Shadow geometry of Kerr MOG naked singularity and analysis of accretion disk luminosity

TL;DR

This work investigates shadows and thin-disk emission around Kerr-MOG naked singularities (KMNS) inModified Gravity. It employs analytic null geodesic methods with Hamilton-Jacobi separation to obtain unstable spherical photon orbits and the shadow boundary, expressed via the conserved quantities $\tilde{E}$, $L_z$, and Carter's constant $\mathcal{C}$ and the corresponding impact parameters $\xi$, $\zeta$. It also computes disk observables, including the energy flux $F(r)$, temperature $T(r)$, and spectral luminosity $L_{\nu,\infty}$ under a blackbody disk assumption, revealing that increasing the spin $a$ and the MOG deformation $\alpha$ enhances inner-disk emission and frame-dragging effects. The results show that KMNS shadows can be open, closed, or vanish depending on $a$, $\alpha$, and the inclination $i$, while causality-violating regions grow with spin and deformation, offering potential observational signatures to distinguish KMNS from Kerr naked singularities or Kerr black holes in future high-resolution imaging and spectroscopy.

Abstract

Naked singularities are hypothetical astrophysical entities featuring gravitational singularities without event horizons. In this study, we analyze the shadow properties of Kerr Modified Gravity (Kerr MOG) naked singularities (KMNSs). We show that the KMNS shadow can appear closed, open, or even vanish, depending on the dimensionless spin parameter a, the modified gravity parameter alpha, and the observer's inclination angle. We identify the critical conditions under which the KMNS shadow develops a gap, a unique feature not present in BH shadows. We analyze the properties of a thin accretion disk surrounding a KMNS, within the framework of MOG characterized by the parameter alpha. The study includes a detailed examination of the spacetime geometry and the equations of motion for test particles. In addition, we adopt a simplified model for the disk's radiative flux, temperature distribution, and spectral luminosity. Our analysis primarily focuses on the flux distribution of the accretion disk around KMNS with identical mass but varying spin and MOG deformation parameters. This allows us to explore how modifications in rotation and the MOG parameter alpha influence the radiative properties of the disk. Further, these observational signatures may serve as effective tools for clearly distinguishing KMNS from standard Kerr naked singularities (KNSs), where the MOG parameter alpha = 0

Figures (13)

• Figure 1: The causality violation/preservation regions of the KMNS are analyzed here. We have fixed $\mathcal{M} = 1$ and considered different values of the spin parameter $a$ along with the corresponding values of $\alpha$.
• Figure 2: This figure demonstrates the behavior of $\zeta$ and $\xi$ for the KMNS as functions of the radial coordinate $r$, with parameters $a = 1.38$ and $\mathcal{M} = 1$. The left panel corresponds to the MOG case with $\alpha = -0.40$, while the right panel shows the GR limit with $\alpha = 0$. The plots illustrate how the presence of the MOG parameter $\alpha$ alters the radial profiles of $\zeta$ and $\xi$.
• Figure 3: This figure illustrates the forbidden region for photon motion in the Kerr-MOG spacetime for the parameter values $a = 1.02$ and $\mathcal{M} = 1$. The black-shaded region represents the zone where the effective potential $\mathcal{V}_{\text{eff}} > 0$, indicating that photon propagation is dynamically forbidden in these areas.
• Figure 4: Shadows of a KMNS with observational inclinations angles $"\mathrm{i}"$, spin $"a"$ and MOG parameter $\alpha$. The plots from left to right are associated with $\mathrm{i} = 15^{\circ}$ and $\mathcal{M}=1$. As $\alpha$ increases, the shadow size contracts noticeably due to enhanced MOG corrections, especially at higher spin. The low inclination angle reduces asymmetry, highlighting the radial effect of $\alpha$ on the photon region.
• Figure 5: Shadows of a KMNS at an inclination angle of $i = 15^\circ$ for various spin parameters $a$ and MOG parameters $\alpha$, with $\mathcal{M} = 1$. Panels (a) and (b) show the evolution of the shadow as the spin parameter $a$ increases from 1 to 2, with corresponding values of $\alpha$. Panel (c) shows the shadow as the spin parameter $a$ increases from 1 to 1.7 with corresponding values of $\alpha$. Panel (d) illustrates the effect of increasing spin from $a = 1$ to $a = 1.42$ for fixed $\alpha =-0.02833$. The shadow becomes increasingly deformed and ultimately disappears beyond $a = 1.42$.