Spherical orbits around a Kerr black hole

Abstract

A special class of orbits known to exist around a Kerr black hole are spherical orbits—orbits with constant coordinate radii that are not necessarily confined to the equatorial plane. Spherical time-like orbits were first studied by Wilkins almost 50 years ago. In the present paper, we perform a systematic and thorough study of these orbits, encompassing and extending previous works on them. We first present simplified forms for the parameters of these orbits. The parameter space of these orbits is then analysed in detail; in particular, we delineate the boundaries between stable and unstable orbits, bound and unbound orbits, and prograde and retrograde orbits. Finally, we provide analytic solutions of the geodesic equations, and illustrate a few representative examples of these orbits.

Appendices

A Horizon-skimming orbits

In [14], Wilkins pointed out the existence of a class of so-called horizon-skimming orbits, which appear to lie on the event horizon of the extremal Kerr black hole with $a=M$. They arise by taking the $r\rightarrow r_1$ limit of the solution $(E_\text {a},\Phi _\text {a})$, and have the energy and angular momentum

$$\begin{aligned} E=\frac{\sqrt{M^2+Q}}{\sqrt{3}M},\qquad \Phi =2ME, \end{aligned}$$

(A.1)

where Q takes the range $0\le Q<\infty$. These orbits are represented by the black line on the left edge of the parameter space of Fig. 1. The limit $Q\rightarrow \infty$ corresponds to taking the null limit of these orbits [24].

The fact that the radii of these orbits coincide with that of the event horizon, is due to the well-known fact that the extremal Kerr black hole has an infinite throat in this region of the space-time [25]. Points along this throat share the same coordinate radius $r=M$, even though they might be (finitely or even infinitely) separated in space. Thus the horizon-skimming orbits remain above the event horizon; in fact, they also remain above the prograde circular photon orbit at $r_1$.

To resolve the throat region, we introduce a new parameter $\epsilon$ defined by

$$\begin{aligned} \epsilon =\sqrt{1-\frac{a^2}{M^2}}. \end{aligned}$$

(A.2)

The extremal limit is then taken as $\epsilon \rightarrow 0$. We would like to understand the region of the parameter space near $r_1$ as this limit is taken. In particular, we focus on the marginally stable and marginally bound orbits in this region. Our results are consistent with those recently obtained in [8].

Recall that marginally stable orbits are described by the curve $Q=Q_\text {ms}$. Substituting an expansion of the form $r=M+A\epsilon ^p+\cdots$ into the right-hand side of this equation, and requiring that the lowest-order term is zeroth order in $\epsilon$, implies that $p=2/3$. The coefficient A can then be determined in terms of Q, and we obtain [8]

$$\begin{aligned} r=M\Bigg [1+\bigg (\frac{M^2+Q}{M^2/2-Q}\bigg )^{1/3}\,\epsilon ^{2/3}\Bigg ]+O(\epsilon ^{4/3}). \end{aligned}$$

(A.3)

This parameterises the radii of these marginally stable orbits in terms of Q, which takes the range $0\le Q<M^2/2$. The energy and angular momentum of these orbits are given by

$$\begin{aligned} E&=\frac{\sqrt{M^2+Q}}{\sqrt{3}M}\Bigg [1+\bigg (\frac{\sqrt{2}(M^2-2Q)\epsilon }{M^2+Q}\bigg )^{2/3}\Bigg ]+O(\epsilon ^{4/3}), \end{aligned}$$

(A.4a)

$$\begin{aligned} \Phi&=2E+O(\epsilon ^{4/3}). \end{aligned}$$

(A.4b)

On the other hand, recall that marginally bound orbits are described by the curve $Q=Q_\text {mb}$. Again, substituting an expansion of the form $r=M+A\epsilon ^p+\cdots$ into the right-hand side of this equation, and requiring that the lowest-order term is zeroth order in $\epsilon$, implies that now $p=1$. The coefficient A can then be determined in terms of Q, and we obtain [8]

$$\begin{aligned} r=M\bigg (1+\frac{2M\epsilon }{\sqrt{2M^2-Q}}\bigg )+O(\epsilon ^2). \end{aligned}$$

(A.5)

This parameterises the radii of these marginally bound orbits in terms of Q, which takes the range $0\le Q<2M^2$. The angular momentum of these orbits is given by [8]¹⁰

$$\begin{aligned} \Phi =2M+\sqrt{2M^2-Q}\,\epsilon +O(\epsilon ^2). \end{aligned}$$

(A.6)

Fig. 4

The (r, Q) parameter space near $r_1$ when $a=0.999995M$ (corresponding to $\epsilon \simeq 0.003$). The blue and red curves are, as in Fig. 1, the $Q=Q_\text {ms}$ and $Q_\text {mb}$ curves, respectively (colour figure online)

The (r, Q) parameter space near $r_1$ is shown in Fig. 4 for the case when $a=0.999995M$, corresponding to $\epsilon \simeq 0.003$. The blue and red curves are, as in Fig. 1, the $Q=Q_\text {ms}$ and $Q_\text {mb}$ curves, respectively. They terminate on the r-axis at $r_\text {ms}$ and $r_\text {mb}$, respectively, if we borrow the notation of [25] in the equatorial limit. We would now like to understand what happens to this part of the parameter space, and in particular the two curves, when we take $\epsilon \rightarrow 0$.

We begin with the red curve corresponding to marginally bound orbits. We have seen that the part of this curve for which $0\le Q<2M^2$ is approximated by (A.5) when $\epsilon$ is small. In the limit $\epsilon \rightarrow 0$, this part of the red curve gets “flattened” onto the black line on the left edge of Fig. 1, between $Q=0$ and $2M^2$. Thus, we see that although the red curve appears to terminate at the non-zero value $Q=2M^2$ in Fig. 1, it actually continues down to $Q=0$ along the black line.

A similar situation happens for the blue curve corresponding to marginally stable orbits. The part of this curve for which $0\le Q<M^2/2$ is approximated by (A.3) when $\epsilon$ is small. In the limit $\epsilon \rightarrow 0$, this part of the curve gets “flattened” onto the same black line in Fig. 1, but now between $Q=0$ and $M^2/2$. Thus, the blue curve does not actually terminate at $Q=M^2/2$ in Fig. 1, but continues down to $Q=0$ along the black line.

It follows that the class of horizon-skimming orbits consists of at least a family of marginally bound orbits, and a family of marginally stable orbits, all sharing the same coordinate radius $r=M$. However, as mentioned above, these orbits are separated in space along the throat of the extremal Kerr black hole. In fact, it can be shown that the distance between $r_\text {mb}$ and $r_\text {ms}$ becomes infinite in the limit $\epsilon \rightarrow 0$ [25]. The distance between $r_\text {ms}$ and the far regions of the space-time also becomes infinite in this limit. This is a manifestation of the fact that the throat is divided into distinct regions, as depicted in Fig. 2 of [25] (see also Fig. 1 of [5]). The marginally bound orbits belong to one region (together with the photon orbit at $r_1$ and the event horizon itself), while the marginally stable orbits belong to another region. Other spherical orbits can also exist in these throat regions, and their locations relative to the marginally bound and marginally stable orbits are determined by the dependence of their radii on $\epsilon$. Geodesic motion in these throat regions have been the focus of recent attention in [5, 8].

B Spherical photon orbits

In this appendix, we provide analytic solutions of the geodesic equations for the spherical photon orbits found in [24]. Recall that the null case corresponds to setting $\mu =0$ in (2.4) and (2.5). This case can also be recovered from the time-like case $\mu =1$, by taking the limit $E\rightarrow \infty$ of the solution $(E_\text {a},\Phi _\text {a})$ when $r_1<r<r_2$. As mentioned in Sect. 4.4, the ratios $\Phi /E$ and $Q/E^2$ remain finite in this limit. If we redefine $\Phi /E\rightarrow \Phi$ and $Q/E^2\rightarrow Q$, we arrive at the solution that was obtained in [24]:

$$\begin{aligned} \Phi&=-\,\frac{r\Delta -M(r^2-a^2)}{a(r-M)}, \end{aligned}$$

(B.1a)

$$\begin{aligned} Q&=-\,\frac{r^3\** }{a^2(r-M)^2}. \end{aligned}$$

(B.1b)

With these values of $\Phi$ and Q, $w_{1,2}$ can be calculated using

$$\begin{aligned} w_{1,2}=\frac{1}{2a^2}\bigg [a^2-Q-\Phi ^2\pm \sqrt{\big (a^2-Q-\Phi ^2\big )^2+4a^2Q}\bigg ]. \end{aligned}$$

(B.2)

The coordinates $(u,\phi ,t)$ of the geodesic can then be expressed in terms of the Mino parameter $\lambda$ as

$$\begin{aligned} u&=\sqrt{-w_2}\,k{{\,\mathrm{sd}\,}}\big (a\sqrt{w_1-w_2}\,\lambda ,k\big ),\end{aligned}$$

(B.3a)

$$\begin{aligned} \phi&=\frac{\Phi }{a(1-w_2)\sqrt{w_1-w_2}}\big [F(\psi ,k)-w_2\Pi \big (\psi ,k^2(1-w_2),k\big )\big ]\nonumber \\&\quad +\frac{a}{\Delta }(2Mr-a\Phi )\lambda ,\end{aligned}$$

(B.3b)

$$\begin{aligned} t&=-\,\frac{a}{\sqrt{w_1-w_2}}\big [(1-w_2)F(\psi ,k)+w_2\Pi \big (\psi ,k^2,k\big )\big ]\nonumber \\&\quad +\frac{1}{\Delta }\big [(r^2+a^2)^2-2Mra\Phi \big ]\lambda , \end{aligned}$$

(B.3c)

where

$$\begin{aligned} \psi ={{\,\mathrm{am}\,}}\big (a\sqrt{w_1-w_2}\,\lambda ,k\big ), \end{aligned}$$

(B.4)

and k is given by (5.17b).

It follows that u is a periodic function of $\lambda$, with period

$$\begin{aligned} \Delta \lambda =\frac{4K(k)}{a\sqrt{w_1-w_2}}. \end{aligned}$$

(B.5)

The change in $\phi$ and t for one period $\Delta \lambda$ are

$$\begin{aligned} \Delta \phi&=\frac{4}{\sqrt{w_1-w_2}}\bigg [\frac{\Phi }{a(1-w_1)}\,\Pi \Big (-\frac{w_1}{1-w_1},k\Big )+\frac{2Mr-a\Phi }{\Delta }\,K(k)\bigg ],\end{aligned}$$

(B.6a)

$$\begin{aligned} \Delta t&=\frac{4}{\sqrt{w_1-w_2}}\bigg [-a\big ((1-w_2)K(k)+w_2\Pi \big (k^2,k\big )\big )\nonumber \\&\quad +\frac{(r^2+a^2)^2-2Mra\Phi }{a\Delta }\,K(k)\bigg ]. \end{aligned}$$

(B.6b)

We note that the result for $\Delta \phi$ agrees with that obtained in [24].