Spherical Accretion on a Schwarzschild-MOG Black Hole

TL;DR

This work studies spherical accretion onto Schwarzschild–MOG black holes within Modified Gravity. By deriving the SH-MOG metric with $f(r)=1 - \dfrac{2(1+α)M}{r} + \dfrac{α(1+α)M^2}{r^2}$ and applying steady, spherically symmetric conservation laws to isothermal fluids, it characterizes sonic points via $a_*^2$ and $(u^r_*)^2$, and analyzes four EoS cases with $k \in \{1,1/2,1/3,1/4\}$. A Hamiltonian formulation yields phase-space trajectories showing subsonic and supersonic branches, and a closed-form accretion rate $\dot{M}$ that increases with radius $r$ and the MOG parameter $α$. Setting $α=0$ recovers the classical Schwarzschild accretion, validating the construction. The results demonstrate that modified gravity enhances accretion efficiency and alters sonic-point structure, with potential observational implications.

Abstract

In this paper we have examined spherical accretion onto Schwarzschild MOG Black Holes within the framework of Modified Gravity. Using isothermal test fluids, we analyze the behavior of the flow near the critical (sonic) point for various values of the equation of state parameter $k$. Depending on the fluid type, the flow exhibits either subsonic or supersonic behavior, with ultra-stiff and ultra-relativistic fluids allowing both regimes, while radiation and sub-relativistic fluids show more restricted dynamics. Phase space analysis helps visualize these transitions. We also compute the mass accretion rate and find that it increases with both radial distance and the MOG parameter $α$, highlighting the role of modified gravity in enhancing accretion processes around black holes

Figures (5)

• Figure 1: Phase space trajectories for different values of the Hamiltonian $\mathcal{H}$. The black curves indicate the solutions at the critical point where $\mathcal{H} = \mathcal{H}_{c}$. The green and red curves show solutions with higher Hamiltonian values, specifically $\mathcal{H} = \mathcal{H}_{\ast} + 50.004$ and $\mathcal{H} = \mathcal{H}_{\ast} + 100.007$, respectively. In contrast, the magenta and blue curves represent lower values, with $\mathcal{H} = \mathcal{H}_{\ast} - 50.004$ and $\mathcal{H} = \mathcal{H}_{\ast} - 100.007$.
• Figure 2: Phase space trajectories of the solutions to Eq. (\ref{['51ab']}) for $\alpha = 0.5$, $M = 1$, and $k = 1/2$. The black curve represents the critical solution ($\mathcal{H} = \mathcal{H}_c$), while the green, magenta, red and blue curves correspond to increasing values of the Hamiltonian: $\mathcal{H}_\ast + 30.04$, $\mathcal{H}_\ast + 60.09$, $\mathcal{H}_\ast + 90.04$, and $\mathcal{H}_\ast + 120.09$, respectively. Supersonic motion appears in the region $v > 0$, and subsonic motion in $v < 0$. Vertical lines highlight regions of unphysical fluid behavior.
• Figure 3: Phase space trajectories of the solutions to Eq. (\ref{['55ab']}) for $\alpha = 0.5$, $M = 1$, and $k = 1/3$. The black curve corresponds to the solution near the critical point with $\mathcal{H} = \mathcal{H}_{c} + 10.04$, while the red, green, magenta, and blue curves represent solutions for $\mathcal{H} = \mathcal{H}_{\ast} + 15.04$, $\mathcal{H} = \mathcal{H}_{\ast} + 18.09$, $\mathcal{H} = \mathcal{H}_{\ast} + 22.04$, and $\mathcal{H} = \mathcal{H}_{\ast}$, respectively.
• Figure 4: Phase trajectories corresponding to solutions of Eq. (\ref{['59ab']}) for $\alpha = 0.5$, $M = 1$, and $k = 1/4$. The black curve shows the critical solution where $\mathcal{H} = \mathcal{H}_c$, while the red and green curves represent lower Hamiltonian values ($\mathcal{H}_\ast - 1.04$ and $\mathcal{H}_\ast - 1.09$). The magenta and blue curves correspond to higher values ($\mathcal{H}_\ast + 1.04$ and $\mathcal{H}_\ast + 1.09$). The figure highlights the transition between supersonic ($v > v_c$) and subsonic ($v < v_c$) flow regimes.
• Figure 5: The plot illustrates the variation in mass accretion rate for different values of the MOG parameter $\alpha$, considering an USF characterized by $k = 1$. The other parameters are fixed and set to $M = A_0 = A_1 = 1$. The plot demonstrates that the accretion rate increases with both the radial distance and the MOG parameter, indicating the enhanced influence of modified gravity on the accretion process.