Born-Infeld Electrogravity and Dyonic Black Holes

TL;DR

This work formulates a determinantal Born-Infeld-like coupling of gravity and electromagnetism in Palatini form, deriving field equations where the gravitational sector reduces to Einstein gravity with a torsion-free, metric-compatible connection and the electromagnetic sector to BI-type dynamics in an effective geometry $\mathcal{G}_{\mu\nu}$ or in the physical metric $g_{\mu\nu}$. A two-picture equivalence is established, linking standard BI electrodynamics in $\mathcal{G}_{\mu\nu}$ to an anomalous BI form in $g_{\mu\nu}$, with the two descriptions related by mappings of invariants and volume elements. The paper analyzes spherically symmetric dyonic black holes, finding RN-like asymptotics but partial smoothing of curvature divergences: in the anomalous picture the curvature diverges at a finite radius $r_{0}$ (or at $\rho=1$ nondimensionally), while with an imaginary BI parameter the electric field can be finite at the origin but curvature may still diverge there or be softened depending on the mass parameter. Horizon structure can yield two, one (extremal), or no horizons, indicating a rich phenomenology akin to RN solutions. Overall, the framework preserves the equivalence principle within the two-picture formulation and suggests a route to mitigating spacetime singularities within a unified determinantal gravity–electromagnetism theory.

Abstract

Born-Infeld electrogravity is defined through a Lagrangian that couples gravity and electromagnetism within a single determinantal structure. The field equations are derived in Palatini's formalism, where the metric, connection, and vector potential are varied independently in the action. As a result, the gravitational sector reduces to Einstein's equations with a torsion-free, metric-compatible connection. The electrodynamic sector, in turn, admits two equivalent interpretations or $pictures$: it can be seen either as a standard Born-Infeld electrodynamics in an effective background geometry, or as an $anomalous$ Born-Infeld electrodynamics in the physical metric. We illustrate the dynamics by analyzing the horizon structure and extremality conditions of spherically symmetric dyonic solutions. A comparison with the Reissner-Nordström geometry shows that Born-Infeld electrogravity softens but does not eliminate curvature divergences, and geodesic incompleteness persists.

Figures (1)

• Figure 1: Plot of the function $f(\rho)$ when $C=C_{\text{extr}}$ (Eq. \ref{['eq solucion dionico']}) and when small deviations of that value are considered. It was chosen $\lambda = 1$ and $\alpha = 0.1$.