Equatorial circular motion in Kerr spacetime

We analyze the properties of circular orbits of test particles on the equatorial plane of a rotating central mass whose gravitational field is described by the Kerr spacetime. For rotating black holes and naked singularities we explore all the spatial regions where circular orbits can exist and analyze the behavior of the energy and the angular momentum of the corresponding test particles. In particular, we find all the radii at which a test particle can have zero angular momentum due to the repulsive gravity effects generated by naked singularities. We classify all the stability zones of circular orbits. It is shown that the geometric structure of the stability zones of black holes is completely different from that of naked singularities.

Figure 1

The outer horizon $r_{+}$ (dashed curve), the inner horizon $r_{-}$ (dotted curve), and $r_a$ (solid thin black curve), $r_{c2}$ (solid gray curve), and $r_{\gamma }$ (solid thick black curve) are plotted as a function of the black hole intrinsic angular momentum $a/M$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 2

The energy $E_{-}^{(+)}\equiv E(L_{-})$ (upper plot) and the angular momentum $L_{-}$ (bottom plot) of circular orbits in a rotating Kerr black hole with angular momentum $0<a\le M$, for selected values of $a/M$ in the interval $r>r_{\gamma }$. For $a\ne M$ the energy $E_{-}^{(+)}$ is always positive and diverges as $r$ approaches $r_{\gamma }$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 3

The energy $E_{+}^{(-)}\equiv E(-L_{+})$ and angular momentum $L=-L_{+}$ of circular orbits in a Kerr black hole with angular momentum $0<a\le M$, for selected values of $a/M$ in the region $r>r_a$. The energy $E_{+}^{(-)}$ is always positive and diverges as $r$ approaches $r_a$.

Figure 4

The effective potential $V/\mu$ for a neutral particle of mass $\mu$ in a Kerr black hole with $a/M=0.5$ is plotted as a function of $r/M$ in the range [1.71, 10] for the radial coordinate and in the range $[-10,10]$ for the angular momentum $L/(\mu M)$. The outer horizon is situated at $r_{+}=(1+1/\sqrt{2})M$ (see text). Circular orbits exist for $r>2[1+\sin (\pi /18)]M\approx 2.347M$. The solid curve represents the location of circular orbits (stable and unstable). Stable (unstable) circular orbits are minima (maxima) of the effective potential function. The last stable circular orbits are represented by a point. The minima are located at $r=4.233M$ with $L=2.903/(M\mu )$ and $E=0.918\mu$, and at $r=7.554M$ with $L=-3.884/(M\mu )$ and $E=0.955\mu$.

Figure 5

The outer horizon $r_{+}$ (dashed curve), the inner horizon $r_{-}$ (dotted curve), and $r_a$ (solid thin black curve), $r_{c2}$ (solid gray curve), $r_{\gamma }$ (solid thick black curve), last stable circular orbits $r_{\mathrm{lsco}}^{+}$ (thick dash-dotted black curve) and $r_{\mathrm{lsco}}^{-}$ (thick dash-dotted gray curve) are plotted as functions of the black hole intrinsic angular momentum $a/M$. The curves $r_{\mathrm{lsco}}^{+}$ and $r_{\mathrm{lsco}}^{-}$ represent the radius of the last stable circular orbit for particles with angular momentum $-L_{+}$ and $L_{-}$, respectively. Circular orbits with $L=L_{-}$ exist for $r/M>r_{\gamma }$ and with $L=-L_{+}$ for $r>r_a$. The line $\tilde{a}\approx 0.638285M$ is also plotted. The thin dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 6

The energy $E_{\mathrm{lsco}}/\mu$ and the angular momentum $L_{\mathrm{lsco}}^{\pm }/(M\mu )$ of the last stable circular orbit as functions of the ratio $a/M\le 1$ of a Kerr black hole.

Figure 7

The energy $E/\mu$ (left plot) and the angular momentum $L/(\mu M)$ (right plot) of circular orbits in a Kerr black hole with $a=0.5M$ as functions of the radial coordinate $r/M$. The energy $E_{+}^{(-)}\equiv E(-L_{+})$ and the angular momentum $-L_{+}$ are represented by thick solid black curves, and the energy $E_{-}^{(+)}\equiv E(L_{-})$ and the angular momentum $L_{-}$ by thin solid black curves. In $r_{\gamma }<r<r_a$ there are unstable circular orbits with $L_{-}$. For $r_a<r<r_{\mathrm{lsco}}^{-}$ there are unstable circular orbits with $L_{-}$ and $-L_{+}$. For $r_{\mathrm{lsco}}^{-}<r<r_{\mathrm{lsco}}^{+}$ there are stable circular orbits with $L_{-}$ and unstable with $-L_{+}$. Finally for $r>r_{\mathrm{lsco}}^{+}$ there are stable circular orbits with $L_{-}$ and $-L_{+}$. The radii $r_{+}=1.70711M$, $r_{\gamma }=2.3473M$, $r_a=3.53209M$ and $r_{\mathrm{lsco}}^{-}=4.233M$, and $r_{\mathrm{lsco}}^{+}=7.55458M$ are also plotted. It is evident that $E(-L_{+})>E(L_{-})$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 8

The effective potential $V/\mu$ of a Kerr black hole with $a=0.5M$ as a function of $r/M$. The radii $r_{+}=1.70711M$, $r_{\gamma }=2.3473M$, $r_a=3.53209M$, $r_{\mathrm{lsco}}^{-}=4.233M$, and $r_{\mathrm{lsco}}^{+}=7.55458M$ are also plotted. The left upper plot shows the effective potential with orbital angular momentum $L=L_{-}=3.29806M\mu$ for which we find a minimum (stable orbit) at $r=7.85256M$ with energy $E_{-}/\mu =0.942949$ and a maximum (unstable orbit) at $r=3M$ with $E_{-}/\mu =0.979181$. The right upper plot corresponds to an effective potential with orbital angular momentum $L=L_{-}=2.90877\mu$ (solid black curve) and $L=-L_{+}=-6.45235M\mu$ (solid gray curve). For $L=L_{-}$ there is a minimum (stable orbit) at $r=4.49925M$ with $E_{-}=0.918487\mu$ and a maximum (unstable orbit) at $r=4M$ with $E_{-}=0.918559\mu$. For $L=-L_{+}$ there is a maximum (unstable orbit) at $r=4M$ with $E_{+}^{(-)}=1.23744\mu$. The bottom plot is for an effective potential with orbital angular momentum $L=L_{-}=4.09649\mu$ (solid black curve) and $L=-L_{+}=-4.36042M\mu$ (solid gray curve). For $L=L_{-}$ there is a minimum (stable orbit) at $r=14M$ with $E_{-}=0.96609\mu$ and a maximum (unstable orbit) at $r=2.65996M$ with $E_{-}=1.134\mu$. For $L=-L_{+}$ there is a maximum (unstable orbit) at $r=5.07411M$ with $E_{+}^{(-)}=0.991686\mu$ and a minimum (stable orbit) at $r=14M$ with $E_{+}^{(-)}=0.968052\mu$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 9

The energy $E/\mu$ (left plot) and the angular momentum $L/(\mu M)$ (right plot) of circular orbits in a Kerr black hole with $a=0.7M$ as functions of $r/M$. The energy $E_{+}^{-}\equiv E(-L_{+})$ and the angular momentum $-L_{+}$ are represented by thick solid black curves, and the energy $E_{-}^{+}\equiv E(L_{-})$ and the angular momentum $L_{-}$ by thin solid black curves. The radii $r_{+}=1.54772M$, $r_{\gamma }=2.01333M$, $r_a=3.72535M$, $r_{\mathrm{lsco}}^{-}=3.39313M$, and $r_{\mathrm{lsco}}^{+}=8.14297M$ are also plotted. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$. In $r_{\gamma }<r<r_{\mathrm{lsco}}^{-}$ there are unstable circular orbits with $L_{-}$; in $r_{\mathrm{lsco}}^{-}<r<r_a$ there are stable orbits with $L_{-}$; in $r_a<r<r_{\mathrm{lsco}}^{+}$ there are stable circular orbits with $L_{-}$ and unstable with $-L_{+}$; finally, for $r>r_{\mathrm{lsco}}^{+}$ there are stable circular orbits with $L_{-}$ and $-L_{+}$. It is clear that $E(-L_{+})>E(L_{-})$.

Figure 10

The effective potential $V/\mu$ of a Kerr black hole spacetime with $a=0.7M$ as function of $r/M$. The radii $r_{+}=1.54772M$, $r_{\gamma }=2.01333M$, $r_a=3.72535M$, $r_{\mathrm{lsco}}^{-}=3.39313M$, and $r_{\mathrm{lsco}}^{+}=8.14297M$ are plotted. The left upper plot represents the effective potential with orbital angular momentum $L=L_{-}=2.61948M\mu$ for which we find a minimum (stable orbit) at $r=3.9473M$ with $E_{-}=0.900551\mu$, and a maximum (unstable orbit) at $r=3M$ with $E_{-}=0.901712\mu$. The right upper plot shows an effective potential with orbital angular momentum $L=L_{-}=2.59216M\mu$ for which there exists a minimum (stable orbit) at $r=3.6M$ with $E_{-}=0.897167\mu$, and a maximum (unstable orbit) at $r=3M$ with $E_{-}=0.897167\mu$. The left bottom plot corresponds to effective potentials with orbital angular momenta $L=L_{-}=2.91563\mu$ (solid black curve) and $L=-L_{+}=-4.2694M\mu$ (solid gray curve). For $L=L_{-}$ there is a minimum (stable orbit) at $r=6.M$ with $E_{-}=0.925818\mu$, and a maximum (unstable orbit) at $r=2.50052M$ with $E_{-}=0.960213\mu$. For $L=-L_{+}$ there is a maximum (unstable orbit) at $r=6M$ with $E_{+}^{(-)}=0.973034\mu$. The right bottom plot is for effective potentials with orbital angular momenta $L=L_{-}=4.05058\mu$ (solid black curve) and $L=-L_{+}=-4.42036M\mu$ (solid gray curve). For $L=L_{-}$ there is a minimum (stable orbit) at $r=14M$ with $E_{-}=0.965775\mu$, and a maximum (unstable orbit) at $r=2.1819M$ with $E_{-}=1.23283\mu$. For $L=-L_{+}$ there is a maximum (unstable orbit) at $r=5.6208M$ with $E_{+}^{(-)}=0.98443\mu$, and a minimum (stable orbit) at $r=14M$ with $E_{+}^{(-)}=0.968527\mu$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 11

The energy $E/\mu$ (left plot) and the angular momentum $L/(\mu M)$ (right plot) of circular orbits in an extreme Kerr black hole ($a=M$) as functions of the radial coordinate $r/M$. The energy $E(-L_{+})$ and the angular momentum $-L_{+}$ are represented by thick solid black curves, and the energy $E(L_{-})$ and the angular momentum $L_{-}$ by thin solid black curves. The radii $r_{+}=r_{\gamma }=M$ (dashed curve) and $r_a=4M$ (solid black curve) are also plotted. There exist circular orbits with $L=L_{-}$ in $r>r_{\gamma }$. The energy $E(L_{-})$ is always positive and decreases as $r$ approaches $r_{\gamma }$. Circular orbits with $L=-L_{+}$ exist also in $r>r_{+}$. The energy $E(-L_{+})$ is always positive and increases as $r$ approaches $r_a$. It is evident that $E(-L_{+})>E(L_{-})$.

Figure 12

The effective potential $V/\mu$ of an extreme Kerr black hole for a test particle with a fixed orbital angular momentum as function of $r/M$. The radii $r_{+}=r_{\gamma }=M$ (dashed curve) and $r_a=4M$ (solid black curve) are plotted. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$. The left plot shows the effective potentials with orbital angular momenta $L=L_{-}$ (solid black curve) and $L=-L_{+}$ (solid gray curve). For $L=L_{-}$ there is a minimum (stable orbit) at $r=5M$ with $L_{-}=2.53075M\mu$ and $E_{-}=0.906154\mu$. For $L=-L_{+}$ there is a maximum (unstable orbit) at $r=5M$ with $-L_{+}=-5.79614M\mu$ and $E_{+}^{-}=1.08576\mu$. There exist circular orbits with $L=L_{-}$ in the region $r>r_{\gamma }$, and orbits with $L=-L_{+}$ in the region $r>r_{+}$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 13

Angular momentum and energy of test particles in a Kerr naked singularity with $a\ge (3\sqrt{3}/4)M$. The angular momentum$L=L_{-}$ (left plot) and the energy $E_{-}^{(+)}\equiv E(L_{-})$ (right plot) of circular orbits are plotted as functions of $r>0$ and $a\ge (3\sqrt{3}/4)M$. The particle’s energy is always positive. It is possible to note a region of minima for the energy corresponding to the minima of $L_{-}$.

Figure 14

Angular momentum and energy of test particles in a Kerr naked singularity with $a\ge (3\sqrt{3}/4)M$. The angular momentum $L=-L_{+}$ (left plot) and the energy $E_{+}^{(-)}\equiv E(-L_{+})$ (right plot) of circular orbits are plotted as functions of $r>r_a$ and $a\ge (3\sqrt{3}/4)M$. The particle’s energy is always positive. It is possible to note a region of minima for the energy corresponding to the minima of $-L_{+}$.

Figure 15

The graphic shows the radius $r_a/M$ (left plot) as a function of the intrinsic angular momentum parameter $a/M$ in the interval $(3\sqrt{3}/4,100)$, and the particle orbital angular momentum $L(r_a)/(\mu M)$ (right plot) as a function of $a/M$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 16

The energy of circular orbits in a Kerr naked singularity with source angular momentum $a\ge (3\sqrt{3}/4)M$ is plotted in terms of the radial coordinate $r/M$ for selected values of $a/M$. The left plot corresponds to particles with $L=L_{-}$, and the right plot is particles with $L=-L_{+}$. The energy is always positive and diverges as the limiting radius is approached. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 17

Radius of the last stable circular orbits for test particles in a Kerr naked singularity with $a\ge \frac{3\sqrt{3}}{4}M$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$. The radius ${\overline{r}}_{\mathrm{lsco}}$ ($r_{\mathrm{lsco}}^{+}$) is the limiting minimum radius of stability for particles with $L=L_{-}$ ($L=-L_{+}$).

Figure 18

Orbits’ stability in a Kerr naked singularity with $\frac{3\sqrt{3}}{4}M\le a\lesssim 9$. The different radii of the last stable circular orbits $r_{\mathrm{lsco}}$ are plotted in terms of the intrinsic angular momentum $a/M$.

Figure 19

The angular momentum and the energy of test particles in the field of a Kerr naked singularity with $a=3\sqrt{3}/4M\approx 1.30M$ are plotted as functions of the radial coordinate $r/M$. The dots represent the last stable circular orbits; numbers close to the points denote the energy $V/\mu$ of the last stable circular orbit. For $r>r_a\approx 4.2592M$ there exist circular orbits with angular momentum $L=-L_{+}$, and for all $r>0$ with $L=L_{-}$ (see text). For $r=0.75M$ the particle has $L=0$ and energy $E\approx 0.333M$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 20

The effective potential of a naked singularity with $a=\frac{3\sqrt{3}}{4}M$ for fixed values of the particle angular momentum $L/(M\mu )$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$. The radius $r_a$ is also plotted (see text). The dots denote the critical points of the potential. Numbers close to the dots denote the energy $V/\mu$ of the maxima and minima of the effective potential.

Figure 21

Orbits’ stability in a Kerr naked singularity with $a\gtrsim 9M$. The radii $r_{\mathrm{lsco}}$ of the last stable circular orbits are plotted as functions of the intrinsic angular momentum $a/M$. The radius $r=r_a$ is also plotted.

Figure 22

The angular momentum and the energy of circular orbits in a Kerr naked singularity with $a=2M$, as functions of the radial distance $r/M$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$. The dots denote the position of the last stable circular orbits, and the numbers close to the dots indicate the value of the corresponding energy $V/\mu$ or angular momentum $L/(M\mu )$. In the range $r>r_a\approx 4.822M$ there exist circular orbits with $L=-L_{+}$, and in $r>0$ with $L=L_{-}$ (see text).

Figure 23

The effective potential of a naked singularity with $a=2M$ for fixed values of the particle angular momentum $L/(M\mu )$. The radius $r_a$ is also plotted (see text). The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$. The dots represent the critical points of the potential. Numbers close to the dots indicate the energy $V/\mu$ of the maxima and minima of the effective potential.

Figure 24

Circular motion around a naked singularity with $M<a<(3\sqrt{3}/4)M$. The angular momentum $L=-L_{+}$ (left plot) and the energy $E_{+}^{(-)}\equiv E(-L_{+})$ (right plot) for circular orbits are plotted as functions of $a$ in the range $1<a/M<3\sqrt{3}/4$ and $r$ in the range $r>r_a$. The particle energy is always positive with a region of minima corresponding to the minima of $-L_{+}$.

Figure 25

Circular motion around a naked singularity with $M<a<(3\sqrt{3}/4)M$. The angular momentum $L=L_{-}$ (left plot) and the energy $E_{-}\equiv E(L_{-})$ (right plot) of circular orbits are plotted as functions of $a$ in the range $1<a/M<3\sqrt{3}/4$ and $r$ in the intervals $r>{\widehat{r}}_{+}$ and $0<r<{\widehat{r}}_{-}$. The region ${\widehat{r}}_{-}<r<{\widehat{r}}_{+}$ is represented as a dark region. As $r/M$ approaches the singularity, the particle energy and angular momentum diverge. As $r/M$ approaches ${\widehat{r}}_{-}$ from the left and ${\widehat{r}}_{+}$ from the right, the particle energy and angular momentum decrease.

Figure 26

Circular motion around a naked singularity with $M<a<(3\sqrt{3}/4)M$. The angular momentum $L=-L_{-}$ (left plot) and the energy $E_{-}^{-}\equiv E(-L_{-})$ (right plot) of circular orbits are plotted as functions of $a$ in the range $1<a/M<3\sqrt{3}/4$ and of $r$ in the interval ${\widehat{r}}_{-}<r<{\widehat{r}}_{+}$. The solid black curves represent the radii ${\widehat{r}}_{-}$ and ${\widehat{r}}_{+}$. The presence of negative values for the particle energy is evident.

Figure 27

The radii $r_a$ and ${\widehat{r}}_{\pm }$ are plotted as functions of $a/M$. Circular orbits with angular momentum $L=-L_{+}$ exist for $r>r_a$, with $L=L_{-}$ in $0<r<{\widehat{r}}_{-}$ and $r\ge {\widehat{r}}_{+}$, and with $L=-L_{-}$ in ${\widehat{r}}_{-}<r<{\widehat{r}}_{+}$ (see text). The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 28

The energy $E(L_{-})$ of test particles is plotted for selected values of $a$ in the range $M<a<(3\sqrt{3}/4)M$ and for $r>{\widehat{r}}_{+}$ and $0<r<{\widehat{r}}_{-}$.

Figure 29

The energy $E(-L_{+})$ is plotted for selected values of $a$ in the range $M<a<(3\sqrt{3}/4)M$ and for $r>r_a$. The energy $E(-L_{+})$ is always strictly positive and increases as the angular momentum $a/M$ increases.

Figure 30

The angular momentum $\overline{a}=-(r-2M)\sqrt{r/M}$ is plotted as a function of $r$. The energy vanishes, $E(-L_{-})=0$, for $a=\overline{a}$, and is negative, $E(-L_{-})<0$, for $1<a<\sqrt{32/27}M$ in the interval ${\overline{r}}_1<r<{\overline{r}}_2$. For $a>\sqrt{32/27}M$ the energy $E(-L_{-})$ is always strictly positive. For a naked singularity with momentum $a=1.02M$ the energy $E(-L_{-})=0$ at $r=0.41M$ and $r=0.96M$, and $E(-L_{-})<0$ in $0.41M<r<0.96M$. For $a=\sqrt{32/27}M$ the energy $E(-L_{-})=0$ at $r=2/3M$, whereas $E(-L_{-})>0$ for $a=1.1M$. In the bottom plot, the energy $E(-L_{-})$ is plotted for selected values of $a/M$ in the interval ${\widehat{r}}_{-}<r<{\widehat{r}}_{+}$.

Figure 31

The radii $r_{\mathrm{lsco}}^{\pm }$ of the last stable circular orbits are plotted as functions of the intrinsic angular momentum $a$ in the interval $M<a<\frac{3\sqrt{3}}{4}M$. The radii $r_a$ and ${\widehat{r}}_{\pm }$ are also plotted. The particle angular momentum $L_{\pm }$ is also denoted for some particular radii.

Figure 32

Stability of circular orbits in a Kerr naked singularity with $M<a<\frac{3\sqrt{3}}{4}M$. The radii $r_{\mathrm{lsco}}^{+}$ and ${\tilde{r}}_{\mathrm{lsco}}^{-}$ of the last stable circular orbits as functions of the ratio $a/M$. The special radii $r_a$ and ${\widehat{r}}_{\pm }$ are also plotted.

Figure 33

Angular momentum and energy of circular orbits in a Kerr naked singularity with $a=1.1M$. The dots denote the position of the last stable circular orbits, and the numbers close to the dots indicate the value of the energy $V/\mu$ or the angular momentum of the last stable circular orbits. In $r>r_a\approx 4.088M$, the particles have angular momentum $L=-L_{+}$; in $0<r<{\widehat{r}}_{-}\approx 0.378M$ and $r\ge {\widehat{r}}_{+}\approx 0.989M$, there exist particles with $L=L_{-}$; in ${\widehat{r}}_{-}<r<{\widehat{r}}_{+}$, there exist particles with $L=-L_{-}$.

Figure 34

The effective potential of a naked singularity with $a=1.1M$ for fixed values of the particle angular momentum $L/(M\mu )$. The radii $r_a$ and ${\widehat{r}}_{\pm }$ are also plotted. The dots denote the critical points of the potential. Numbers close to the dots indicate the energy $V/\mu$ of the maxima and minima of the effective potential. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 35

The effective potential of a Kerr naked singularity with angular momentum parameter $a=1.1M$ and $a=2M$ is plotted for the particle orbital angular momentum $L/(M\mu )=0$ as a function of the radius $r/M$. The radii $r_a$ and ${\widehat{r}}_{\pm }$ are also plotted for both cases (see text). The dots represent the critical points of the potential, and the numbers close to the dots indicate the energy $V/\mu$ of the maxima and minima of the effective potential. In the case $a=2M$ no extreme points are observed in the potential.

Figure 36

Location of particles with $L=0$ in a Kerr naked singularity with $1<a/M\le 3\sqrt{3}/4$. The picture plots the locus of the critical points of the effective potential $V/\mu$ with (particle) angular momentum $L/(M\mu )=0$. The radius of these “circular” orbits is plotted as a function of the source angular momentum $a/M$. Numbers close to the dots indicate the value of the energy $V/\mu$.

Figure 37

The energy particles with $L=0$ in a naked singularity with $1<a/M\le 3\sqrt{3}/4$. The orbits are located on the radii $r={\widehat{r}}_{+}$ (stable) and $r={\widehat{r}}_{-}$ (unstable). The energies $E({\widehat{r}}_{+})$ (solid black curve) and $E({\widehat{r}}_{-})$ (solid gray curve) are plotted as functions of the intrinsic angular momentum $a/M$. It is possible to see that $E({\widehat{r}}_{+})<E({\widehat{r}}_{-})$ for $1<a/M<3\sqrt{3}/4$, and $E({\widehat{r}}_{+})=E({\widehat{r}}_{-})$ for $a/M=3\sqrt{3}/4$.

Figure 38

The radii $r_{\mathrm{lsco}}$ of the turning points of the effective potential and the special radii $r_a$ and ${\widehat{r}}_{\pm }$ are plotted as functions of the ratio $a/M$ within the interval [1, 1.7]. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.

Figure 39

Behavior of the $E_{\mathrm{lsco}}/\mu$ and the angular momentum for the last stable circular orbits as functions of the intrinsic angular momentum of the naked singularity.

Figure 40

Arrangement of the radii determining the properties of circular orbits around a rotating central mass. The upper plot is for black holes and naked singularities with intrinsic angular momentum $a/M\in [0,1.7]$. The bottom plot is for rotating naked singularities with $a/M\in [1.7,14]$. The dash-dotted gray line represents the outer boundary of the ergosphere $r_{+}^0=2M$.