Einstein–Born–Infeld–Dilaton Model

• The EBID model integrates Einstein gravity with a dilaton field and a square-root Born–Infeld electromagnetic sector, leading to modified field equations and non-standard asymptotics.
• It deforms classical black hole thermodynamics by altering horizon properties, critical phenomena, and phase transitions while preserving mean-field critical exponents.
• Analytic and perturbative methods reveal rich dynamics in static, rotating, and holographic extensions, with dilaton coupling reshaping both local fields and global phase structures.

The Einstein–Born–Infeld–Dilaton model is a class of gravitational theories in which Einstein gravity is coupled simultaneously to a scalar dilaton and a nonlinear Born–Infeld electromagnetic sector. Across the literature, the model appears in several closely related realizations: higher-dimensional topological black holes in extended phase space, asymptotically flat rotating solutions, four-dimensional non-asymptotically flat black holes with Liouville potentials, conformally related Brans–Dicke constructions, and five-dimensional bottom-up holographic backgrounds (Dehghani et al., 2016, Allahverdizadeh et al., 2014, Hendi et al., 2015, Destounis et al., 2018, Jena et al., 2024). In all of these settings, the defining structural feature is the replacement of linear Maxwell electrodynamics by a square-root Born–Infeld sector, together with an exponential or otherwise nontrivial coupling of that sector to a dilaton field. This coupling modifies the field equations, deforms the asymptotics, changes the horizon thermodynamics, and, in several settings, preserves qualitative mean-field critical behavior while altering the detailed equation of state and response functions (Dehghani et al., 2016).

1. Defining structure of the model

A representative Einstein–Born–Infeld–Dilaton action in $(n+1)$ dimensions is

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$

where $R$ is the Ricci scalar, $\Phi$ is the dilaton field, $V(\Phi)$ is a dilaton potential, and $L(F,\Phi)$ is the Born–Infeld Lagrangian coupled to the dilaton (Dehghani et al., 2016). In this formulation,

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$

with

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$

Here $\alpha$ is the dilaton–gauge coupling and $\beta$ is the Born–Infeld parameter controlling the strength of the nonlinearity (Dehghani et al., 2016).

In the Maxwell limit $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$0, the square root expands and the Born–Infeld sector reduces to the usual exponential dilaton–Maxwell coupling,

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$1

so the model interpolates continuously between Einstein–Maxwell–dilaton gravity and genuinely nonlinear electrodynamics (Dehghani et al., 2016, Allahverdizadeh et al., 2014). In the five-dimensional rotating theory studied perturbatively around extremal Myers–Perry backgrounds, the same Born–Infeld–dilaton structure appears in the action

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$2

with

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$3

again making the exponential dilaton dressing of the Born–Infeld square root explicit (Allahverdizadeh et al., 2014).

A four-dimensional realization used in perturbation and scattering analyses takes the form

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$4

with

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$5

where $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$6 is the Born–Infeld parameter and $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$7 is the dilaton coupling constant (Destounis et al., 2018, Panotopoulos et al., 2017). Although the notation differs, the physical content is the same: Einstein gravity, a dilaton, and nonlinear electrodynamics coupled through a dilaton-dependent prefactor.

A distinct development concerns the conformal relation between Einstein-frame and Jordan-frame theories. One study argues that the older Einstein-frame Born–Infeld–dilaton Lagrangian is not conformally related to Brans–Dicke–Born–Infeld theory and therefore introduces a new Einstein-frame Born–Infeld–dilaton sector,

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$8

This construction is presented as the conformally consistent Einstein-frame counterpart of Brans–Dicke–Born–Infeld gravity (Hendi et al., 2015). This suggests that “Einstein–Born–Infeld–Dilaton model” is not a single universally fixed Lagrangian, but a family of closely related theories distinguished by frame choice, scalar couplings, and the role assigned to conformal equivalence.

2. Static black-hole sectors and geometric structure

A standard static topological ansatz is

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$9

where the dilaton modifies the transverse radius through

$R$0

For the higher-dimensional topological solutions,

$R$1

The transverse area element is therefore dressed by a nontrivial power of $R$2, and the resulting spacetimes are generally neither asymptotically flat nor AdS in the usual sense (Dehghani et al., 2016). In the four-dimensional spherical solutions used in the reentrant-transition analysis, the same mechanism appears as

$R$3

so the effective area radius is $R$4 rather than $R$5 (Hendi et al., 2017).

The corresponding metric function typically contains a curvature term, a mass term, a cosmological or Liouville term, and a Born–Infeld hypergeometric correction. In the $R$6-dimensional topological case,

$R$7

The event horizon is the largest root $R$8 of $R$9 (Dehghani et al., 2016). The existence of the solutions and well-behaved thermodynamics requires $\Phi$0, while the later critical analysis imposes the stricter condition $\Phi$1 for physical criticality (Dehghani et al., 2016).

A different four-dimensional branch, used in exact quasinormal-mode and greybody-factor calculations, adopts the string-inspired choice $\Phi$2, for which

$\Phi$3

and the metric becomes

$\Phi$4

The horizon lies at $\Phi$5, and the spacetime is neither asymptotically flat nor asymptotically (A)dS (Destounis et al., 2018, Panotopoulos et al., 2017). This branch is geometrically unusual because the area of the two-sphere grows linearly with $\Phi$6, not quadratically.

The conformally reconstructed Einstein-frame model also admits static topological black holes,

$\Phi$7

with radial electric field

$\Phi$8

The geometry has a curvature singularity at $\Phi$9 and horizons at positive roots of $V(\Phi)$0 (Hendi et al., 2015).

3. Rotating and extremal realizations

The Einstein–Born–Infeld–Dilaton model also supports rotating solutions, although exact higher-dimensional rotating configurations are difficult to obtain analytically because rotation, Born–Infeld nonlinearity, and dilaton coupling jointly complicate the field equations (Allahverdizadeh et al., 2014). A five-dimensional perturbative construction begins from an extremal Myers–Perry seed with equal angular momenta,

$V(\Phi)$1

and expands the solution in a small electric charge parameter $V(\Phi)$2, holding the angular momentum fixed and imposing extremality and regularity order by order (Allahverdizadeh et al., 2014).

The metric ansatz is

$V(\Phi)$3

with gauge potential

$V(\Phi)$4

All functions depend only on $V(\Phi)$5, and the perturbative expansions are organized so that metric and dilaton fields contain even powers of $V(\Phi)$6, while gauge potentials contain odd powers (Allahverdizadeh et al., 2014).

Several physical quantities can then be extracted asymptotically. The electric charge and angular momentum are

$V(\Phi)$7

while the mass, dilaton charge, and magnetic dipole moment are expanded as

$V(\Phi)$8

$V(\Phi)$9

$L(F,\Phi)$0

The gyromagnetic ratio

$L(F,\Phi)$1

is modified by both $L(F,\Phi)$2 and $L(F,\Phi)$3, and for sufficiently small $L(F,\Phi)$4 it can become negative (Allahverdizadeh et al., 2014).

The horizon radius and angular velocity are likewise deformed: $L(F,\Phi)$5

$L(F,\Phi)$6

The perturbative solutions are constructed to be extremal, so the surface gravity vanishes,

$L(F,\Phi)$7

and hence the temperature is zero (Allahverdizadeh et al., 2014).

A generalized Smarr relation holds,

$L(F,\Phi)$8

showing explicitly that the Born–Infeld parameter contributes a thermodynamic work term and the dilaton contributes through its charge $L(F,\Phi)$9 (Allahverdizadeh et al., 2014). A plausible implication is that the extended thermodynamic role of the Born–Infeld parameter, which appears in static black-hole chemistry as a vacuum-polarization variable, has an analogous structural role in rotating extremal sectors.

4. Extended thermodynamics and black-hole chemistry

One of the central developments in the Einstein–Born–Infeld–Dilaton literature is the extension of the phase space by treating the cosmological constant as a thermodynamic pressure and, in several formulations, also treating the Born–Infeld parameter as a thermodynamic variable (Dehghani et al., 2016, Bhamidipati et al., 2016). Because the dilaton modifies the effective stress tensor and the asymptotics, the pressure is generally not the standard AdS expression $L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$0. In the higher-dimensional topological model,

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$1

which reduces to the usual AdS relation only when $L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$2 (Dehghani et al., 2016). In the four-dimensional reentrant-transition study, the pressure is

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$3

again displaying explicit dilaton dressing (Hendi et al., 2017). In the conformally reconstructed EBID model, the Einstein-frame pressure is

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$4

whereas the Jordan-frame pressure differs by a distinct dilaton exponent (Hendi et al., 2015).

The thermodynamic volume is correspondingly non-geometric. In the $L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$5-dimensional topological model,

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$6

while in the four-dimensional reentrant-transition model

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$7

Both reduce to the usual geometric behavior only in the nondilatonic limit (Dehghani et al., 2016, Hendi et al., 2017).

The Born–Infeld parameter $L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$8 may also be promoted to a thermodynamic variable. In the higher-dimensional topological model its conjugate is defined by

$L(F,\Phi)=4\beta^{2} e^{\frac{4\alpha\Phi}{n-1}\mathcal{L}(Y),\qquad \mathcal{L}(Y)=1-\sqrt{1+Y},$9

and interpreted as a “vacuum polarization” associated with nonlinear electrodynamics (Dehghani et al., 2016). In the four-dimensional reentrant-transition model the conjugate is

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$0

and enters the first law and Smarr relation (Hendi et al., 2017). In the rotating extremal setting, the same structural role appears through the term $Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$1 in the generalized Smarr formula (Allahverdizadeh et al., 2014).

The extended first law in the static EBID setting takes the form

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$2

or, in the four-dimensional notation,

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$3

with the mass interpreted as enthalpy rather than internal energy (Dehghani et al., 2016, Hendi et al., 2017). The corresponding Smarr relation in the higher-dimensional topological case is

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$4

while in the four-dimensional reentrant-transition model it is

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$5

These formulas make clear that the dilaton changes the scaling weights of the thermodynamic variables (Dehghani et al., 2016, Hendi et al., 2017).

The same thermodynamic structure underlies black-hole heat engines. Using charged dilatonic Born–Infeld black holes as a working substance, one can define a rectangular cycle in the $Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$6–$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$7 plane for which the efficiency is

$Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$8

with $Y=\frac{e^{-\frac{8\alpha\Phi}{n-1}F^{2}{2\beta^{2},\qquad F^{2}=F_{\mu\nu}F^{\mu\nu}.$9 interpreted as enthalpy at the corners of the cycle (Bhamidipati et al., 2016). In that analysis, the dilaton and Born–Infeld couplings both modify the efficiency, with the dilaton coupling having the more pronounced effect (Bhamidipati et al., 2016).

5. Criticality, reentrant transitions, and universality

In the canonical ensemble, with charge fixed, Einstein–Born–Infeld–Dilaton black holes admit an equation of state analogous to that of a Van der Waals fluid (Dehghani et al., 2016, Hendi et al., 2015, Hendi et al., 2017). In the higher-dimensional topological model, after introducing a specific volume proportional to the horizon radius,

$\alpha$0

the equation of state can be written as $\alpha$1, with nonlinear Born–Infeld and dilaton corrections entering explicitly (Dehghani et al., 2016). Critical points are determined by the usual inflection conditions

$\alpha$2

The physical existence of criticality is constrained. In the higher-dimensional analysis, physical critical values of pressure, temperature, and volume exist only when the horizon is spherical, $\alpha$3, and the dilaton coupling satisfies

$\alpha$4

Planar and hyperbolic horizons do not exhibit the relevant Van der Waals oscillation in this ensemble (Dehghani et al., 2016). The critical ratio

$\alpha$5

is modified by both the dilaton and the Born–Infeld parameter. In the large-$\alpha$6 regime,

$\alpha$7

which reduces to the classic $\alpha$8 value when $\alpha$9 and $\beta$0 (Dehghani et al., 2016). In the deep Born–Infeld regime,

$\beta$1

showing that Born–Infeld nonlinearity deforms the universal ratio even while preserving the mean-field form of the critical exponents (Dehghani et al., 2016).

The corresponding Gibbs free energy is

$\beta$2

For $\beta$3, its graph as a function of $\beta$4 exhibits a swallowtail, indicating a first-order transition between small and large black holes; at $\beta$5, the swallowtail terminates in a critical point (Dehghani et al., 2016). This same phenomenology appears in the conformally reconstructed Einstein-frame and Jordan-frame theories, although the numerical values of $\beta$6, $\beta$7, and $\beta$8 differ because the pressure and volume definitions differ between frames (Hendi et al., 2015).

A more intricate four-dimensional phenomenon is the reentrant phase transition. In that setting, the model exhibits not only the standard first-order small/large black-hole transition, but also zeroth-order transitions and a narrow region in which a large–small–intermediate sequence occurs as a control parameter is varied monotonically (Hendi et al., 2017). The analysis identifies small black hole, large black hole, and intermediate black hole branches through the multivalued structure of $\beta$9 and $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$00. In the reentrant window, the system can undergo a first-order large-to-small transition followed by a zeroth-order small-to-intermediate transition. The same study also finds that the conventional first-order small/large transition can be replaced, below the critical point, by a zeroth-order transition in which the Gibbs free energy itself jumps discontinuously (Hendi et al., 2017).

The influence of the dilaton is decisive in these phenomena. Very small $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$01 produces the more familiar reentrant behavior already known in Born–Infeld AdS systems, whereas intermediate and larger $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$02 generate additional zeroth-order segments and modify the topology of the coexistence lines (Hendi et al., 2017). This suggests that the scalar sector is not a mild perturbation of Born–Infeld black-hole chemistry but a structural component that reorganizes the phase diagram.

Despite these deformations, the critical exponents remain mean-field. Using reduced variables and an expansion about the critical point, the higher-dimensional topological analysis derives

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$03

The entropy depends only on volume, so $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$04, giving $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$05. Maxwell’s equal-area construction then yields $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$06, the isothermal compressibility gives $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$07, and the critical isotherm gives $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$08 (Dehghani et al., 2016). The authors emphasize that although $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$09, $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$10, $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$11, and $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$12 depend on $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$13, $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$14, dimension, charge, and horizon data, the exponents do not. This places Einstein–Born–Infeld–Dilaton black holes in the same universality class as the Van der Waals fluid and Reissner–Nordström–AdS black holes (Dehghani et al., 2016).

6. Perturbations, scattering, and holographic extensions

Beyond equilibrium thermodynamics, the Einstein–Born–Infeld–Dilaton model has been used as a laboratory for dynamical stability, wave scattering, and holographic spectroscopy. In the four-dimensional string-inspired branch with metric

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$15

a minimally coupled neutral massless scalar satisfies

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$16

After introducing a tortoise coordinate and transforming to Schrödinger form, the effective potential tends to a positive constant at infinity rather than decaying to zero (Destounis et al., 2018, Panotopoulos et al., 2017). This already signals dynamics different from Schwarzschild or Reissner–Nordström backgrounds.

The quasinormal-mode problem in this branch is exactly solvable in terms of hypergeometric functions. Imposing ingoing behavior at the horizon and outgoing behavior at infinity yields the exact spectrum

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$17

The frequencies are purely imaginary and independent of the mass $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$18 (Destounis et al., 2018). For the fundamental overtone,

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$19

for $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$20, which in the paper’s time convention implies an instability. The resulting instability of a neutral scalar resembles phenomena more commonly associated with charged scalar perturbations of Reissner–Nordström black holes, even though the perturbing field here is electrically neutral (Destounis et al., 2018).

The same geometry also permits an analytic treatment of greybody factors. For a minimally coupled massless scalar, the reflection coefficient can be written in closed form as

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$21

leading to an absorption cross section

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$22

and a Hawking temperature

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$23

These quantities depend on the Born–Infeld parameter and the Liouville parameter through $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$24, but not directly on the mass (Panotopoulos et al., 2017).

A more recent development places the model in a five-dimensional bottom-up holographic setting designed to mimic QCD in a magnetic field. The action is

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$25

or equivalently

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$26

Here the magnetic field is implemented by

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$27

and the geometry is reconstructed dynamically using a warp factor $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$28 (Jena et al., 2024).

In this holographic EBID model, the nonlinear Born–Infeld structure is used to couple an external magnetic field to quarkonium spectral functions without introducing back-reacting charged flavor branes. The model supports a Hawking–Page transition between thermal AdS and a black hole background, and the deconfinement temperature decreases with increasing magnetic field, giving inverse magnetic catalysis in the thermodynamic sector (Jena et al., 2024). Spectral functions for vector fluctuations reveal anisotropic quarkonium melting. In the polarization parallel to the magnetic field, the melting temperature first decreases and then increases with $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$29, suggesting a switch from inverse magnetic to magnetic catalysis, while perpendicular modes melt more readily (Jena et al., 2024). This suggests that the Einstein–Born–Infeld–Dilaton framework has become useful not only in black-hole thermodynamics but also in phenomenological gauge/gravity modeling of strongly coupled matter.

7. Variants, conformal issues, and broader theoretical context

A recurring issue in the literature is that the precise form of the Einstein–Born–Infeld–Dilaton action is not unique. One line of work uses the older exponential Born–Infeld–dilaton coupling and studies static and thermodynamic properties directly (Dehghani et al., 2016, Hendi et al., 2017, Bhamidipati et al., 2016). Another line argues that this older Einstein-frame Lagrangian is not conformally related to Brans–Dicke–Born–Infeld theory and therefore constructs a new Einstein-frame Lagrangian whose dilaton exponents are fixed by conformal consistency (Hendi et al., 2015). The difference is not merely formal: the Einstein-frame and Jordan-frame theories agree on several quantities, including temperature, entropy, and mass, but assign different pressures and volumes, so their critical quantities differ numerically even when the qualitative phase structure is similar (Hendi et al., 2015).

Another broader extension is the inclusion of an axion. A Born–Infeld axion–dilaton model has been formulated with couplings chosen to reproduce electromagnetic duality and $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$30 symmetries in the weak-field limit (Burton et al., 2010). In that construction, the full nonlinear Einstein-frame Lagrangian contains the dilaton $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$31, the axion $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$32, and a Born–Infeld square root depending on the invariants

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$33

with

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$34

That theory goes beyond the pure EBID model by adding an axion, but it clarifies how dilaton couplings, duality, and Born–Infeld structure can coexist in a nonlinear setting (Burton et al., 2010). A plausible implication is that some properties often attributed to “the” EBID model depend sensitively on whether one insists on duality, conformal equivalence, or scalar–tensor interpretation.

The relation to generalized Born–Infeld electrodynamics also matters conceptually. A two-parameter flat-space generalization with independent scales $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$35 and $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$36 exhibits vacuum birefringence when $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$37, while ordinary Born–Infeld theory is recovered when $$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$38 (0909.1032). The trace of the Belinfante energy–momentum tensor is nonzero,

$$S=\frac{1}{16}\int d^{n+1}x\sqrt{-g}\left( R-\frac{4}{n-1}(\nabla\Phi)^{2} - V(\Phi) + L(F,\Phi)\right) ,$$39

showing that the intrinsic Born–Infeld scale breaks dilatation symmetry (0909.1032). Although this result is derived in flat-space nonlinear electrodynamics rather than in EBID gravity, it provides context for why dilaton couplings are natural: the dilaton can be viewed as coupling to the scale set by the nonlinear electromagnetic sector.

Across these variants, several robust themes recur. The model interpolates between Einstein–Maxwell–dilaton and fully nonlinear electrodynamics. The dilaton typically deforms the effective transverse radius and spoils standard AdS asymptotics. The Born–Infeld parameter can be treated thermodynamically and modifies both local field regularity and global phase structure. Critical exponents often remain mean-field despite strong deformations of the equation of state. Rotating sectors require perturbative control, and dynamical sectors can display exact spectra, analytic greybody factors, or anisotropic holographic response (Dehghani et al., 2016, Allahverdizadeh et al., 2014, Destounis et al., 2018, Panotopoulos et al., 2017, Jena et al., 2024).

In that sense, the Einstein–Born–Infeld–Dilaton model is best understood not as a single isolated theory but as a broad framework for studying how scalar couplings and nonlinear electrodynamics reshape gravitational solutions, thermodynamic phases, and real-time dynamics.