Shadow of F(R)-EH Black Hole and Constraints from EHT Observations

TL;DR

This work analyzes a static, charged black hole in $F(R)$ gravity coupled to Euler-Heisenberg nonlinear electrodynamics by deriving the constant-curvature solution, constructing the photon-propagation (effective) geometry, and studying null geodesics to characterize photon spheres and shadows. It then confronts the theoretical shadow predictions with the Event Horizon Telescope data for M87$^\ast$, deriving constraints on the parameters $q$, $a$, $f_{R_0}$, and $R_0$ in both de Sitter and anti-de Sitter backgrounds, with AdS being disfavored for small charge and $f_{R_0}>-1$. The paper also links the shadow radius to the high-energy emission rate via the cross-section $\sigma_{\text{lim}}\approx\pi R_{\text{sh}}^2$ and the Hawking temperature $T_H$, showing how electromagnetic and gravitational modifications influence black hole evaporation differently in $dS$ and $AdS$ spacetimes. Overall, the results demonstrate that horizon-scale imaging, together with nonlinear electrodynamics and $F(R)$ gravity, provides a potent probe of beyond-GR physics and constrains the parameter space of viable models for supermassive black holes like M87$^\ast$.

Abstract

This work investigates the optical properties of a static, spherically symmetric, electrically charged black hole in f(R) gravity coupled to Euler-Heisenberg(EH) nonlinear electrodynamics(NLED). By analyzing photon trajectories in this background spacetime, we show how the model parameters affect light propagation, leading to wider ranges of lensed trajectories and photon rings. We identify regions of parameter space that admit physically consistent black hole shadows, characterized by the existence of a photon sphere located outside the event horizon and a shadow formed beyond it. These viable regions expand with increasing electric charge and increasing fR0, illustrating the interplay between gravitational and electromagnetic effects. By constraining the model using Event Horizon Telescope observations of M87*, we find that de Sitter black hole solutions remain compatible with the observational data, whereas anti-de Sitter solutions are disfavored for low electric charge and fR0 > -1. Finally, an analysis of the energy emission rate shows that higher electric charge enhances black hole evaporation, while stronger nonlinear electrodynamics effects and larger values of fR0 suppress it.

Figures (9)

• Figure 1: The number of photon orbits $n$ (left column) and trajectories of photons (right column) as a function of the impact parameter $b$ for the F(R)-EH-AdS black hole with $a = 0.1$. The red lines, blue lines, and green lines correspond to $n < \frac{3}{4}$, $\frac{3}{4} < n < \frac{5}{4}$, and $n > \frac{5}{4}$, respectively. The black disk and the dashed curves denote the event horizon and photon sphere.
• Figure 2: Top row: The constraint $\frac{r_{p}}{r_{eh}} > 1$ (shaded areas) for the F(R)-EH-dS black hole is shown in: (a) $(R_{0},q)$ plane for $f_{R_{0}} = 0.5$ (solid), $f_{R_{0}} = 1.0$ (dash-dotted), and $f_{R_{0}} = 1.5$ (dashed); (b) $(f_{R_{0}},q)$ plane for $R_{0} = 0.5$, $1.0$, and $1.5$; (c) $(f_{R_{0}},R_{0})$ plane for $q = 0.1$, $0.3$, and $0.5$. Bottom row: The constraint $\frac{r_{p}}{r_{eh}} > 1$ for the F(R)-EH-AdS black hole is shown in: (d) $(R_{0},q)$ plane for $f_{R_{0}} = 0.5$, $1.0$, and $1.5$; (e) $(f_{R_{0}},q)$ plane for $R_{0} = -0.5$, $-1.0$, and $-1.5$; (f) $(f_{R_{0}},R_{0})$ plane for $q = 0.5$, $0.6$, and $0.7$. The unshaded region corresponds to $\frac{r_{p}}{r_{eh}} < 1$, which is physically forbidden. Parameters are set as $M = 1$, $a = 0.2$.
• Figure 3: Top row: The constraint $\frac{r_{sh}}{r_{p}} > 1$ (shaded areas) for the F(R)-EH-dS black hole is shown in: (a) $(R_0, q)$ plane for $f_{R_0} = 0.5$ (solid), $f_{R_0} = 1.0$ (dash-dotted), and $f_{R_0} = 1.5$ (dashed); (b) $(f_{R_0}, q)$ plane for $R_0 = 0.5$, $1.0$, and $1.5$; (c) $(f_{R_0}, R_0)$ plane for $q = 0.1$, $0.3$, and $0.5$. Bottom row: The constraint $\frac{r_{sh}}{r_{p}} > 1$ for the F(R)-EH-AdS black hole is shown in: (d) $(R_0, q)$ plane for $f_{R_0} = 0.5$, $1.0$, and $1.5$; (e) $(f_{R_0}, q)$ plane for $R_0 = -0.5$, $-1.0$, and $-1.5$; (f) $(f_{R_0}, R_0)$ plane for $q = 0.5$, $0.6$, and $0.7$. The unshaded (colorless) regions correspond to $\frac{r_{sh}}{r_{p}} < 1$, which violates the condition. Parameters: $M = 1$, $a = 0.2$.
• Figure 4: The boundary of the black hole shadow for different values of the parameters $q$, $a$, $f_{R_0}$, and $R_0$.
• Figure 5: Constraints on parameters of the F(R)-EH-dS black hole based on the EHT observations of $M87^*$.