Circular orbits of Charged Particles around a Weakly Charged and Magnetized Schwarzschild Black Hole

The Schwarzschild metric admits four Killing vectors. Two of them are the temporal and azimuthal Killing vectors, since the metric is temporally and azimuthally symmetric. They respectively read

$$\xi^{\mu}_{(t)}=\delta^{\mu}_{t},\>\>\>\xi^{\mu}_{(\phi)}=\delta^{\mu}_{\phi}.$$

(3)

Let a test particle of mass $m$ be moving with four-velocity $u^{\mu}$ in the Schwarzschild background. There are two constants of the particle’s motion associated with the two Killing symmetries:

	$\displaystyle{\cal{E}}$	$\displaystyle=$	$\displaystyle-p_{\mu}\xi^{\mu}_{(t)}/m=f(r)\dot{t},$		(4)
	$\displaystyle{\cal{L}}$	$\displaystyle=$	$\displaystyle p_{\mu}\xi^{\mu}_{(\phi)}/m=r^{2}\sin^{2}{\theta}\dot{\phi}.$		(5)

Here $p^{\mu}=mu^{\mu}$ is the particle’s four-momentum. The two constants of motion ${\cal{E}}$ and ${\cal{L}}$ are the specific energy and specific azimuthal angular momentum, respectively. Using them along with the normalization $u_{\mu}u^{\mu}=-1$, we reduce the radial equation of motion in the equatorial submanifold ($\theta=\pi/2$, $\dot{\theta}=0$) to quadrature:

$$\dot{r}^{2}={\cal{E}}^{2}-V(r).$$

(6)

The overdot denotes differentiation with respect to the particle’s proper time. The effective potential $V(r)$ reads

$\displaystyle V(r)=\left(1-\frac{{\cal{L}}^{2}}{r^{2}}\right)f(r).$

(7)

The radial motion is invariant under the transformations

$$\phi\rightarrow-\phi,\hskip 8.53581pt\dot{\phi}\rightarrow-\dot{\phi},\hskip 8.53581pt{\cal{L}}\rightarrow-{\cal{L}}.$$

(8)

Therefore, there is only one mode of radial motion. Without loss of generality, we will consider ${\cal{L}}>0$.

The two conditions of circular motion $V(r)={\cal{E}}$ and $V^{\prime}(r)=0$ give us

		$\displaystyle{\cal{E}}^{2}$	$\displaystyle=\left(1-\frac{{\cal{L}}^{2}}{r^{2}}\right)f(r),$		(9)
		$\displaystyle{\cal{E}}^{2}$	$\displaystyle(r-M)+(2r-3M)r^{2}=0.$		(10)

We will use $r_{o}$, ${\cal{E}}_{o}$ and ${\cal{L}}_{o}$ to denote quantities corresponding to circular orbits from here on. Solving the above two equations for ${\cal{E}}_{o}$ and ${\cal{L}}_{o}$ for future-directed orbits yields

	$\displaystyle{\cal{E}}_{o}$	$\displaystyle=$	$\displaystyle\frac{(r_{o}-2M)}{r_{o}^{1/2}\sqrt{r_{o}-3M}},$		(11)
	$\displaystyle{\cal{L}}_{o}$	$\displaystyle=$	$\displaystyle\frac{Mr_{o}}{\sqrt{r_{o}-3M}}.$		(12)

The value of ${\cal{E}}_{o}$ is always positive with a value of $1$ far away from the black hole. A circular orbit is the ISCO when $V^{\prime\prime}(r_{o})$ vanishes. This condition gives us that $r_{\text{ISCO}}=6M$. It is worth mentioning that ${\cal{E}}_{\text{ISCO}}=2\sqrt{2}/3$ and ${\cal{L}}_{\text{ISCO}}=2\sqrt{3}M$. Therefore, a particle ending in the ISCO can release an energy of $1-2\sqrt{2}/3\>(\approx 0.057)$ of its rest energy.

This is identical to the source-free Maxwell equations for a four-potential $A^{\mu}$ in the Lorentz gauge ($A^{\mu}_{\>\>\>;\mu}=0$),

$$A^{\mu\>\>\>\>;\nu}_{\>\>\;;\nu}=0.$$

(14)

Therefore, any linear combination of the Killing vectors the spacetime admits is automatically a solution to the Maxwell equations.

Consider the electromagnetic potential constructed of the temporal and azimuthal Killing vectors of the Schwarzschild geometry:

$$A^{\mu}=-\frac{Q}{2M}\xi^{\mu}_{(t)}+\frac{B}{2}\xi^{\mu}_{(\phi)}.$$

(15)

Lowering the index and removing the constant term give

$$A_{\mu}=-\frac{Q}{r}\>\delta_{\mu}^{t}+\frac{B}{2}r^{2}\sin^{2}\theta\>\delta_{\mu}^{\phi}.$$

(16)

This potential describes the electromagnetic fields around a charged black hole immersed in an axisymmetric magnetic field. The black holes’s charge $Q$ is given by

$$Q=\frac{1}{4\pi}\int_{\sigma}F^{\mu\nu}d\sigma_{\mu\nu},$$

(17)

where $\sigma$ is a 2D surface surrounding the black hole and $F^{\mu\nu}$ is the electromagnetic field tensor (see Eq. 19). The magnetic field is axisymmetric with a strength of $B$ asymptotically [6, 18, 19]. This is the potential that we will use in this paper.

The electromagnetic field tensor $F_{\;\;\nu}^{\mu}$ is given by

$$F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}.$$

(19)

In the frame of an observer with four-velocity $u_{\mu}^{\text{obs}}$, the electric and magnetic fields are, respectively

	$\displaystyle E^{i}$	$\displaystyle=$	$\displaystyle F^{i\nu}u_{\nu}^{\text{obs}},$		(20)
	$\displaystyle B^{\mu}$	$\displaystyle=$	$\displaystyle-\frac{1}{2}\frac{\varepsilon^{\mu\nu\lambda\sigma}}{\sqrt{-g}}F_{\lambda\sigma}u_{\nu}^{\text{obs}},$		(21)

where $g=\mbox{det}(g_{\mu\nu})$, $\varepsilon_{0123}=+1$ and $i=1,2,3$. For a stationary observer ($u_{\mu}^{obs}=-f^{1/2}\delta^{t}_{\mu}$), the electric and magnetic fields are

	$\displaystyle E^{i}$	$\displaystyle=$	$\displaystyle\frac{Qf^{1/2}}{r^{2}}\delta_{r}^{i},$		(22)
	$\displaystyle B^{i}$	$\displaystyle=$	$\displaystyle Bf^{1/2}\left(\cos{\theta}\>\delta^{i}_{r}-\frac{\sin\theta}{r}\>\delta^{i}_{\theta}\right),$		(23)

The generalized four-momentum of the particle is

$$P_{\mu}=mu_{\mu}+eA_{\mu}.$$

(24)

The Lie derivatives of $A_{\mu}$ with respect to $\xi_{(t)}^{\mu}$ and $\xi_{(\phi)}^{\mu}$ identically vanish:

	$\displaystyle{\cal L}_{\xi^{\nu}_{(t)}}A^{\mu}$	$\displaystyle=$	$\displaystyle 0,$		(25)
	$\displaystyle{\cal L}_{\xi^{\nu}_{(\phi)}}A^{\mu}$	$\displaystyle=$	$\displaystyle 0.$		(26)

The energy and azimuthal angular momentum of a charged particle are therefore constants of motion. Eqs. 4 and 5 are generalized to

	$\displaystyle{\cal E}$	$\displaystyle=$	$\displaystyle-P_{\mu}\xi^{\mu}_{(t)}/m=\frac{q}{r}+f(r)\dot{t},$		(27)
	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle P_{\mu}\xi^{\mu}_{(\phi)}/m=r^{2}(b+\dot{\phi})\sin^{2}{\theta}.$		(28)

where $q={eQ}/{m}$ and $b={eB}/{2m}$. We can straightforwardly obtain the charged particle version of Eq. 6 by combining Eqs. 27 and 28 with $u_{\mu}u^{\mu}=-1$. The radial equation of motion in the equatorial submanifold then reads

$$r^{2}\dot{r}^{2}=\left({\cal{E}}r-q\right)^{2}-\left[r^{2}+\left({\cal{L}}-br^{2}\right)^{2}\right]f(r).\>\>\>$$

(29)

It is more convenient to recast it as

$$\dot{r}^{2}=\left({\cal{E}}-V_{+}\right)\left({\cal{E}}-V_{-}\right),$$

(30)

where

$$V_{\pm}(r)=\frac{q}{r}\pm\sqrt{\left[1+\left(\frac{{\cal{L}}}{r}-br\right)^{2}\right]f(r)}.$$

(31)

It is $V_{+}(r)$ that corresponds to future-directed orbits. Eq. 29 is invariant under the symmetry transformations

$$\phi\rightarrow-\phi,\;\;\;\dot{\phi}\rightarrow-\dot{\phi},\;\;\;{\cal{L}}\rightarrow-{\cal{L}},\;\;\;b\rightarrow-b.$$

(32)

As in the previous section, we will keep ${\cal{L}}>0$ without any loss of generality. When $b>0$ ($b<0$), the magnetic force is radially out (in). Likewise, $q>0$ ($q<0$) corresponds to Coulomb repulsion (attraction). Therefore, there four different modes of radial motion, in general.

In conventional units, the weak field approximation fails when

$$Q\sim\frac{G^{1/2}M}{k^{1/2}}\sim 10^{20}\frac{M}{M_{\odot}}\;\text{coulomb},$$

(34)

or

$$B\sim\frac{k^{1/2}c^{3}}{G^{3/2}M}\sim 10^{19}\frac{M_{\odot}}{M}\;\text{gauss},$$

(35)

where $M_{\odot}$ is the solar mass. The typical magnetic field strength near a black hole’s horizon has been estimated to be $\sim 10^{8}$ gauss ($10^{-15}\>\text{meter}^{-1}$) for stellar mass black holes and $\sim 10^{4}$ gauss ($10^{-19}\>\text{meter}^{-1}$) for supermassive black holes [9, 10, 11, 12, 13]. According to Ref. [3], the charge of Sgr A* is estimated to be in the range $10^{8}$–$10^{15}$ ($10^{-9}$–$10^{-2}$ meter). These estimates validate ignoring corrections to the metric due to the presence of the electromagnetic fields.

In spite of the fact that the electromagnetic fields are geometrically insignificant, their effects on the dynamics of charged particles can be significant since $e/m=2.04\times 10^{21}\;(1.11\times 10^{18})$ for electrons (protons). For electrons and protons near a black hole with $Q=10^{8}\;\text{coulomb}$ and $B=10^{4}\;\text{gauss}$, for example,

$$q_{\text{e}}\sim 10^{12}\;\text{meter},\;\;\;q_{\text{p}}\sim 10^{9}\;\text{meter},$$

(36)

and

$$b_{\text{e}}\sim 10^{3}\;\text{meter}^{-1},\;\;\;b_{\text{p}}\sim 10^{-1}\;\text{meter}^{-1}.$$

(37)

The subscripts ”e” and ”p” refer to electrons and protons, respectively.

Now, we write the expressions similar to Eqs. 11 and 12 for a charged particle orbiting a charged Schwarzschild black hole and study its circular orbits and ISCOs. Setting $b=0$ in Eq. 31 gives

$$V_{+}(r)=\frac{q}{r}+\sqrt{\left(1+\frac{{\cal{L}}^{2}}{r^{2}}\right)\left(1-\frac{2M}{r}\right)}.$$

(38)

The two conditions for circular orbits ($V(r)={\cal{E}}$, $V^{\prime}(r)=0$) respectively yield

$${\cal{E}}_{o}=\frac{q}{r_{o}}+\sqrt{\left(1+\frac{{\cal{L}}_{o}^{2}}{r_{o}^{2}}\right)\left(1-\frac{2M}{r_{o}}\right)},$$

(39)

and

	$\displaystyle{\cal{L}}_{o}^{2}=$		$\displaystyle\frac{2Mr_{o}^{2}}{r_{o}-3M}+\Bigg{[}\frac{qr_{o}\left(r_{o}-2M\right)}{\left(r_{o}-3M\right)^{2}}$		(40)
			$\displaystyle\left(q-\sqrt{4r_{o}\left(r_{o}-3M\right)+q^{2}}\right)\Bigg{]}.$

The condition for a circular orbit to be an ISCO $V^{\prime\prime}(r_{o})=0$ reads

	$\displaystyle[r_{o}\left({\cal{L}}_{o}^{2}+r_{o}^{2}\right)(r_{o}-2M)]^{1/2}\left({\cal{L}}_{o}^{2}-2Mr_{o}\right)+\;\;\;\;\;\;\;$
	$\displaystyle q\left[{\cal{L}}_{o}^{2}(r_{o}-M)+r^{2}(2r_{o}-3M)\right]=0.$		(41)

Figs. 1, 2 and 3 show how the radius, azimuthal specific angular momentum and specific energy of the ISCO vary with $q$, respectively. For $q>0$, $r_{\text{ISCO}}$ increases very steeply and approaches infinity as $q$ approaches $M$, in agreement with Refs. [2, 21]. This is possibly due to that the Coulomb repulsion makes the circular orbits less stable. When $q<0$, $r_{\text{ISCO}}$ also increases as the magnitude of $q$ increases, but more smoothly. Overall, the black hole charge pushes away $r_{\text{ISCO}}$ beyond $r=6M$ for both signs of $q$.

At $q=M$, ${\cal{L}}_{\text{ISCO}}$ has its minimum of $M$. It then increases monotonically as $q$ decreases. On the other hand, ${\cal{E}}_{\text{ISCO}}$ has its maximum of $1$ when $q=M$. As $q$ decreases, ${\cal{E}}_{\text{ISCO}}$ monotonically decreases and approaches $0$ as $q$ approaches $-\infty$. Therefore, the efficiency of energy liberation of a charged particle at the ISCO can be close to 100% of the particle’s rest energy.

In the case when $q<<-M$, we can write approximate expressions for $r_{\text{ISCO}}$, ${\cal{L}}_{\text{ISCO}}$ and ${\cal{E}}_{\text{ISCO}}$ as

	$\displaystyle r_{\text{ISCO}}$	$\displaystyle\approx$	$\displaystyle\sqrt[3]{2Mq^{2}},$		(42)
	$\displaystyle{\cal{L}}_{\text{ISCO}}$	$\displaystyle\approx$	$\displaystyle-q,$		(43)
	$\displaystyle{\cal{E}}_{\text{ISCO}}$	$\displaystyle\approx$	$\displaystyle 3\sqrt[3]{-{M}/{4q}}.$		(44)

The results of this section, and the relationship between $r_{\text{ISCO}}$ and $q$ in particular, demonstrates that even a trace charge on a black hole can have profound astrophysical implications.

In this section, we review the circular orbits and the ISCOs of a charged particle near a magnetized Schwarzschild black hole. Eq. 31 with $q=0$ reads

$$V_{+}(r)=\sqrt{\left[1+\left(\frac{{\cal{L}}}{r}-br\right)^{2}\right]f(r)}.$$

(45)

The two conditions for circular orbits give us that

	$\displaystyle{\cal{E}}_{o}$	$\displaystyle=$	$\displaystyle\sqrt{\left[1+\left(\frac{{\cal{L}}_{o}}{r_{o}}-br_{o}\right)^{2}\right]\left(1-\frac{2M}{r_{o}}\right)},$		(46)
	$\displaystyle{\cal{L}}_{o}$	$\displaystyle=$	$\displaystyle\frac{r_{o}\sqrt{b^{2}r_{o}^{2}(r_{o}-2M)^{2}+M(r_{o}-3M)}}{r_{o}-3M}\;\;\;\;\;\;\;\;$		(47)
			$\displaystyle-\frac{bMr_{o}^{2}}{r_{o}-3M}.$

For a circular orbit to be an ISCO, it must satisfy the condition

$$b^{2}r_{o}^{3}(5r_{o}-4M)-4b{\cal{L}}_{o}Mr-{\cal{L}}_{o}^{2}+2Mr_{o}=0.$$

(48)

Fig. 4 shows $r_{\text{ISCO}}$ vs $b$. The effect of the magnetic field is always to bring the ISCO inward closer than $r=6M$. As $b$ approaches $\infty$ and $-\infty$, $r_{\text{ISCO}}$ approaches $2M$ and $(\sqrt{13}+5)M/2$, respectively.

When both $q$ and $b$ are very large and positive ($q>>M$, $b>>1/M$), $r_{\text{ISCO}}$ approaches $[(3+\sqrt{3})/2]M\approx 2.366\>M$ and $q_{max}$ approaches $(\sqrt{135+78\sqrt{3}}/2)bM^{2}\approx 8.217\>bM^{2}$. However, when $q$ and $|b|$ are very large, $q$ is positive, but $b$ is negative ($q>>M$, $b<<-1/M$), $r_{\text{ISCO}}$ approaches $3M$ and $q_{max}$ approaches $-6\sqrt{3}bM^{2}\approx-10.39\>bM^{2}$.

We can see in the three plots of Fig. 7 that the positive $b$ and negative $b$ curves always cross at $r_{\text{ISCO}}=6M$, the neutral particle value. This finding was noted in Ref. [23]. It maybe tempting to think that the radial component of the electromagnetic force ($eF^{r}_{\;\;\rho}u^{\rho}$) vanishes where the crossing occurs, but this is not the case. The corresponding values of ${\cal{L}}_{\text{ISCO}}$ and ${\cal{E}}_{\text{ISCO}}$ are different from the neutral particle values.

Fig. 11 shows how the radius of the circular orbit $r_{o}$ changes with ${\cal{L}}_{o}$ when $b=-1/M$ and $q=10M$. The inner (true) ISCO is at $r_{o}=2.682M$. Between $r_{o}=2.914M$ (the value when ${\cal{L}}_{o}=0$) and $r_{o}=3.314M$ (the larger $r_{\text{ISCO}}$), no stable circular orbits exist.

Figs. 12 and 13 show ${\cal{L}}_{\text{ISCO}}$ and ${\cal{E}}_{\text{ISCO}}$ that correspond to $r_{\text{ISCO}}$ shown in Fig. 7. It is compelling that ${\cal{E}}_{\text{ISCO}}$ becomes negative when $b>0$ after $q$ falls behind a certain negative value, call it $q_{o}$. Fig. 14 shows how $q_{o}$ changes with $b$. The parameter $q_{o}$ approaches $0$ ($-\infty$) as $b$ approaches $\infty$ ($0$). We can write approximate expressions for $q_{o}$ when $b<<1/M$ or $b>>1/M$. They read

	$\displaystyle q_{o}$	$\displaystyle\approx$	$\displaystyle-\sqrt{\frac{8\sqrt{3}M}{9b}}=-1.241\sqrt{\frac{M}{b}}\hskip 2.84526pt(b>>1/M),\;\;\;\;\;\;\;$		(55)
	$\displaystyle q_{o}$	$\displaystyle\approx$	$\displaystyle-\frac{1}{2b}\hskip 99.58464pt(b<<1/M).$		(56)

This finding may have important astrophysical consequences. The energy liberated by a charged particle ending in such an ISCO can be several order of magnitudes greater than its rest energy! Similar ’superbound’ stable circular orbits were found near magnetized Kerr black holes in Ref. [20]. However, we are not aware of any negative-energy orbits outside the event horizon of the Schwarzschild black hole.