Equatorial circular orbits of neutral test particles in the Kerr-Newman spacetime

We present a detailed analysis of the orbital circular motion of electrically neutral test particles on the equatorial plane of the Kerr-Newman spacetime. Many details of the motion in the cases of black hole and naked singularity sources are pointed out. We identify four different types of orbital regions, which depend on the properties of the orbital angular momentum, and define four different kinds of naked singularities, according to the values of the charge-to-mass ratio of the source. It is shown that the presence of a particular type of counterrotating test particle is sufficient to uniquely identify naked singularities. It is pointed out that the structure of the stability regions can be used to differentiate between black holes and naked singularities.

Figure 1

The angular momentum of circular orbits is plotted as a function of the orbital radius for different source charge–mass ratios $Q/M$. Circular orbits in the KN spacetime with $a=1.1M$ are explored. Right: The energy $V/\mu$ of circular orbits as a function of $r/M$ for selected values of $Q/M$.

Figure 2

Circular orbits in the KN spacetime with $a=2M$ are explored. The angular momentum of circular orbits is plotted as a function of the orbital radius for different source charge–mass ratios $Q/M$. Right: The energy $V/\mu$ of circular orbits as a function of $r/M$ for selected values of $Q/M$.

Figure 3

Left: The radii $r_1$ (black surface), $r_2$ (white surface), $r_3$ (gray, black-shaded, surface), and $r_4$ (black, white-shaded surface) plotted as functions of $Q/M\in (0,2)$ and $a/M\in (0,2)$; it is $r_1<r_2<r_3<r_4$. Center: Surfaces $(r_1,r_2,r_3,r_4)$ in the region $Q/M\in \{0,1.1\}$, $a/M\in \{0,1\}$, $r/M\in \{0,4\}$. The black region represents $r<r_{+}$, where $r_{+}$ is the outer horizon of the black hole. Right: The region $Q/M\in \{1,1.07\}$, $a/M\in \{0,0.251\}$, $r/M\in \{1,2.35683\}$ corresponds to the case of a naked singularity.

Figure 4

Left: Plot of $a_T$ and $a_s$ as a function of $Q/M$. Right: Plot of $Q_4$ (black curve), $Q_s$ (gray curve), $Q_5$ (dashed curve), $Q_6$ (black thick curve) as functions of $a/M$. Black regions are forbidden: no solution of Eq. (29) exists. In the white regions, Eq. (29) is satisfied for $r=r_3$ and $r=r_4$, and in the light gray regions, it is satisfied for $r=r_1$, $r=r_2$, $r=r_3$ and $r=r_4$. In particular, for $Q=0$ and $a=0$ the solution is $r=3M$; for $0<a<M$ it is $r=r_2$, $r=r_3$ and $r=r_4$. For $a=M$ it is $r=M$ and $r=4M$; finally for $a>M$ it is $r=r_3=r_4$.

Figure 5

Black regions are forbidden: no solution of Eq. (29) exists. Left: Plot of $a_{r_1}$ and $a_{r_2}$ as functions of $r/M$. The solutions $Q_{\mathrm{T}}^{\pm }$ are shown in the light-gray region and $Q_{\mathrm{T}}^{+}$ in the white regions. Right: Plot of $r_{{\mathrm{M}}_1}$ (gray curve), $r_{{\mathrm{M}}_2}$ (dashed black curve), $r_{{\mathrm{M}}_3}$ (dotted-dashed black curve), $r_{{\mathrm{M}}_{-}}$ (gray curve), $r_{{\mathrm{M}}_{+}}$ (gray dashed curve) as functions of $a/M$. Black regions are forbidden: no solution of Eq. (29) exists. In the white regions, the solution is $Q=Q_{\mathrm{T}}^{+}$; in the light gray regions the solutions are $Q=Q_{\mathrm{T}}^{\pm }$.

Figure 6

Left: Plot of $r_{\gamma }^{+}$ (thick black curve), $r_{\gamma }^{-}$ (thick gray curve), $r_{*}$ (black curve), $r_{ϵ}$ (dotted-dashed curve) as functions of $Q/M$. The solution of Eq. (29) is $a_i$ in the white regions. Center: Plot of $Q_i$ and $Q_{*}/M=\sqrt{r/M}$ as functions of the radius $r/M$. Dotted-dashed curve represents $r_{ϵ}$. In the white regions, Eq. (29) is satisfied for $a=a_i$. The plot is therefore divided in the regions $0<r<M$, $M<r\le 3M$, $r>3M$. In particular, at $r=M$ and $Q=1$, a solution exists for $a\ge 0$. The dotted-dashed line is $Q=M$, the dashed line is $r=M$ and the dotted line is $r=3M$. Right: The black hole angular momentum $a_i$ is plotted as a function of $r/M$, and the charge-to-mass ratio $Q/M$ of the black hole. The outer horizon plane $r_{+}$ (gray) is also plotted. The black region is $r<r_{+}$. The curve $r=r_{\gamma }^{+}$ (black thick) is also plotted.

Figure 7

Left: Plots of $\tilde{r}$ as functions of $Q/M$ and $a/M$. The surfaces $r=r_{+}$ and $r=r_{-}$, delimited by the outer and inner horizon respectively are also plotted. The region $r<r_{+}$ is shaded. Center: Plot of ${\tilde{r}}_Q$, that is $\tilde{r}$ in the case $Q=0$ and $Q=M$ (inset plot). Right: Plot of $r_x$ in the range $[0,3M)$ as a function of $a/M$. The inset plot shows a zoom for $a\le M$.

Figure 8

Left: The outer horizon $r_{+}=M+\sqrt{M^2-a^2-Q^2}$, the outer ergosphere radius on the equatorial plane $r_{ϵ}^{+}=M+\sqrt{M^2-Q^2}$ and the radius $r_d^{+}=2r_{ϵ}^{+}$ are plotted as functions of the black hole charge-to-mass ratio $Q/M$, for different values of the intrinsic angular momentum $a/M$. Center and right: The outer horizon $r_{+}=M+\sqrt{M^2-a^2-Q^2}$, the outer ergosphere radius on the equatorial plane $r_{ϵ}^{+}=M+\sqrt{M^2-Q^2}$ and the radius $r_d^{\pm }=2r_{ϵ}^{\pm }$ are plotted as functions of $r/M$ and the black hole charge-to-mass ratio $Q/M$. The angular momentum $a_s=\sqrt{M^2-Q^2}$ is plotted (gray curve); for the black hole case it is $a<a_s$. Center: $Q=0.5M$, $a_s=0.866M$, $r_{ϵ}^{+}=1.866M$, $r_d^{+}=3.732M$, $r_d^{-}=0.268M$. Right: $Q=0.9M$, $a_s=0.436M$, $r_{ϵ}^{+}=1.436M$, $r_d^{+}=2.872M$, $r_d^{-}=1.1282M$. The outer horizon $r_{+}=M+\sqrt{M^2-a^2-Q^2}$, the outer ergosphere radius on the equatorial plane $r_{ϵ}^{+}=M+\sqrt{M^2-Q^2}$ and the radius $r_d^{\pm }=2r_{ϵ}^{\pm }$ are plotted as functions of $r/M$ and the black hole charge-to-mass ratio $Q/M$. The angular momentum $a_s=\sqrt{M^2-Q^2}$ is plotted (gray curve); for the black hole case it is $a<a_s$. Shaded regions are forbidden: the horizon constraint of the KN black hole implies $0\le a\le a_s\le 1$ and $r>r_{+}$.

Figure 9

Black hole case: circular orbital regions in the plane $(Q/M,r/M)$ for fixed values of $a/M$. The angular momentum of the orbiting test particle is shown in each spatial region. The curves $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ are also plotted. See also Table . The horizons $r_{\pm }$ are represented as white curves. The dotted-dashed curve is the outer ergosphere $r_{ϵ}^{+}$.

Figure 10

Black hole case: circular orbital regions in the plane $(a/M,r/M)$ for fixed $Q/M$. The angular momentum of the orbiting test particle is shown in each spatial region. The intrinsic KN angular momenta ${\tilde{a}}_{-}/M$ (gray curve), ${\tilde{a}}_{+}/M$ (black curve) and $a_i/M$ (dashed gray curve) are also plotted as functions of $r/M$. The horizons $r_{\pm }$ are represented as white curves. See also Table . The dotted-dashed line is the outer ergosphere $r_{ϵ}^{+}$.

Figure 11

Left: The outer horizon (black surface), $r_{\mathrm{lsco}}^{RN}$ (gray surface), $r_{\mathrm{lsco}}^{K-}$ (light-gray surface), $r_{\mathrm{lsco}}^{K+}$ (white-black, thick-meshed surface), the surface $r_{\mathrm{lsco}}(L_{-})$ (light-gray, meshed surface) and $r_{\mathrm{lsco}}(-L_{+})$ (gray surface with black thick mesh). Note that $r_{\mathrm{lsco}}^{K+}=r_{\mathrm{lsco}}^{K-}$ at $r=6M$ and $r_{\mathrm{lsco}}^{K-}=r_{\mathrm{lsco}}^{RN}$ along the curve $Q=Q_k$ (see text) and $r_{\mathrm{lsco}}(-L_{+})<r_{\mathrm{lsco}}(L_{-})$. Center: Last stable circular orbit radius $r_{\mathrm{lsco}}$ for a Kerr black hole ($Q=0$), $r_{\mathrm{lsco}}^{\mathrm{K}-}$ (gray curve) and $r_{\mathrm{lsco}}^{\mathrm{K}+}$ (black curve), the outer horizon $r_{+}$, and the radius $r_{\beta }$ (dotted-dashed curve) are plotted as functions of the KN black hole spin $a/M$. The dashed line represents $a=0.430075M$. Right: The charge $Q_{\mathrm{K}}$ (black curve) of a KN black hole is plotted as a function of the momentum $a/M$, where $r_{\mathrm{lsco}}^{\mathrm{RN}}=r_{\mathrm{lsco}}^{\mathrm{K}-}$ (see text). A naked singularity exists in the region $Q>Q_s$.

Figure 12

Left: The radii $r_{\mathrm{lsco}}^{+}\equiv r_{\mathrm{lsco}}(-L_{+})$ and $r_{\mathrm{lsco}}^{-}\equiv r_{\mathrm{lsco}}(L_{-})$ of the last stable circular orbits for a KN black hole, with fixed charge-to-mass ratio $Q/M$, as a function of the black hole spin $0\le a\le a_s$. The horizon $r+$, the radius $r_{\gamma }^{+}$ (dotted line), $r_d^{+}$ (dashed line), $r_3$ (black curve), $r_4$ (gray curve) are also plotted. The black region ($r<r_{+}$) is forbidden. The value of $a_c$, the crossing point between $r_3$ and $r_{\mathrm{lsco}}^{-}$, is represented by a dashed line. The dotted-dashed line is the outer ergosphere $r_{ϵ}^{+}$. Center and right: Last stable circular orbits $r_{\mathrm{lsco}}(L_{-})$ (thick black curve), $r_{\mathrm{lsco}}(-L_{-})$ (thick black dashed curve) and $r_{\mathrm{lsco}}(-L_{+})$ (thick black dotted curve) as functions of $Q/M$, for increasing values of the source spin. The angular momentum of an orbiting test particle is shown in each region. Black regions are forbidden. In the region $r<r_{\mathrm{lsco}}$ ($r>r_{\mathrm{lsco}}$), circular orbits are unstable (stable). The curves $L=-L_{-}$ exist in a region located in $r<r_{-}$.

Figure 13

The effective potential $V/\mu$ for a neutral test particle with angular momentum $L=0$ in the field of a Kerr-Newman naked singularity with different values of $Q/M$ and $a/M$. See also Table . We also plot the radii ${\tilde{r}}_1$, ${\tilde{r}}_2$ and ${\tilde{r}}_3$ (dashed gray line).

Figure 14

Plots of the intrinsic KN angular momenta ${\tilde{a}}_{-}/M$ (gray curve), ${\tilde{a}}_{+}/M$ (black curve) and $a_i/M$ (dashed gray curve) as functions of $r/M$, for increasing values of the charge-to-mass ratio $Q/M$. The angular momentum of the orbiting test particle is shown in each region. Black regions are forbidden: circular motion is not possible there. Dotted lines represent the radii $r_{*}$, $r_{\gamma }^{\pm }$ and $r_y$.

Figure 15

Plots of the intrinsic KN angular momentum ${\tilde{a}}_{-}/M$ (gray curve), ${\tilde{a}}_{+}/M$ (black curve) and $a_i/M$ (dashed gray curve) as functions of $r/M$, for increasing values of the charge-to-mass ratio $Q/M=0.1$, 0.5, 0.866025. The region $a>a_s$ is explored: a naked singularity occurs for $Q<M$ and $a>a_s$. The angular momentum of the orbiting test particle is shown in each region. Black region $r<r_{*}$ is forbidden; no circular motion is possible there. The dotted-dashed line is the outer ergosphere $r_{ϵ}^{+}$.

Figure 16

The charge-to-mass ratios $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ and $Q_{c_2}$ as functions of $r/M$, for increasing values of the source spin. The angular momentum of the orbiting test particle is denoted in each region. Black regions are forbidden. The radius of the last stable circular motion $r_{\mathrm{lsco}}$ is plotted: $r_{\mathrm{lsco}}(L_{-})$ (thick black curve), $r_{\mathrm{lsco}}(-L_{-})$ (thick black dashed curve) and $r_{\mathrm{lsco}}(-L_{+})$ (thick black dotted curve). In the region $r<r_{\mathrm{lsco}}$ ($r>r_{\mathrm{lsco}}$), circular orbits are unstable (stable). A naked singularity occurs for $a>M$ and $Q>Q_s$; then, $Q_s=0.994987M$ for $a=0.1M$, $Q_s=0.707107M$ for $a=1/\sqrt{2}M$ and $Q_s=0$ for $a=M$. The dotted-dashed curve is the ergosphere boundary $r_{ϵ}$.

Figure 17

Left: The radii $r_{\mathrm{lsco}}^{+}\equiv r_{\mathrm{lsco}}(-L_{+})$, $r_{\mathrm{lsco}}^{-}\equiv r_{\mathrm{lsco}}(L_{-})$, and $r_{\mathrm{lsco}}^{-}\equiv r_{\mathrm{lsco}}(-L_{-})$ of the last stable circular orbit in the Kerr spacetime as functions of the source angular momentum $a$. The horizon $r_{+}$ and the radii $r_{\gamma }^{+}$ (dotted line), $r_d^{+}$ (dashed line), $r_3$ (black curve), and $r_4$ (gray curve) are also plotted. The black region ($r<r_{+}$) is forbidden. The value $a_c$, at the crossing point between $r_3$ and $r_{\mathrm{lsco}}^{-}$, is represented by a dashed line. The gray banned regions correspond to the naked singularity case (NS), the white ones to the black hole case (BH). Right: The last stable circular orbit radius $r_{\mathrm{lsco}}$ in the Reissner-Nordström spacetime for a black hole $Q\le M$ and a naked singularity $Q>M$ with different charge-to-mass ratios $Q/M$.

Figure 18

The case: $a=0.1M$. Properties of circular orbits in a KN spacetime, including black holes and naked singularities. The charge-to-mass ratios $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ and $Q_{c_2}$ are plotted as functions of $r/M$. The value of the angular momentum of the orbiting test particle is shown in each region. Black regions are forbidden; no circular motion is possible there. Dotted lines represent the radii ${\tilde{r}}_0$, $r_{{\mathrm{M}}_2}$, and $r_{*}$. Upper left shows the range $r/M\in [0,4]$. The remaining plots magnify the intervals where nontrivial behaviors are observed. The solid (dashed) white curve is the outer (inner) horizon $r_{+}$ ($r_{-}$). We denoted with ${\tilde{r}}_0^{-}$ the lowest value of $\tilde{r}(Q=0)$. The horizontal thick black line represents $Q=Q_s$. A naked singularity occurs for $a>M$ and $Q>Q_s$, i.e., $Q_s=0.994987M$ in this case. The black hole (BH) and naked singularity (NS) regions are denoted explicitly. The dotted-dashed curve represents the ergosphere boundaries $r_{ϵ}^{\pm }$.

Figure 19

The case: $a=0.8M$. Properties of circular orbits in a KN spacetime, including black holes and naked singularities. The charge-to-mass ratios $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ and $Q_{c_2}$ are plotted as functions of $r/M$. The angular momentum of the orbiting test particle is shown in each region. Black regions are forbidden; no circular motion is possible there. Dotted lines represent the radii ${\tilde{r}}_0$, $r_{{\mathrm{M}}_2}$, $r_{{\mathrm{M}}_3}$, $r_{*}$, $r_x$. Upper left shows the range $a/M\in [0,5]$. The remaining plots magnify the intervals where nontrivial behaviors are observed. The solid (dashed) white curve is the outer (inner) horizon $r_{+}$ ($r_{-}$). We denoted with ${\tilde{r}}_0^{-}$ the lowest value of $\tilde{r}(Q=0)$. The horizontal thick black line represents $Q=Q_s$. A naked singularity occurs for $a>M$ and $Q>Q_s$, i.e., $Q_s=0.6M$ in this case. The black hole (BH) and naked singularity (NS) regions are denoted explicitly. The dotted-dashed curve represents the ergosphere boundaries $r_{ϵ}^{\pm }$.

Figure 20

The case: $a=M$. Properties of circular orbits in a KN spacetime, including black holes and naked singularities. The charge-to-mass ratios $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ and $Q_{c_2}$ are plotted as functions of $r/M$. The dotted-dashed curve represents the ergosphere boundaries $r_{ϵ}^{\pm }$. The angular momentum of the orbiting test particle is shown in each region. Black regions are forbidden; no circular motion is possible there. Dotted lines represent the radii ${\tilde{r}}_0$, $r_{*}$, $r_x$. Upper left shows the range $a/M\in [0,6]$. The remaining plots magnify the intervals where nontrivial behaviors are observed. The solid (dashed) white curve is the outer (inner) horizon $r_{+}$ ($r_{-}$). We denoted with ${\tilde{r}}_0^{-}$ the lowest value of $\tilde{r}(Q=0)$. A naked singularity occurs for $a>M$ and $Q>Q_s$, i.e., $Q_s=M$ in this case.

Figure 21

The charge-to-mass ratio as a function of the radial coordinate for increasing values of the source angular momentum. The ratios $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ and $Q_{c_2}$ are also plotted. The angular momentum of the orbiting test particle is shown in each region. Black regions are forbidden; no circular motion is possible there. Dotted lines represent the radii ${\tilde{r}}_0$, $r_{*}$, and $r_x$. We denoted with ${\tilde{r}}_0^{-}$ the lowest value of $\tilde{r}(Q=0)$. The horizontal thick black line represents $Q=Q_s$. A naked singularity occurs for $a>M$ and $Q>Q_s$, i.e., $Q_s=0.994987M$ for $a=0.1M$, $Q_s=0.707107M$ for $a=1/\sqrt{2}M$, $Q_s=0.43589M$ for $a=0.9M$ and $Q_s=0$ for $a=M$. The black hole (BH) and naked singularity (NS) regions are denoted explicitly. The dotted-dashed curve represents the ergosphere boundaries $r_{ϵ}^{\pm }$.

Figure 22

The intrinsic KN angular momenta ${\tilde{a}}_{-}/M$ (gray curve), ${\tilde{a}}_{+}/M$ (black curve) and $a_i/M$ (dashed gray curve) as functions of $r/M$, for increasing values of the charge-to-mass ratio $Q/M$. The angular momentum of the orbiting test particle is shown in each region. Black regions are forbidden; no circular motion is possible there. Dotted lines represent the radii $r_{*}$, $r_{\gamma }^{\pm }$ and $r_z$. The dotted-dashed white curve is the outer horizon $r_{+}$. The dotted-dashed black curve represents the ergosphere boundaries $r_{ϵ}^{\pm }$. The thick black horizontal line is $a=a_s$; a naked singularity occurs for $Q<M$ and $a>a_s$. The black hole (BH) and the naked singularity (NS) regions are explicitly denoted. In this case, $a_s=0.994987M$ for $Q=0.1M$, $a_s=0.866025M$ for $Q=0.5M$, and $a_s=0.5$ for $Q=\sqrt{3}/2M$.

Figure 23

The charge-to-mass ratios $Q_{\mathrm{T}}^{\pm }/M$, $Q_{*}/M$ and $Q_{c_2}$ as functions of $r/M$, for increasing values of the source spin. The angular momentum of the orbiting test particles is shown in each region. Black regions are forbidden. The radius of the last stable circular motion $r_{\mathrm{lsco}}$ is plotted: $r_{\mathrm{lsco}}(L_{-})$ (thick black curve), $r_{\mathrm{lsco}}(-L_{-})$ (thick black dashed curve) and $r_{\mathrm{lsco}}(-L_{+})$ (thick black dotted curve). In the region $r<r_{\mathrm{lsco}}$ ($r>r_{\mathrm{lsco}}$), circular orbits are unstable (stable). The horizontal thick black line represents $Q=Q_s$. A naked singularity occurs for $a>M$ and $Q>Q_s$, i.e., $Q_s=0.994987M$ for $a=0.1M$, $Q_s=0.707107M$ for $a=1/\sqrt{2}M$, $Q_s=0.43589M$ for $a=0.9M$ and $Q_s=0$ for $a=M$. The black hole (BH) and naked singularity (NS) regions are denoted explicitly. The dotted-dashed curve represents the ergosphere boundaries $r_{ϵ}^{\pm }$.

Figure 24

Classification of KN compact objects according to the charge-to-mass ratio $Q/M$ and the spin $a/M$ of the source. Black holes can have regions of type $\mathbf{I}$ and $\mathbf{I}\mathbf{I}$ and only naked singularities can accommodate regions of all four types. The type NS-III (with $*$) has two forbidden orbital regions while NS-II has only one forbidden orbital region.

Figure 25

Left: Plot of $r_{*}/2$, $r_{*}$, $r_{\pi }$, ${\overline{r}}_{y_1}$, ${\overline{r}}_{y_2}$ and ${\overline{r}}_{y_3}$ as functions of $Q/M$. Right: Plot of $r_{{\rho }_{\pm }}$, ${\overline{r}}_{{\rho }_{\pm }}$, $r_{\varphi }$ and $r_p^{\pm }$ as functions of $Q/M$.

Figure 26

Circular orbits in Kerr-Newman naked singularity for $Q<M$ are explored. We plot the parameters $a_1$, $a_2$ and $a_0$ as functions of the charge $Q/M$.