Black holes in $f(R,T)$ gravity coupled with Euler-Heisenberg electrodynamics

TL;DR

The paper investigates static, spherically symmetric black holes in $f(R,T)$ gravity coupled to Euler–Heisenberg nonlinear electrodynamics, deriving modified field equations for the linear coupling $f(R,T)=R+2\beta T$ and obtaining magnetic, electric, and dyonic solutions. Magnetic solutions acquire a nonlinear $1/r^6$ correction, while electric solutions develop an effective cosmological constant $\Lambda_{\text{eff}}=\beta/(3\pi)$, producing AdS or dS asymptotics; dyons incorporate QED vacuum polarization corrections that break duality and introduce higher-order terms. Energy conditions are analyzed, showing region-dependent satisfaction (e.g., WEC/SEC outside horizons) and highlighting how the $f(R,T)$ coupling alters horizon structures and curvature. The study then analyzes effective metrics, photon geodesics, and ISCOs across the three charge configurations, presenting geodesic images and constraining parameter space with EHT observations of M87*, illustrating potential observational signatures of matter–geometry coupling in strong gravity. Overall, the work reveals how $f(R,T)$ gravity with EH electrodynamics reshapes BH shadows, photon rings, and ISCOs, providing a framework to test these modified gravity ideas with future high-precision black hole observations.

Abstract

We investigate the scenario of black holes coupled with the Euler-Heisenberg nonlinear electromagnetic field in the framework of $f(R,T)$ gravity. The black hole solutions for electrically charged, magnetically charged and the dyonic case are separately analyzed, and we discuss the scalar curvature and the energy conditions of the black hole spacetime. In the magnetic charge solution, the $f(R,T)$ correction appears in the $1/r^6$ Euler-Heisenberg electromagnetic field correction term, while the electrically charged black hole solution exhibits an $r^2$ term in the metric function, corresponding to an effective cosmological constant $Λ_\text{eff} = β/(3π)$, inducing asymptotically anti-de Sitter or de Sitter spacetimes depending on the sign of the coupling parameter $β$. The dyonic solution is obtained through vacuum polarization quantum electrodynamics corrections, where electromagnetic duality is broken and the solution contains higher-order correction terms. The relationship between the event horizon and charge of dyonic extreme black holes is studied. Furthermore, we investigate the effective metric, photon trajectories, and innermost stable circular orbit under nonlinear electromagnetic effects, providing images of photon geodesic for varying electric and magnetic charge strengths and $f(R,T)$ coupling parameter.

Figures (7)

• Figure 1: The relationship between the outer event horizon $r_+$(Solid line), inner event horizon $r_-$(dashed line) and the charge $Q_m$ is shown in the figure. With $M = 1$ fixed, $a$ and $\beta$ vary. When $a=0$ the solution reduce to RN black hole.
• Figure 2: The relationship between the outer event horizon $r_+$(Solid line), inner event horizon $r_-$(dashed line) and the charge $Q_e$ are shown in the figure (a) and (b). For the case where $\beta$ is negative (dS spacetime), the plots of A(r) are shown in (c) and (d). All figures have $M =1$ fixed, $a$ and $\beta$ vary.
• Figure 3: Phase diagram in the $(Q_e, \beta)$ parameter space for $M=1, a=2$. The central line ($\beta=0$) divides the spacetime into the AdS($\beta>0$) and dS($\beta<0$) regimes. In both regions, darker shades denote the Black Hole (BH) phase (characterized by the existence of event horizons $r_{\pm}$), and lighter shades represent the Naked Singularity (NS) phase (where $r_{\pm}$ are absent or degenerate). The boundary between the BH and NS regions is defined by the Extremal Black Hole locus ($r_{-} = r_{+}$).
• Figure 4: The relationship between the outer event horizon $r_+$(blue), inner event horizon $r_-$(pink), extremal black hole horizon $r_e$(black curve) and the charge $Q_e, Q_m$ is shown in the figure. With $M = 1$ fixed, $a=0.001768$ and $\beta$ vary.
• Figure 5: Geodesics structure of magnetically charged black hole in $f(R,T)$ gravity coupled with EH field. Upper line are the number of orbits $n$ with respect to impact parameter $b$. Lower line are the geodesics figure. All fixed $a=2$ and $M=1$.