Circular motion of neutral test particles in Reissner-Nordström spacetime

We investigate the motion of neutral test particles in the gravitational field of a mass $M$ with charge $Q$ described by the Reissner-Nordström (RN) spacetime. We focus on the study of circular stable and unstable orbits around configurations describing either black holes or naked singularities. We show that at the classical radius, defined as $Q^2/M$, there exist orbits with zero angular momentum due to the presence of repulsive gravity. The analysis of the stability of circular orbits indicates that black holes are characterized by a continuous region of stability. In the case of naked singularities, the region of stability can split into two nonconnected regions inside which test particles move along stable circular orbits.

Figure 1

In this graphic the radii $r_{{\gamma }_{+}}\equiv [3M+\sqrt{(9M^2-8Q^2)}]/2$ and $r_{\pm }=M+\sqrt{M^2-Q^2}$, and $r_{*}=Q^2/M$ are plotted. Timelike circular orbits exist only for $r>r_{{\gamma }_{+}}$, whereas $r=r_{{\gamma }_{+}}$ represents a null hypersurface. Circular motion inside the regions $r<r_{*}$ and $r\in (r_{{\gamma }_{-}},r_{{\gamma }_{+}})$ is possible only along spacelike geodesics.

Figure 2

The effective potential $V$ for a neutral particle of mass $\mu$ in a RN black hole of charge-to-mass ratio $Q/M=0.5$ is plotted as a function of $r/M$ in the range [1.87, 8], and the angular momentum $L/\mu$ in [0, 10]. In this case the outer horizon $r_{+}=1.87M$, and $r_{{\gamma }_{+}}=2.823M$ (see text). Circular orbits exist for $r>2.823M$. The solid line represents the location of circular orbits (stable and unstable). The last circular orbit is represented by a point. The number close to the plotted point represents the radius $r/M=5.6$, the angular momentum $L/\mu =3.34$, and the energy $E/M=0.939$ of the last stable circular orbit.

Figure 3

The effective potential $V$ for a neutral particle of mass $\mu$ in a RN black hole of charge $Q$ and mass $M$ is plotted in terms of the radius $r/M$ for different values of the angular momentum $L^{*}\equiv L/(M\mu )$ and $Q=0.5M$. The outer horizon is located at $r_{+}\approx 1.87M$. The effective potential has a minimum, $V_{\min }\approx 0.93$, at $r_{\min }\approx 5.60M$, for $L^{*}\approx 3.34$. The plotted points represent local extrema.

Figure 4

The radius of the last stable orbit $r_{\mathrm{lsco}}/M$ is plotted as a function of the ratio $Q/M$. Numbers close to the points represent the value of the energy $E/\mu$ and of the angular momentum $L/(\mu M)$ (underlined numbers) of the corresponding orbit.

Figure 5

The pictures show (a) the energy $E/\mu$ and (b) the angular momentum $L_{+}/(M\mu )\equiv L^{*}$ of a circular orbit as a function of $r/M$. Notice that for $Q=0$, the relevant radii are $r_{+}=2M$ and $r_{{\gamma }_{+}}=3M$, and the minima are $E_{\min }/\mu \approx 0.943$ and $L_{\min }^{*}\approx 3.46$, with $r_{\min }=6M$. Furthermore, for $Q=0.5M$, we obtain $r_{+}\approx 1.87M$ and $r_{{\gamma }_{+}}\approx 2.83M$, so that the minima are $E_{\min }/\mu \approx 0.939$ and $L_{\min }^{*}\approx 3.34$, with $r_{\min }\cong 5.61M$. The energy and angular momentum diverge as the limiting radius $r_{\gamma +}$ is approached.

Figure 6

The energy $E/\mu$ of a test particle on a circular orbit as a function of $r/M$ and charge-to-mass ratio $Q/M\in [0,1]$. The radius of the last stable circular orbit is also plotted (thick line). Numbers close to the points correspond to the energy and the angular momentum (underlined numbers) of the last stable circular orbit.

Figure 7

Plots of the energy $E_{\mathrm{lsco}}/M$ (a), and the angular momentum $L_{\mathrm{lsco}}/(\mu M)$ (b) of the last stable circular orbit in terms of the ratio $Q/M$ of the black hole.

Figure 8

The effective potential $V$ for a neutral particle of mass $\mu$ in the field of an extreme RN black hole is plotted as a function of the radius $r/M$ for different values of the angular momentum $L^{*}\equiv L/(M\mu )$. The outer horizon is located at $r_{+}=M$. For $L^{*}\approx 2.83$ the effective potential has a minimum $V_{\min }\approx 0.91$ at $r_{\min }=4M$. There is a maximum $V_{\max }\approx 1.35$ at $r\approx 2.19M$ for $L^{*}=5$, and for $L^{*}=10$ the maximum $V_{\max }\approx 2.55$ is located at $r\approx 2.04M$.

Figure 9

Plots of (a) the energy $E/\mu$ and (b) the angular momentum $L_{+}/(M\mu )\equiv L^{*}$ of a neutral test particle in circular motion around an extreme RN black hole. The outer horizon is $r_{+}=M$ and $r_{{\gamma }_{+}}=2M$. At $r=4M$ the minimum values of the energy $E_{\min }/\mu \approx 0.91$ and angular momentum $L_{\min }^{*}\approx 2.83$ are reached.

Figure 10

Plot of the energy $E/\mu$ of a neutral particle in circular motion around a RN naked singularity as a function of $r/M$ and $Q/M$ in the interval [1, 1.6] (a) and in the interval [1.1, 2] (b). The line $r_{*}=Q^2/M$ is also plotted. Numbers close to the points represent the energy and the angular momentum (underlined numbers) of the last stable circular orbits.

Figure 11

The effective potential $V$ for a neutral particle of mass $\mu$ in a RN naked singularity with $Q/M=1.1$ is plotted as a function of the radius $r/M$ for different values of the angular momentum $L^{*}\equiv L/(M\mu )$. The classical radius $r_{*}\equiv Q^2/M=1.21M$ is represented by a dashed line. For $L^{*}=0$ the effective potential presents a minimum $V_{\min }\approx 0.42$ at $r_{\min }=r_{*}$. For $L^{*}\approx 3.37$ the minimum $V_{\min }/\mu \approx 0.95$ is situated at $r_{\min }\approx 8.97M$.

Figure 12

The effective potential $V$ for a neutral particle of mass $\mu$ in a RN naked singularity. Plot (a) is for $Q/M=1.5$ so that $r_{*}=Q^2/M=2.25M$. For the lowest value $L^{*}=0$ at $r=r_{*}$, there is global minimum with $V_{\min }\approx 0.74$. The value of $V_{\min }$ increases as $L^{*}$ increases. Plot (b) corresponds to $Q/M=1.06$, and shows the characteristic radii $r_{*}=1.12M$, $r_{{\gamma }_{+}}\equiv [3M+\sqrt{(9M^2-8Q^2)}]/2\approx 5.55M$ and $r_{{\gamma }_{-}}\equiv [3M-\sqrt{(9M^2-8Q^2)}]/2\approx 1.44M$. For $L^{*}=0$ the global minimum is at $r=r_{*}$ with $V_{\min }\approx 0.33$. As $L^{*}$ increases, the value of $V_{\min }$ increases, and at $L^{*}\approx 2.7$ a second local minimum appears. The first one with $V_{\min }\approx 0.81$ is at $r_{\min }\approx 1.29M$ and the second one with $V_{\min }\approx 0.91$ is located at $r_{\min }\approx 3.5M$.

Figure 13

The pictures show (a) the energy $E/\mu$ and (b) the angular momentum $L_{+}/(M\mu )\equiv L^{*}$ of a neutral particle moving on a circular orbit around a RN naked singularity with $Q/M=1.06<\sqrt{9/8}$. The shaded regions are forbidden, i.e., circular orbits can exist only in the regions defined by $r_{*}<r<r_{{\gamma }_{-}}$ and $r>r_{{\gamma }_{+}}$, where $r_{*}=Q^2/M\approx 1.12M$, $r_{{\gamma }_{-}}\equiv [3M-\sqrt{(9M^2-8Q^2)}]/2\approx 1.44M$, and $r_{{\gamma }_{+}}\equiv [3M+\sqrt{(9M^2-8Q^2)}]/2\approx 1.55M$ (see text). A minimum for the energy and the angular momentum is located at $r_{\min }\approx 3.55M$, where $L_{\min }^{*}\approx 2.69$ and $E_{\min }/\mu \approx 0.91$. Moreover, $L^{*}(r_{*})=0$ and $(E_{\min }/\mu )(r_{*})\approx 0.33$.

Figure 14

The effective potential $V$ for a neutral particle of mass $\mu$ in a RN naked singularity with $Q/M=\sqrt{9/8}$ is plotted as a function of the radius $r/M$ for different values of the angular momentum $L^{*}\equiv L/(M\mu )$. In this case, $r_{*}=Q^2/M=1.12M$ and $r_{{\gamma }_{\pm }}\equiv [3M\pm \sqrt{(9M^2-8Q^2)}]/2=1.5M$.

Figure 15

Behavior of (a) the energy $E/\mu$ and (b) the angular momentum $L_{+}/(M\mu )\equiv L^{*}$ of a neutral particle moving along a circular orbit around a RN naked singularity with $Q/M=\sqrt{9/8}$. Shaded regions are forbidden for timelike particles.

Figure 16

The pictures show (a) the energy $E/\mu$ and (b) the angular momentum $L_{+}/(M\mu )\equiv L^{*}$ of a neutral particle moving along a circular orbit in a RN naked singularity with $Q/M=1.1$ as a function of $r/M$. The shaded regions are forbidden. Circular orbits can exist only for $r>r_{*}\approx 1.21M$. A minimum of the energy and the angular momentum is located at $r_{\min }\approx 3.08M$, where $L_{\min }^{*}\approx 2.57$ and $E_{\min }/\mu \approx 0.90$. Moreover, $L^{*}(r_{*})=0$ and $(E_{\min }/\mu )(r_{*})\approx 0.33$.

Figure 17

The pictures show (a) the energy $E/\mu$ and (b) the angular momentum $L_{+}/(M\mu )$ of a test particle of mass $\mu$ in the field of a RN naked singularity as a function of $r/M$ and for different values of the ratio $Q/M$, satisfying the condition $(Q/M{)}^2>9/8$ (see text). Circular orbits can exist only for $r>r_{*}\equiv Q^2/M$.

Figure 18

The radius of the last stable circular orbit (solid line) of a neutral particle moving in the field of a RN naked singularity. The ratio $Q/M$ varies in the interval [1, 2.2] in plot (a), and in the interval [1, 1.15] in plot (b). The dashed curve represents the classical radius $r_{*}=Q^2/M$. The dotted curve corresponds to photon orbits with radius $r_{{\gamma }_{-}}=[3M-\sqrt{(9M^2-8Q^2)}]/2$, whereas the dot-dashed curve denotes the photon orbits with radius $r_{{\gamma }_{+}}=[3M+\sqrt{(9M^2-8Q^2)}]/2$. Shaded regions are forbidden. In the interval $1<Q/M<1.061$ circular orbits can exist only for $r_{*}<r<r_{{\gamma }_{-}}$ (all stable) and $r>r_{{\gamma }_{+}}$ (unstable for $r_{{\gamma }_{+}}<r<r_{\mathrm{lsco}}^{+}$ and stable for $r>r_{\mathrm{lsco}}^{+}$). For $Q/M>1.061$ and $r>r_{*}$ the region of stability divides into two separated regions: $r_{*}<r<r_{\mathrm{lsco}}^{-}$ and $r>r_{\mathrm{lsco}}^{+}$. The numbers close to the plotted points denote the energy $E/\mu$ and the angular momentum $L/(\mu M)$ (underlined numbers) of the last stable circular orbits.

Figure 19

The effective potential $V$ for a neutral particle of mass $\mu$ in a RN naked singularity is plotted as a function of the radius $r/M$ and of the angular momentum $L^{*}\equiv L/(M\mu )$. Solid lines represent the radii of circular orbits. The last stable circular orbit radius is denoted by a point. Plot (a) is for $Q=1.1M$ so that $r_{{\gamma }_{-}}=1.04M$, $r_{{\gamma }_{+}}=1.95M$. The classical radius is $r_{*}\equiv Q^2/M=1.21M$, where $L^{*}=0$ and $V_{\min }\approx 0.42$. Unstable circular orbits are in $r<3.94M$ and at $r=3.94M$, the values are $L^{*}\approx 2.81$ and $V/\mu \approx 0.91$. Plot (b) is for $Q=1.08M$ so that $r_{{\gamma }_{\pm }}$ do not exist and $r_{*}\equiv Q^2/M=1.16M$ with $V_{\min }(r_{*})\approx 0.37$. Stable orbits are in the interval $r_{*}<r<1.68M$ and at the boundary, $r=1.68M$, the values are $L^{*}=3.55$ and $V/\mu =1.10$. The interval $1.68M<r<3.35M$ corresponds to unstable orbits with the boundary values $L^{*}=2.64$ and $V/\mu =0.90$ at $r=3.35M$. Stable orbits are located at $r>3.35M$. Plot (c) is for $Q=2M$ and $r_{*}=4M$. Stable circular orbits exist for $r>4M$ and at the boundary $V(4M)=0.86\mu$.

Figure 20

The pictures show the energy $E/\mu$ (a) and the angular momentum $L_{+}/(\mu M)$ (b) of a test particle moving along a circular orbit around a RN source as a function of $r/M$ and the charge-to-mass ratio $Q/M$ in the range [0, 2]. The curve $r=r_{*}\equiv Q^2/M$ is also plotted. The numbers close to the points denote the energy and the angular momentum (underlined numbers) of the last stable circular orbits.

Figure 21

The radius $r/M$ of circular orbits for a neutral particle of mass $\mu$ in a RN geometry of charge $Q$ and mass $M$ is plotted as a function of the angular momentum $L/(\mu M)$ for different values of the charge-to-mass ratio $Q/M$ in the interval [0, 2]. The radius of circular orbits for the Newtonian limit (dashed curve) and for the Schwarzschild spacetime (dot-dashed curve) are also plotted. The plotted points represent the last stable circular orbits. Black curves correspond to stable circular orbits, and gray curves denote unstable circular orbits. The dotted line represents the classical radius $r=r_{*}\equiv Q^2/M$.

Figure 22

The radius $r/M$ of the last stable circular orbit of a neutral particle of mass $\mu$ in a RN geometry of charge $Q$ and mass $M$ is plotted as a function of the ratio $Q/M$ in the interval [0, 1.5]. Here $r_{\gamma \pm }\equiv [3M\pm \sqrt{(9M^2-8Q^2)}]/2$ and $r_{\pm }\equiv M\pm \sqrt{M^2-Q^2}$ (see text). In the Schwarzschild case, $Q=0$, we find $r_{\mathrm{lsco}}=6M$, $r_{+}=2M$, and $r_{{\gamma }_{+}}=3M$. For an extreme black hole, $Q=M$, we find $r_{\mathrm{lsco}}=4M$ with $r_{+}=M$ and $r_{{\gamma }_{+}}=2M$. The numbers close to the plotted points denote the energy $E/\mu$ and the angular momentum $L/(\mu M)$ (underlined numbers) of the last stable circular orbit.