Einstein-Born-Infeld Theory

• Einstein-Born-Infeld theory is a deformation paradigm that couples Einstein gravity with nonlinear electrodynamics or modifies the gravitational action to recover standard behavior in weak fields while altering strong-field regimes.
• It employs square-root and determinantal actions that regularize electric fields, modify black hole horizons, and reshape the stability profiles of charged shells, wormholes, and rotating black holes.
• The theory offers practical insights into singularity avoidance, modified thermodynamics, and bouncing cosmological models, laying groundwork for quantum gravity extensions.

Searching arXiv for recent and foundational papers on Einstein-Born-Infeld theory to ground the article in the literature. Einstein-Born-Infeld theory denotes, in its most common usage, Einstein gravity coupled to Born-Infeld nonlinear electrodynamics, and, in a broader Born-Infeld program, also refers to Born-Infeld-type deformations of the gravitational action itself. The unifying idea is the replacement of a linear field theory by a square-root or determinantal structure that recovers Einstein or Einstein-Maxwell behavior in an appropriate low-field limit while modifying the strong-field regime, often with the explicit aim of softening divergences, altering horizon structure, or constraining the propagating spectrum (Eiroa et al., 2011, Escobar, 2012, Fiorini et al., 2016, Gullu et al., 2015).

1. Action principles and sectoral definitions

In the Einstein-plus-electrodynamics sense, the four-dimensional theory is described by an action of the form

$$S = \int d^4x \sqrt{-g} \left[\frac{1}{16\pi} (R - 2\Lambda) + L(F)\right],$$

with Born-Infeld electromagnetic Lagrangian

$$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$

where $b$ controls the nonlinearity of the electromagnetic sector (Richarte et al., 2020). A more invariant formulation used for rotating black holes is

$$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$

with $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$ (Cheng et al., 1 Jul 2025).

The nonlinear electrodynamic constitutive relations encode the distinction from Maxwell theory. In the electrostatic case one finds

$$\mathbf{D} = \frac{\mathbf{E}}{\sqrt{1 - |\mathbf{E}|^2/\beta^2}},\qquad \mathbf{E} = \frac{\mathbf{D}}{\sqrt{1 + |\mathbf{D}|^2/\beta^2}},$$

so the true electric field is bounded by the Born-Infeld scale while the displacement field retains the Coulombic interpretation (Falciano et al., 2021). This bounded-field structure is the direct descendant of the original Born-Infeld motivation to resolve the infinite self-energy problem of point charges, a motivation explicitly retained in Einstein-Born-Infeld shell and wormhole studies (Eiroa et al., 2011).

A recurrent technical source of confusion is the normalization of the Born-Infeld parameter. In different subliteratures summarized here, the Maxwell limit is written either as $b\to 0$ or as $b\to \infty$, depending on the normalization chosen for the square-root action (Eiroa et al., 2011, Richarte et al., 2020). This suggests that cross-paper comparisons require attention to conventions rather than to the symbol $b$ alone.

2. Nonlinear electrodynamics in curved spacetime

The static, spherically symmetric Einstein-Born-Infeld geometry is typically written as

$$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$0

or equivalently with lapse function $$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$1, where the nonlinear electromagnetic sector modifies the Reissner-Nordström structure through square-root and elliptic or hypergeometric terms (Eiroa et al., 2015, Falciano et al., 2021). In the appropriate Maxwell limit, these solutions reduce to Reissner-Nordström, whereas for finite Born-Infeld coupling the near-origin behavior and horizon structure are altered (Eiroa et al., 2012).

The black-hole electric field and electrostatic potential are correspondingly deformed. One formulation gives

$$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$2

together with potential

$$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$3

so the displacement field remains Coulomb-like while the physical electric field is regularized by the Born-Infeld scale (Falciano et al., 2021). In the Einstein-Born-Infeld-Yang-Mills extension, the metric function contains both the Born-Infeld electric contribution and a Yang-Mills magnetic term $$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$4, and several degenerated limits recover Einstein-Born-Infeld, Einstein-Maxwell-Yang-Mills, or pure Yang-Mills black holes (Meng et al., 2017).

The nonlinear electromagnetic sector also admits exact finite-energy solutions for continuous electric, magnetic, and dyonic source distributions. In particular, the finiteness of the total prescribed charges implies finiteness of the total field energy, and for continuous dyonic distributions the “charge mixing” anomaly of the point-dyon case does not occur (Yang, 2024). A plausible implication is that Einstein-Born-Infeld theory has a mathematically better-behaved matter sector available as a source for gravitating configurations than linear Maxwell theory in the same static-source regime.

3. Singularities, causal structure, and lower-dimensional solutions

One of the clearest singularity-resolution results in the broader Born-Infeld program appears not in the electrodynamic coupling but in Born-Infeld gravity itself. For the cosmic string spacetime, the Einstein solution

$$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$5

has a conical singularity at $$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$6 and closed timelike curves for $$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$7, with $$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$8. In Born-Infeld gravity the metric becomes

$$L(F) = 4b^2 \left( 1 - \sqrt{1 + \frac{F}{2b^2}} \right),$$9

where $b$0 diverges at the finite minimal radius $b$1, making the proper distance to that circle infinite. The resulting vacuum solution is free of conical singularity and closed timelike curves; the space ends at a minimal circle where the curvature invariants vanish, but this circle cannot be reached in finite proper time (Ferraro et al., 2010).

The regularity mechanism is geometric rather than merely algebraic. In the regular cosmic-string solution,

$b$2

and as $b$3, $b$4 while $b$5, $b$6, and $b$7 all vanish (Ferraro et al., 2010). This realizes, in explicit form, the Born-Infeld-gravity aim of excluding singular regions from the spacetime manifold.

Lower-dimensional Einstein-Born-Infeld solutions show that regularization is not automatic in every setting. A static circularly symmetric $b$8-dimensional solution without cosmological constant has no horizon and two singular points, but the point at the origin is not in the physical region because reality of the metric requires $b$9. In that physical region the weak energy condition is satisfied, whereas the causal energy condition is violated (Kuniyasu, 2014). This provides an important corrective to the common expectation that Born-Infeld nonlinearities universally remove pathologies: the outcome depends on dimension, matter content, and whether the Born-Infeld structure modifies the electromagnetic or the gravitational sector.

4. Thin shells, exotic matter, and wormhole stability

The thin-shell sector of Einstein-Born-Infeld theory is formulated with the Darmois-Israel formalism. Two manifolds are cut at radius $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$0 and sewn together on a timelike hypersurface $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$1, with the Lanczos equations

$$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$2

determining the shell stress tensor (Eiroa et al., 2011). For spherical symmetry the shell dynamics reduces to an effective one-dimensional problem,

$$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$3

and linearized stability of a static configuration at $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$4 requires $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$5 and $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$6 (Eiroa et al., 2011, Eiroa et al., 2015).

For ordinary charged shells made of normal matter, the principal result is that increasing charge broadens the stability domain. Eiroa and Simeone found that as the charge increases, the shells can be stable for a wider range of the parameters, and for $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$7 negative values of the equation-of-state parameter $$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$8 can also yield stable configurations (Eiroa et al., 2011). In this setting, Born-Infeld nonlinearity alters the horizon structure and effective potential in a way that enlarges the set of mechanically stable charged shells relative to the Einstein-Maxwell case.

Thin-shell wormholes exhibit a more model-dependent pattern. When the throat is supported by a generalized Chaplygin gas,

$$\mathcal{S}=\frac{1}{16\pi}\int d^4x \sqrt{-g}\left[R+4\mathcal{L}(s,p)\right],\qquad \mathcal{L}(s,p) = \frac{1}{a} \left(1 - \sqrt{1 - 2as - a^2p^2} \right),$$9

the stability region in parameter space reduces and then disappears as the value of the Born-Infeld parameter is modified in the sense of a larger departure from Maxwell theory (Eiroa et al., 2012). The same qualitative conclusion reappears in a later treatment: for small $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$0 the stability pattern is close to Reissner-Nordström, increasing $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$1 generally shrinks the stable region, and for large $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$2 stable wormhole solutions disappear (Eiroa et al., 2015).

At the same time, the amount of exotic matter at the throat need not increase with Born-Infeld nonlinearity. For spherically symmetric thin-shell wormholes, Richarte and Simeone found that for certain values of the Born-Infeld parameter the amount of exotic matter on the shell can be reduced in relation to the Maxwell case, especially for finite and moderate or large charge and moderate or small $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$3 (Richarte et al., 2020). The combined picture is therefore not contradictory but model-specific: Born-Infeld nonlinearity can reduce the exotic matter content required to support a wormhole while nonetheless shrinking or eliminating the region of linearized stability for a particular equation of state.

5. Rotating black holes, thermodynamics, and holographic probes

The rotating sector of Einstein-Born-Infeld theory has recently been developed beyond perturbative approximations. Numerical stationary axisymmetric black-hole solutions with electric charge and rotation show two distinct regimes: when nonlinear electromagnetic effects are weak, rotating Born-Infeld black holes with fixed spin approach the extremal limit as the electric charge increases; when nonlinear effects are strong, the solution family terminates at configurations corresponding to naked singularities (Cheng et al., 1 Jul 2025). In the same family, nonlinear electrodynamics enhances the gyromagnetic ratio relative to Kerr-Newman, and both prograde and retrograde ISCO radii are consistently smaller than those of Kerr-Newman black holes (Cheng et al., 1 Jul 2025).

The extremal rotating limit displays additional Born-Infeld-specific phenomena. Exact near-horizon solutions for extremal rotating black holes reveal configurations with vanishing charge but nontrivial electric and magnetic fields, a feature identified as a direct consequence of the nonlinearities of Born-Infeld theory. The same analysis finds that extremal rotating black holes do not exist for sufficiently small charge and angular momentum, and argues by analogy with the static case that the full rotating solutions in these regions may provide examples of rotating black holes without Cauchy horizons (Hale et al., 16 Sep 2025). This is a substantial departure from Kerr-Newman expectations.

Thermodynamic analyses retain familiar structures while modifying their detailed realization. In Einstein-Born-Infeld-Yang-Mills theory, the relevant thermodynamic quantities satisfy the first law, and in extended phase space the system exhibits van der Waals-like phase transition behavior between small and large black holes (Meng et al., 2017). In the entropy-bound problem, however, Einstein-Born-Infeld theory departs from a standard universality claim: the absorption of a charged test particle by an Einstein-Born-Infeld black hole violates the standard Bekenstein entropy bound, and the modified upper bound depends explicitly on the maximum field parameter of the Born-Infeld theory (Falciano et al., 2021).

The same nonlinearities also leave signatures in holography. In AdS Einstein-Born-Infeld backgrounds, the evolution of holographic subregion complexity during a quench depends sensitively on both the nonlinear parameter and the charge: increasing the nonlinear parameter accelerates saturation and lowers the maximal complexity, while sufficiently large charge changes the evolution pattern and can wash out the stage of linear growth (Ling et al., 2018). In the thermodynamic-topological classification of critical points, Born-Infeld corrections alone alter the topological charge of critical points of charged AdS black holes, but in the combined 4D novel Einstein-Gauss-Bonnet plus Born-Infeld system the total topological charge remains unchanged relative to the Gauss-Bonnet case (Yerra et al., 2022).

6. Born-Infeld gravity, cosmology, and higher-curvature generalizations

A distinct line of development replaces or deforms the Einstein-Hilbert action itself by a Born-Infeld-type gravitational action. In one $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$4-type construction, the Einstein-Hilbert Lagrangian is replaced by

$s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$5

or, more generally,

$s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$6

with stability conditions imposed to avoid Dolgov-Kawasaki instability, tachyon instability, and negative effective gravitational coupling (Escobar, 2012). The linearized spectrum contains the usual massless graviton together with a spin-0 scalar associated with the $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$7 term, and the theory admits a scalar-tensor representation in the Einstein frame with explicit potential (Escobar, 2012).

A more geometrically distinctive construction is Born-Infeld determinantal gravity in Weitzenböck spacetime, where the action is built from the vielbein and a symmetric tensor quadratic in the torsion two-form,

$s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$8

This theory reduces to Einstein gravity in the low-field limit $s = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$9, yields second-order field equations for the vielbein in $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$0 dimensions, and, under fairly general circumstances, its equations of motion can be written as Einstein equations plus a purely geometrical effective energy-momentum tensor determined by the spacetime parallelization (Fiorini et al., 2016). In a complementary program, an $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$1-dimensional Born-Infeld gravity was constructed so as to possess a unique maximally symmetric vacuum and a single massless unitary spin-2 graviton; the Gauss-Bonnet combination plays a non-trivial role in that construction, and the infinite-dimensional limit gives rise to an exponential gravity (Gullu et al., 2015).

Cosmological applications divide sharply by model. In Eddington-inspired Born-Infeld theory,

$$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$2

the Einstein static universe cannot be stable against both homogeneous and inhomogeneous scalar perturbations simultaneously in the spatially flat and closed cases, so the emergent-universe scenario is not viable there; it survives only in the spatially open case, with $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$3 and $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$4 (Li et al., 2017). In flux compactifications, Born-Infeld nonlinearity does not significantly modify the qualitative result of Einstein-Maxwell theory, does not generate de Sitter vacua without a bulk cosmological constant, and, in the Einstein-Higgs Born-Infeld setting with $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$5, renders all compactifications unstable against semiclassical tunneling to nothing (Ramadhan et al., 2015).

More recent Born-Infeld-type modified gravity extends the construction to a ghost-free $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$6 theory,

$$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$7

obtained by embedding Born-Infeld electrodynamics in a five-dimensional pure modified gravity and exploiting the correspondences $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$8 and $$p = -\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$9. The theory reduces to Einstein gravity in the low-energy limit and admits a vast space of bouncing cosmological solutions, including multiple-bounce solutions, although positive spatial curvature is required (Aldabergenov et al., 27 Apr 2026). On brane worldvolumes, a Born-Infeld-Einstein action

$$\mathbf{D} = \frac{\mathbf{E}}{\sqrt{1 - |\mathbf{E}|^2/\beta^2}},\qquad \mathbf{E} = \frac{\mathbf{D}}{\sqrt{1 + |\mathbf{D}|^2/\beta^2}},$$0

provides an alternative description of braneworld dynamics by summing all powers of the worldvolume Ricci tensor in a determinant structure (Rojas, 2010).

Taken together, these results show that “Einstein-Born-Infeld theory” names not a single model but a family of closely related strong-field deformations. In the electrodynamic sector, the theory modifies charged black holes, shells, wormholes, entropy bounds, and holographic probes through nonlinear constitutive relations. In the gravitational sector, Born-Infeld constructions target second-order equations, unique vacua, massless gravitons, singularity avoidance, or non-singular cosmologies. The literature summarized here therefore supports a precise but plural definition: Einstein-Born-Infeld theory is best understood as a Born-Infeld deformation paradigm applied either to the matter sector of general relativity or to gravity itself, with the physical outcome determined by which sector is deformed and by the specific determinantal or square-root action employed.