Einstein–ModMax Gravity Overview

• Einstein–ModMax gravity is a modification of Einstein gravity coupled to nonlinear ModMax electrodynamics that retains conformal invariance and electric–magnetic duality in four dimensions.
• The theory supports screened Reissner–Nordström black holes, exact multi-black-hole configurations, and holographic transport phenomena through a hyperbolic deformation of standard Maxwell theory.
• Extensions with higher-curvature, scalar, and Lorentz-violating sectors offer novel observational signatures, modified thermodynamics, and unique integrability properties distinct from the Einstein–Maxwell framework.

Einstein–ModMax-type gravity denotes Einstein gravity coupled to the ModMax nonlinear electromagnetic sector, and, in broader usage, the family of extensions that retain the same ModMax constitutive structure while adding matter, higher-curvature terms, or symmetry-breaking sectors. In four dimensions, ModMax is the unique one-parameter deformation of Maxwell electrodynamics that preserves both continuous electric–magnetic duality and conformal invariance; its deformation parameter $\gamma$ is dimensionless and is taken to satisfy $\gamma\ge 0$. Across the current literature, this framework supports screened Reissner–Nordström-type black holes, exact multi-black-hole and Melvin-type solutions, nontrivial holographic transport, and several extended theories whose nonlinear effects range from simple charge rescalings to branches with no Maxwell analogue (Bokulić et al., 8 Jan 2025, Barrientos et al., 3 Jun 2025, Bixano et al., 15 Apr 2026).

1. Core formulation and invariance structure

A representative Einstein–ModMax action takes the form

$$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$

with electromagnetic invariants defined, up to normalization conventions, by

$$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$

and ModMax Lagrangian

$$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$

Equivalent conventions using $(\mathcal S,\mathcal P)$ or $(\mathcal F,\mathcal G)$ differ by factors of $2$ or $4$, but the defining structure is the same: a single hyperbolic deformation of Maxwell theory that remains duality invariant and traceless in $D=4$ (Bokulić et al., 8 Jan 2025, Siahaan, 2024).

The field equations consist of Einstein equations sourced by a traceless nonlinear electromagnetic stress tensor and generalized Maxwell equations written in terms of a constitutive tensor or excitation two-form. In index notation one may write

$\gamma\ge 0$0

with $\gamma\ge 0$1 a linear combination of $\gamma\ge 0$2 and $\gamma\ge 0$3 whose coefficients depend nonlinearly on the invariants. The tracelessness of $\gamma\ge 0$4 is the direct expression of conformal invariance in four dimensions, while the continuous $\gamma\ge 0$5 duality rotates the field strength and its excitation without changing the equations of motion (Bokulić et al., 8 Jan 2025, Bokulić et al., 22 Jul 2025).

A recurrent simplification is that purely electric or purely magnetic sectors become Maxwell-like with $\gamma\ge 0$6-dependent effective couplings. In the static sectors emphasized in several constructions, the electromagnetic contribution to the metric is screened by an overall factor $\gamma\ge 0$7, so nonlinear effects often enter through the replacement $\gamma\ge 0$8 or $\gamma\ge 0$9 (Siahaan, 2024, Sucu et al., 8 Aug 2025). This underlies much of the solution theory, but it does not exhaust the theory’s content: time-dependent perturbations, momentum relaxation, rotating branches, and certain dilatonic sectors exhibit genuinely non-Maxwell behavior (Barrientos et al., 3 Jun 2025, Bixano et al., 15 Apr 2026).

2. Static black holes, screening, and thermodynamic structure

The simplest Einstein–ModMax black holes are static and spherically symmetric, with line element

$$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$0

and RN-like lapse

$$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$1

in dyonic conventions, or its AdS analogue with a cosmological term (Sucu et al., 8 Aug 2025, Bokulić et al., 8 Jan 2025). In these sectors, the horizon equation and extremality bound are deformed only by the screened charge invariant. For the asymptotically flat dyonic case this yields

$$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$2

so increasing $$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$3 weakens the electromagnetic backreaction and drives the geometry toward Schwarzschild (Sucu et al., 8 Aug 2025).

The thermodynamic quantities inherit the same screening. Hawking temperature, entropy, and electric potential take the RN form with the replacement $$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$4, while in AdS the same pattern persists with the cosmological contribution added to $$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$5 (Siahaan, 2024, Panah, 2024). A particularly sharp statement appears in the holographic AdS$$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$6 dyonic sector: for static backgrounds the ModMax action becomes on-shell equivalent to an effective Maxwell action with a renormalized Newton constant, and static thermodynamics coincides with Einstein–Maxwell after screening by $$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$7; genuine nonlinear signatures then require perturbations or momentum relaxation (Barrientos et al., 3 Jun 2025).

Extremal multi-center solutions exhibit a distinctive ModMax charge-to-mass relation. In the extremal Majumdar–Papapetrou limit, the geometry is isometric to the usual MP spacetime, but the balance condition becomes

$$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$8

so the individual black holes have a nonunitary charge-to-mass ratio due to screening (Bokulić et al., 8 Jan 2025). This is one of the clearest examples where the spacetime geometry remains formally familiar while the physical interpretation of charge balance changes.

Thermodynamic extensions modify this picture in different ways. In Einstein-dyonic-ModMax black holes with Generalized Uncertainty Principle corrections, the corrected tunneling temperature satisfies $$S=\int d^4x\,\sqrt{-g}\,\Big[\frac{R-2\Lambda}{16\pi G}+L_{\rm MM}\Big],$$9, slowing evaporation and permitting remnants (Sucu et al., 8 Aug 2025). In regularized four-dimensional Einstein–Gauss–Bonnet gravity, the exact ModMax black hole has

$$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$0

with extremal radius $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$1 and minimum mass $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$2, so the combined higher-curvature and ModMax screening effects generate stable remnants (Hamil, 6 Jan 2026).

3. Exact solution spaces, Harrison maps, and multi-black-hole configurations

Static axisymmetric Einstein–ModMax solutions admit a Weyl reduction closely paralleling Einstein–Maxwell theory. For purely electric fields with vanishing second invariant, the reduced equations become Maxwell-like and can be linearized by a harmonic map, which allows the construction of exact multi-black-hole spacetimes from vacuum Weyl data (Bokulić et al., 8 Jan 2025). The resulting nonextremal colinear configurations are direct ModMax analogues of multi–Reissner–Nordström and generically require conical singularities to balance the mutual attraction.

The extremal limit is structurally different. The conical defects disappear, the geometry becomes the standard Majumdar–Papapetrou multi-center metric, and the electromagnetic sector carries the screened ModMax charge described above (Bokulić et al., 8 Jan 2025). Magnetic and dyonic generalizations follow immediately from the exact $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$3 duality symmetry, and Kastor–Traschen-type multi-black-hole solutions with $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$4 also persist, with the extremality condition modified by the same screened charge invariant (Bokulić et al., 8 Jan 2025).

A parallel development uses Ernst-like methods. In the purely electric and purely magnetic sectors, generalized Harrison transformations preserve sector closure and act fractionally linearly on the Ernst potentials, reducing to the standard Einstein–Maxwell Harrison maps when $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$5 (Bokulić et al., 22 Jul 2025). These transformations generate ModMax Reissner–Nordström-type black holes from vacuum seeds, ModMax Melvin universes, and magnetized embeddings of static black holes. They also produce a balanced black dihole: two oppositely magnetically charged extremal ModMax black holes in equilibrium inside a ModMax Melvin background, with the background field tuned so that the strut disappears (Bokulić et al., 22 Jul 2025).

Exact electromagnetized backgrounds extend further. The ModMax analogue of the Melvin–Bonnor universe, Schwarzschild–Melvin–Bonnor black holes, ModMax C-metrics, Taub–NUT–ModMax geometries, and a vortex-like swirling background have all been constructed explicitly (Barrientos et al., 2024). An important structural point is that some of these solutions admit Kerr–Schild representations that shift both metric and gauge potential, not only the metric. This suggests that Einstein–ModMax retains a nontrivial integrability structure beyond the strictly static sector.

The symmetry analysis has recently been systematized in Einstein–ModMax–scalar systems. In the real potential space $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$6, the visible symmetries form a solvable algebra, Ehlers symmetry survives in the gravito-rotational sector, and electric and magnetic Harrison maps coexist only when

$$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$7

which selects the frozen ModMax sector $$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$8 (Bixano et al., 18 May 2026). In that regime the Maxwell sectorial transformations survive, but with explicit ModMax deformations in the electromagnetic block.

4. Holography, AdS reductions, and transport

Einstein–ModMax theory has a particularly developed holographic realization in asymptotically AdS$$X=\frac14 F_{\mu\nu}F^{\mu\nu},\qquad Y=\frac14 F_{\mu\nu}\tilde F^{\mu\nu},$$9 backgrounds dual to $$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$0-dimensional field theories. In the dyonic black-brane model with linear axions,

$$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$1

the axions break translational invariance and permit finite DC transport (Barrientos et al., 3 Jun 2025). The background remains analytically tractable, and the conductivities can be expressed in terms of horizon data through a Donos–Gauntlett-type construction.

Without momentum relaxation, the DC electric conductivity is purely Hall. With axions turned on, both longitudinal and Hall conductivities become nonzero, and the Hall angle obeys

$$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$2

at low temperature, reproducing the anomalous strange-metal scaling while leaving the ModMax parameter $$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$3 to suppress the overall magnitude through the factor $$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$4 in the denominator (Barrientos et al., 3 Jun 2025). In the large-$$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$5 regime the Hall response is quenched, and transport becomes predominantly longitudinal.

The same model yields a nontrivial Nernst signal. For small magnetic field, $$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$6 is bell-shaped and reminiscent of high-$$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$7 cuprates; at larger field it crosses over to a negative tail associated with quasiparticle contributions (Barrientos et al., 3 Jun 2025). The deformation parameter $$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$8 controls the amplitude and the onset and critical temperatures of the dome-like profile, while very large $$L_{\rm MM}(X,Y;\gamma)=-X\cosh\gamma+\sqrt{X^2+Y^2}\,\sinh\gamma.$$9 drives the system into a broad negative-Nernst regime interpreted as an exotic quasiparticle-dominated normal state.

A lower-dimensional holographic reduction leads to a rather different conclusion. Starting from four-dimensional Einstein–ModMax theory and reducing to two dimensions, one obtains a JT-like dilaton gravity model with a projected ModMax sector. The parent $(\mathcal S,\mathcal P)$0 theory remains duality and conformally invariant, but the $(\mathcal S,\mathcal P)$1 projected theory is neither conformal nor duality invariant, and the boundary central charge is corrected perturbatively as

$(\mathcal S,\mathcal P)$2

to quadratic order in the gauge and ModMax couplings (Rathi et al., 2023). This establishes a useful distinction: conformal and duality invariance are robust in four-dimensional ModMax itself, but they do not automatically survive dimensional reduction.

5. Higher-curvature, scalar, and Lorentz-violating extensions

Much of the recent literature uses “Einstein–ModMax-type gravity” in a broader sense that includes higher-curvature, scalar, and Lorentz-violating sectors while keeping the ModMax electrodynamics intact. In $(\mathcal S,\mathcal P)$3 gravity with constant curvature $(\mathcal S,\mathcal P)$4, purely electric ModMax black holes satisfy

$(\mathcal S,\mathcal P)$5

and the entropy becomes

$(\mathcal S,\mathcal P)$6

so the $(\mathcal S,\mathcal P)$7 sector rescales both the effective charge term and the Wald entropy (Panah, 2024). Thermodynamic stability can then be analyzed through heat capacity and HPEM thermodynamic geometry, whose Ricci scalar diverges at both the physical limitation points and the heat-capacity poles (Panah, 2024).

Adding quintessence to the $(\mathcal S,\mathcal P)$8-ModMax system introduces a Kiselev-type term $(\mathcal S,\mathcal P)$9 in the lapse function and substantially enlarges the optical shadow. In the purely electric sector, ModMax again collapses to Maxwell up to the factor $(\mathcal F,\mathcal G)$0, but higher-curvature and quintessence effects enlarge the photon sphere and shadow more strongly than charge shrinks them (Al-Badawi et al., 13 May 2026). A related $(\mathcal F,\mathcal G)$1-ModMax model with a cloud of strings yields analytical photon-sphere and shadow radii, with the string-cloud parameter $(\mathcal F,\mathcal G)$2 giving the dominant shadow amplification (Al-Badawi, 26 Oct 2025).

Regularized four-dimensional Einstein–Gauss–Bonnet gravity provides the higher-curvature extension in which the thermodynamic and quasinormal-mode sectors have been studied most explicitly (Hamil, 6 Jan 2026). By contrast, the exact rotating dilatonic branch discovered in Einstein–ModMax–dilaton theory does not admit continuation to Maxwell except at the isolated coupling $(\mathcal F,\mathcal G)$3. It belongs to the nonlinear sector with constant $(\mathcal F,\mathcal G)$4, carries both $(\mathcal F,\mathcal G)$5 and $(\mathcal F,\mathcal G)$6, possesses a NUT-free asymptotically flat limit, and has a prolate black-hole regime where the null energy condition holds throughout the exterior region (Bixano et al., 15 Apr 2026). This is the clearest current example showing that Einstein–ModMax-type gravity is not always reducible to Einstein–Maxwell with screened charge.

Lorentz-violating versions add further branch structure. In Kalb–Ramond–ModMax black holes the metric becomes

$(\mathcal F,\mathcal G)$7

with branch selector $(\mathcal F,\mathcal G)$8. The ordinary branch $(\mathcal F,\mathcal G)$9 admits extremal and nonextremal configurations and finite-radius tidal inversion, whereas the phantom branch $2$0 generically has a single horizon and no tidal inversion (Sucu et al., 27 Jan 2026). Bumblebee gravity with a cloud of strings produces another Lorentz-violating deformation,

$2$1

together with modified thermodynamics, shadow radius, and greybody factors (Belchior et al., 20 May 2026).

6. Optical, orbital, and observational signatures

Because many Einstein–ModMax black holes differ from Einstein–Maxwell primarily through screened charge terms, their observational signatures often organize themselves around the competition between $2$2 suppression and whatever extra sector accompanies ModMax. In weakly magnetized Einstein–ModMax black holes, Wald-type magnetization generalizes with

$2$3

and charged particle motion acquires an effective potential in which $2$4 is replaced by $2$5 (Siahaan, 2024). The resulting Larmor and anti-Larmor branches shift the ISCO and orbital energetics in the expected direction, while the nonlinear effect itself remains an exponential screening.

This weak-field regime has also been pushed into nonlinear dynamics. For purely magnetically charged black holes in a weak uniform external field, charged-particle motion becomes nonintegrable once the magnetic field is turned on; Shannon entropy and mutual information for particle pairs separate regular from chaotic orbits efficiently, and EHT shadow bounds constrain the viable $2$6 region (Liu et al., 23 Apr 2026). A notable conclusion is that chaos is much more sensitive to the particle constants of motion $2$7 than to the model parameters $2$8 and $2$9 themselves (Liu et al., 23 Apr 2026).

Accretion-disk and QPO phenomenology show a similar pattern. In weakly magnetized dyonic Einstein–ModMax black holes surrounded by quintessence, the epicyclic frequencies, ISCO, and Novikov–Thorne flux profiles depend on the screened combination $4$0, the electric and magnetic Lorentz couplings, and the quintessence parameters. Markov Chain Monte Carlo fits to high-frequency QPO data place upper bounds on the ModMax coupling $4$1, magnetic interaction, and dyonic charge, and find that the observations remain close to the Einstein–Maxwell limit (Rehman et al., 6 Jan 2026).

Optical propagation has been developed most extensively for Einstein-dyonic-ModMax black holes. In vacuum, the weak deflection angle begins as

$4$2

so charge-dependent corrections are exponentially damped as $4$3 grows (Sucu et al., 8 Aug 2025). In homogeneous plasma, the deflection becomes frequency dependent and still carries $4$4-suppressed charge terms; in axion–plasmon media, an additional resonant structure appears through the refractive index, again with the dyonic contribution exponentially screened (Sucu et al., 8 Aug 2025). The same work shows that the heat-capacity critical points shift with $4$5, while larger $4$6 reduces the region of near-horizon energy-condition violation (Sucu et al., 8 Aug 2025).

A recurring misconception is therefore only partially correct: many static observables in Einstein–ModMax systems do look like Einstein–Maxwell with $4$7, but the full theory has a much wider phenomenology. Exact rotating branches without Maxwell analogue exist (Bixano et al., 15 Apr 2026), nonlinear transport in holography cannot be recovered from simple charge screening (Barrientos et al., 3 Jun 2025), and higher-curvature or Lorentz-violating sectors produce branch-dependent shadows, lensing, tidal forces, and remnants that are specific to Einstein–ModMax-type models rather than to screened electrovacuum alone (Sucu et al., 27 Jan 2026, Hamil, 6 Jan 2026).