Revisiting critical orbits of test particles traveling in a black hole background

TL;DR

The paper provides a unified, detailed treatment of critical orbits for test particles traveling in Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman backgrounds. By examining the root structures of the radial potential $R(r)$ (and its angular counterpart $\Theta$ in axisymmetric spacetimes), it derives explicit relationships among energy, angular momentum, charge-to-mass ratio, and critical radii, distinguishing null and timelike cases. The analysis clarifies how photon spheres and unstable circular orbits define black hole shadows and accretion boundaries, and it presents extensive analytic expressions complemented by numerical illustrations for both single- and multi-root degeneracies (double and triple roots). The work also lays groundwork for a 3+1 formalism describing Vlasov gas accretion onto Kerr–Newman black holes, highlighting practical implications for shadow observations and high-energy astrophysical processes.

Abstract

This paper systematically revisits the critical orbits of test particles moving in various black hole backgrounds, including the Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman spacetimes. We identify the critical orbit cases directly from the root structure of the radial equation, and provide explicit expressions relating the relevant parameters -- energy, angular momentum, and charge-to-mass ratio -- to the critical radius, as well as explicit expressions for the critical orbits in each scenario. Special attention is given to the relationship between the photon spheres, black hole shadows and the critical null geodesics. Extensive numerical results are also provided.

Figures (12)

• Figure 1: All possible root configurations of $f(u)=0$. In Case 1, where there are no real roots for $u>0$, a photon incident from infinity will be absorbed by the black hole. In Case 2, characterized by two distinct real roots within $u>0$, the photon will be scattered. Case 3 corresponds to the critical orbit, where a double root occurs at $u_c>0$, In this critical case, as we will discuss later, the photon is neither absorbed nor scattered but instead approaches a circular orbit at a finite radius $r_c=\frac{1}{u_c}$.
• Figure 2: The critical orbits for the null geodesic in the Schwarzschild metric. The mass is chosen to be $M=1$ and the constant $\varphi_0=0$. The dashed line is the location of the photon sphere.
• Figure 3: The critical orbits for the time-like geodesic in the Schwarzschild metric. The parameters are chosen to be $M=1,\varphi_0=0,u_c=\frac{7}{24}$. The dashed line is the location of the critical radius $r_c=\frac{1}{u_c}$.
• Figure 4: The critical orbits of case (i) for the time-like world line in the Reissner–Nordström metric. The parameters are chosen to be $u_c=\frac{2}{3M},e=1.5m,Q=\frac{3 m M}{\sqrt{e^2+8 m^2}},M=1,\varphi_0=0$. The dashed line is the location of the critical radius $r_c=\frac{1}{u_c}$.
• Figure 5: The shadow of the Kerr black hole as seen by a distant observer in the equatorial plane. The Mass of the black hole is chosen to be $M=1$.