Aschenbach effect and circular orbits in static and spherically symmetric black hole backgrounds

Next, let us focus on the TCO, which requires

where the prime denotes the derivative with respect to $r$. For a given radius of an existing TCO, the corresponding energy and angular momentum can be solved from (11) Cardoso

Due to the spherically symmetry, the angular momentum of a TCO is symmetric under $l\leftrightarrow-l$, so we only concern the positive one. Moreover, a stable or unstable TCO corresponds to positive or negative $V_{\text{eff}}^{\prime\prime}$, where

The existence of TCOs indicates that the denominator in the above expression (14) is positive. As a result, one has

where $\mathcal{C}=3ff^{\prime}-2rf^{\prime 2}+rff^{\prime\prime}$. Meanwhile, for the marginally stable circular orbit, $V_{\text{eff}}^{\prime\prime}=0$ gives

Such particular marginally stable circular orbit denotes the stable circular orbit with the smallest radius that can be continuously connected to spatial infinity by a set of stable TCOs Delgado . In general, it coincides with the innermost stable circular orbit (ISCO), while if novel branch of TCO emerges near the black hole horizon, they will be different. In what follows, we would like to address some properties of TCOs by employing Eqs. (13) and (12).

The first property is that the energy and angular momentum diverge at the radius of the photon sphere (PS). Starting with $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$ and performing the above similar calculation, one easily finds that the radius of the photon sphere should satisfies $2f-rf^{\prime}=0$, which indicates that the denominators of $l$ and $E$ vanish. So this property is obvious.

Second, the radius of a TCO is limited in several separated ranges in the radial distance. Positive energy (12) requires $f>0$ and $2f-rf^{\prime}>0$. The former one is naturally satisfied outside the black hole horizon. Note that $f^{\prime}<0$ shall lead to imaginary angular momentum for positive energy case. Thus a real $l$ requires both $f^{\prime}>0$ and $2f-rf^{\prime}>0$. Combining with them, we find that the radius of a TCO must simultaneously satisfy $f^{\prime}>0$ and $2f-rf^{\prime}>0$, which limits the existence of TCOs in several possible separated ranges in $r$.

It was pointed out in Refs. Weia ; Weib that, for a fixed angular momentum $l$, the stable and unstable TCOs always come in pairs both in rotating and nonrotating black hole backgrounds if they exist. Moreover, the topological argument also suggests that the outermost and innermost ones are radial stable or unstable, respectively. In particular, for each pair TCOs, the stable one and unstable one can be connected or separated. For example, the stable and unstable TCOs for the Schwarzschild black hole is connected by the ISCO. However, if $f^{\prime}<0$ at such points determined by $V_{\text{eff}}=V_{\text{eff}}^{\prime}=V_{\text{eff}}^{\prime\prime}=0$, this pair of TCOs will be separated. Note that different pairs of TCOs are separated as expected from the second property referred above due to the existence of multiple photon spheres.

There is another particular TCO, named as the static point Weic with vanished angular momentum or angular velocity. Obviously, it requires $f^{\prime}=0$, and the corresponding energy is $E=\sqrt{f}$ Weic ; Lehebel .

One also worths noting that the energy and angular momentum of a TCO are unbounded regardless that the energy is bounded below at radial infinity. Moreover, previous study indicates that the outer pair of stable and unstable TCOs is connected, but the inner pair can be either connected or disconnected. For spinning black holes, we refer to the work in Refs. Delgado ; Lehebel ; Collodel , where the corresponding properties of TCOs and static rings were discussed.

Now, let us turn to the Aschenbach effect. In Ref. Aschenbach , Aschenbach found for the first time that, when the Kerr black hole spin is beyond $a/M=0.9953$, the angular velocity of the TCO has a nonmonotonic behavior for a zero-angular-momentum observer or Bardeen observer, while such behavior is absent for a distant observer. Up to now, this Aschenbach effect has not been observed for a nonrotating black hole. In this section, we aim to analyze the conditions of the Aschenbach effect and show that it can exist even for a distant observer in a nonrotating black hole background.

The angular velocity of a TCO for a distant observer can be obtained by inserting the angular momentum (12) and energy (13) into Eq. (10):

As we have shown above, $f^{\prime}=0$ is the condition for a static point. If this case occurs, one can see that $\Omega_{CO}$ vanishes at a finite distance. Further considering that it approaches $0^{+}$ at infinity, one can easily conclude that there must exist at least one extremal point of $\Omega_{CO}$, and a branch of TCOs with positive radial gradient of $\Omega_{CO}$ should emerge. A simple algebraic calculation gives

So the extremal point is at $rf^{\prime\prime}-f^{\prime}=0$. The radial range with monotonically increasing angular velocity should have

On the other hand, at some certain values of $r$, the TCOs are absent, so we need to check the TCO can exist even when condition (20) is required. Here, we reformulate the quantity $\mathcal{C}$

If the TCO exists, the first term always positive as mentioned above. Thus the condition (20) leads to $\mathcal{C}>0$. This result states that the Aschenbach effect only occurs in stable TCO branch. While the monotonically decreasing $\Omega_{CO}$ can be both in the stable and unstable TCO branches.

Here we would like to investigate whether the presence of multiple photon spheres is an indicator of the Aschenbach effect. It has been verified that there exists at least one unstable photon sphere or light ring in asymptotically flat spacetime Cunha2020 ; Weid . If multiple photon spheres appears, there must be one stable one. For the convenience of discussion, we define a function

Obviously, $b$ vanishes for the photon spheres. Moreover, we have $b^{\prime}>0$ ($<0$) for the unstable (stable) photon spheres. Let mainly focus on the stable one, the following relation must be hold near it

or

which is exactly the requirement (20). Therefore, near the stable photon spheres, $\Omega_{CO}^{\prime}$ could be positive and indicates the existence of the Aschenbach effect.

Another thing worths pointing out is that $\Omega_{CO}$ always maintains finite values at the location of the photon spheres regardless of the infinite energy and angular momentum.

Since the Schwarzschild black hole is the most known black hole, we now take it as the first example. The line element can be described in the form (1), while with the metric function given by

where $M$ is the black hole mass. The corresponding effective potential will be immediately obtained

For the TCO, we can express its radius in term of the energy or angular momentum. Alternatively, the angular momentum and energy can be expressed as

In order to guarantee a positive energy, we easily see that $r_{CO}>r_{PS}=3M$, where $r_{PS}$ is the radius of the photon sphere. Furthermore, by solving the condition (11), one shall find the ISCO orbit locating at $r_{ISCO}=6M$. Thus the radius of the unstable TCOs is in (3$M$, 6$M$), and the stable ones in (6$M$, $\infty$). Obviously, they are connected by the ISCO. On the other hand, the angular velocity of the massive particle along the TCO will be

Obviously, with the increase of $r_{CO}$, $\Omega_{CO}$ decreases. For clarity, we show the angular velocity $\Omega_{CO}$ in Fig. 1. The unstable and stable TCO regions are marked in light blue and red colors. The angular velocity $\Omega_{CO}$ in both the regions is a monotonically decreasing function. This indicates that the Aschenbach effect does not exist for the Schwarzschild black hole.

When an electric field is included in, one will obtain the charged Reissner-Nordström (RN) black hole black hole solution by solving the Einstein field equation. The line element can still be described by the form (1), and the metric function reads

where $M$ and $Q$ are the mass and charge of the black hole, respectively. The event horizon locates at $r_{h}=M+\sqrt{M^{2}-Q^{2}}$ obtained by solving $f(r_{h})=0$. It is easy to find when the charge $Q$ tends to $M$, there only leaves an extremal RN black hole. And a naked singularity presents for $Q>M$. Therefore, we require $0\leq Q/M\leq 1$ for the RN black hole.

Following (18), the angular velocity is

When the charge $Q$ vanishes, the result of the Schwarzschild black hole will be recovered. However, for nonvanishing charge $Q$, $\Omega_{CO}$ has a chance to exhibit a non-monotonic behavior. Such behavior is determined by the extremal point satisfying $\partial_{r}\Omega_{CO}=0$, which gives

Therefore, $\Omega_{CO}^{\prime}$ crosses zero at $r_{*}$, and behaves differently. As the first condition, we require that $r_{*}$ should larger than the radius of the event horizon, i.e., $r_{*}\geq r_{h}$, leading to

This can also be observed in Fig. 2, where $r_{*}$ and $r_{h}$ are described in purple and black colors, respectively. For small $Q/M$, $r_{*}$ is below $r_{h}$, while this result reverses when $Q/M$ is beyond $\sqrt{15}/4$. As another requirement, the stable or unstable TCOs should be oustside the photon sphere even when the condition (33) is satisfied. We also plot the radii $r_{PS}$ and $r_{ISCO}$ in Fig. 2. These TCOs with $r_{PS}<r_{CO}<r_{ISCO}$ (light blue regions) are unstable while these with $r_{ISCO}<r_{CO}<\infty$ (light red regions) are stable. Here, we list the expressions of these two radii

From Fig. 2, we observe that $r_{*}$ is always smaller than $r_{PS}$, and out of the light blue and red regions. So, the Aschenbach effect will not appear even for a large black hole charge.

For the sake of clarity, we show the angular velocity in Fig. 3 for $Q/M$=0.9 and 0.98, respectively. When $Q/M$=0.9, we find that $\Omega_{CO}$ decreases with $r$. Even in the range ($r_{h}$, $r_{PS}$), where the TCO does not exists, $\Omega_{CO}$ also is a monotonically decreasing function. However, for $Q/M$=0.98, there is a distinct difference. Despite the monotonically decreasing behavior shown in both the stable and unstable TCO regions, there is a monotonic increasing behavior quite near the horizon, ($1.20M$, $1.28M$), which indicates there might exist the Aschenbach effect. Nevertheless, no TCO can survive in this region. Therefore, the Aschenbach effect is still absent. Moreover, a simple algebra shows that, for the extremal black hole with $Q/M=1$, we have $r_{PS}=2M>r_{*}=4M/3$. This furhter implies that the Aschenbach effect is absent for the RN black hole. Meanwhile, there is only one pair of TCOs, and the stable and unstable ones are connected.

As shown above, both the Schwarzschild and RN black holes do not exhibit the Aschenbach effect. Here as the third characteristic example, we would like to consider the dyonic black hole given in Ref. Liu2020 . Previous work has shown that there may exist intriguing properties for the motion of particles. There allows the existence of three photon spheres Liu2020 , and static point orbits Weic . This, in some certain, indicates the Aschenbach effect in such black hole background.

To illustrate such result, let us consider the dyonic black hole. The action includes a quasi-topological electromagnetic action term Liu2020

where the field strength is $F^{2}=F^{\mu\nu}F_{\mu\nu}$ and $F^{(4)}=F^{\mu}_{\;\;\nu}F^{\nu}_{\;\;\rho}F^{\rho}_{\;\;\sigma}F^{\sigma}_{\;\;\mu}$. The coupling parameters $\alpha_{1}$ and $\alpha_{2}$ are for the standard Maxwell and quasi-topological electromagnetic actions, respectively. When $\alpha_{1}=1$ and $\alpha_{2}=0$, it recovers the standard Maxwell theory. In particular, under the ansatz of global polarization, this quasi-topological term has no influence on the Maxwell equation and the energy-momentum tensor.

The static spherically symmetric black hole solution to the field equations following from (36) can also described by the line element (1). The corresponding metric function is given by Liu2020

with $~{}_{2}F_{1}$ the hypergeometric function. The electric and magnetic charges are $q$ and $p/\alpha_{1}$, respectively.

Aiming at uncovering the Aschenbach effect, we take $\alpha_{1}=1,\;\alpha_{2}/M^{2}=2.76,\;p/M=0.14$. Further taking $q/M$=1, we plot the angular velocity in Fig. 4. For this case, we have $r_{h}=1.09M$, $r_{PS}=1.97M$, and $r_{ISCO}=3.94M$. In the range ($r_{h}$, $r_{PS}$), no any TCO can exist. Both in the connected unstable and stable TCO regions, $\Omega_{CO}$ decreases monotonically. Thus the Aschenbach effect is absent for this case.

In Ref. Weic , we find that there shall be a pair static point orbits when the black hole charge $q/M\in$ (1.0143, 1.0305). As we shown above, the presence of the static point orbit must be an indictor of the Aschenbach effect. For the purpose, we take $q/M$=1.02 as an example. In previous examples, one easily finds that there exists only one pair unstable and stable regions. However, for this case, beside the outer pair, a novel inner pair near the horizon emerges. These two pairs of TCOs are separated. Moreover, we observe from Fig. 5 that the outer stable and unstable TCO regions are connected, but for the narrow inner pair, the stable and unstable TCO regions are separated. The main reason is that there exists a region where these TCOs have complex angular momentum and thus are excluded. Therefore, the stable and unstable TCO regions are separated as expected. This is also a novel phenomenon that not observed before.

We display the angular velocity $\Omega_{CO}$ in Fig. 5. These regions marked in light blue and red colors are for the unstable and stable TCO regions, respectively. The $\Omega_{CO}$ is described with blue dashed curve in unstable TCO regions, and red solid curve in stable TCO regions. In these two unstable TCO regions and outer stable TCO region, $\Omega_{CO}$ decreases, while in the stable TCO region (0.90$M$, 0.93$M$), $\Omega_{CO}$ starts at zero and ends at a finite value. This is clearly an increasing behavior, and thus the Aschenbach effect indeed exists to a distant observer in the static, spherically symmetric black hole background. Furthermore, we have two unstable photon spheres locate at $r=0.36M$ and 1.88 $M$, and one stable photon sphere at $r$=0.93$M$. The ISCO is at $r=0.90M$.

In order to examine the effect of the black hole charge on the Aschenbach effect, we show the allowed range of radial coordinate as a function of the charge in Fig. 6. When the charge satisfies $q/M\geq 1.0143$, the Aschenbach effect appears. With the increase of the charge, the range of the radial coordinate expands until the maximal charge $q/M=1.0305$ is reached, and beyond which, there only exists the naked singularity.