---
title: ModMax Black Hole Solutions
url: https://www.emergentmind.com/topics/modmax-black-hole
type: topic
---

# ModMax Black Hole Solutions

A ModMax black hole is a charged black-hole solution sourced by Modified Maxwell (ModMax) nonlinear electrodynamics, either in Einstein gravity or in extensions such as \(F(R)\) gravity, Einstein–Gauss–Bonnet gravity, dRGT-like massive gravity, Kalb–Ramond gravity, and backgrounds containing quintessence, perfect-fluid dark matter, or a cloud of strings. In the purely electric, static, spherically symmetric sector, the characteristic deformation is that the Coulomb term is screened by a factor such as \(e^{-\gamma}\), so many exact metrics take a Reissner–Nordström-like form with \(Q^2\to e^{-\gamma}Q^2\); the same sector supports detailed analyses of horizons, thermodynamics, phase structure, photon spheres, shadows, scattering, and quasinormal modes [2308.12572, 2402.12492, 2601.02717].

## 1. Field-theoretic basis and the canonical Einstein–ModMax solution

ModMax is presented as the unique one-parameter nonlinear electrodynamics that preserves conformal invariance and electromagnetic duality. In one standard Einstein–ModMax formulation, the action is
\[
S=\int d^4x \sqrt{-g}\,\Bigl[\frac{1}{16\pi}R+\mathcal{L}_{\rm ModMax}(F,G)\Bigr],
\]
with
\[
\mathcal{L}_{\rm ModMax}
=-\frac{\cosh\gamma}{4}F+\frac{\sinh\gamma}{4}\sqrt{F^2+G^2},
\]
where \(F\equiv F_{\mu\nu}F^{\mu\nu}\) and \(G\equiv F_{\mu\nu}{}^\star F^{\mu\nu}\) [2308.12572]. A closely related convention writes the Lagrangian in terms of the invariants \(\mathcal{S}\) and \(\mathcal{P}\), again with a single real, dimensionless ModMax parameter \(\gamma\) [2409.12336].

For a static, spherically symmetric black hole with line element
\[
ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2),
\]
the Einstein–ModMax field equations admit the exact solution
\[
f(r)=1-\frac{2M}{r}+\frac{e^{-\gamma}Q^2}{r^2},
\]
with ADM mass \(M\) and electric charge \(Q\) [2308.12572]. Several later treatments use equivalent parameterizations, such as \(e^{-\eta}Q^2\) or \(\widetilde Q^2=Q^2e^{-\lambda}\), but the structural role of the ModMax parameter is the same: the electric sector is screened by an exponential factor [2511.07487, 2605.26131].

A recurrent point in the literature is that, within the purely electric static sector, ModMax often reduces Maxwell electrodynamics up to a rescaling of the electric charge. This is stated explicitly for higher-order curvature gravity with quintessence, where “the ModMax theory effectively reduces Maxwell electrodynamics up to a rescaling of the electric charge,” so that the resulting solution is a consistent subset of the broader nonlinear theory [2605.13908]. This does not eliminate genuinely nonlinear effects in propagation or thermodynamic diagnostics, but it explains why many exact metrics retain Reissner–Nordström-type algebraic form.

## 2. Horizon structure, extremality, and representative geometries

For the canonical Einstein–ModMax black hole, the horizons are the roots of \(f(r)=0\):
\[
r_\pm=M\pm\sqrt{M^2-e^{-\gamma}Q^2},
\]
so the extremal limit is
\[
M^2=e^{-\gamma}Q^2.
\]
As \(\gamma\) increases, the charge term is suppressed and the geometry approaches Schwarzschild [2308.12572, 2506.17489].

Representative ModMax black-hole geometries in the literature are summarized below.

| Framework | Metric function | Distinctive deformation |
|---|---|---|
| Einstein–ModMax | \(f(r)=1-\frac{2M}{r}+\frac{e^{-\gamma}Q^2}{r^2}\) | Screened RN term |
| \(F(R)\)–ModMax | \(g(r)=1-\frac{m_0}{r}-\frac{R_0}{12}r^2+\frac{q^2e^{-\gamma}}{(1+f_{R_0})r^2}\) | Constant-curvature \(F(R)\) sector |
| Topological Mod(A)Max AdS | \(f(r)=k-\frac{m}{r}-\frac{\Lambda r^2}{3}+\eta\frac{q^2e^{-\gamma}}{r^2}\) | Horizon topology \(k=\{+1,0,-1\}\) |
| KR + PFDM | \(f(r)=\frac1a-\frac{2M}{r}+\frac{\widetilde Q^2}{a^2r^2}+\frac{\beta}{ar}\ln\!\bigl(\frac{r}{|\beta|}\bigr)\) | Lorentz violation and logarithmic PFDM tail |
| 4D EGB–ModMax | \(f(r)=1+\frac{r^2}{2\alpha}\Bigl(1-\sqrt{1+\frac{4\alpha}{r^2}\bigl(\frac{2M}{r}-\frac{Q^2}{r^2}\bigr)}\Bigr)\) | Gauss–Bonnet square-root branch |

These examples illustrate two generic patterns. First, the electromagnetic contribution almost always enters through \(q^2e^{-\gamma}\) or an equivalent screened combination. Second, the horizon polynomial is altered by curvature, topology, or environmental terms. In \(F(R)\)–ModMax theory, the horizon equation is generically quartic; for \(R_0>0\) three real roots can appear (inner, event, cosmological), while for \(R_0<0\) one finds two, one extremal, or none [2402.12492]. In topological AdS constructions the discrete index \(k\) changes the horizon geometry directly, while in perfect-fluid dark matter or string-cloud backgrounds additional logarithmic or deficit-angle terms modify the root structure [2512.22654, 2605.26131, 2605.20570].

Several limiting relations are standard. The Einstein–ModMax metric reduces to Reissner–Nordström for \(\gamma\to0\) [2308.12572]. In \(F(R)\)–ModMax theory, the joint limit \(\gamma\to0\), \(f_{R_0}\to0\), and \(R_0=4\Lambda\) recovers Reissner–Nordström–(A)dS [2402.12492]. In higher-order curvature gravity with quintessence, taking \(c\to0\) removes the quintessence term, while \(f_{R_0}\to0\) returns the general-relativistic sector and \(\gamma\to0\) returns Maxwell electrodynamics [2605.13908].

## 3. Thermodynamics, stability, and thermodynamic geometry

For the asymptotically flat Einstein–ModMax black hole, the event-horizon radius \(r_h\) determines the mass, temperature, and entropy through
\[
M(r_h)=\frac{r_h}{2}+\frac{Q^2e^{-\gamma}}{2r_h},
\qquad
T=\frac{1}{4\pi r_h}\Bigl(1-\frac{Q^2e^{-\gamma}}{r_h^2}\Bigr),
\qquad
S=\pi r_h^2.
\]
The first law takes the standard form
\[
dM=T\,dS+\Phi\,dQ,
\]
and the fixed-\(Q\) heat capacity is
\[
C_Q=\frac{2\pi r_h^2\bigl(1-\frac{Q^2e^{-\gamma}}{r_h^2}\bigr)}
{\bigl(-1+\frac{3Q^2e^{-\gamma}}{r_h^2}\bigr)}.
\]
Its zeros occur at the physical-limitation point \(r_h^2=Q^2e^{-\gamma}\), while its divergence at \(r_h^2=3Q^2e^{-\gamma}\) signals a second-order phase transition [2507.05864]. In the extended phase space, the same family satisfies
\[
dM=T\,dS+\Phi\,dQ+V\,dP,
\qquad
M=2TS+\Phi Q-2PV,
\]
and the isoperimetric ratio is exactly \(\mathcal R=1\) [2507.05864].

In \(F(R)\)–ModMax theory, the thermodynamic quantities are modified by the constant-curvature factor \(1+f_{R_0}\). The entropy becomes
\[
S=\pi(1+f_{R_0})r_+^2,
\]
the ADM mass is
\[
M=\frac12(1+f_{R_0})m_0,
\]
and the Hawking temperature is
\[
T=\frac{1}{4\pi r_+}-\frac{R_0r_+}{16\pi}
-\frac{q^2e^{-\gamma}}{4\pi(1+f_{R_0})r_+^3}.
\]
The first law
\[
dM=T\,dS+\Phi\,dQ
\]
is verified explicitly. Local stability is controlled by the heat capacity \(C_Q\), while global stability is analyzed through the Helmholtz free energy \(F_H=M-TS\). The HPEM thermodynamic Ricci scalar diverges exactly at the physical-bound point and at the phase-transition points given by the poles of \(C_Q\) [2402.12492].

Topological Mod(A)Max AdS black holes extend the same structure to nontrivial horizon topology and pressure. For
\[
f(r)=k-\frac{m}{r}-\frac{\Lambda r^2}{3}+\eta\frac{q^2e^{-\gamma}}{r^2},
\]
one has
\[
S=\frac{\Sigma_k r_+^2}{4},\qquad
T=\frac{1}{4\pi}\Bigl(\frac{k}{r_+}-\Lambda r_+ -\eta\frac{q^2e^{-\gamma}}{r_+^3}\Bigr),
\]
together with the extended first law and Smarr relation. For \(k=+1\), the critical point satisfies
\[
r_c^2=6\,\eta\,q^2e^{-\gamma},
\]
whereas for \(k=0\) or \(k=-1\) no Van der Waals-type critical point exists in the physical region [2512.22654].

Thermodynamic geometry is used widely in the ModMax literature. In the quintessence-corrected higher-curvature case, the Ruppeiner scalar diverges precisely at the zeros of the heat capacity, while the Weinhold curvature does not generally coincide with those singularities [2605.13908]. In \(F(R)\)–ModMax theory, the HPEM curvature distinguishes bound-point and critical-point singularities through different sign changes [2402.12492].

## 4. Geodesics, birefringence, photon spheres, and shadows

Null propagation in ModMax is not exhausted by the background metric. For general nonlinear electrodynamics, wave fronts propagate according to an effective metric
\[
g_{\rm eff}^{\mu\nu}(a)=g^{\mu\nu}-4\lambda_a t^{\mu\nu},
\]
and for ModMax one finds
\[
\lambda_1=4\tanh\gamma\,[\sqrt{F^2+G^2}+F\tanh\gamma],\qquad \lambda_2=0.
\]
Hence one polarization sees the background geometry, while the other sees a distinct optical geometry [2308.12572]. In the static spherically symmetric ModMax black-hole background, the equatorial optical metrics are
\[
ds_{\rm eff(1)}^2=-e^{-\gamma}f(r)\,dt^2+e^{\gamma}f(r)^{-1}dr^2+e^{\gamma}r^2d\phi^2,
\]
\[
ds_{\rm eff(2)}^2=ds_{\rm background}^2.
\]
This yields birefringence and branch-dependent lensing, redshift, and shadow observables [2308.12572].

For the Einstein–ModMax background, the unstable circular photon orbit is located at
\[
r_c=\frac{3M}{2}\Bigl[1\pm\sqrt{1-\frac{8}{9}\frac{e^{-\gamma}Q^2}{M^2}}\Bigr].
\]
The critical impact parameters are
\[
b_{c1}=e^{-\gamma}\frac{r_c}{\sqrt{f(r_c)}},
\qquad
b_{c2}=\frac{r_c}{\sqrt{f(r_c)}},
\]
so the branch-1 shadow is reduced by an additional factor \(e^{-\gamma}\) relative to branch 2 [2308.12572]. For scalar-wave absorption, the low-frequency limit is universal,
\[
\sigma_{\rm abs}(\omega\to0)=A_h=4\pi r_h^2,
\]
while the high-frequency limit approaches the capture cross section
\[
\sigma_{\rm geo}=\pi b_c^2.
\]
Numerically, increasing \(\gamma\) screens the charge contribution and causes cross-section curves for different \(Q\) to coalesce [2506.17489].

In matter-dressed or modified-gravity backgrounds, the photon-sphere condition usually retains the form
\[
2f(r_s)-r_s f'(r_s)=0,
\]
but the resulting shadow radius depends sensitively on the extra sector. In higher-order curvature gravity with quintessence, exact analytic expressions for \(r_{\rm ph}\) are available for \(\omega=-1,-2/3,-1/3\); higher-order curvature corrections and quintessence significantly enhance the shadow size, whereas the electric charge has the opposite effect, and quintessence has a more pronounced impact on the shadow than the charge [2605.13908]. In perfect-fluid dark-matter backgrounds, the shadow radius \(R_{\rm sh}=r_s/\sqrt{f(r_s)}\) also controls the eikonal quasinormal frequency through \(\Omega_c=1/R_{\rm sh}\) [2602.07806].

## 5. Modified-gravity and matter-dressed ModMax black holes

The phrase “ModMax black hole” now denotes a broad class of exact solutions obtained by combining the same screened electrodynamic sector with modified gravitational dynamics or environmental sources.

In constant-curvature \(F(R)\) gravity, the exact electrically charged black hole is
\[
g(r)=1-\frac{m_0}{r}-\frac{R_0}{12}r^2+\frac{q^2e^{-\gamma}}{(1+f_{R_0})r^2},
\]
with entropy and mass rescaled by \(1+f_{R_0}\) [2402.12492]. In the same \(F(R)\) setting with quintessence dark energy, the lapse function becomes
\[
f(r)=1-\frac{m_0}{(1+f_{R_0})r}+\frac{q^2e^{-\gamma}}{(1+f_{R_0})r^2}-\frac{R_0r^2}{12}-\frac{c}{r^{3\omega+1}},
\]
so the ModMax screening competes directly with the quintessence tail \(c\,r^{-3\omega-1}\) [2605.13908].

In 4-dimensional Einstein–Gauss–Bonnet gravity, the negative branch of the exact solution is
\[
f(r)=1+\frac{r^2}{2\alpha}\Bigl(1-\sqrt{1+\frac{4\alpha}{r^2}\bigl(\frac{2M}{r}-\frac{Q^2}{r^2}\bigr)}\Bigr),
\qquad Q^2=q^2e^{-\gamma}.
\]
The horizon condition implies
\[
M=\frac{r_H}{2}\Bigl(1+\frac{\alpha+Q^2}{r_H^2}\Bigr),
\]
and extremality gives
\[
r_{\rm ext}^2=Q^2+\alpha,\qquad M_{\rm ext}=\sqrt{Q^2+\alpha}.
\]
The entropy acquires the Gauss–Bonnet correction
\[
S=\pi r_H^2+4\pi\alpha\ln r_H,
\]
and the minimum mass
\[
M_{\min}=\sqrt{\alpha+Q^2}
\]
is interpreted as a stable remnant [2601.02717].

Lorentz-violating extensions are especially active. In Kalb–Ramond gravity with perfect-fluid dark matter, the lapse is
\[
f(r)=\frac1a-\frac{2M}{r}+\frac{\widetilde Q^2}{a^2r^2}
+\frac{\beta}{ar}\ln\!\Bigl(\frac{r}{|\beta|}\Bigr),
\]
where \(a\equiv1-\alpha\) and \(\widetilde Q^2=Q^2e^{-\lambda}\). The perfect-fluid dark matter term introduces an additional logarithmic correction that is subleading at large \(r\) but significant at intermediate scales [2605.26131]. In Einstein–bumblebee gravity with a cloud of strings, one has
\[
A(r)=1-\alpha-\frac{2M}{r}+\frac{\beta}{r^2},
\qquad
B(r)=\frac{1+\ell}{A(r)},
\]
with
\[
\beta=\frac{2(1+\ell)Q^2e^{-\gamma}}{2+\ell},
\]
so Lorentz violation, strings, and ModMax screening enter nontrivially and independently [2605.20570]. In the dyonic Kalb–Ramond plus string-cloud solution,
\[
f(r)=\frac{1-\alpha}{1-\ell}-\frac{2M}{r}
+\frac{e^{-\gamma}(Q_e^2+Q_m^2)}{(1-\ell)^2r^2},
\]
and all shadow-size observables inherit the conical-deficit factor \(\sqrt{(1-\alpha)/(1-\ell)}\) [2603.11312].

Massive-gravity versions preserve the same screened electric term while adding massive couplings. A representative AdS ModMax–dRGT black hole has
\[
f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^2+\frac{e^{-\gamma}q^2}{r^2}
+C m_g^2\Bigl(Cc_2+\frac{c_1r}{2}\Bigr),
\]
so the screening factor directly enters the enthalpy, equation of state, thermodynamic geometry, and heat-engine efficiency [2606.28468].

## 6. Perturbations, quasinormal modes, and current interpretation

Linear perturbations of ModMax black holes are typically reduced to Schrödinger-type equations in the tortoise coordinate. For topological ModMax (A)dS black holes, the scalar, electromagnetic, and Dirac effective potentials are
\[
V_s=f(r)\Bigl[\frac{\ell(\ell+1)}{r^2}+\frac{f'(r)}{r}\Bigr],
\qquad
V_{EM}=f(r)\frac{\ell(\ell+1)}{r^2},
\qquad
V_\pm=W^2\pm\frac{dW}{dx},
\]
with \(W=\xi\sqrt{f(r)}/r\). In the eikonal limit,
\[
\omega(\ell\gg1)=\Omega_c\,\ell-i\Bigl(n+\frac12\Bigr)\lambda_L,
\]
where \(\Omega_c=\sqrt{f(r_c)}/r_c\) and \(\lambda_L\) is the Lyapunov exponent of the unstable null orbit [2411.02907]. In that model, both \(\Re\omega\) and \(|\Im\omega|\) decrease monotonically as \(\gamma\) increases from zero [2411.02907].

The dependence on \(\gamma\) is not universal across all embeddings. In the Kalb–Ramond phantom-sector construction, Padé-averaged WKB and time-domain analyses show that increasing either \(\ell\) or \(\gamma\) raises \(\Re\omega\) and \(|\Im\omega|\), with the strongest sensitivity in the phantom branch \(\zeta=-1\) [2507.19088]. This model dependence is important: the screened charge term is common, but the surrounding gravitational sector controls how the perturbation barrier is reshaped.

Emission-rate calculations likewise tie thermodynamic and optical sectors together. For topological ModMax (A)dS black holes,
\[
\frac{d^2E}{d\omega\,dt}
=\frac{2\pi^3R_{\rm sh}^2\omega^3}{e^{\omega/T_H}-1},
\]
so the shadow radius and Hawking temperature jointly determine the high-frequency emission profile [2411.02907]. In dyonic Kalb–Ramond black holes with a string cloud, the same geometric-optics limit gives \(\sigma_{\rm lim}\approx \pi R_{\rm sh}^2\), and the shadow radius carries the same conical-deficit prefactor that appears in asymptotic observables [2603.11312].

A plausible implication is that “ModMax black hole” names a robust RN-like core plus a large family of curvature- and matter-dressed exact solutions. The core sector is governed by the screened combination \(e^{-\gamma}Q^2\); the surrounding theory then determines whether that screening primarily shifts horizons, alters entropy and criticality, produces birefringence, changes shadow size, or reshapes the quasinormal spectrum.

Source: https://www.emergentmind.com/topics/modmax-black-hole