Bound orbits around charged black strings

Particle trajectories in black string can be classified in terms of orbital types and regions. Denoting $r_{EH}$ as the radius of outer event horizon, the three possible orbit types and their definitions are listed as below, in which their visualizations are shown in Fig. 1. Here we use similar convention as in Grunau Grunau:2013oca .

1. 1.

   Terminating orbit (TO): particles arrive at a periapsis $r_{p}$ and falls into singularity.

2. 2.

   Escape orbit (EO): particles come from $\infty$, approach a periapsis $r_{p}$, and return to $\infty$.

3. 3.

   Bound orbit (BO): particles oscillate between its periapsis and apoapsis under the condition of $r_{EH}<r_{p}<r_{a}<\infty$.

Based on the effective potential, the regions can be classified using the Hackman formalism Hackmann:2010tqa . Solving the Eq. (5) results in various scenarios based on the numbers of its positive real solutions, which are shown in Table 1. To help understanding the regions better, the dependence of the orbits on the energy levels is shown in Fig. 2.

A typical $V_{eff}$ we study here generically have three turning points, as we shall see later, only two of which are outside the horizon. The local maxima represent the unstable circular orbit (UCO)¹¹1For timelike case, this local maxima is called the marginally bound orbit (MBO) Ono:2015jqa ., while the local minima identify the stable circular orbit (SCO). Both circular orbits (CO) satisfy

which translates into

Both CO depend on the angular momenta and black hole’s charge. For a given black hole there exists a critical circular radius beyond which it no longer supports bound orbits. For timelike case this critical value is called the innermost stable circular orbit (ISCO), and this is achieved when UCO and SCO radii merge. Mathematically, this is the inflection point of the effective potential,

This adds the following condition,

into (III.1).

Here we briefly review the scheme of Levin and Perez-Giz in assigning each orbital solution to a distinct rational number Levin:2008mq . Any (positive) rational number can always be cast as

where $s\geq 0$ is an integer and $1\leq m\leq n-1$ are (relatively) prime numbers. This taxonomy draws correspondence between $q$ and the topological features of orbit. The claim is that each periodic orbit is completely characterized by three integers $(z,w,v)$ that can be expressed as rational number, Eq. 1. Each of them corresponds to a specific topological trait of a periodic orbit. The first and most tangible number is $z$, which is the number of leaves (or ”zooms”) of the orbit. The second integer, $w$, is the number of whirls which the particle performs in its path from apastron to periastron to the next apastron. Note that every particle performs at least a full $2\pi$ trip. The number of whirls is the additional integer number of full $2\pi$ accomplished beyond this. The last number $v$, the vertex number, differentiate between orbits that have equal $z$ and $w$ but are geometrically distinct. Particle can skip leaves in its motion from apastron to apastron. One important thing to address is the degeneracy that arises when the quotient $v=z$ is a reducible fraction. This is solved by requiring $v$ and $z$ to be relatively prime. The bounds on v then can be written as

The proof of such claim is based on the fact that for a periodic orbit, the accumulated angle between one apastron to another apastron can be expressed as

where $\Delta\varphi\equiv z\Delta\varphi_{r}$ is the total accumulated angle in a complete orbit. We may also define the radial period as the (affine) time taken by particle to return to the same radius upon starting at the apsis $r_{a}$ (where $\mathbb{E}^{2}=V_{eff}$) Levin:2009sk ; Babar:2017gsg . If within this period, the evolution of $\varphi$ is an integer multiple of $2\pi$, we have a periodic orbit. Therefore the accumulated radial angle can be written as

where $r_{p}$ is the periastron radius. The rational number in Eq.(13) can be related to the orbital angular frequency by the following arguments. Every eccentric equatorial orbit has two types of frequencies: the radial ($\omega_{r}$) and angular ($\omega_{\varphi}$) frequencies. They are given by

where $T_{r}$ is the period of one radial cycle. Any periodic orbit must then satisfy

The existence of CO and ISCO for timelike geodesic in $4d$ RN black hole has been widely discussed in Pugliese:2010ps ; Schroven:2020ltb and the references therein. As stated earlier, the condition for CO, given by Eq. (III.1), for the RN string translates into

where $r_{CO}$ is the circular radii. Simple algebra shows that those equations can be solved simultaneously to obtain one-parameter class of solutions parametrized by $r_{CO}$,

In the extremal case we have

When $J\rightarrow 0$ they reduce to the circular motion conditions for neutral test particles around $4d$ RN black hole Pugliese:2010ps .

For any given $L$ the ISCO is the smallest radius of circular orbit before the particle plunges into the black hole. It is the inflection point of $V_{eff}$. From condition (9), the additional equation to be solved is

Since the extra-dimensional signature always appears multiplying $L$, i.e., $(1+J)L$, then from the expression of $L$ in (IV.1) it is obvious that the ISCO radii satisfy

The same thing happens for the extremal black strings. The ISCO radii becomes

with solutions

the same as the extremal $4d$ RN ISCO; the timelike ISCO is oblivion to the existence of extra dimension (i.e., independent of $J$). Notice that this is precisely the same condition for ISCO in $4d$ RN black hole Pugliese:2010ps .

For null geodesic ($\epsilon=0$) it is customary to define impact parameter $b$, where $b^{2}\equiv 1/\mathbb{E}^{2}L$. The geodesic equation (5) becomes

where $j\equiv JL$ and we subsequently rescale the proper time $\tau\rightarrow\tau/L$. It is well-known that $4d$ RN spacetime does not support stable CO, except in the extremal limit. To be precise, for any given charge $Q/M$ the $\partial_{r}V_{eff}=0$ condition for null gives us a quadratic equation whose roots are

One can easily observe that $r^{-}_{CO}$ is at the local minima of $V_{eff}$ and is between the two horizons. Thus, the only observable CO, called the photon sphere, $r^{+}_{CO}$ is unstable Pradhan:2010ws ; Khoo:2016xqv . At $M=Q$, the CO radii shifts further into

the stable $r_{CO}$ coincides with the extremal horizon $r_{extr}$.

The situation is rather different in this $5d$ black string. The additional term that multiplies $j$ in $V_{eff}$ gives qubic equation for the $\partial_{r}V_{eff}=0$,

i.e., in general we have an additional local minima of $V_{eff}$. By adjusting $j$ we can have a stable photon sphere outside the outer horizon, $r_{ps}>r_{+}$. For extremal case, Eq. (28) has three roots

one of which lies on the extremal horizon, $r^{e}_{CO}=r_{extr}=M$. The other two ($r_{\pm}$) are real and located outside $r_{extr}$ provided the following condition is satisfied,

The $r_{CO}^{+}$ ($r_{CO}^{-}$) act as the stable (unstable) radii. As $j\rightarrow 0$ we have $r_{CO}^{\pm}\rightarrow\{2M,\infty\}$ and we recover the condition (27).

Studying the shape of $V_{eff}$ can only tell us to so much about the qualitative types of orbits. To know their specific shapes we must solve the geodesic equation. Using the black string metric above and setting $r=r(\phi)$, the geodesic equation (5) can be cast into

This can be simplified by rescaling $r\rightarrow r/M$, $Q\rightarrow Q/M$, $L\rightarrow L/M^{2}$, and defining:
$v\equiv L(\mathbb{E}^{2}-(J+\epsilon)),\ w\equiv 2L(J+\epsilon),\ x\equiv-LQ^{2}(J+\epsilon)-1,\ y\equiv 2,$ and $z\equiv-Q^{2},$ such that Eq. (31) now reads

Further simplification by expanding $r$ around (one of its zeros) $r=s+r_{0}$ and defining: $a_{0}\equiv v,\ a_{1}\equiv 4r_{0}v+w,\ a_{2}\equiv 6r_{0}^{2}v+3r_{0}w+x,$ and $a_{3}\equiv 4r_{0}^{3}v+3r_{0}^{2}w+2r_{0}x+y$, transform Eq. (32) into

The degree of the corresponding polynomial can be reduced by setting $r\equiv 1/u$, and further by $u\equiv 4y/a_{3}-a_{2}/3a_{3}$. We then obtain the Weierstrass form

where $g_{2}$ and $g_{3}$ are given as

Eq. (31) admits analytical solution in terms of Weierstrass $\wp$-function Hackmann:2010tqa :

where the initial angle $\phi_{in}$ is only dependent on $\phi_{0}$ and $y_{0}$:

Making use of (II) we have the equation for $\omega$ as

which gives the solution of $\omega$ in terms of $\phi$ as

Eq. (40) is later to play a significant role in enabling null bound orbit outside the string’s horizons.

As discussed above, the black string’s $V_{eff}$ is quartic in $r$, so in general it gives three local extrema; an additional local minimum outside the horizon. In this work we limit ourselves only to the case of physical orbits; that is, which lies outside the horizon and can thus be observable. We avoid discussing the tricky case of horizon-crossing orbits despite the premise of many-world orbits by Grunau and Kagramanova, in which the orbit is viewed as infinite continuity of the patches of the spacetime Grunau:2010gd . Armed with the aforementioned taxonomy in III.2, we carefully examine the bounded domain of region IV (Fig. 2) in RN black string potential and present the periodic orbits up to 3 zooms and 2 whirls. The whole existence of stable null orbits in our case is genuine. The $J$-factor leads to the existence of periodic orbits, as opposed to the $4d$ case where no such stable orbits present Misra:2010pu ; Pradhan:2010ws . The plots are shown in Fig. 5.

Levin in Levin:2009sk shows that, since rational numbers are discrete so are the energy levels that produce the orbits. The existence of bound orbits are bounded by UCO and SCO, so we can quantify the level energy by the deviation from them. We define the corresponding deviation $\eta$ as

In Table 2 we list the energy level of various periodic orbits according to their energy level, for massive particles, in both extremal and two-horizon RN black string. Similar table for light particles is shown in Table 3. The table shows increasing number of rational number as it goes down the list. One behavior that is observed in both cases is the standard deviations, while fluctuate, tends to decrease as energy level rises to UCO. If we calculate the average of deviation for timelike and null condition, we obtain the value $6.36\%$ and $6.94\%$, respectively. The reasoning of why we separate the table by horizon case becomes clear when we pay attention to each deviation average: $2.03\%$ for extremal case and $0.86\%$ for two horizon one, which is way smaller than the numbers earlier. This finding means that the distance between energy levels is more consistent in each specific horizon case.

For the timelike case we obtain a qualitatively similar pattern of orbits found in RN black hole ”zoo” (see Misra:2010pu ), as shown in Fig. 6. One might also notice that the whirl is stronger near the radius of UCO ($r_{UCO}$), which is the same behaviour observed in the $4d$ RN. The significant difference is that the test particle in RN black string requires higher energy for the orbit to exist ($\mathbb{E}\approx 2$). We can also calculate the accumulated radial angle of black strings’ rational orbit for any given $q$. The Eq. (14) reads

This equation can be integrated numerically. In Table 4 and 5 we show the accumulated angle for extremal and two-horizon RN black strings, respectively. For each case we compare their values with their corresponding $4d$ counterpart from Misra:2010pu . The accumulated radial angle for black strings is higher than for black holes.

Applying the condition (III.1) for the black string yields

One can easily verify that upon $J\rightarrow 0$ they reduce to the MBO conditions for $4d$ Bardeen black hole Gao:2020wjz . Solving them for $L$ and $\mathbb{E}^{2}$ yield

where $A\equiv\sqrt{Q^{2}+r_{CO}^{2}}$.

The ISCO condition gives the additional condition:

In the extremal case, $Q^{2}=16/27$, the condition becomes

which can be determined numerically.

Null geodesic in $4d$ Bardeen spacetime has been discussed widely, for example in Eiroa:2010wm ; He:2021htq . In $5d$ by defining $b^{2}$ and $j$, and by rescaling the proper time the effective potential can be written as

One can see that the appearance of $j$-term implies the existence of another turning point in $V_{eff}$ outside the horizon, as can be seen in Fig. 10. The CO radii are determined by the roots of the following equation

For the extremal case, the condition becomes

Both conditions are polynomial equations of degree six, and the roots can best be found numerically.

Unlike the RN case, we have been unable to cast Eq. (5) into a form that can be solved analytically,

To solve it numerically in Mathematica, it is better to rewrite it in terms of the $2^{nd}$-order ODE:

We obtain the numerical solutions of periodic orbits, both for timelike as well as null conditions. They are shown up to $q=1{2\over 3}$ in Figs. 11-12, respectively. For each case we are able to obtain the extremal and two-horizon solutions. What is interesting is that the orbits found in extremal and two-horizon case have almost identical form for each equivalent counterpart.

To create a more comprehensive view of periodic orbits in the nonsingular string, we provide similar tables as in IV.4. Tables 6 and 7 show various energy levels and each of their corresponding position in extremal and two-horizon cases, respectively. A quick calculation of average of standard deviation of each case would give $0.09\%$ for extremal and $0.07\%$ for two horizon one. This is way smaller than the ones found in RN black string, which means the prediction for the position of each periodic orbit will be more precise in the nonsingular black string.

Another point to be noted is that the average of deviation for timelike and null condition is $2.54\%$ and $2.45\%$, respectively. This is a similar manner found in RN black string: the distance between energy levels is more consistent when we compare each specific horizon case, which means this behavior exists generally on charged black holes. By carefully inspecting the numbers from both table, we also find that the orbits in two-horizon condition are more dense toward the local maximum (point of UCO). This behavior is observed in both timelike and null orbits. Nevertheless, one might notice that in the two-horizon nonsingular black string the shift is less drastic compared to similar behavior in two-horizon RN IV.4 where the orbits become highly condensed near the maximum.

As in the timelike RN, the accumulated radial angle can be calculated as follows:

We calculate numerically the accumulated angle and show it in Table 8. We compare the angles we calculate with the corresponding angle from the $4d$ counterpart. As in the RN case, the nonsingular black strings exhibits higher values for the accumulated angle. From the Table 8 the $4d$ angles reach maximum at $q=1{1\over 2}$, then decreases. the $5d$ counterpart, however, keeps increasing as $q$ increasing.