Circular orbits in the extreme Reissner-Nordstrøm dihole metric

We study the motion of neutral test particles in the gravitational field of two charged black holes described by the extreme Reissner-Nordstrøm dihole metric where the masses and charges of the black holes are chosen such that the gravitational attraction is compensated by the electrostatic repulsion. We investigate circular orbits in the equatorial plane between the two black holes with equal masses as well as the case of circular orbits outside this symmetry plane. We show that the first case reduces to an effective two-body problem with a behavior similar to a system described by the Reissner-Nordstrøm spacetime. The main focus is directed to the second case with circular orbits outside the equatorial plane.

Figure 1

Effective potential $V_{\mathrm{eff}}(z)$ from Eq. (12) for three different combinations of the mass parameters $M_1$ and $M_2$. The local maximum is always in the interval $(-1;+1)$ and marks the point of equilibrium $z_{\mathrm{equ}}$.

Figure 2

Point of equilibrium $z_{\mathrm{equ}}(q)$ from Eq. (14) and the singularity $z_{\mathrm{sing}}(q)$ from Eq. (34). It is $z_{\mathrm{equ}}(q=0)=+1$ and $z_{\mathrm{equ}}(q\to \infty )=-1$. The bigger one mass is than the other, the closer $z_{\mathrm{equ}}$ lies to that mass. The quantity $z_{\mathrm{sing}}$ has exactly opposite properties and becomes important in Sec. 4.

Figure 3

Effective potentials for a lightlike particle with different angular momenta $L_z$ for three different values of $M$. (a) $M=1$: There are no extremal points. (b) $M=\overline{M}\approx \sqrt{27/8}$: There is one photon orbit. (c) $M=2.5$: There are two photon orbits. The radius ${\rho }_{*}$ with $V_{\mathrm{eff}}({\rho }_{*})=V_{\mathrm{eff}}({\rho }_{\mathrm{p}1})$ and ${\rho }_{*}\ne {\rho }_{\mathrm{p}1}$ is also shown.

Figure 4

Photon orbits ${\rho }_{\mathrm{p}1/2}(M)$ in the equatorial plane. They do not exist for $M<\overline{M}$ and coincide for $M=\overline{M}$.

Figure 5

A light ray is emitted in the equatorial plane from ${\rho }_0$ at an angle $\xi$ and asymptotically approaches the photon orbit ${\rho }_{\mathrm{p}1}$.

Figure 6

The angle $\xi ({\rho }_0)$ from Eq. (26) in degrees for $M=\overline{M}=\sqrt{27/8}$ and $M=2.5$.

Figure 7

Effective potentials for a timelike particle in the case $M=1$ with different angular momenta $L_z$.

Figure 8

Effective potentials for a timelike particle in the case $M=1.5$ with different angular momenta $L_z$. In this and the following figures, the radii ${\rho }_{\mathrm{lsco}}$ and ${\rho }_{\mathrm{luco}}$ stand for the last radially stable and unstable circular orbits, and are obtained from Eq. (29). Further information on their nomenclature is in the text.

Figure 9

Effective potentials for a timelike particle in the case $M=\overline{M}=\sqrt{27/8}\approx 1.837$ with different angular momenta $L_z$.

Figure 10

Effective potentials for a timelike particle in the case $M=2.5$ with different angular momenta $L_z$.

Figure 11

The functions $E^2(\rho )/2$ and $L_z^2(\rho )/2$ for $M=1$ as an example of the interval $0<M\le M_{*}\approx 1.029$ [see Eq. (30)]. For all $L_z^2/2$, there is one circular orbit (see also Fig. 7). For $M=M_{*}$, there would be a reflection point in the diagrams. In this and the following figures, the stability refers to the radial direction.

Figure 12

The functions $E^2(\rho )/2$ and $L_z^2(\rho )/2$ for $M=1.5$ as an example of the interval $M_{*}<M<\overline{M}$. There can be one, two, or three circular orbits. The area for $L_z^2/2$ with three circular orbits is bounded (see also Fig. 8). These boundaries are given by the radii ${\rho }_{\mathrm{lsco}}$ and ${\rho }_{\mathrm{luco}}$, which follow from solving Eq. (29).

Figure 13

The functions $E^2(\rho )/2$ and $L_z^2(\rho )/2$ for $M=\overline{M}=\sqrt{27/8}\approx 1.837$. There can be one, two, or three circular orbits. The area for $L_z^2/2$ with three circular orbits no longer has an upper boundary (see also Fig. 9). The singularity is given by the two degenerate photon orbits ${\rho }_{\mathrm{p}1/2}$ and the local minimum by ${\rho }_{\mathrm{lsco}}$, which follows from solving Eq. (29).

Figure 14

The functions $E^2(\rho )/2$ and $L_z^2(\rho )/2$ for $M=2.5$ as an example of the interval $M>\overline{M}$. The behavior is qualitatively similar to the case $M=\overline{M}$. The corresponding effective potential is shown in Fig. 10. Between the two photon orbits ${\rho }_{\mathrm{p}1}$ and ${\rho }_{\mathrm{p}2}$, there cannot be any circular orbits. The local minimum is given by ${\rho }_{\mathrm{lsco}}$ and follows from solving Eq. (29).

Figure 15

The function $f(\eta )$ from Eq. (29) for different mass values. The number of roots depends on $M$.

Figure 16

Numerical solution of Eq. (29). Solutions exist for $M\ge M_{*}$. For $M\ge \overline{M}$, the dash-dotted branch would lie between the two photon orbits ${\rho }_{\mathrm{p}1/2}$ (see also Fig. 4) but would have no physical meaning.

Figure 17

Number of circular timelike orbits with dependence on the parameters $M$ and $L_z^2/2$. The different regions are colored uniformly. In parentheses, we indicate whether the orbits are radially stable (s) or unstable (u). The order is given by their appearance with increasing $\rho$ in the effective potential from Eq. (17).

Figure 18

Local velocity $\beta (\rho )$ on a circular orbit with radius $\rho$. This velocity vanishes for $\rho =0$, and becomes $\beta =1$ at the photon orbits ${\rho }_{\mathrm{p}1/2}$.

Figure 19

Radius $\rho (z)$ of a circular orbit at height $z$ above the equatorial plane for several mass ratios $q$; see Eq. (33). In the definition range $z\in (-1;+1)$, there is an additionally forbidden domain for $q\ne 1$ between $z_{\mathrm{equ}}$ and $z_{\mathrm{sing}}$.

Figure 20

The function $f_{\mathrm{e}}(z)$ from Eq. (36) for several masses $M$. The heights of the photon orbits are given by the roots. Their number depends on the concrete value of $M$.

Figure 21

Numerical analysis of Eq. (36). Shown are (a) the heights $z_{\mathrm{p}\pm }$ and the corresponding radii ${\rho }_{\mathrm{p}\pm }$ obtained from Eq. (33) depending on (b) the heights $z$ and (c) the mass $M$. Because of the equatorial symmetry, we have $z_{\mathrm{p}+}=-z_{\mathrm{p}-}$ and ${\rho }_{\mathrm{p}+}={\rho }_{\mathrm{p}-}$.

Figure 22

The function $f_{\mathrm{u}}(z)$ from Eq. (40) for several combinations of the mass values $M_1$ and $M_2$. The heights of the photon orbits are given by the roots. Their number varies between two and four.

Figure 23

Numerical analysis of Eq. (40) for fixed $M_2=1$ and variable $M_1$. Shown are (a) the heights $z_{\mathrm{p}1/2}$ and the corresponding radii ${\rho }_{\mathrm{p}1/2}$ obtained from Eq. (33) depending on (b) the heights $z$ and (c) the mass $M_1$. For all $M_1$, there are two photon orbits.

Figure 24

Numerical analysis of Eq. (40) for fixed $M_2=1.8$ and variable $M_1$. Shown are (a) the heights $z_{\mathrm{p}1/2/3/4}$ and the corresponding radii ${\rho }_{\mathrm{p}1/2/3/4}$ obtained from Eq. (33) depending on (b) the heights $z$ and (c) the mass $M_1$. There is a range of $M_1$ with four photon orbits (shaded domain).

Figure 25

Local velocity $\beta (z)$ from Eq. (42) for a circular orbit at height $z$. Shown are plots for equal masses $M≔M_1=M_2$. The domain with existing photon orbits is bounded by the photon orbits discussed in Sec. 4a.

Figure 26

Local velocity $\beta (z)$ from Eq. (42) for a circular orbit at height $z$. Shown are plots for several combinations of the masses $M_1$ and $M_2$. The domain with existing photon orbits is bounded by the photon orbits discussed in Sec. 4a and the characteristic heights $z_{\mathrm{equ}}$ and $z_{\mathrm{sing}}$.

Figure 27

Edge-on view of circular orbits for $M_1=M_2=1.5$. The photon orbits are located at $z_{\mathrm{p}\pm }\approx \pm 0.80324$ with ${\rho }_{\mathrm{p}\pm }\approx 0.95502$. In the plane $z=z_{\mathrm{sing}}=z_{\mathrm{equ}}=0$, there are no photon orbits but an arbitrary number of timelike orbits (compare with Secs. 3a, 3b).

Figure 28

Edge-on view of circular orbits for $M_1=2$ and $M_2=1.5$. The photon orbits are located at $z_{\mathrm{p}1}\approx 0.832490$ with ${\rho }_{\mathrm{p}1}\approx 0.80124$ and $z_{\mathrm{p}2}\approx 0.35591$ with ${\rho }_{\mathrm{p}2}\approx 1.89345$. The plane with $z=z_{\mathrm{sing}}\approx 0.14286$ marks the orbit with infinite radius and vanishing velocity. The point on the $z$ axis at $z=z_{\mathrm{equ}}\approx -0.07180$ is the circular orbit degenerated to a point.

Figure 29

Edge-on view of circular orbits for $M_1=2$ and $M_2=1.8$. There are four photon orbits with heights $z_{\mathrm{p}1}\approx -0.72907$, $z_{\mathrm{p}2}\approx 0.08139$, $z_{\mathrm{p}3}\approx 0.16681$, and $z_{\mathrm{p}4}\approx 0.55673$, and radii ${\rho }_{\mathrm{p}1}\approx 1.00515$, ${\rho }_{\mathrm{p}2}\approx 2.72863$, ${\rho }_{\mathrm{p}3}\approx 1.81819$, and ${\rho }_{\mathrm{p}4}\approx 1.31671$. The plane with $z=z_{\mathrm{sing}}\approx 0.05263$ marks the orbit with infinite radius and vanishing velocity. The point on the $z$ axis at $z=z_{\mathrm{equ}}\approx 0.02633$ is the circular orbit degenerated to a point. In this case, there are three $z$ domains, where timelike circular orbits are possible inside.

Figure 30

A neutral test particle with mass $m$ is attracted by the forces ${\mathit{F}}^{(1)}$ and ${\mathit{F}}^{(2)}$ of the two fixed masses $M_1$ and $M_2$. For describing a circular orbit, the $z$ components of the attracting forces have to compensate each other and the $\rho$ components have to act as centripetal force ${\mathit{F}}^Z=F_{\rho }^Z{\mathit{e}}_{\rho }$.

Figure 31

Classical velocity $v(z)$ from Eq. (b3) for a circular orbit at height $z$ for (a) equal masses $M≔M_1=M_2$ and (b)–(d) several combinations of $M_1$ and $M_2$ with $M_1\ne M_2$. As in the relativistic case, the domain between the characteristic heights $z_{\mathrm{equ}}$ and $z_{\mathrm{sing}}$ is forbidden. In (b)–(d), the ratios $M_1/M_2$ correspond to those from Fig. 26. The speed of light ($v=1$) is also shown, but represents, in this classical case, no upper boundary of the velocity.