Charged Particles Capture Cross-Section by a Weakly Charged Schwarzschild Black Hole

Astrophysicists generally assume black holes are electrically neutral. This is because they would quickly attract oppositely charged matter to balance out any access charge. However, there are compelling reasons why weakly charged black holes might exist as discussed in Al Zahrani (2021, 2022); Zajaček & Tursunov (2019); Zajaček et al. (2019, 2018); Carter (1973) and the references therein. The differences in how a black hole accretes electrons and protons within its plasma environment, influenced by radiation, could render it charged. Also, the spin of a black hole in the presence of a magnetic field can induce the accretion of charged particles. In fact, using the EHT observations, it was inferred that Sgr A* and M87 can be charged Ghosh & Afrin (2023); Kocherlakota et al. (2021). The black hole’s charge is weak in the sense that it has no tangible effect on spacetime, but its effect on charged particle dynamics is prominent.

Here, we review the dynamics of charged particles near a weakly charged black hole. The spacetime geometry around a black hole of mass $M$ and charge $Q$ is described by the Schwarzschild Reissner-Nordströn metric which reads Misner et al. (1973)

where $h=1-r_{S}/r+Q^{2}/r^{2}$ and $r_{S}=2M$ is the Schwarzschild radius. The electromagnetic 4-potential is

However, when the charge is weak we can ignore the curvature due to it and use the Schwarzschild metric, which reads Misner et al. (1973)

where $f=1-r_{S}/r$ and $r_{S}=2M$ is the Schwarzschild radius. This weak charge approximation is valid unless the charge creates curvature comparable to that due to the black hole’s mass. This happens when

In conventional units, the weak charge approximation fails when

This charge is way greater than the greatest estimated change on any black hole. Although the black hole charge is tiny, its effect on charged particles dynamics is profound because it is multiplied by the charge-to-mass ratio of these particles ($\sim 10^{21}\>{\text{m}}^{-1}$ for electrons and $\sim 10^{18}\>{\text{m}}^{-1}$ for protons).

The Lagrangian describing a charged particle of charge $q$ and mass $m$ in a spacetime described by a metric $g_{\mu\nu}$ and an electromagnetic field produced by a 4-potential $A^{\mu}$ reads Chandrasekhar (1983)

where $u^{\mu}\equiv dx^{\mu}/d\tau$ is the particle’s 4-velocity and $\tau$ is its proper time. In our case, the Lagrangian becomes

This Lagrangian is cyclic in $t$ and $\phi$, which means that the particle’s energy and azimuthal angular momentum are constants of motion. The specific energy and azimuthal angular momentum are, respectively, given by

Combining these equations with the normalization condition $g_{\mu\nu}u^{\mu}u^{\nu}=-1$ and solving for $dr/d\tau$ give

In the equatorial plane where $\theta=\pi/2$, the equation becomes

Let us rewrite the last equation in a dimensionless form. We first introduce the following dimensionless quantities:

Equation 11 then becomes

where

The parameter $\alpha$ represent the relative strength of the electromagnetic force to the Newtonian gravitational force. We can rewrite Eq. 13 as

where

is an effective potential. It is $V_{+}$ that corresponds to physical, future-directed motion and hence will be used in all of the analyses below. Without loss of generality, we will consider ${\cal L}>0$ only.

Before we tackle the main problem, let us find the capture cross-cross section for neutral particles first. Setting $\alpha=0$, the effective potential $V_{+}$ reduces to

Capture occurs whenever the particle’s energy is greater than the maximum of $V_{+}$. The function $V_{+}$ is at an extremum when $dV_{+}/d\rho=0$ or

which gives the position of the extrema in terms of ${\cal L}$ as

where ${\cal L}\in[\sqrt{3},\infty)$. When ${\cal L}=\sqrt{3}$ ($\equiv{\cal L}_{\text{min}}$), $\rho_{+}$ and $\rho_{-}$ meet at a saddle point. Inspecting $d^{2}V_{+}/d\rho^{2}$ reveals that $\rho_{-}$ corresponds to the position of the local maximum of $V_{+}$. In terms of ${\cal L}$, the escape condition ${\cal E}=V_{+}|_{\rho=\rho_{-}}$ becomes

Inverting this equation gives

The impact parameter $b$ is defined as the perpendicular distance between the center of force and the incident velocity Goldstein et al. (2001). It can be written as

where $\cal P$ is the specific linear momentum. The maximum impact parameter for capture $b_{\text{max}}$ is given by

The capture cross-section ${\sigma}_{\text{cap}}$ is given by

Figure 6 shows how $b_{\text{max}}$ depends on ${\cal E}$ for several positive values of $\alpha$ greater than $0.5$. Generally, $b_{\text{max}}$ becomes smaller as $\alpha$ increases. This is expected because the greater the Coulomb repulsion the more difficult it is for a charged particle to be captured. In fact, there is a threshold energy ${\cal E}_{\text{thr}}$ below which capture cannot occur. It is given by

This equation is valid for $\alpha\geq 0.5$ only. Fig. 7 shows how ${\cal E}_{\text{thr}}$ vary with $\alpha$.
 
The capture cross-section $\sigma_{\text{cap}}$ corresponding to Figs. 4, 5 and 6 is shown in Figs. 8, 10 and 9, respectively. In all cases, $\sigma_{\text{cap}}$ vs. ${\cal E}$ curves inherent the features of the $b_{\text{min}}$ vs. ${\cal E}$ curves.

We have studied the capture cross-section of charged particles by a weakly charged Schwarzschild black hole. We have shown that a trace charge on the black hole can have prominent effects.

When the Coulomb force between a charged particle and the black hole is attractive, it enlarges the capture cross-section significantly. This is expected since the Coulomb attraction enhances the capture of charged particles. However, when the Coulomb force between a charged particle and the black hole is repulsive, it shrinks the capture cross-section significantly. When the electromagnetic coupling strength is below a critical value, capture is possible for all values of the particle’s energy. When the electromagnetic coupling strength is above the critical value, there is a minimum value of the particle’s energy below which capture is impossible. This is because the Coulomb repulsion surpasses the gravitational attraction unless the particle’s radial momentum is large enough.

Our results emphasizes the assertion that charged black holes will favorably accretes charges of the opposite sign. However, it is still possible for the black hole charge to grow if the plunging charged particles are energetic enough to the limit that the capture cross-section becomes independent of the sign of the charges. Moreover, the fact that the electromagnetic coupling constant is three orders of magnitudes greater for electrons than protons suggests that it is relatively easier for a black hole to accumulate positive charge than negative charge.

It will be an astrophysically interesting to study the energies of charged particles near an astrophysical black hole to understand better how the black hole’s charge evolves. The problem can be astrophyically more viable when other astrophysical black holes, such as rotating black holes, are studied (in progress).