Motion of charged test particles in Reissner-Nordström spacetime

We investigate the circular motion of charged test particles in the gravitational field of a charged mass described by the Reissner-Nordström spacetime. We study in detail all the spatial regions where circular motion is allowed around either black holes or naked singularities. The effects of repulsive gravity are discussed by finding all the circles at which a particle can have vanishing angular momentum. We show that the geometric structure of stable accretion disks, made of only test particles moving along circular orbits around the central body, allows us to clearly distinguish between black holes and naked singularities.

Figure 1

Effective potential for a charged test particle with $ϵ=-2$ moving in a Coulomb potential with $Q/M=2$ for different values of the momentum $L^{*}\equiv L/(\mu M)$. The points indicate the minima of the potential. In particular, for $L^{*}=|ϵQ|/M$ the potential vanishes on the origin $r=0$ (see text).

Figure 2

The effective potential as a function of $r/M$ for a charged particle of charge-to-mass ratio $ϵ\equiv q/\mu$ moving in a Reissner–Nordström black hole of charge $Q$ and mass $M$. The graphic shows the positive $E^{+}/\mu$ (black curves) and negative roots $E^{-}/\mu$ (gray curves) of the effective potential for $Q/M=0.5$, $ϵ=-2$, and different values of the momentum $L^{*}\equiv L/(M\mu )$. The outer horizon is located at $r_{+}\equiv M+\sqrt{M^2-Q^2}\approx 1.87M$. Note the presence of negative energy states for the positive roots.

Figure 3

The effective potential $V_{+}$ for a charged particle of charge-to-mass ratio, $ϵ=q/\mu$, moving in a Reissner-Nordström spacetime of charge $Q$ and mass $M$ with charge-to-mass ratio $Q/M=0.5$ is plotted as a function of the radial coordinate $r/M$ for different values of the angular momentum $L^{*}\equiv L/(M\mu )$. The outer horizon is located at $r_{+}\approx 1.87M$. In the graphic on the left with $ϵ=0.1$, the effective potential for $L^{*}\approx 3.5$ has a minimum $V_{\min }\approx 0.954$ at $r_{\min }\approx 8.84M$. In the graphic on the right with $ϵ=-0.1$, the minimum $V_{\min }\approx 0.94$ is located at $r_{\min }\approx 7.13M$ for $L^{*}\approx 3.5$.

Figure 4

The effective potential $V_{+}$ is plotted as a function of $r/M$ for a charged test particle with specific charge $ϵ=q/\mu$ moving in the field of a Reissner-Nordström extreme black hole ($Q=M$). Here $L/(M\mu )=4$, and the effective potential is plotted for different values of $ϵ$. The outer horizon is located at $r_{+}\equiv M+\sqrt{M^2-Q^2}=M$. Note the presence of negative energy states for particles with negative $ϵ$.

Figure 5

The positive solution of the linear velocity ${\nu }_{ϵ}^{+}$ is plotted as a function of the radial distance $r/M$ for the parameter choice $Q/M=0.6$ and $ϵ=-3$ so that $r_{\gamma }^{+}/M\approx 2.74$ and the outer horizon is located at $r_{+}/M=1.8$. The geodesic velocity ${\nu }_g$ is also shown (gray curve). The shaded region ($r<r_{\gamma }^{+}$) is forbidden.

Figure 6

The positive solution of the linear velocity ${\nu }_{ϵ}^{+}$ is plotted as a function of the radial distance $r/M$ for the parameter choice $Q/M=0.6$ and different values of $ϵ=-5$ (black curve), $ϵ=-3$ (thick black curve), and $ϵ=-0.5$ (dashed curve). The geodesic velocity ${\nu }_g$ for $ϵ=0$ is also shown (gray curve). The choice of parameters implies that $r_{\gamma }^{+}/M\approx 2.74$ and the outer horizon is located at $r_{+}/M=1.8$. The shaded region is forbidden. For $r>r_{\gamma }^{+}$ it holds that ${\nu }_{ϵ}^{+}>{\nu }_g$.

Figure 7

The energy $E/\mu$ and angular momentum $L^{*}\equiv L/(\mu M)$ of a charged particle of charge-to-mass ratio $ϵ$ moving in the field of a RN black hole with charge $Q$ and mass $M$ are plotted as functions of the radial distance $r/M$ for the parameter choice $Q/M=0.6$ and $ϵ=-3$, with $r_{\gamma }^{+}/M\approx 2.74$ and the outer horizon located at $r_{+}/M=1.8$. The shaded region is forbidden.

Figure 8

The energy $E/\mu$ and angular momentum $L^{*}\equiv L/(\mu M)$ of a charged particle of charge-to-mass ratio $ϵ$ moving in the field of a RN black hole with charge $Q$ and mass $M$ are plotted as functions of the radial distance $r/M$ for the parameter choice $Q/M0.6$ and $ϵ=-3$ (solid curves), $ϵ=-5$ (dashed curves), $ϵ=-0.5$ (dotted curves). Here $r_{\gamma }^{+}/M\approx 2.74$ and the outer horizon is located at $r_{+}/M=1.8$. The shaded region is forbidden. The energy and angular momentum decrease as $|ϵ|$ decreases.

Figure 9

The positive solution of the linear velocity ${\nu }_{ϵ}^{\pm }$ is plotted as a function of the radial distance $r/M$ in the region [1.8, 5] (left graphic) and [2.5, 3] (right graphic). Here $Q/M=0.6$ and $ϵ=1.2$, so that $r_l/M=2.68$ and $r_{\gamma }^{+}=2.737M$. For the chosen parameters we have that $\widetilde{ϵ}=3.25$ and ${ϵ}_l=4.12$. The region within the interval $[r_l,r_{\gamma }^{+}]$ is forbidden for neutral particles.

Figure 10

The energy $E/\mu$ and angular momentum $L^{*}\equiv L/(\mu M)$ of a charged particle of charge-to-mass ratio $ϵ$ moving along circular orbits in a Reissner–Nordström black hole of charge $Q$ and mass $M$ are plotted in terms of the radial distance $r/M$ in the range [2.6, 3.8] (left graphic) [2.68, 2.74] (right graphic). Here $Q/M=0.6$ and $ϵ=1.2$, so that $r_l/M=2.68$ and $r_{\gamma }^{+}/M=2.737$. For the chosen parameters we have that $\tilde{\epsilon }=3.25$ ${ϵ}_l=4.12$. The shaded region is forbidden for any particles.

Figure 11

Radius $r_s=r_s^{+}$ (black curve) and $r_l=r_l^{+}$ (gray curve), are plotted as function of $ϵ$ for $Q=0.6M$. $r_s^{+}=r_l^{+}$ for $ϵ=\widetilde{ϵ}\approx 3.25$.

Figure 12

Left graphic: The positive solution of the linear velocity $\nu$ is plotted as a function of the radial distance $r/M$. Right graphic: The energy $E/\mu$ and angular momentum $L^{*}\equiv L/(\mu M)$ of a charged particle of charge-to-mass ratio $ϵ$ are plotted in terms of $r/M$. The parameter choice is $Q/M=0.6$ and $ϵ=2.1$. Then, $r_l/M=2.56$, $r_{\gamma }^{+}/M=2.737$, and $r_s/M=3.99$. Moreover, for this choice $\widetilde{ϵ}=3.25$ and ${ϵ}_l=4.12$. The shaded region is forbidden.

Figure 13

Left graphic: The positive solution of the linear velocity $\nu$ is plotted as a function of the radial distance $r/M$. Right graphic: The energy $E/\mu$ and angular momentum $L^{*}\equiv L/(\mu M)$ of a charged particle of charge-to-mass ratio $ϵ=3.8$ moving in a RN spacetime with $Q/M=0.6$ are plotted in terms of the radial distance $r/M$. For this choice of parameters the radii are $r_l/M=1.98$, $r_s/M=2.11$, and $r_{\gamma }^{+}/M=2.737$ whereas the charge parameters are $\widetilde{ϵ}=3.25$ and ${ϵ}_l=4.12$.

Figure 14

Left graphic: The positive solution of the linear velocity $\nu$ is plotted as a function of the radial distance $r/M$. Right graphic: The energy $E/\mu$ and angular momentum $L^{*}\equiv L/(\mu M)$ of a charged particle of charge-to-mass ratio $ϵ=7$, moving in the field of a RN black hole with $Q/M=0.6$, are plotted in terms of the radial distance $r/M$. For this parameter choice $r_s/M=1.88$, $\widetilde{ϵ}=3.25$, and ${ϵ}_l=4.12$.

Figure 15

The radius of the last stable circular orbit $r_{\mathrm{lsco}}$ in a RN black hole of mass $M$ and charge $Q$ for a particle with ratio $ϵ=-0.2$ (left plot) and $ϵ=-1.5$ (right plot). Numbers close to the point represent the energy $E/\mu$ of the last stable circular orbits at that point. Underlined numbers represent the corresponding angular momentum $L/(M\mu )$. Stable orbits are possible only for $r>r_{\mathrm{lsco}}$. For comparison we also include the curves for the radii $r_{+}$ and $r_{\gamma }^{+}$.

Figure 16

The radius $r_{\mathrm{lsco}}$ of the last stable circular orbit in a RN black with charge-to-mass ratio $Q/M$ for selected values of the charge-to-mass ratio $ϵ$ of the test particle. Only the case $ϵQ\le 0$ is illustrated. Stable orbits are possible only for $r>r_{\mathrm{lsco}}$.

Figure 17

The radius of the last stable circular orbit $r_{\mathrm{lsco}}$ (solid curve) for a charged test particle with $ϵ=7$, in a RN black hole with charge $Q$ and mass $M$, is plotted as a function of the ratio $Q/M$. Other curves are the outer horizon radius $r_{+}=M+\sqrt{M^2-Q^2}$ and the radii $r_{\gamma }^{+}\equiv [3M+\sqrt{(9M^2-8Q^2)}]/2$, $r_s\equiv \frac{Q^2}{{ϵ}^2{Q}^2-M^2}[ϵ\sqrt{M^2-Q^2}\sqrt{{ϵ}^2-1}+M({ϵ}^2-1)]$, $r_l\equiv \frac{3M}{2}+\frac{1}{2}\sqrt{9M^2-8Q^2-Q^2{ϵ}^2}$. Shaded and dark regions are forbidden for timelike particles. Stable orbits are possible only for $r>r_{\mathrm{lsco}}$.

Figure 18

The radius of the last stable circular orbit $r_{\mathrm{lsco}}$ (solid curve) for a charged test particle with $ϵ=0.5$, in a RN black hole with charge $Q$ and mass $M$, is plotted as a function of the ratio $Q/M$. Other curves are the outer horizon radius $r_{+}$ and the radius $r_{\gamma }^{+}$. Shaded and dark regions are forbidden for timelike particles. Stable orbits are possible only for $r>r_{\mathrm{lsco}}$.

Figure 19

The effective potential for a charged particle with charge-to-mass ratio $ϵ$ in a RN naked singularity of charge $Q$ and mass $M$ is plotted as a function of the radius $r/M$ for fixed values of the angular momentum $L^{*}\equiv L/(\mu M)$. Black curves represent the positive solution $V_{+}$ while gray curves correspond to $V_{-}$. The boldfaced points denote the minima of the potentials. In upper left plot, the parameter choice is $Q/M=2$ and $ϵ=-1.5$; the upper right plot is for $Q/M=2$ and $ϵ=-5$ while the bottom plot corresponds to the choice $Q/M=1.5$ and $ϵ=-0.2$.

Figure 20

The effective potential of a RN naked singularity with $Q/M=2$ for a particle with charge-to-mass ratio $ϵ$ in the range $[-10,-1]$ and angular momentum $L/(M\mu )=4$.

Figure 21

The effective potential of a RN naked singularity with $Q/M=3/2$ for a particle with charge-to-mass ratio $ϵ$ in the range $[-2,+2]$ and angular momentum $L/(M\mu )=4$.

Figure 22

The effective potential $V_{+}$ of a RN naked singularity with $Q/M=3/2$ for a charged particle is plotted for different values of the angular momentum $L^{*}\equiv L/(M\mu )$. The left plot corresponds to the ratio $ϵ=0.1$ while the right one is for $ϵ=-0.1$. For $ϵ=0.1$ there is a minimum, $V_{\min }\approx 0.81$, at $r_{\min }\approx 2.52M$ for $L^{*}=0$, and a minimum, $V_{\min }\approx 0.96$, at $r_{\min }\approx 29M$ for $L^{*}=5$. For $ϵ=-0.1$ the minimum, $V_{\min }\approx 0.67$, is located at $r_{\min }\approx 2.02M$ for $L^{*}=0$, and at $r_{\min }\approx 20.8M$ with $V_{\min }\approx 0.97$ for $L^{*}=5$.

Figure 23

The charge parameters ${ϵ}_l$ (black solid curve), ${\widetilde{ϵ}}_{-}$ (gray solid curve), ${\widetilde{ϵ}}_{+}$ (dashed curve), ${\widetilde{\tilde{ϵ}}}_{-}$ (dotted curve), and ${\widetilde{\tilde{ϵ}}}_{+}$ (dotdashed curve) as functions of the charge-to-mass ratio of the RN naked singularity. The special lines $Q/M=5/(2\sqrt{6})\approx 1.02$, $Q/M=3\sqrt{6}/7\approx 1.05$, and $Q/M=\sqrt{9/8}\approx 1.06$ are also plotted.

Figure 24

The case $ϵ>1$. The energy (black curve) and angular momentum (gray curve) for a test particle with charge-to-mass ratio $ϵ=2$ in a RN naked singularity with $Q=1.06M$. Circular orbits exist in the interval $r_{\gamma }^{-}<r<r_{\gamma }^{+}$, where $r_{\gamma }^{-}=1.04196M$ and $r_{\gamma }^{+}=1.95804M$.

Figure 25

Case: $M<Q\le 5/(2\sqrt{6})M$ and ${\widetilde{ϵ}}_{+}<ϵ\le {ϵ}_l$. Parameter choice: $Q=1.01M$ and $ϵ=0.902$. Then ${ϵ}_l=0.907$, ${\widetilde{ϵ}}_{+}=0.8963$, $r_s^{+}=1.44942M$, $r_{\gamma }^{-}=1.04196M$, $r_{\gamma }^{+}=1.95804M$, $r_l^{-}=1.45192M$, and $r_l^{+}=1.548077M$. Circular orbits exist with angular momentum $L=L^{+}$ (gray curves) and energy $E=E^{+}$ (black curves) in $r_{\gamma }^{-}<r<r_s^{+}$ (upper left plot); $L=L^{\pm }$ in $r_s^{+}\le r<r_l^{-}$ (upper right plot) and $r_l^{+}\le r<r_{\gamma }^{+}$ (bottom left plot); $L=L^{-}$ in $r\ge r_{\gamma }^{+}$ (bottom right plot).

Figure 26

Case: $ϵ<-1$ and $Q>\sqrt{9/8}M$. Parameter choice: $Q=2M$ and $ϵ=-2$. Circular orbits exist with angular momentum $L=L^{+}$ (gray curve) and energy $E=E^{+}$ (black curve).

Figure 27

Case: $ϵ<-1$ and $M<Q\le \sqrt{9/8}M$. Parameter choice: $Q=1.01M$ and $ϵ=-2$. Then, $r_{\gamma }^{-}=1.04196M$ and $r_{\gamma }^{+}=1.95804M$. Circular orbits exist with angular momentum $L=L^{+}$ (gray curve) and energy $E=E^{+}$ (black curve) in $0<r<r_{\gamma }^{-}$ and $r>r_{\gamma }^{+}$.

Figure 28

Case: $M<Q\le \sqrt{9/8}M$ and $-M/Q<ϵ<0$. Parameter choice: $Q=1.05M$ and $ϵ=-0.2$. Then $r_{\gamma }^{-}=1.28787M$, $r_{\gamma }^{+}=1.71213M$, and $r_s^{-}=1.03487M$. Circular orbits exist with angular momentum $L=L^{+}$ (gray curve) and energy $E=E^{+}$ (black curve) in $r_s^{-}<r<r_{\gamma }^{-}$ (left plot) and in $r>r_{\gamma }^{+}$ (right plot). For $r=r_s^{-}$, $L=0$.

Figure 29

The radius of the last stable circular orbit $r_{\mathrm{lsco}}$ (gray curve) of a charged particle with ratio $ϵ=+7$ in a RN naked singularity with ratio $Q/M\in [1,1.2]$. The radii $r_{*}=Q^2/M$ and $r=r_{\gamma }^{\pm }\equiv [3M\pm \sqrt{9M^2-8Q^2}]/2$ are also plotted. Circular orbits exist only in the interval $1<Q/M<\sqrt{9/8}$. The shaded region is forbidden for timelike particles. Stable orbits are located in the region $r>r_{\mathrm{lsco}}$.

Figure 30

The radius of the last stable circular orbit $r_{\mathrm{lsco}}^{\pm }$ (black curves) of a charged particle with ratio $ϵ=-7$ in a RN naked singularity with ratio $Q/M\in [1,1.8]$. The radii $r_{*}$, and $r_{\gamma }^{\pm }$ are also plotted for comparison. In the shaded region no circular orbits can exist. Stable circular orbits are situated outside the region with boundaries $r_{\mathrm{lsco}}^{+}$, $r_{\mathrm{lsco}}^{-}$, and the vertical axis $Q/M=1$.

Figure 31

The radius of the last stable circular orbit $r_{\mathrm{lsco}}^{\pm }$ (black and grey curves) of a charged particle with ratio $ϵ=0.5$ in a RN naked singularity with ratio $Q/M\in [1,1.1]$. Also plotted: $r_{\gamma }^{\pm }\equiv [3M\pm \sqrt{(9M^2-8Q^2)}]/2$, $r_s^{+}\equiv \frac{Q^2}{{ϵ}^2{Q}^2-M^2}[\sqrt{{ϵ}^2({ϵ}^2-1)(M^2-Q^2)}M({ϵ}^2-1)]$, $r_l^{\pm }\equiv \frac{3M}{2}\pm \frac{1}{2}\sqrt{9M^2-8Q^2-Q^2{ϵ}^2}$, and $r_{*}=Q^2/M$. Regions of stability are: for $Q>\sqrt{9/8}M$ in $r>r_s$, for $(3\sqrt{6}/7)M<Q<\sqrt{9/8}M$ exist stable orbits in $r_{\gamma }^{-}<r$, for $(5/(2\sqrt{6}))M<Q<(3\sqrt{6}/7)M$ exist stable orbits in $r_{\gamma }^{-}<r$. For $M<Q<(5/(2\sqrt{6}))M$ stable orbits are located in $r>r_{\mathrm{lsco}}^{+}$.

Figure 32

The radius of the last stable circular orbit $r_{\mathrm{lsco}}^{\pm }$ (black and grey curves) of a charged particle with ratio $ϵ=-0.5$ in a RN naked singularity with ratio $Q/M\in [1,1.2]$. Also plotted: $r_{\gamma }^{\pm }\equiv [3M\pm \sqrt{(9M^2-8Q^2)}]/2$, $r_s^{+}\equiv \frac{Q^2}{{ϵ}^2{Q}^2-M^2}[\sqrt{{ϵ}^2({ϵ}^2-1)(M^2-Q^2)}M({ϵ}^2-1)]$, $r_l^{\pm }\equiv \frac{3M}{2}\pm \frac{1}{2}\sqrt{9M^2-8Q^2-Q^2{ϵ}^2}$, and $r_{*}=Q^2/M$. Shaded regions are forbidden. Regions of stability are: for $Q>(\sqrt{9/8})M$ stable circular orbits exist in $r_s^{+}<r<r_{\mathrm{lsco}}^{-}$, and $r>r_{\mathrm{lsco}}^{+}$. For $M<Q<(\sqrt{9/8})M$ stable circular orbits exist in $r_s^{+}<r<r_{\gamma }^{-}$, and $r>r_{\gamma }^{+}$.

Figure 33

The positive solution of the linear velocity ${\nu }_{ϵ}^{+}$ is plotted as a function of the radial distance $r/M$ for different values of the ratios $Q/M$ and $ϵ$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $Q/M=3$ and $ϵ=2$, with $r_{*}\equiv Q^2/M=9M$. In (b) the parameter choice is $Q/M=2$ and $ϵ=3$, with $r_{*}\equiv Q^2/M=4M$. In (c) the parameter choice is $Q/M=1.04$ and $ϵ=2$, with $r_{*}\equiv Q^2/M\approx 1.08M$, $r_{\gamma }^{+}\equiv [3M+\sqrt{9M^2-8Q^2}]/2\approx 1.79M$, and $r_{\gamma }^{-}\equiv [3M-\sqrt{9M^2-8Q^2}]/2\approx 1.201M$. In (d) the parameter choice is $Q/M=\sqrt{9/8}$ and $ϵ=2$, with $r_{*}\equiv Q^2/M=(9/8)M$, $r_{\gamma }^{\pm }\equiv [3M\pm \sqrt{9M^2-8Q^2}]/2=(3/2)M$.

Figure 34

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=1.01$ and different values of the ratio $ϵ$. In this case $r_{\gamma }^{+}=1.96M$, $r_{\gamma }^{-}=1.042M$ with $r_{*}\equiv Q^2/M=1.02M$, ${\widetilde{ϵ}}_{-}=0.31$, ${\widetilde{ϵ}}_{+}=0.9$, ${ϵ}_l\approx 0.91$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=0.2$ with $r_s^{+}=1.05M$, and $r_l^{+}=1.95M$, $r_l^{-}=1.05M$. In (b) the parameter choice is $ϵ=0.5$. Here $r_s^{+}=1.11M$, and $r_l^{+}=1.88M$, $r_l^{-}=1.12M$. In (c) the parameter choice is $ϵ=0.9$. Here $r_s^{+}=1.11M$, and $r_l^{+}=1.88M$, $r_l^{-}=1.12M$. In (d) the parameter choice is $ϵ=0.95$. Here $r_s^{+}=1.79M$, and $r_l^{\pm }$ do not exist.

Figure 35

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=1.05$ and different values of the ratio $ϵ$. In this case $r_{\gamma }^{+}=1.71M$, $r_{\gamma }^{-}=1.29M$ with $r_{*}\equiv Q^2/M\approx 1.102M$, ${ϵ}_l\approx 0.40$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=0.7$. Here $r_s^{+}=1.61M$, and $r_l^{\pm }$ are not defined. In (b) the parameter choice is $ϵ=0.2$. Here $r_s^{+}=1.2M$, and $r_l^{+}=1.68M$, $r_l^{-}=1.32M$.

Figure 36

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=\sqrt{9/8}$ and different values of the ratio $ϵ$. In this case $r_{\gamma }^{+}=r_{\gamma }^{-}=3/2M$ with $r_{*}\equiv 9/8M$, ${ϵ}_l=0$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=0.2$ with $r_s^{+}=1.2M$. In (b) the parameter choice is $ϵ=0.7$ with $r_s^{+}=1.72M$.

Figure 37

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=2$ and different values of the ratio $ϵ$. In this case $r_{*}\equiv Q^2/M\approx 4M$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=0.7$ with $r_s^{+}=1.48M$. In (b) the parameter choice is $ϵ=0.2$ with $r_s^{+}=6.2M$.

Figure 38

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=1.01$ and different values of the ratio $ϵ$. In this case $r_{\gamma }^{+}=1.96M$, $r_{\gamma }^{-}=1.042M$ $r_{*}\equiv Q^2/M=1.02M$, ${\widetilde{ϵ}}_{-}=0.31$, ${\widetilde{ϵ}}_{+}=0.9$, and ${ϵ}_l\approx 0.91$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=-0.95$. Here $r_s^{+}=1.79M$, and $r_l^{\pm }$ do not exist. In (b) the parameter choice is $ϵ=-0.9$. Here $r_s^{+}=1.11M$, $r_l^{+}=1.88M$, and $r_l^{-}=1.12M$. In (c) the parameter choice is $ϵ=-0.5$. Here $r_s^{+}=1.11M$, $r_l^{+}=1.88M$, and $r_l^{-}=1.12M$. In (d) the parameter choice is $ϵ=-0.2$. Here $r_s^{+}=1.05M$, $r_l^{+}=1.95M$, and $r_l^{-}=1.05M$.

Figure 39

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=1.05$ and different values of the ratio $ϵ$. In this case $r_{\gamma }^{+}=1.71M$, $r_{\gamma }^{-}=1.29M$, $r_{*}\equiv Q^2/M\approx 1.102M$, and ${ϵ}_l\approx 0.40$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=-0.2$. Here $r_s^{+}=1.2M$, $r_l^{+}=1.68M$, and $r_l^{-}=1.32M$. In (b) the parameter choice is $ϵ=-0.7$. Here $r_s^{+}=1.61M$, and $r_l^{\pm }$ are not defined.

Figure 40

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=\sqrt{9/8}$ and different values of the ratio $ϵ$. In this case $r_{\gamma }^{+}=r_{\gamma }^{-}=3/2M$, $r_{*}\equiv 9/8M$, and ${ϵ}_l=0$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=-0.7$ with $r_s^{+}=1.72M$. In (b) the parameter choice is $ϵ=-0.2$ with $r_s^{+}=1.2M$.

Figure 41

The positive solution of the linear velocity ${\nu }_{ϵ}$ is plotted as a function of the radial distance $r/M$ for $Q/M=2$ and different values of the ratio $ϵ$. In this case $r_{*}\equiv Q^2/M\approx 4M$. The geodesic velocity ${\nu }_g$ is also shown (dashed curve). Shaded region is forbidden. In (a) the parameter choice is $ϵ=-0.2$. Here $r_s^{+}=6.2M$. In (b) the parameter choice is $ϵ=-0.7$. Here $r_s^{+}=1.48M$.