Topology of equatorial timelike circular orbits around stationary black holes

Topology provides a powerful tool to understand the light rings (LRs) around compact objects and black holes. It was shown that LRs can be well-explored by calculating the topological charge while ignoring where the LRs are located Cunhaa ; Cunhab . Due to the fact that the LRs are closely related to the ringdown stage of a black hole merger and formation of black hole shadow, further study of the topology approach may peek into the modern observations of the LIGO/Virgo Collaboration Abbott and EHT Collaboration Akiyama1 ; Akiyama2 .

In 2017, Cunha, Berti, and Herdeiro Cunhaa constructed a special vector by using the null geodesics and found that the LRs are exactly at the zero points of the vector. Then in the vector space, the winding number of a zero point can provide us with some interesting topological results. Meanwhile, summing the winding numbers of all the zero points, one can obtain the topological charge for the LRs. Via such a topological approach, they originally proposed that under a stationary and axisymmetric equilibrium ultracompact object, the topological charge of the LRs always vanishes. This indicates that if the LRs exist, they must come in pairs for such ultracompact objects. Moreover, if one of the LRs is stable, the other one must be unstable Crispino ; Hoda .

Shortly afterwards, the study was generalized to the black hole backgrounds Cunhab . The result suggests that for a stationary, axisymmetric, asymptotically flat black hole spacetime in four dimensions, it possesses at least one standard LR outside the horizon for each rotation sense. Other related issues including the stability of angular motion were discussed in Refs. Guo ; Junior ; Ghosh . In particular, it was found that there exists a topological phase transition for the dilatonic Melvin spacetime and Schwarzschilld-dilatonic-Melvin black hole at a certain value of the dilatonic parameter JuniorYang .

Combining this with Duan’s topological current $\phi$-mapping theory Duan , we also applied the study to the nonrotating black hole case Wei2019 . The inner structure of the topological charge, as well as the bifurcation phenomena was investigated. For the Dyonic black hole solution Lu , the previous study showed that there exist several photon spheres (PSs), which are the counterpart of LRs in spherically symmetric spacetime. This provides us with a good opportunity to test the nontrivial topological argument. After calculating the topological charge, we found that its value is always -1 independent of other parameters. This implies that the standard PSs are always one more than the exotic ones. So the topological approach holds even when multi PSs are presented. The black hole spin will also not affect the value of the topological charge.

Inspiring by the study, we proposed a topological approach for the black hole thermodynamics Wei2022 . Starting with the first law of black hole thermodynamics, we constructed a vector. The critical point of the phase transition was found to exactly locate at the zero point of the vector. By making use of the Duan’s topological current $\phi$-mapping theory, we endowed each critical point with a winding number. Summing the winding numbers for all the critical points, we shall obtain the topological charge, which remains unchanged for the system. Unexpectedly, we found that besides the conventional critical point of negative winding number, there is another novel critical point with positive winding number. Combining with the black hole thermodynamics, we pointed out that only the conventional critical point acts as an indicator of the first order black hole phase transition. Such a topological study was also extended to the Gauss-Bonnet black hole Yerra , where three critical points appear. Nevertheless, the total topological charge remains unchanged with the black hole electric charge.

Considering that the black hole solutions with the critical points are very limited, we introduced another topological approach by the free energy of the black hole system Weito . Each actual black hole solution corresponds to the zero point of a constructed vector. Thus, it will be endowed with a winding number in the vector space. Taking the inverse Hawking temperature as the control parameter, the total topological number is universal and independent of the black hole temperature and other parameters. By taking advantage of such topological properties, different black hole systems are divided into different topological classes. For example, the Schwarzschild, Reissner-Nordström, and Reissner-Nordström-AdS black holes, respectively, have topological number -1, 0, 1, which indicates they belong in different topological classes. All these results indicate that the topological approach is a promising way to study the LRs and thermodynamic properties of the black hole systems.

Comparing with the photons, the massive particles can also go around the black hole along the timelike circular orbits (TCOs). In certain astronomical processes, such orbits serve as the endpoints of dynamical evolution due to the dissipative effects Peters ; Rodrigo . In particular, there is the innermost stable circular orbit (ISCO), below which the massive particles shall quickly plunge into the black hole. Therefore, the stable accretion disk can only be formed above this bound, and provide a light source to illuminate the black hole, which may cast potential features of the black hole shadow in the observation of EHT Akiyama1 ; Akiyama2 . For a particle falls towards the black hole at a large radius, it will pass through a series of TCOs by losing angular momentum, and converting its energy into radiation, and finally reaches the ISCO before it plunges into the black hole. In the extremal Kerr black hole background, a particle could convert its 42% energy, or achieve a higher efficiency in compact bosonic objects Delgado . Via this process, which leads to the possible fact that black holes could be the powerful luminosity sources to trigger the energetic astronomical phenomena in strong gravity regime.

On the other hand, very recently, the general properties of the TCOs have been explored in Refs. Delgado ; Cardoso . It was found that the stable and unstable LRs delimit the region of stable and unstable TCOs. However, the topological approach for the TCOs remains to be established despite the great success gained for the LRs. In the backgrounds of compact objects or black holes, the locations of LRs are only dependent of the parameters of the compact objects or black holes, while independent of the energy and angular momentum of the photon. This states that the LRs are intrinsic properties of the spacetime. However, not all the photons can surround the black hole along these LRs. Considering that these photons come from far away from the black hole, some of them will fall into and some of them will fly past it. Only those with appropriate energy and angular momentum will surround the black hole along the LRs. Moreover, if the background is fixed, the relation of the energy and angular momentum of the photon is linear.

Now we wonder whether a massive test particle with certain energy and angular momentum can feel the TCOs. Similar to the photon, there must be some conditions. As is well-known, the radius of the TCO depends both on the black hole parameters and the massive particles themselves. For a given TCO, the relation of the energy and angular momentum of these particles orbiting along the black hole is not linear anymore. Due to these facts, it seems that we cannot construct a well-behaved topological approach for the TCOs as expected, or it is impossible to obtain the corresponding topological information of the TCOs.

However, it is worthwhile pointing out that the greatest advantage of the topological approach is that it ignores the specific details, such as the relation of the energy and angular momentum, as well as the locations of the TCOs. Keeping this in mind, in order to establish a well-behaved topological approach, we just require that, in certain parameter regions, the asymptotic behavior of the constructed vector does not change. Motivated by this idea, we manage to construct a topological approach for the TCOs for the first time. We shall show that when taking the angular momentum as the control parameter, the energy of the massive particle does not change the asymptotic behavior of the considered vector, indicating a well-behaved topology. In such approach, the marginally stable circular orbit (MSCO) acts as a bifurcation point of the zero points of the vector. More importantly, we find that the TCOs with the same angular momentum (or energy less than one) always come in pairs for the Kerr-like black holes.

The present paper is organized as follows. In Sec. II, we start with the geodesics and construct a vector with its zero points corresponding to the TCOs. The asymptotic behavior of the vector is examined when we take the angular momentum as the control parameter, which means that the TCOs evolve with the angular momentum. In Sec. III, we calculate the global and local topological charges for the TCOs. Meanwhile, the topological property of the MSCO is also discussed. We then apply this approach to the Kerr black hole in Sec. IV as an example. Adopting energy as the control parameter, we investigate the corresponding topology in Sec. V. Finally, we summarize and discuss our results in Sec. VI.

We consider a four-dimensional stationary, axisymmetric, asymptotically-flat black hole spacetime possessing two Killing vectors, $\xi^{\mu}=(\partial_{t})^{\mu}$ and $\psi^{\mu}=(\partial_{\varphi})^{\mu}$. Moreover, we assume the metric is circular, i.e., $\xi^{\mu}R_{\mu}^{[\nu}\xi^{\rho}\psi^{\sigma]}=\psi^{\mu}R_{\mu}^{[\nu}\xi^{\rho}\psi^{\sigma]}=0$, and has a north-south $\mathcal{Z}_{2}$ symmetry. As a result, the circular orbit motion can only be on the equatorial plane. Then under these conditions, the most generic metric reads

assuming a signature (-, +, +, +). In addition to the Kerr black hole, most known black holes can also be described by this metric. For a black hole, we suppose that its horizon is located. at a constant (positive) radial coordinate $r=r_{\text{h}}$. So in the exterior region of the black hole, one should have $r_{\text{h}}<r<\infty$. We also have $g_{rr}>0$, $g_{\theta\theta}>0$, and $g_{\varphi\varphi}>0$ outside the horizon. The angular coordinates $\theta\in[0,\pi]$ and $\varphi\in[0,2\pi]$. According to the $\mathcal{Z}_{2}$ symmetry, the equatorial plane is at $\theta=\pi/2$. Although we only concern the equatorial TCOs, here we still expect to leave $\theta$ as a free variable. Considering the condition det$(-g)>0$, one easily obtains

At the horizon, we have $B(r_{\text{h}},\theta)=0$. Under the metric (1), the motion of a test particle can be described by the Lagrangian

where the dots denote the derivative with respect to an affine parameter, and $\mu^{2}=$1, 0, -1 are for the timelike, null, and spacelike geodesics, respectively. The conjugate momenta can be calculated as

Therefore, the Hamiltonian for the test particle reads

Rearranging it, we get

Following the treatment for the null geodesics Grover ; Herdeiro , we denote the kinetic term $\mathcal{K}$ and potential term $\mathcal{V}$ as

Therefore, the motion of the test particle (6) turns to

Note that the kinetic term $\mathcal{K}\geq 0$, and the inequality is only saturated at $\dot{r}=\dot{\theta}=0$. Then the motion of the particle is completely governed by the effective potential $\mathcal{V}$. On the other hand, these two Killing vectors, $\xi^{\mu}$ and $\psi^{\mu}$, are related to two constants, the energy and orbital angular momentum of the test particle, along each geodesic

where $u^{\mu}$ denotes the tangent vector of the geodesics. Solving the above two equations, one can obtain

Plugging them into the potential (8), one has

When $\mu^{2}=0$, this exactly reduces to the potential of a photon Grover ; Herdeiro . Similarly, we can reexpress this potential as

where $H_{\pm}$ are only dependent of the black hole metric and are given by

For a photon with $\mu^{2}=0$, it is quite clear that the LR related with $\mathcal{V}=\partial_{r}\mathcal{V}=0$ satisfies Grover ; Herdeiro

This is an amazing result that the radius $r_{LR}$ of the LR (or the photon sphere for the static spherically black hole) of the black hole can be solved by Eq. (18) and is independent the properties of the photon, indicating that it is an intrinsic structure of the black hole just like the horizon. Then plugging it into (17), one shall get $E/l=H_{\pm}(r_{LR})$ for these photons orbiting along the LR. In Ref. Cunhab , by defining a new vector $v=((\partial_{r}H_{\pm})/\sqrt{g_{rr}},(\partial_{\theta}H_{\pm})/\sqrt{g_{\theta\theta}})$, one finds the location of the LR is exactly at the zero point of $v$ from (18). As a result, by calculating the winding number in the vector space, we can obtain some properties of the LR. In particular, considering all the regions of ($r$, $\theta$), the global property of the LR in a black hole background will be obtained by examining the asymptotic behaviors of $v$ near the boundary. This treatment mainly takes advantage of the result that $H_{\pm}$ are independent of the energy and angular momentum of the photon. However, due to the presence of the extra term $\mu^{2}$ in (15), the case becomes unclear.

In order to establish the topology for the TCO, we reexpress the effective potential (14) in the following form

where

Obviously, $\mathcal{V}=0$ leads to the energy of the test particle $E=e_{1,2}$. Considering the condition (2) and $g_{\varphi\varphi}>0$, it is easy to obtain $e_{2}<g_{t\varphi}(|l|-l)/g_{\varphi\varphi}$. For $l\geq 0$, it is easy to obtain $e_{2}<0$, and for $l<0$, $e_{2}<-2lg_{t\varphi}/g_{\varphi\varphi}<0$ due to negative $g_{t\varphi}$. Therefore, it is notable that for any value of $l$, $e_{2}$ takes a negative value. As a result, we abandon it. Then, the conditions $\mathcal{V}=\partial_{r}\mathcal{V}=0$ for determining the TCO become

Quite different from the LR, $\partial_{r}e_{1}$ here not only depends on $r$, but also on the angular momentum $l$ of the test particle, so it seems that we cannot construct the topology for the TCO as expected.

However, we would like to propose a new viewpoint while is still similar to that for the photon. For each given angular momentum $l$, we can obtain the radius of the TCO by solving (23). Whether the equation has solution or not, the angular momentum $l$ of the test particle does not affect our treatment. After obtaining the radius of the TCO, we can substitute it into (22), and then the energy of the test particle will be obtained.

In order to give a global topology, we only require that the values of the angular momentum do not change the asymptotic behavior of $\partial_{r}e_{1}$ at the boundary of the ($r$, $\theta$) plane. Aiming at it, we define a new vector $\phi$=($\phi^{r}$, $\phi^{\theta}$) by analog to Ref. Cunhab for the photon,

in a flat vector space. In order to be consistent with the notations of Ref. Duan , we write these indices with up ones. Obviously, $\phi=0$ corresponds to the TCO and $\theta=\pi/2$ as expected. Next, we turn to the asymptotic behavior of $\phi$ at the boundary of the ($r$, $\theta$) plane; this shall further confirm that the construction of the topology for the TCO is meaningful.

For our case, we should examine four asymptotic behaviors of $\phi$ near the horizon $r\rightarrow r_{\text{h}}$, infinity $r\rightarrow\infty$, and $\theta\rightarrow 0$, $\pi$.

i) Axis limits. Here we shall consider the axis limits of the vector $\phi$. In order to clearly show them, we here follow Ref. Cunhab and give a detailed calculation. Defining a local coordinate $\rho=\sqrt{g_{\varphi\varphi}}$, one can find when $\theta\rightarrow 0$ or $\pi$, $d\rho/d\theta$ is positive or negative. Since the metric functions are independent of the particles, one has the same result as that of Ref. Carot by expanding the metric function at small $\rho$

where $g_{tt}^{0}$, $g_{\rho\rho}^{0}$, $b_{0}$ are constants, and $n\in N$. Assuming $C^{2}$ smoothness and regularity close to the axis, one shall have $n\geq 2$ Carot . Plugging them into Eq. (20), we obtain

At zero order in $\rho$, one has $g_{\rho\rho}d\rho^{2}\sim g_{\theta\theta}d\theta^{2}$, and thus

for nonvanishing and vanishing $l$, respectively. Therefore, it is clear that $\phi^{\theta}$ diverges at $\rho\to 0$ and the angular momentum do not affect the behavior of $\phi^{\theta}$ at small $\rho$. Further combining with $\phi^{r}\sim\rho^{-1}$, we have $\phi^{\theta}\gg\phi^{r}$. This indicates that the direction of the vector is vertical near $\theta$=0 and $\pi$. Moreover, considering the fact that $d\rho/d\theta$ is positive and negative as $\theta\rightarrow$0 and $\pi$, we find the vector is outward at $\theta$=0 and $\pi$.

ii) Horizon limit. Here we would like to consider the behavior of the vector $\phi$ when $r\to r_{h}$. As discussed in Ref. Medved , we can set a local radial coordinate $x$ such that $g_{xx}=1$, and $x=0$ at the black hole horizon. Near the horizon, one can obtain these metric functions Medved

where $\omega=-g_{t\varphi}/g_{\varphi\varphi}$ and $\kappa$ denotes the surface gravity of the black hole. $g_{\varphi\varphi}^{H}$ and $\omega_{H}$ are zero-order terms of the expansions in $x$. Then

By using $(1/\sqrt{g_{xx}})(\partial/\partial x)=(1/\sqrt{g_{rr}})(\partial/\partial r)$, one shall obtain

For a nonextremal black hole with $\kappa>0$, we see that $\phi^{r}$ takes a positive value. In order to determine the direction of the vector $\phi$ at the horizon, one needs to calculate $\phi^{\theta}$. However, no matter whether $\phi^{\theta}$ is positive or negative, we can claim that the direction of the vector $\phi$ must be to the right, with at most an inclination.

iii) Asymptotic limit. Here let us consider the asymptotic behavior of the vector at infinity. Here we require the spacetime tends to a flat one in standard spherical coordinates, namely

Parameter $M$ is a positive constant and can be treated as the black hole mass in the asymptotically flat spacetime. Therefore, we have

on the equatorial plane by taking $\theta=\pi/2$, which is small but a positive value. As a result, by ignoring the specific value of $\phi^{\theta}$, the direction of the vector $\phi$ is to the right similar to that in the horizon limit.

In summary, we see that the angular momentum $l$ of the test massive particle do not affect the behavior of the vector $\phi$ at the boundary of the ($r$, $\theta$) plane in our above approach. This result strongly implies that we can construct the topology for the TCOs of the massive particle in a black hole background.

Based on the asymptotic behavior of $\phi$ at the boundary of the ($r$, $\theta$) plane, we here investigate the global and local topological properties for the TCOs. Moreover, we also examine the topology for the MSCO and ISCO.

In the previous section, we constructed the topology for the TCOs in a generic black hole background. In this section, we manage to apply it to the Kerr black hole and to confirm whether the above study is applicable. This shall uncover the detailed properties for a specific example.

As shown above, we take the angular momentum $l$ as the control parameter to construct the topology. Since the energy $E$ is also an alternative parameter for the massive test particle, it is interesting to consider this case.

Here we need to reformulate the effective potential $\mathcal{V}$ as

where $l_{1,2}$ are given by