolev regularity. For the metric to admit a weak extension across $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$0, the scalar fluctuation must belong to $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$1 there. If a mode behaves as $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$2 near $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$3, then it lies in $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$4 only if $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$5. Equivalently,

$f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$6

SCC is violated if the least-damped quasinormal mode obeys $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$7, whereas the existence of any mode with $f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$8 makes the curvature blow up strongly enough to render the Cauchy horizon inextendible (Sang et al., 2022).

The perturbation problem is posed by separating variables as

$f(\phi)=\frac{1}{2\beta}(1-e^{-\beta\phi^2})$9

which yields

$|D_\mu\psi|^2$0

with effective potential

$|D_\mu\psi|^2$1

Quasinormal modes satisfy purely ingoing boundary conditions at the event horizon and purely outgoing ones at the cosmological horizon. The paper states that one may compute them numerically by pseudospectral Chebyshev discretization, direct integration with matching of near-horizon series expansions, or higher-order WKB for large $|D_\mu\psi|^2$2 (Sang et al., 2022).

The main result is parameter-dependent restoration of SCC. For generic non-extremal RNdS black holes, $|D_\mu\psi|^2$3 and SCC is never threatened. In the near-extremal regime, however, modes with $|D_\mu\psi|^2$4 can occur unless the Gauss-Bonnet coupling is sufficiently large. At $|D_\mu\psi|^2$5, a critical coupling $|D_\mu\psi|^2$6 is found: for $|D_\mu\psi|^2$7, the maximum $|D_\mu\psi|^2$8 over all near-extremal charges drops below $|D_\mu\psi|^2$9, whereas for $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$0 a window of near-extremal charges still violates SCC. The same repair mechanism is not universal across all de Sitter parameters; for $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$1, even $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$2 is too small to push $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$3 below $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$4 (Sang et al., 2022).

5. Extended solution landscape: wormholes and NUT-charged geometries

Beyond ordinary spherical black holes, EMSGB-type models support more exotic geometries. One exact construction gives a static, spherically symmetric, asymptotically flat traversable wormhole in Einstein-scalar-Gauss-Bonnet theory coupled to a power-Maxwell nonlinear electrodynamics. In Morris-Thorne form,

$D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$5

the solution sets $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$6 and uses two parameters $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$7 and $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$8. For $D_\mu\psi=\nabla_\mu\psi-iqA_\mu\psi$9, the throat is at

$\mathcal L(F)=(-\tilde\kappa F)^{3/2}$0

with $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$1, $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$2, and no horizon because $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$3. A central point of that solution is that the null energy condition is violated entirely by the scalar-Gauss-Bonnet sector, while the NLED matter itself does not violate the NEC. In the uncharged limit $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$4, the solution reduces to the Ellis wormhole supported by a phantom scalar (Cañate et al., 2019).

Another extension introduces NUT twist in a four-dimensional theory with action

$\mathcal L(F)=(-\tilde\kappa F)^{3/2}$5

The resulting geometries depend on the electric charge $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$6, scalar charge $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$7, NUT charge $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$8, and mass $\mathcal L(F)=(-\tilde\kappa F)^{3/2}$9, and are obtained numerically from asymptotic and near-horizon expansions. The far-field behavior is

$f(\phi)=\tfrac12\alpha\phi^2$0

$f(\phi)=\tfrac12\alpha\phi^2$1

The paper reports that increasing $f(\phi)=\tfrac12\alpha\phi^2$2 enhances the scalar amplitude, larger $f(\phi)=\tfrac12\alpha\phi^2$3 increases the scalar hair, increasing $f(\phi)=\tfrac12\alpha\phi^2$4 strongly suppresses the scalar amplitude, and nonzero $f(\phi)=\tfrac12\alpha\phi^2$5 generically suppresses both $f(\phi)=\tfrac12\alpha\phi^2$6 and $f(\phi)=\tfrac12\alpha\phi^2$7 compared to $f(\phi)=\tfrac12\alpha\phi^2$8. In the limits $f(\phi)=\tfrac12\alpha\phi^2$9 one recovers the Reissner-Nordström-NUT solution, while $A_\mu$00 gives the pure scalar-Gauss-Bonnet NUT-hairy black hole (Butler et al., 2023).

These solution sectors correct two common oversimplifications. First, EMSGB interactions do not only deform RN-type black holes; they also support exact wormholes and NUT-charged families. Second, exoticity need not be sourced by ordinary matter alone: in the traversable wormhole construction, the Gauss-Bonnet-scalar interaction is the only responsible term for the negative energy density necessary for traversability (Cañate et al., 2019, Butler et al., 2023).

6. Holographic realizations and thermodynamic phase structure

In asymptotically AdS spacetime, EMSGB theory provides a setting in which the charged-scalar instability of holographic superconductors and curvature-induced spontaneous scalarization coexist. The bulk action is

$A_\mu$01

with

$A_\mu$02

In the associated effective Schrödinger problem,

$A_\mu$03

the standard holographic superconducting instability is driven by the $A_\mu$04 term, while scalarization is driven by the Gauss-Bonnet term. Numerically, $A_\mu$05 in units $A_\mu$06. For $A_\mu$07, one obtains holographic superconductors with scalar condensation below $A_\mu$08; for $A_\mu$09, scalarized black holes exist even at $A_\mu$10. Near the transition curve the two hairy phases are largely indistinguishable in radial profile and effective potential, but the Gibbs free energy shows that the passage between them is smooth but first order rather than an identification of the two solutions (Guo et al., 2024).

The AdS phase diagram is organized in the grand-canonical ensemble $A_\mu$11. For small $A_\mu$12, RN-AdS is stable at high temperature and cooling below $A_\mu$13 produces a second-order transition into the superconducting branch. For larger $A_\mu$14 or larger $A_\mu$15, the scalarization branch becomes thermodynamically preferred through a first-order transition. The intersection of the two transition lines is not a true multi-critical point in the $A_\mu$16-plane, but the endpoint of the superconducting line where it meets the scalarization curve (Guo et al., 2024).

A distinct thermodynamic program has been carried out for asymptotically flat charged black holes in the canonical ensemble. There the entropy is given by the Wald formula,

$A_\mu$17

and the Helmholtz free energy is

$A_\mu$18

The reported phase structure contains three coupling regimes. For weak coupling, a second-order phase transition coincides with the second bifurcation point at which the scalarized branch reconnects with the RN branch and scalar hair is spontaneously shed. At intermediate coupling, this transition becomes zeroth order, the scalarized branch shrinks, and a fish-like structure develops in $A_\mu$19, yielding up to three phase transitions. In the strong-coupling limit, the scalarized branch becomes Schwarzschild-like and eventually disappears as the RN phase becomes the sole thermodynamically preferred configuration (Erices et al., 28 Apr 2026).

Taken together, these results show that EMSGB theory supports two distinct but interacting notions of phase structure. In AdS, the relevant competition is between superconducting and scalarized hairy phases in a dual-CFT grand-canonical ensemble. In asymptotically flat space, curvature-induced scalarization alone is sufficient to generate bona fide second-order, zeroth-order, and multi-transition thermodynamic phenomena without invoking AdS boundaries or extended thermodynamic variables (Guo et al., 2024, Erices et al., 28 Apr 2026).