Modified light-cylinder and centrifugal acceleration in Schwarzschild geometry

TL;DR

This paper extends magnetocentrifugal particle acceleration to Schwarzschild geometry by deriving a reduced 2D metric along a magnetic field line, which reveals a modified light cylinder consisting of inner and outer radii $r_{in}$ and $r_{out}$ separated by an unstable orbit $r_{cr}$. Between these radii, an ergosphere-like region emerges where lab-time dynamics drive the particle toward the boundaries, with the Lorentz factor diverging as $r\to r_{in,out}$ in the absence of losses. The authors compare energy gain to losses from curvature radiation, synchrotron emission, and inverse-Compton scattering (KN regime), concluding that curvature radiation is the primary limiter for black holes in the mass range $0.1-1$ M_sun, while KN cooling is not constraining and synchrotron losses are subdominant. This work generalizes the light-cylinder concept to a gravitational setting and provides a tractable framework for studying acceleration and plasma behavior near compact, rotating spacetimes, with future extensions to Kerr geometry and self-consistent field treatments.

Abstract

We examine the motion of an electron constrained to follow a magnetic field line near a primordial sub-stellar mass black hole. Earlier studies treated the problem in flat (Minkowski) spacetime, yielding qualitatively correct results and introducing a light cylinder (LC), a hypothetical surface where the linear velocity of rotation equals the speed of light. However, this picture changes significantly when gravity is included. By analyzing the electron's dynamics in the Schwarzschild metric, we obtain a modified light cylinder (MLC) whose geometry no longer resembles a cylinder. We then determine the maximum energies attainable by the electrons under the limiting effects of inverse Compton scattering, curvature radiation, and synchrotron radiation.

Figures (5)

• Figure 1: : Distance r from the center of a black hole as a function of lab time t.
• Figure 2: :Ergo sphere for different cases 1)$r_s = 0$ 2)$r_s<\frac{2c}{3\sqrt{3}\Omega}$ 3)$r_s=\frac{2c}{3\sqrt{3}\Omega}$ 4)$r_s>\frac{2c}{3\sqrt{3}\Omega}$
• Figure 3: : Lorentz factor vs distance r for electrons released from both inside and outside the critical radius.
• Figure 4: : Lorentz Factors(Curvature radiation) vs Black Hole Mass
• Figure 5: : Lorentz Factors(synchrotron radiation ) vs Black Hole Mass