Einstein–Born–Infeld Black Holes

• Einstein–Born–Infeld black holes are solutions in Einstein gravity coupled to nonlinear electrodynamics, exhibiting modified extremality and horizon structures relative to Schwarzschild and Reissner–Nordström black holes.
• They display rich phenomena including static, rotating, higher-dimensional, and hairy configurations, where the Born–Infeld parameter controls nonlinear electromagnetic corrections.
• Their unique features influence thermodynamics, phase transitions in AdS spaces, and stability against perturbations, offering insights into energy extraction and nonlinear electromagnetic shielding.

Searching arXiv for recent and foundational papers on Einstein–Born–Infeld black holes to ground the article in current literature. I’ll retrieve a compact set of arXiv papers covering static, rotating, thermodynamic, stability, and near-horizon aspects of Einstein–Born–Infeld black holes. Einstein–Born–Infeld black holes are black-hole solutions of Einstein gravity coupled to Born–Infeld nonlinear electrodynamics. They reduce to Schwarzschild in the uncharged limit and to Einstein–Maxwell charged black holes in the weak-nonlinearity limit, but the Born–Infeld square-root structure modifies the near-horizon field configuration, extremality conditions, horizon multiplicity, thermodynamic phase structure, and perturbative dynamics. The subject now includes asymptotically flat electric and magnetic solutions, AdS families with extended thermodynamics, rotating and higher-dimensional generalizations, and scalarized or hairy branches (Myung et al., 2012, He et al., 2019, Cheng et al., 1 Jul 2025).

1. Field equations and Born–Infeld parameterizations

A common four-dimensional formulation uses the action

$$\mathcal{S}=\frac{1}{16\pi}\int d^4x\,\sqrt{-g}\left[ R+4\mathcal{L}(s,p)\right],$$

with

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$

and electromagnetic invariants

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$

In this convention the Maxwell limit is $a\to 0$ (Cheng et al., 1 Jul 2025). A second standard convention writes the Lagrangian 4-form as

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$

with $\mathcal{F}=F_{ab}F^{ab}$ and $\bm F=d\bm A$ (He et al., 2019). A third convention uses

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$

with $F=F_{\mu\nu}F^{\mu\nu}$ (Pereira et al., 2015). These parameterizations are notationally different but serve the same role: the Born–Infeld parameter controls departure from Maxwell electrodynamics.

The nonlinear constitutive relation is one of the main structural differences from Einstein–Maxwell theory. In the Iyer–Wald analysis of electrically charged EBI black holes, the constitutive tensor is

$\bm G=h'(\mathcal F)\,\bm F,$

and the conserved electric charge is defined by

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$0

Thus the displacement-like tensor $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$1, rather than $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$2, enters the conserved-charge and variational identities (He et al., 2019).

The Born–Infeld parameter is interpreted in several closely related ways across the literature: as the scale controlling the nonlinearity of the electromagnetic sector, as the maximum field intensity in the purely electric theory, and, in string-motivated conventions, through

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$3

The weak-field limit recovers Einstein–Maxwell theory. In many conventions this is $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$4 or $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$5; in the convention of the extremal-superradiance analysis it is $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$6, because that paper uses $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$7 (Breton, 2016, Hale et al., 16 Sep 2025, Wu et al., 2024).

2. Static geometries, horizons, and extremality

The standard static, spherically symmetric line element is

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$8

or equivalent sign conventions. For electrically charged asymptotically flat EBI black holes one representative form is

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$9

with

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$0

The event horizon is the largest root of $$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$1, and extremality is characterized by

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$2

In that same formulation,

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$3

so extremality is more complicated than in Reissner–Nordström even though it is still determined by the vanishing of the minimum of $$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$4 (He et al., 2019).

For magnetically charged asymptotically flat EBI black holes, the metric function can be written as

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$5

with

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$6

Its large-radius expansion is

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$7

which makes the asymptotically Reissner–Nordström-like behavior explicit (Myung et al., 2012).

The electromagnetic field is softened relative to Maxwell theory. In a purely electric sector one finds, for example,

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$8

or equivalently

$$s=-\frac14 F^{\mu\nu}F_{\mu\nu},\qquad p=-\frac18 \epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$9

so the electric field is finite at the origin, unlike in Maxwell theory (Meng et al., 2017, Pereira et al., 2015). This softening does not generally remove the spacetime singularity. The asymptotically flat EBI solutions discussed in the rotating and extraction literature still have a curvature singularity at $a\to 0$0, and recent numerical work on rotating EBI black holes likewise finds a divergent Kretschmann scalar in the static seed solution (Breton, 2016, Cheng et al., 1 Jul 2025).

The horizon structure is richer than in Reissner–Nordström. One exact energy-decomposition analysis shows that EBI black holes split into two families according to

$a\to 0$1

For $a\to 0$2, the mass formula is monotonic in the irreducible mass, there is no minimum irreducible mass, and the black hole always has a single nondegenerate horizon. For $a\to 0$3, the mass function develops a minimum at

$a\to 0$4

and the black hole has an RN-like two-horizon sector up to a transitional irreducible mass, after which a single-horizon BI sector reappears (Pereira et al., 2015). A closely related analysis of energy extraction states that, for fixed $a\to 0$5 and $a\to 0$6, the EBI outer horizon is always larger than the RN outer horizon, which the authors interpret as a nonlinear electromagnetic shielding effect (Breton, 2016).

3. Hair, quasilocal energy, and extractable mass

A recurrent interpretation treats the EBI black hole as a nonlinear-electromagnetic hairy black hole. In the magnetic solution, the metric is rewritten as

$a\to 0$7

which defines a quasilocal mass function outside the horizon. The photon sphere at $a\to 0$8 is fixed by

$a\to 0$9

and the exterior Born–Infeld hair is decomposed into a near-horizon shell and a far region by

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$0

The quantity of interest is the ratio $$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$1, because Hod’s conjecture states that

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$2

For the representative values

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$3

the paper finds

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$4

so more than half of the total hair mass lies beyond the photon sphere (Myung et al., 2012).

The same nonlinear structure underlies generalized Christodoulou–Ruffini decompositions. In the spherically symmetric EBI setting,

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$5

and the total mass becomes an explicit function $$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$6 rather than the Maxwellian

$$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$7

This decomposition is the basis for the two-family classification by $$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$8, the existence of minimum-energy extremal states when $$\bm{L}=\frac{1}{16\pi}\bigl[R-h(\mathcal{F})\bigr]\bm{\epsilon},\qquad h(\mathcal{F})=\frac{\beta^2}{4}\left(1-\sqrt{1+\frac{\mathcal{F}}{2\beta^2}}\right),$$9, and the evaporation result that such black holes approach the zero-temperature endpoint only asymptotically in infinite time (Pereira et al., 2015).

The irreducible-mass viewpoint also controls extraction processes. For spherically symmetric black holes in nonlinear electrodynamics,

$\mathcal{F}=F_{ab}F^{ab}$0

and the extractable energy is

$\mathcal{F}=F_{ab}F^{ab}$1

For fixed nonextremal $\mathcal{F}=F_{ab}F^{ab}$2 and $\mathcal{F}=F_{ab}F^{ab}$3, the EBI irreducible mass is always greater than or equal to the RN irreducible mass, hence

$\mathcal{F}=F_{ab}F^{ab}$4

The effective ergoregion for charged particles is likewise smaller in EBI than in RN. The extremal case is subtler: the formal EBI expression can yield more extractable energy than extremal RN, but those extremal EBI solutions do not possess a linear electromagnetic black-hole limit, because the corresponding Maxwell limit is overcharged and horizonless (Breton, 2016).

4. Thermodynamics, AdS criticality, and holographic action growth

In Einstein gravity, the thermodynamic horizon data have the standard form. For static charged EBI black holes,

$\mathcal{F}=F_{ab}F^{ab}$5

and the entropy is the Bekenstein–Hawking value

$\mathcal{F}=F_{ab}F^{ab}$6

The Iyer–Wald formalism then yields the physical-process inequality

$\mathcal{F}=F_{ab}F^{ab}$7

under the null energy condition, and this same framework underlies second-order stability analyses (He et al., 2019).

AdS extensions preserve the characteristic Born–Infeld square-root and hypergeometric structures while enlarging the phase diagram. In four-dimensional Einstein–Born–Infeld–Yang–Mills theory, the metric function is

$\mathcal{F}=F_{ab}F^{ab}$8

Setting $\mathcal{F}=F_{ab}F^{ab}$9 recovers the usual EBI black hole in (A)dS space. In extended phase space,

$\bm F=d\bm A$0

and the first law becomes

$\bm F=d\bm A$1

The solution exhibits the standard AdS charged-black-hole structure: Van der Waals-like small/large black-hole transitions, a swallow-tail Gibbs free energy for $\bm F=d\bm A$2, and additive charge contributions in appropriate limits (Meng et al., 2017).

Topological classifications of Born–Infeld AdS criticality sharpen this picture. In the 4D Einstein–Gauss–Bonnet–Born–Infeld AdS system, the ordinary BI-AdS limit is obtained by $\bm F=d\bm A$3. In that BI-AdS limit there can be two critical points in the reentrant regime with topological charges

$\bm F=d\bm A$4

so the total topological charge is $\bm F=d\bm A$5. In the combined 4D EGB+BI system, by contrast, one can obtain either one critical point or three critical points together with a triple point, yet the total topological charge remains

$\bm F=d\bm A$6

This distinguishes the ordinary Einstein–Born–Infeld AdS critical set from its Gauss–Bonnet deformation (Yerra et al., 2022).

Holographic action-growth calculations reveal a further electric–magnetic asymmetry. For purely electric EBI black holes with two horizons, the late-time action growth has the RN-like form

$\bm F=d\bm A$7

For purely magnetic double-horizon EBI black holes, however,

$\bm F=d\bm A$8

Adding a matter boundary term restores a magnetic contribution and interpolates between electric and magnetic ensembles through a parameter $\bm F=d\bm A$9 (Meng, 2018).

5. Cosmic censorship, superradiance, and scalarization

The static, electrically charged EBI black hole has been tested directly against overcharging in the Sorce–Wald framework. Using the first- and second-order variational identities, the analysis finds that optimal first-order perturbations satisfy

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$0

but second-order backreaction enforces

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$1

and the minimum of the blackening function remains negative. The conclusion is that static charged EBI black holes cannot be overcharged by infalling charged matter once second-order effects are included (He et al., 2019).

Extremal EBI black holes are nevertheless not generically superradiantly stable. For charged scalar perturbations of charged extremal EBI black holes, the superradiant threshold is

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$2

and the numerical quasi-bound-state analysis finds unstable modes with $L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$3 for arbitrarily small nonzero Born–Infeld coupling. In the convention of that work, the Maxwell limit is $L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$4, so the result is interpreted as a new example of the statement that the superradiant stability of Reissner–Nordström is a fine-tuned result (Wu et al., 2024).

In AdS, the instability structure becomes twofold. Small, spherical Born–Infeld black holes suffer superradiant instability in a finite frequency range, and the larger the BI coupling parameter, the slower the growth rate. The proposed endpoint is a small hairy black hole with a charged scalar condensate floating near the horizon. Large or planar BI black holes exhibit a different instability, associated with a tachyonic near-horizon scalar condensation; in that setting, the BI coupling has the opposite effect on the critical temperature compared with the small spherical case. The small hairy black hole is found to be stable at the critical temperature and is therefore proposed as the endpoint of the superradiant instability (Rahmani et al., 2020).

Nonminimal scalar couplings to the BI sector generate a broader scalarization problem. In the Einstein–Born–Infeld–scalar model with

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$5

there are two types of scalarized black-hole solutions: scalarized RN-like and scalarized Schwarzschild-like solutions. The RN-like scalarized family resembles the Einstein–Maxwell–scalar case, but the Schwarzschild-like family has additional structure: the domain of existence contains a region composed of two branches, one of which is a disconnected family of scalarized solutions that does not bifurcate from scalar-free black holes. Depending on parameters, one or both scalarized Schwarzschild-like branches can be entropically disfavored over comparable scalar-free black holes (Wang et al., 2020).

6. Rotating and higher-dimensional families

An older rotating literature studies a Boyer–Lindquist-type EBI metric in which

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$6

with an effective radial charge function

$L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$7

In that framework, for fixed $L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$8, there is a critical spin $L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right),$9 separating nonextremal, extremal, and naked-singularity sectors, with

$F=F_{\mu\nu}F^{\mu\nu}$0

The nonrotating shadow remains circular and is slightly smaller than the Reissner–Nordström shadow (Atamurotov et al., 2015).

More recent work constructs stationary, axisymmetric rotating EBI black holes directly from the full coupled Einstein–nonlinear-electromagnetic equations. The metric ansatz employs six functions

$F=F_{\mu\nu}F^{\mu\nu}$1

of $F=F_{\mu\nu}F^{\mu\nu}$2 and $F=F_{\mu\nu}F^{\mu\nu}$3, with the horizon fixed at $F=F_{\mu\nu}F^{\mu\nu}$4. These numerical solutions show a sharp distinction between weakly and strongly nonlinear regimes. For sufficiently small $F=F_{\mu\nu}F^{\mu\nu}$5, increasing charge at fixed spin drives the black hole toward extremality, in close analogy with Kerr–Newman. For sufficiently large $F=F_{\mu\nu}F^{\mu\nu}$6, the charged rotating branch terminates at configurations interpreted as naked singularities rather than extremal horizons. Throughout the charged families studied, the gyromagnetic ratio satisfies

$F=F_{\mu\nu}F^{\mu\nu}$7

and both prograde and retrograde ISCO radii are smaller than in Kerr–Newman at the same $F=F_{\mu\nu}F^{\mu\nu}$8 and $F=F_{\mu\nu}F^{\mu\nu}$9 (Cheng et al., 1 Jul 2025).

Exact analytic control is available in the extremal near-horizon regime. The near-horizon metric

$\bm G=h'(\mathcal F)\,\bm F,$0

and gauge field

$\bm G=h'(\mathcal F)\,\bm F,$1

admit exact Born–Infeld solutions in terms of elliptic integrals. Generalized Komar integrals then define the electric charge and angular momentum. Two genuinely nonlinear effects follow. First, there are regular extremal rotating near-horizon geometries with vanishing total charge but nontrivial electric and magnetic fields. Second, there is a forbidden region in the $\bm G=h'(\mathcal F)\,\bm F,$2 plane where no regular extremal near-horizon geometry exists. By analogy with the static case, this suggests the possibility of rotating black holes without Cauchy horizons in the full theory (Hale et al., 16 Sep 2025).

Higher-dimensional rotating EBI black holes are analytically accessible in perturbation theory. In five dimensions, starting from the extremal Myers–Perry solution with equal angular momenta, the charge parameter $\bm G=h'(\mathcal F)\,\bm F,$3 is used as a perturbative expansion parameter up to $\bm G=h'(\mathcal F)\,\bm F,$4. The BI parameter $\bm G=h'(\mathcal F)\,\bm F,$5 first appears at $\bm G=h'(\mathcal F)\,\bm F,$6 in the gauge potential and at $\bm G=h'(\mathcal F)\,\bm F,$7 in the metric. The horizon radius

$\bm G=h'(\mathcal F)\,\bm F,$8

is independent of $\bm G=h'(\mathcal F)\,\bm F,$9 up to the computed order, while the mass, electrostatic potential, magnetic moment, and gyromagnetic ratio all acquire $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$00-dependent corrections. For sufficiently small $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$01 within the perturbative formulas, both the magnetic moment and the gyromagnetic ratio can pass through zero and become negative (Allahverdizadeh et al., 2013).

7. Matter-coupled and curvature-corrected extensions

Einstein–Born–Infeld black holes also appear as limiting cases of broader matter-coupled systems. In Einstein–Born–Infeld–Yang–Mills theory, the action contains both a Born–Infeld $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$02 field and an $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$03 Yang–Mills field, and the exact metric function is

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$04

The term $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$05 is the non-Abelian contribution, and setting $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$06 reduces exactly to the standard EBI black hole in (A)dS space (Meng et al., 2017).

Curvature corrections change the gravitational sector without removing the characteristic Born–Infeld matter terms. In the 4D Einstein–Gauss–Bonnet–Born–Infeld solution, the gauge potential is

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$07

and the metric function has the Gauss–Bonnet branch form

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$08

Only the minus branch has the correct Einstein limit. The entropy is no longer purely area-law but becomes

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$09

and the paper argues that BI charge can be a more natural charged extension than Maxwell charge of the positive-$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$10 Schwarzschild-AdS-like black hole in that theory. The singularity, however, remains physically accessible (Yang et al., 2020).

Lower-dimensional models admit exact scalar-haired Born–Infeld black holes. In $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$11 dimensions, Einstein gravity coupled to BI electrodynamics and a nonminimally coupled self-interacting scalar field yields

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$12

with an exact, but tuned, metric function and scalar potential. In that convention,

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$13

so $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$14 is the Maxwell limit rather than $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$15. The electric field is finite at the origin, the curvature singularity is softened but not removed, and a dynamic family can be constructed from the tuned static solution (Mazharimousavi et al., 2014).

Matter environments introduce further deformations while leaving the EBI core intact. In AdS with perfect fluid dark matter, the metric function becomes

$$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$16

where $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$17 is a hypergeometric BI contribution. The first law holds in both ordinary and extended phase space, the Ehrenfest equations give a Prigogine–Defay ratio $$\mathcal{L}(s,p)=\frac{1}{a}\left(1-\sqrt{1-2as-a^2p^2}\right),$$18, and the critical behavior is therefore second order. In the orbital sector, the paper finds that perfect fluid dark matter modifies the ISCO and photon sphere more strongly than BI corrections do (Panah et al., 15 Mar 2026).

Taken together, these constructions show that Einstein–Born–Infeld black holes form a large but structurally coherent class. The defining Born–Infeld features are the nonlinear constitutive relation, finite electric field in the strong-field regime, square-root and hypergeometric corrections to the metric and potential, and a deformation of the Reissner–Nordström horizon and extremality structure. Around that core, the theory supports hairy and scalarized branches, AdS critical phenomena, rotating families with enhanced gyromagnetic response, higher-dimensional perturbative solutions, and non-Abelian, lower-dimensional, or curvature-corrected generalizations.