--- title: Einstein–Born–Infeld Black Holes url: https://www.emergentmind.com/topics/einstein-born-infeld-black-holes type: topic --- # Einstein–Born–Infeld Black Holes 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 [1201.1173] [1912.05217] [2507.00879]. ## 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 \(a\to 0\) [2507.00879]. 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 \(\bm F=d\bm A\) [1912.05217]. A third convention uses \[ L_{BI}=b^2\left(1-\sqrt{1+\frac{F}{2b^2}}\right), \] with \(F=F_{\mu\nu}F^{\mu\nu}\) [1503.02441]. 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 \[ Q=\frac{1}{8\pi}\int_\infty \epsilon_{ebcd}G^{eb}. \] Thus the displacement-like tensor \(G^{ab}\), rather than \(F^{ab}\), enters the conserved-charge and variational identities [1912.05217]. 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 \[ a=(2\pi\alpha')^2. \] The weak-field limit recovers Einstein–Maxwell theory. In many conventions this is \(b\to\infty\) or \(\beta\to\infty\); in the convention of the extremal-superradiance analysis it is \(\beta\to 0\), because that paper uses \(\beta=1/b\) [1611.08584] [2509.13099] [2404.02977]. ## 2. Static geometries, horizons, and extremality The standard static, spherically symmetric line element is \[ ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega_2^2, \] or equivalent sign conventions. For electrically charged asymptotically flat EBI black holes one representative form is \[ f(r)=1-\frac{2M}{r}+\frac{q^2(r)}{r^2}, \] with \[ q^2(r)=\frac{2\beta^2 r^4}{3}\left(1-\sqrt{1+\frac{Q^2}{\beta^2 r^4}}\right)+\frac{4Q^2\mathcal K(r)}{3}, \qquad \mathcal K(r)={}_2F_1\!\left(\frac14,\frac12,\frac54,-\frac{Q^2}{\beta^2 r^4}\right). \] The event horizon is the largest root of \(f(r_h)=0\), and extremality is characterized by \[ f(r_e)=0,\qquad f'(r_e)=0. \] In that same formulation, \[ M=\frac{r_e}{3}+\left(\frac{2r_e}{3}+\frac{1}{6r_e\beta^2}\right)\mathcal K(r_e), \qquad Q^2=r_e^2+\frac{1}{4\beta^2}, \] so extremality is more complicated than in Reissner–Nordström even though it is still determined by the vanishing of the minimum of \(f(r)\) [1912.05217]. For magnetically charged asymptotically flat EBI black holes, the metric function can be written as \[ f_{\rm EBI}(r)=1-\frac{2M}{r}+ \frac{2 b^2 r^2}{3}\left(1-\sqrt{1+\frac{Q_g^2}{b^2r^4}}\right)+\frac{4Q_g^2}{3r}G(r), \] with \[ G(r)=\frac{1}{r}F\Big[\frac14,\frac12,\frac54;-\frac{Q_g^2}{b^2r^4}\Big]. \] Its large-radius expansion is \[ f_{\rm EBI}(r)\simeq 1-\frac{2M}{r}+\frac{Q_g^2}{r^2}-\frac{Q_g^4}{20b^2r^6}+\frac{Q_g^6}{72b^4r^{10}}-\frac{5Q_g^8}{832b^6r^{14}}+\cdots, \] which makes the asymptotically Reissner–Nordström-like behavior explicit [1201.1173]. The electromagnetic field is softened relative to Maxwell theory. In a purely electric sector one finds, for example, \[ F^{tr}=\frac{\beta q}{\sqrt{\beta^2r^4+q^2}}, \] or equivalently \[ E_r(r)=\frac{Q}{\sqrt{r^4+\beta^2}}, \] so the electric field is finite at the origin, unlike in Maxwell theory [1701.06837] [1503.02441]. 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 \(r=0\), and recent numerical work on rotating EBI black holes likewise finds a divergent Kretschmann scalar in the static seed solution [1611.08584] [2507.00879]. 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 \[ b|Q|\le \frac12 \qquad\text{or}\qquad b|Q|>\frac12. \] For \(b|Q|\le 1/2\), 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 \(b|Q|>1/2\), the mass function develops a minimum at \[ M_{irr}^{min}=\frac{\sqrt{4b^2Q^2-1}}{4b}, \] 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 [1503.02441]. A closely related analysis of energy extraction states that, for fixed \(M\) and \(Q\), the EBI outer horizon is always larger than the RN outer horizon, which the authors interpret as a nonlinear electromagnetic shielding effect [1611.08584]. ## 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 \[ f_{\rm EBI}(r)\equiv 1-\frac{2m_{\rm EBI}(r)}{r}, \] which defines a quasilocal mass function outside the horizon. The photon sphere at \(r=r_\gamma\) is fixed by \[ 2f_{\rm EBI}(r)-rf_{\rm EBI}'(r)\big|_{r=r_\gamma}=0, \] and the exterior Born–Infeld hair is decomposed into a near-horizon shell and a far region by \[ m_{\rm EBI}^{+}\equiv M-m_{\rm EBI}(r_\gamma),\qquad m_{\rm EBI}^{-}\equiv m_{\rm EBI}(r_\gamma)-m_{\rm EBI}(r_+). \] The quantity of interest is the ratio \(m_{\rm EBI}^{+}/m_{\rm EBI}^{-}\), because Hod’s conjecture states that \[ \frac{m_{\rm hair}^{+}}{m_{\rm hair}^{-}}\ge 1. \] For the representative values \[ M=1.5,\qquad Q_g=1,\qquad b=2,\qquad r_+=2.61,\qquad r_\gamma=4, \] the paper finds \[ \frac{m_{\rm EBI}^{+}}{m_{\rm EBI}^{-}}=1.89, \] so more than half of the total hair mass lies beyond the photon sphere [1201.1173]. The same nonlinear structure underlies generalized Christodoulou–Ruffini decompositions. In the spherically symmetric EBI setting, \[ r_+=2M_{irr},\qquad A=16\pi M_{irr}^2, \] and the total mass becomes an explicit function \(M(M_{irr},Q,b)\) rather than the Maxwellian \[ M=M_{irr}+\frac{Q^2}{4M_{irr}}. \] This decomposition is the basis for the two-family classification by \(b|Q|\), the existence of minimum-energy extremal states when \(b|Q|>1/2\), and the evaporation result that such black holes approach the zero-temperature endpoint only asymptotically in infinite time [1503.02441]. The irreducible-mass viewpoint also controls extraction processes. For spherically symmetric black holes in nonlinear electrodynamics, \[ M_{ir}=\frac{r_+}{2}, \] and the extractable energy is \[ E_{\rm extr}=M-M_{ir}=4\pi\int_{r_+}^{\infty}x^2T^0{}_0(x)\,dx. \] For fixed nonextremal \(M\) and \(Q\), the EBI irreducible mass is always greater than or equal to the RN irreducible mass, hence \[ E_{\rm extr}^{\rm BI}0\) for arbitrarily small nonzero Born–Infeld coupling. In the convention of that work, the Maxwell limit is \(\beta\to0\), so the result is interpreted as a new example of the statement that the superradiant stability of Reissner–Nordström is a fine-tuned result [2404.02977]. 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 [2001.05177]. Nonminimal scalar couplings to the BI sector generate a broader scalarization problem. In the Einstein–Born–Infeld–scalar model with \[ f(\phi)=e^{\alpha\phi^2}, \] 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 [2012.01066]. ## 6. Rotating and higher-dimensional families An older rotating literature studies a Boyer–Lindquist-type EBI metric in which \[ \Delta=r^2-2GMr+a^2+Q^2(r), \] with an effective radial charge function \[ Q^2(r)=\frac{2\beta^{2}r^{4}}{3}\left(1-\sqrt{1+\frac{Q^2}{\beta^2 r^4}}\right)+\frac{4Q^2}{3}\,{}_2F_1\!\left(\frac14,\frac12;\frac54;-\frac{Q^2}{\beta^2 r^4}\right). \] In that framework, for fixed \(M,Q,\beta\), there is a critical spin \(a_E\) separating nonextremal, extremal, and naked-singularity sectors, with \[ a_E \downarrow\ \text{as}\ \beta \uparrow,\qquad r_H^E \uparrow\ \text{as}\ \beta \uparrow. \] The nonrotating shadow remains circular and is slightly smaller than the Reissner–Nordström shadow [1506.03690]. 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 \[ F_0,\;F_1,\;F_2,\;W,\;A_t,\;A_\varphi \] of \(r\) and \(\theta\), with the horizon fixed at \(N=1-r_H/r=0\). These numerical solutions show a sharp distinction between weakly and strongly nonlinear regimes. For sufficiently small \(\tilde a=a/M^2\), increasing charge at fixed spin drives the black hole toward extremality, in close analogy with Kerr–Newman. For sufficiently large \(\tilde a\), the charged rotating branch terminates at configurations interpreted as naked singularities rather than extremal horizons. Throughout the charged families studied, the gyromagnetic ratio satisfies \[ g=\frac{2\mu_M M}{QJ}>2, \] and both prograde and retrograde ISCO radii are smaller than in Kerr–Newman at the same \(q\) and \(\chi\) [2507.00879]. Exact analytic control is available in the extremal near-horizon regime. The near-horizon metric \[ ds^2=\left(x_0^2 y^2+n^2\right)\left(-r^2dt^2+\frac{dr^2}{r^2}\right)+\frac{dy^2}{g(y)} +x_0^2g(y)\left(\frac{d\phi}{\omega}-2nr\,dt\right)^2 \] and gauge field \[ A=h(y)\left(\frac{d\phi}{\omega}-2nr\,dt\right) \] 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 \((Q,J)\) 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 [2509.13099]. 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 \(q\) is used as a perturbative expansion parameter up to \(O(q^4)\). The BI parameter \(\beta\) first appears at \(O(q^3)\) in the gauge potential and at \(O(q^4)\) in the metric. The horizon radius \[ r_H=\sqrt{2}\,\nu+\frac{\sqrt{2}}{24\nu^3}q^2+\frac{11\sqrt{2}}{1152\nu^7}q^4+O(q^6) \] is independent of \(\beta\) up to the computed order, while the mass, electrostatic potential, magnetic moment, and gyromagnetic ratio all acquire \(\beta\)-dependent corrections. For sufficiently small \(\beta\) within the perturbative formulas, both the magnetic moment and the gyromagnetic ratio can pass through zero and become negative [1302.5079]. ## 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 \(U(1)\) field and an \(SU(2)\) Yang–Mills field, and the exact metric function is \[ f(r)=1-\frac{m_0}{r}+\frac{1}{3}(2\beta^2-\Lambda)r^2+\frac{\nu^2}{r^2} -\frac{2\beta}{3}\sqrt{q^2+r^4\beta^2} +\frac{4q^2}{3r^2}\,{}_2F_1\!\left[\frac12,\frac14,\frac54,-\frac{q^2}{r^4\beta^2}\right]. \] The term \(\nu^2/r^2\) is the non-Abelian contribution, and setting \(\nu=0\) reduces exactly to the standard EBI black hole in (A)dS space [1701.06837]. 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 \[ \Phi(r)=\frac{Q}{r}\,{}_2F_1\left(\frac14,\frac12,\frac54,-\frac{Q^2}{\beta^2 r^4}\right), \] and the metric function has the Gauss–Bonnet branch form \[ a(r)=1+\frac{r^2}{2\alpha}\left\{1\pm\left[1+4\alpha\left(\frac{2M}{r^3}-\frac{1}{l^2} -\frac{2\beta^2}{3}\left(1-\sqrt{1+\frac{Q^2}{\beta^2 r^4}}\right)-\frac{4Q}{3r^3}\Phi(r)\right)\right]^{1/2}\right\}. \] Only the minus branch has the correct Einstein limit. The entropy is no longer purely area-law but becomes \[ S=\frac{A_h}{4}+2\pi\alpha\ln\frac{A_h}{A_0}, \] and the paper argues that BI charge can be a more natural charged extension than Maxwell charge of the positive-\(\alpha\) Schwarzschild-AdS-like black hole in that theory. The singularity, however, remains physically accessible [2004.14468]. Lower-dimensional models admit exact scalar-haired Born–Infeld black holes. In \(2+1\) dimensions, Einstein gravity coupled to BI electrodynamics and a nonminimally coupled self-interacting scalar field yields \[ E(r)=\frac{q}{\sqrt{r^2+\beta^2q^2}}, \qquad \psi^2(r)=\frac{r_0}{r+r_0}, \] with an exact, but tuned, metric function and scalar potential. In that convention, \[ \lim_{\beta\to0}L(F)=-F, \] so \(\beta\to0\) is the Maxwell limit rather than \(\beta\to\infty\). 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 [1405.2956]. Matter environments introduce further deformations while leaving the EBI core intact. In AdS with perfect fluid dark matter, the metric function becomes \[ \psi(r)=1-\frac{2m_0}{r}-\frac{\Lambda r^2}{3} +\frac{2\beta^2r^2(1-\mathfrak F_1)}{3} +\frac{b}{r}\ln\left(\frac{r}{|b|}\right), \] where \(\mathfrak F_1\) is a hypergeometric BI contribution. The first law holds in both ordinary and extended phase space, the Ehrenfest equations give a Prigogine–Defay ratio \(\Pi=1\), 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 [2603.14159]. 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. Source: https://www.emergentmind.com/topics/einstein-born-infeld-black-holes