Letelier black hole immersed in an electromagnetic universe

TL;DR

We address how a cloud of strings and an external electromagnetic background modify a Schwarzschild black hole by introducing two parameters, $\alpha$ and $a$, which alter horizon structure, thermodynamics, geodesic motion, scalar perturbations, quasinormal modes, photon sphere, shadow, and weak gravitational lensing. The spacetime is characterized by the lapse function $f(r)=1-\alpha-\frac{2M}{r}+\frac{(1-a^2)M^2}{r^2}$, with horizons at $r_{\pm}=\frac{M}{1-\alpha}\left(1\pm\sqrt{1-(1-\alpha)(1-a^2)}\right)$, interpolating between SBH, SEBH, Letelier BH, and extremal RN limits. Thermodynamics exhibit a modified temperature, entropy, and a heat-capacity divergence at $r_+=r_c$ with $r_c=\sqrt{\frac{3(1-a^2)}{1-\alpha}}\,M$, together with a generalized Smarr relation; ISCO and photon-sphere radii increase with $\alpha$ and are affected by $a$, while QNMs computed via sixth-order WKB show stronger sensitivity to $\alpha$ than to $a$, all indicating stability. Weak-lensing deflections reveal a leading CoS-enhanced term $\propto(1-\alpha)^{-1}$, alongside EMU second-order corrections and pure CoS second-order terms, producing degeneracies in parameter space but offering observational discriminants via shadow measurements and lensing statistics. Overall, the coupled effects of CoS and EMU yield distinctive signatures across horizons, thermodynamics, orbital dynamics, wave dynamics, and lensing, providing a framework to test nontrivial matter and field configurations around black holes.

Abstract

We investigate a static, spherically symmetric black hole solution surrounded by a cloud of strings and immersed in an electromagnetic universe. By deriving the event horizon from the lapse function, we demonstrate that both the string cloud parameter and the electromagnetic background parameter significantly modify the horizon radius compared to the Schwarzschild case. Consequently, thermodynamic quantities-including the Hawking temperature, Bekenstein-Hawking entropy, and heat capacity-become explicit functions of these additional parameters, with the heat capacity exhibiting divergences that signal phase transitions. We analyze the motion of massive test particles in this spacetime, deriving the effective potential and calculating the innermost stable circular orbit radius, which governs the inner edge of accretion disks and influences orbital stability. Scalar perturbations are examined through the associated effective potential, and quasinormal mode frequencies are computed using the sixth-order WKB approximation; the negative imaginary parts confirm the stability of the black hole under such perturbations. We also study the photon sphere structure, black hole shadow radius, and photon trajectories, showing how the interplay between string clouds and the electromagnetic background shapes the optical properties of this spacetime. Finally, we investigate weak gravitational lensing phenomena by deriving the deflection angle for both massive particles and photons using the Gauss-Bonnet theorem applied to the optical geometry. The results exhibit notable deviations from the Schwarzschild geometry, with the string cloud enhancing the deflection through a $(1-α)^{-1}$ factor while the electromagnetic parameter introduces competing corrections at second order.

Figures (16)

• Figure 1: Metric function $f(r)$ for Letelier BHs with CoS immersed in EMU with fixed mass $M=1$. Horizons occur where $f(r_h) = 0$ (black horizontal line). Eight distinct configurations are shown: (i) Blue solid: extremal RN limit ($a=0$, $\alpha=0$) with degenerate horizon at $r_h = 1.0M$; (ii) Red dashed: SEBH with minimal CoS ($a=0$, $\alpha=0.01$) showing split horizons at $r_h = [0.909M, 1.111M]$; (iii) Black dotted: strong EM field with moderate CoS ($a=0.1$, $\alpha=0.1$) yielding $r_h = [0.760M, 1.462M]$; (iv) Purple dash-dot: combined strong effects ($a=0.5$, $\alpha=0.5$) with widely separated horizons at $r_h = [0.419M, 3.581M]$; (v) Orange solid: SBH limit ($a=1$, $\alpha=0$) with $r_h = 2.0M$; (vi) Brown solid: SBH with moderate CoS ($a=1$, $\alpha=0.5$) pushing the horizon to $r_h = 4.0M$; (vii) Magenta solid: near-critical CoS ($a=0$, $\alpha=0.9$) with extremal configuration at $r_h = 0.513M$; (viii) Cyan solid: naked singularity case ($a=1.0$, $\alpha=0.9$) with no horizons. All curves approach $f(r) \to 1 - \alpha$ as $r \to \infty$, reflecting the global effect of CoS on the spacetime geometry.
• Figure 3: Surface gravity $\kappa$ as a function of EMU parameter $a$ for varying CoS parameter $\alpha$ with $M=1$. The surface gravity increases with $a$ (approaching the SBH limit) and decreases with $\alpha$ (stronger CoS effect). All curves show smooth monotonic behavior across the parameter space.
• Figure 4: Effective potential $V_{\rm eff}$ for timelike geodesics in the Letelier BH spacetime immersed in EMU with $M=1$ and $\mathcal{L}=4$. Left panel: Fixed EMU parameter $a=0.5$ with varying CoS parameter $\alpha$. The curves correspond to $\alpha=0.05$ (red), $\alpha=0.10$ (blue), $\alpha=0.20$ (green), and $\alpha=0.30$ (purple). As $\alpha$ increases, the asymptotic value $V_{\rm eff}(r\to\infty)\to 1-\alpha$ decreases, and the potential barrier is progressively suppressed, indicating weaker gravitational confinement. Right panel: Fixed CoS parameter $\alpha=0.1$ with varying EMU parameter $a$. The curves correspond to $a=0.8$ (yellow), $a=0.6$ (green), $a=0.4$ (blue), and $a=0.2$ (red). All curves asymptotically approach $V_{\rm eff}\to 0.9$, reflecting the universal CoS contribution. As $a$ decreases (stronger EM field), the potential peak is suppressed and shifts inward due to the $(1-a^2)M^2/r^2$ term.
• Figure 5: Three-dimensional plot of the ISCO radius as a function of $(\alpha,a)$ with $M=1$. The surface shows that $r_{\rm ISCO}$ increases with both the CoS parameter $\alpha$ and the EMU parameter $a$. For the SBH limit ($a=1$, $\alpha=0$), we recover $r_{\rm ISCO}=6M$. The CoS effect pushes the ISCO outward through the $(1-\alpha)^{-1}$ scaling, while weaker EM fields ($a\to 1$) also expand the ISCO by eliminating the attractive $(1-a^2)$ contribution.
• Figure 6: Scalar perturbation potential $M^2\mathcal{V}_s$ as a function of $r/M$ for the $\ell=1$ mode. Panel (i): Fixed $a=0.5$ with varying CoS parameter $\alpha$. The potential peak decreases and shifts outward as $\alpha$ increases. Panel (ii): Fixed $\alpha=0.1$ with varying EMU parameter $a$. The potential peak increases and shifts inward as $a$ decreases (stronger EM field).