Born-Infeld Inspired Gravity

• Born-Infeld inspired gravity is a modified gravitational theory that replaces the Einstein–Hilbert action with a nonlinear determinantal structure to regularize strong-field singularities.
• It employs a Palatini formulation coupling the metric with an independent connection via algebraic determinant relations, yielding effective bi-metric dynamics.
• This approach offers novel phenomenology in compact stars, cosmic bounces, and black holes, while imposing strict observational bounds on the Born–Infeld parameter.

Born-Infeld inspired gravity encompasses a family of classical modifications to general relativity (GR) in which the gravitational action is regularized by replacing the Einstein–Hilbert Lagrangian with a non-linear, determinantal (“Born–Infeld–type”) structure. The motivation traces to Born–Infeld electrodynamics, where nonlinearity tames classical singularities, and Eddington’s affine gravity proposal, which suggested that spacetime geometry could be encoded in the symmetric part of the Ricci tensor alone. These gravity theories generically demonstrate profound regularizing abilities in strong-field regimes such as the early universe, compact stars, and black holes. Central to their construction are new algebraic relations between the spacetime metric and its connection, typically formulated in the Palatini (metric-affine) approach, and the introduction of a single new parameter (most often denoted κ or ε) that sets the scale of deviations from GR.

1. Theoretical Foundations and Formal Structure

Born-Infeld inspired gravity (hereafter BI gravity) modifies the Einstein–Hilbert action to a determinantal form involving the metric $g_{\mu\nu}$, the Ricci tensor $R_{\mu\nu}(\Gamma)$, and an independent affine connection $\Gamma$: $$S_{\text{BI}} = \frac{1}{\kappa} \int d^4x \left[ \sqrt{-\det(g_{\mu\nu} + \kappa R_{\mu\nu}(\Gamma))} - \lambda \sqrt{-g} \right] + S_M$$ where $\kappa$ (or alternatively $\epsilon$) is the Born–Infeld parameter with dimensions of (length)$^2$, $\lambda$ relates to a cosmological constant, and $S_M$ is the matter action, assumed to couple minimally to $g_{\mu\nu}$ (1201.2544).

A key property is that in vacuum ($T_{\mu\nu}=0$), BI gravity reduces precisely to GR, as the auxiliary connection becomes the Levi–Civita connection. Deviations from Einsteinian dynamics appear only in the presence of matter, where the non-linear structure alters the effective coupling between matter and geometry. Notably, the field equations relate the auxiliary metric $$q_{\mu\nu} = g_{\mu\nu} + \kappa R_{\mu\nu}(\Gamma)$$ to the physical metric and stress–energy content by algebraic determinant and inversion relations, giving rise to a bi-metric, Palatini-type system (Delsate et al., 2013).

The BI approach has inspired both simple Eddington-like (EiBI) models and a wide variety of extensions—e.g., to actions of the form $f(\det(g^{-1} q))$, $f(R)$-extensions (Odintsov et al., 2014, Makarenko et al., 2014), and even multi-field and determinate non-linear couplings that mix gravity and gauge fields (Afonso et al., 2021).

2. Non-Perturbative Modifications and Fundamental Scales

The Born-Infeld parameter $\kappa$ introduces three characteristic "fundamental" scales:

• Length: $R_* = \sqrt{\kappa/G}$
• Mass: $M_* = \sqrt{\kappa} c^2 / G^{3/2}$
• Density: $\rho_* = c^2/\kappa$

These scales set thresholds for modifications to GR. Physical effects include a minimum size for gravitationally bound objects, a maximum allowed density for stable stars, and a critical mass above which compact configurations collapse to black holes. For example, a Jeans-like scale,

$c_{\text{eff}}^2 = \frac{\kappa \bar{\rho}_m}{4}$

appears even for pressureless matter, and leads to a minimum compact object size $\sim R_*$ (1201.2544). Above $M_*$, objects become unstable to gravitational collapse.

Tight bounds on $\kappa$ arise from the requirement that gravitationally bound bodies as small as observed neutron stars (radius $\approx 12$ km) can exist: $$\kappa < G R_{\text{NS}}^2 \sim 10^{-2}\ \mathrm{m}^5\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$$ This is an exceptionally strong constraint—ten orders of magnitude below Big-Bang-nucleosynthesis bounds and seven orders below solar system bounds. In the non-relativistic limit, the modified Poisson equation,

$$\nabla^2 \phi = 4\pi G \rho_m + \frac{\kappa}{4} \nabla^2 \rho_m$$

demonstrates an effective sound speed and minimum object size (1201.2544, 1207.4730).

3. Astrophysical and Cosmological Applications

Astrophysical Structure: BI gravity naturally modifies the structure of compact objects:

• Nonrelativistic Regime: The effective "pressure-like" correction term can stabilize pressureless fluid spheres, unlike in GR, yielding dust solutions and shifting Lane–Emden profiles (Jimenez et al., 2017).
• Relativistic Stars: The Tolman–Oppenheimer–Volkoff–like equations receive corrections via the auxiliary metric, leading to altered mass–radius relations. For $\kappa > 0$, neutron stars are larger and less dense; for $\kappa < 0$, they are more compact. Existing solar and neutron star observations strongly constrain $\kappa$ (Jimenez et al., 2017).
• Surface Singularities: If matter is modeled as a polytropic fluid with sharp boundaries, curvature invariants (e.g., $R_g$) become singular at surfaces for $\Gamma > 3/2$; this is a pathology inherited from the generic structure of Palatini $f(R)$ models, suggesting sensitivity to matter regularity and the need for improved microphysical modeling (Pani et al., 2012).

Cosmology:

• Early Universe: For a wide range of BI-inspired theories (including functional extensions), the initial big bang singularity is replaced by either a regular bounce (for $\epsilon < 0$) or a loitering phase (for $\epsilon > 0$). These results are robust to the inclusion of higher-curvature $f(R)$-type terms: bounces persist, while loitering behavior is more sensitive and can yield inflationary-like plateaus in $H^2$ (Odintsov et al., 2014, Benisty et al., 2021).
• Inflation and Reheating: In Palatini BI gravity, a phase of cosmic inflation can occur even in dust-dominated universes without a scalar inflaton (Jimenez et al., 2015). This phase is generically superinflationary ($\epsilon_1 < 0$), with a vanishing tensor-to-scalar ratio due to the structure of tensor perturbations, in contrast to standard scenarios. Inflation typically ends via decay of unstable dust to radiation, and a cascade of dust species is required for an appropriate reheating epoch. Energy densities self-regulate, and the universe can avoid singularities by bouncing if densities exceed a critical threshold.
• Cosmological Constraints: Big Bang Nucleosynthesis and combined cosmological data (CMB, BAO, SNe) currently constrain $|\epsilon| \lesssim 5 \times 10^{-8}\ \mathrm{m}^2$ (Benisty et al., 2021), strengthening previous bounds by several orders of magnitude.

4. Black Holes, Wormholes, and High-Curvature Solutions

EiBI gravity gives rise to a rich variety of strong-field solutions:

• Charged and Rotating Black Holes: Spherically symmetric and exact rotating black holes have been constructed via a "mapping" from GR solutions coupled to nonlinear electrodynamics (Guerrero et al., 2020). The physical metric differs from Kerr–Newman by $\epsilon$-dependent terms, yielding regularized interiors, horizon structure modifications, and possible wormhole throats for certain parameter branches.
• Wormhole Construction: Exact traversable wormhole solutions with asymptotic flatness can be obtained when certain relations between metric functions and the auxiliary metric are satisfied. These solutions may evade the NEC for the effective energy–momentum tensor, sometimes permitting NEC-satisfying matter to thread the throat within the "apparent" (auxiliary) sector (Harko et al., 2013).
• Multicenter Solutions: Majumdar–Papapetrou-type multicenter solutions, mapped from GR, describe multiple extremal objects in equilibrium. The one-center limit yields either point-like (with repulsive core) or wormhole-type regular objects, depending on the sign of the BI parameter. Geodesics are typically complete even in the presence of curvature divergences at the throat (Olmo et al., 2020).
• Surface and Central Singularities: While BI gravity can regularize cosmological and black hole singularities, singularities may remain in certain regions (e.g., at the "surface" of polytropes or at finite radius in determinantal BI couplings of gravity and electromagnetism). The geodesic completeness of such solutions demands careful analysis (Afonso et al., 2021).

5. Matter Coupling, Bi-Metric Formulation, and Canonical Structure

A distinctive feature of BI gravity is its non-linear, algebraic coupling between matter and geometry:

• Matter–Gravity Coupling: The standard matter–gravity coupling (Einstein tensor directly proportional to $T_{\mu\nu}$) is replaced by a relation in which gravity is sourced by an effective stress–energy tensor, involving "dressed" energy density and an isotropic pressure term, derived from the determinant and inverse relations between $g_{\mu\nu}$ and $q_{\mu\nu}$. This modification is responsible for key regularizing features and a range of effective equations of state for matter (Delsate et al., 2013).
• Bi-Metric Representation: The field equations can be reformulated as Einstein equations for the auxiliary metric $q_{\mu\nu}$, with a matter sector consisting of an effective energy–momentum tensor. The physical metric recovers its GR form in vacuum, while modifications are prominent in matter-rich regions (Odintsov et al., 2014).
• Hamiltonian Formulation: Canonical analysis via the Faddeev–Jackiw mechanism reveals that after integrating out auxiliary variables, the gravity sector retains the standard ADM canonical structure of GR, but the matter Hamiltonian becomes non-trivially coupled: it generically contains corrections that are highly non-linear functions of the matter fields and their momenta. The gravitational constraint algebra remains unchanged, reinforcing that new physical effects stem primarily from the matter couplings (Kluson, 31 Jul 2025).
• Validity of the Continuous Fluid Approximation: At small (nuclear) scales, the interplay between gravity and electromagnetic interactions constrains $\kappa$ even more strongly (e.g., $|κ| < 10^{-3}$ kg$^{-1}$·m$^5$·s$^{-2}$) (1207.4730). Caution is emphasized concerning the use of the continuous fluid approximation for matter sources, as naive smoothing can introduce unphysical artifacts in the presence of curvature terms sensitive to derivatives of the density.

6. Generalizations, Extensions, and Open Issues

Several extensions of BI gravity have broadened its applicability and illuminated outstanding challenges:

• Functional Extensions: Replacing the square root with a general function $f(|\Omega|)$ (where $|\Omega| = \det(g^{-1}q)$) provides a framework wherein the existence of cosmological bounces is robust across a wide family of theories (Odintsov et al., 2014).
• $f(R)$ Extensions and Early Universe Dynamics: The addition of $f(R)$ terms, e.g., $aR^2$, allows tuning of high-curvature corrections as expected from quantum field theory. Bouncing solutions persist in the $\epsilon < 0$ branch, but loitering solutions are sensitive to these modifications and can transition into natural inflationary plateaus, even in radiation-dominated cosmologies (Makarenko et al., 2014).
• Variable Cosmological Constant and Newton’s Constant: Generalized actions with $\kappa$-dependent coefficients promote Newton’s constant and cosmological constant to dynamical variables, whose variation is coupled and controlled by the matter distribution, leading to flexible cosmological phenomenology, potential oscillatory/loitering/bounce cosmologies, and broadening the class of regular solutions (Rao et al., 2019).
• Mixing with Electromagnetism: Determinantal formulations that non-linearly couple gravity and electromagnetic fields, while still recastable as Einstein gravity with a peculiar nonlinear electrodynamics matter sector, can modify the horizon structure and singularity properties of charged black hole solutions, but may not fully regularize field invariants (Afonso et al., 2021).
• Quantum Aspects and Stability: Although the physical metric is regular in BI cosmologies, the auxiliary metric can retain singularities that jeopardize the stability of tensor (gravitational wave) modes. Quantum geometrodynamical analyses employing the Wheeler–DeWitt equation demonstrate that appropriate boundary conditions (e.g., DeWitt’s condition) can neutralize these instabilities at the quantum level, possibly restoring viability (Albarran et al., 2019).

Open challenges include establishing the full viability of BI gravity in the presence of generic matter distributions (especially with sharp gradients), clarifying the microphysics of matter–gravity coupling, understanding the scope of singularity resolution (cosmological and astrophysical), and extracting observational signatures in gravitational wave, neutron star, and black hole phenomenology.

7. Summary Table: Key Scales, Equations, and Phenomenology

Feature	BI Gravity Prediction	Context
Fundamental length	$R_* = \sqrt{\kappa/G}$	Minimum compact object size (1201.2544)
Fundamental mass	$M_* = \sqrt{\kappa} c^2/G^{3/2}$	Critical mass for collapse
Maximum density	$\rho_* = c^2 / \kappa$	Max. stable star density
Modified Poisson eq.	$\nabla^2\phi = 4\pi G\rho + (\kappa/4)\nabla^2\rho$	Effective pressure-like correction
Cosmological singularity	Nonsingular bounce or loitering	Robust to $f(|\Omega|)$, $f(R)$ extensions (Odintsov et al., 2014, Benisty et al., 2021)
Black hole structure	Regularized, modified horizons	Spherically symmetric and rotating cases (Guerrero et al., 2020)
Canonical Hamiltonian	Standard GR gravity; modified matter	Born–Infeld structure modifies only the matter sector (Kluson, 31 Jul 2025)

Born–Infeld inspired gravity thus provides a rich, predictive, and highly constrained framework for exploring strong-field gravity modifications. Its regularizing properties at high curvature, sensitivity to matter couplings, and capacity for ghost-free—but highly non-GR—cosmology and compact object structure motivate its continued paper, both mathematically and via astrophysical and cosmological observations.