Motion of charged particles around a magnetic black hole/topological star with a compact extra dimension

The study of geodesics and particle motion within a spacetime is an important tool to understand the properties of a gravitational system. For instance, the stability of circular orbits is, in specific cases, linked to quasinormal modes of the spacetime [1].²²2However, see Ref. [2] about the subtleties regarding the precise relation between the two. From an astrophysical standpoint, the size of innermost stable circular orbits of charged particles around a black hole may reveal features of magnetic fields in its vicinity [3]. In fact, as the direct imaging of a black hole is now a reality [4], one can seriously consider the optical appearance of the black hole, which can be calculated by considering null geodesics around it [5, 6, 7, 8]. Through this way, models of gravity can be tested or constrained. For instance, this has been done for braneworld gravity [9] and gravity coupled to non-linear electrodynamics [10].

In recent decades, theories of gravity with extra dimensions have received attention due to developments in theoretical physics such as string theory, holographic correspondences, and braneworld scenarios, among many others. A simple and natural candidate of a gravitating source in higher dimensional gravity would be a black hole with a compact fifth dimension. However, these black holes, including electrically charged ones, are known to suffer from Gregory–Laflamme instability [11, 12]. (See, e.g., Refs. [13, 14, 15] for reviews.)

The possibility of stable black holes with compact extra dimensions comes through a topological argument by Stotyn and Mann in [18], where they argued that the presence of a magnetic charge may stabilise the spacetime.³³3On the other hand, one can also have stable black strings or more generally black $p$-branes, by considering Anti-de Sitter black strings or $p$-branes supported by scalar fields [16]. These configurations were recently shown to be perturbatively stable [17]. They introduced a solution to the five-dimensional Einstein–Maxwell equations characterised by two parameters, which are denoted $(\alpha,\beta)$ in the notation of the present paper. If $\alpha>\beta$, the spacetime carries a horizon and hence describes the black hole with the compact extra dimension stabilised by the magnetic charge.

On the other hand, if $\alpha<\beta$, it describes a type of soliton star [18]. In this case, if a certain minimum radial distance is approached the spacetime caps off in a ‘cigar-like’ geometry. Conical singularities may be present, unless the periodicity of the compact fifth dimension is appropriately fixed. In Refs. [19, 20], Bah and Heidmann allow the presence of orbifold fixed points. This introduces topological cycles in the compact fifth direction, and these solutions were called topological stars by the authors. In this case, the magnetic field determines the minimum radial distance where the spacetime caps off. When the magnetic field is zero, this does not happen as the spacetime is simply a direct product between a Schwarzschild/Minkowski spacetime and a circle.

In this paper, we study the motion of charged particles in this family of solutions described in the preceding paragraphs. We shall use the terminology magnetic Kaluza–Klein black hole (KKBH) to refer to the case $\alpha>\beta$, and magnetic topological star (TS) to refer to the case $\alpha<\beta$. Since we always consider a non-zero magnetic field throughout the paper, we will often drop the term ‘magnetic’ since it will be understood that it will be present at all times.

A guiding intuition in understanding the physics of this problem is the fact that the particle is under the influence of two forces. First is the spherically symmetric gravitational attraction towards the KKBH/TS, and second is the Lorentz force due to a spherically symmetric magnetic field. A similar situation occurs for a charged particle around a magnetically charged Reissner–Nordström black hole, which was studied by Grunau and Kagramanova in [21]. There, the authors obtained exact analytical solutions in terms of Weierstraß functions. Further studies of particles in the Reissner–Nordström spacetime were subsequently done by other authors in [22, 23, 24]. A non-relativistic analogue of this problem is the dyon-dyon interaction studied by Schwinger et al. in [25]. (See also [26, 27].) In these similar/analogue problems, the motion of the electric charge is known to move on the surface of a Poincaré cone [28].

It will be shown in this paper that the same is true for charged particles in the non-compact part of the KKBH/TS spacetime. The main reason for this similarity across all the aforementioned problems is that the equations of motion in the angular coordinates are the same. They depend only on the electromagnetic properties of the system which are spherically symmetric, and independent of the other parameters that distinguishes the different systems. Relative to a choice of coordinate axes, the opening angle and orientation of the Poincaré cone depend on the product between the particle charge and the magnetic field strength, as well as its angular momentum.

The radial motion of the particle can be classified into categories depending on whether it has access to the horizon, may escape to infinity, or bound in a finite domain. These depend on the particle’s energy, momenta, and whether a horizon is present in the solution. If a horizon is present (the KKBH case) there is at most one bound domain where the particle orbits the black hole indefinitely. When a particle’s energy exceeds a certain threshold, no bound orbits exist; it could either fall into the horizon or escape to infinity. In this sense, the situation is similar to particles around most spherically symmetric [29, 21] as well as rotating black holes [30]. On the other hand, if no horizon is present (the TS case) the particle may have two disconnected bound domains. Furthermore, a bound domain may still exist when the energy exceeds the aforementioned threshold.

The rest of the paper is organised as follows. In Sec. 2 we describe the spacetime given by [19, 20] and derive the equations of motion for a charged particle in it. The parameter space of conserved quantities of the particle, as well as its domains of motion are studied in Sec. 3. In particular, we obtain the Poincaré cone in Sec. 3.1 and study the domains of radial motion in Sec. 3.2. Conclusions and closing remarks are given in Sec. 5. In this paper, we use geometrical units where the speed of light equals unity and the convention for Lorentzian metric signature is $(-,+,+,+,+)$.

The five-dimensional KKBH/TS spacetime is described by the metric [18, 19, 20]

	$\displaystyle\mathrm{d}s^{2}$	$\displaystyle=-U\mathrm{d}t^{2}+V\mathrm{d}w^{2}+\frac{\mathrm{d}r^{2}}{UV}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2},$		(1a)
	$\displaystyle U$	$\displaystyle=1-\frac{\alpha}{r},\quad V=1-\frac{\beta}{r},$		(1b)

where $\alpha$ and $\beta$ are constants and $w$ is the coordinate representing the compact fifth dimension. If $\alpha>\beta$, the solution describes a KKBH and has a horizon at $r=\alpha$. On the other hand, if $\alpha<\beta$, the solution describes the TS spacetime where the spacetime caps off at $r=\beta$.

In this latter case, the periodicity of $w$ can be appropriately fixed to remove conical singularities, as was done in [18], or such that certain orbifold singularities are allowed, as was done in [19, 20]. Here, we need not choose a particular periodicity for $w$ and mainly focus on the motion of particles in the non-compact directions, namely $(r,\theta,\phi)$.

The gauge potential of this solution is given by

$\displaystyle A=-g\cos\theta\mathrm{d}\phi.$

(2)

The Maxwell tensor is then obtained by taking the exterior derivative, $F=\mathrm{d}A$. For the potential (2), the Maxwell tensor describes a spherically symmetric inverse-square magnetic field, whose strength is parametrised by $g$. The metric (1) with the gauge potential (2) satisfies the Einstein–Maxwell equations in five dimensions provided that $g^{2}=\frac{3}{2}\alpha\beta$.⁴⁴4We normalise our gauge field such that the Einstein–Maxwell action appears as $I\propto\int\mathrm{d}^{5}x\sqrt{-g}\left(R-F_{\mu\nu}F^{\mu\nu}\right)$.

The motion of a test particle of charge per unit mass $e$ is described by a spacetime curve $x^{\mu}(\tau)$, where $\tau$ is an appropriate affine parameter. Here we shall take our choice of parametrisation for $\tau$ such that

$\displaystyle g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1,$

(3)

where over-dots denote derivatives with respect to $\tau$. The motion is governed by the Lagrangian $\mathcal{L}(x,\dot{x})=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+eA_{\mu}\dot{x}^{\mu}$. For the spacetime described by (1) and (2), the Lagrangian is explicitly

$\displaystyle\mathcal{L}(x,\dot{x})$

$\displaystyle=\frac{1}{2}\left(-U\dot{t}^{2}+V\dot{w}^{2}+\frac{\dot{r}^{2}}{UV}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}\theta\dot{\phi}^{2}\right)-eg\cos\theta\dot{\phi}.$

(4)

Since the magnetic field strength and the particle charge always appear together in the equations of motion, it will be convenient to define $q=eg$.

The conjugate momenta are obtained by $p_{\mu}=\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}$. Explicitly, they appear as follows:

	$\displaystyle p_{t}$	$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{t}}=-U\dot{t},$		(5a)
	$\displaystyle p_{w}$	$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{w}}=V\dot{w},$		(5b)
	$\displaystyle p_{\phi}$	$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\sin^{2}\theta\dot{\phi}-q\cos\theta,$		(5c)
	$\displaystyle p_{r}$	$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{r}}=\frac{\dot{r}}{UV},$		(5d)
	$\displaystyle p_{\theta}$	$\displaystyle=\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=r^{2}\dot{\theta}.$		(5e)

Since our spacetime has three Killing vectors $\partial_{t}$, $\partial_{w}$, and $\partial_{\phi}$, the momenta along these directions are constants of motion. We shall denote the constants by

$\displaystyle p_{t}=-E,\quad p_{w}=P,\quad p_{\phi}=L,$

(6)

representing the energy, linear momenta along the $w$-direction, and the angular momentum of the particle, respectively. The evolution of $t$, $w$, and $\phi$ are determined by the first integrals

$\displaystyle\dot{t}=\frac{E}{U},\quad\dot{w}=\frac{P}{V},\quad\dot{\phi}=\frac{L+q\cos\theta}{r^{2}\sin^{2}\theta}.$

(7)

The equations of motion for $r$ and $\theta$ can be obtained from the Euler–Lagrange equations, giving

	$\displaystyle\ddot{\theta}$	$\displaystyle=-\frac{2\dot{r}\dot{\theta}}{r}+\frac{\cos\theta\left(L+q\cos\theta\right)^{2}}{r^{4}\sin^{3}\theta}+\frac{q\left(L+q\cos\theta\right)}{r^{4}\sin\theta},$		(8)
	$\displaystyle\ddot{r}$	$\displaystyle=\frac{1}{2}\left(\frac{U^{\prime}}{U}+\frac{V^{\prime}}{V}\right)\dot{r}^{2}+rUV\dot{\theta}^{2}-\frac{VU^{\prime}E^{2}}{2U}+\frac{UV^{\prime}P^{2}}{2V}+\frac{UV\left(L+q\cos\theta\right)^{2}}{r^{3}\sin^{2}\theta},$		(9)

where primes denote derivatives with respect to $r$.

We also note that Eq. (3) can be regarded as another equation of first integrals. Using Eq. (7) to express $\dot{t}$, $\dot{w}$, and $\dot{\phi}$ in terms of the constants of motion, we have

$\displaystyle\frac{\dot{r}^{2}}{UV}+r^{2}\dot{\theta}^{2}-\frac{E^{2}}{U}+\frac{P^{2}}{V}+\frac{\left(L+q\cos\theta\right)^{2}}{r^{2}\sin^{2}\theta}=-1.$

(10)

Equations. (8) and (9) can be solved numerically while using Eq. (10) as a consistency check. In this work, this is performed by implementing the fourth-order Runge–Kutta algorithm in C.

A deeper analytical insight can be found by considering the Hamilton–Jacobi equation $\mathcal{H}\left(\frac{\partial S}{\partial x},x\right)+\frac{\partial S}{\partial\tau}=0$, where $\mathcal{H}(p,x)=\frac{1}{2}g^{\mu\nu}\left(p_{\mu}-eA_{\mu}\right)\left(p_{\nu}-eA_{\nu}\right)$ is the Hamiltonian obtained from the Legendre transform of the Lagrangian. Explicitly, the Hamilton–Jacobi equation for our present context reads

	$\displaystyle\frac{1}{2}\Bigg{[}-\frac{1}{U}\left(\frac{\partial S}{\partial t}\right)^{2}$	$\displaystyle+\frac{1}{V}\left(\frac{\partial S}{\partial w}\right)^{2}+UV\left(\frac{\partial S}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial S}{\partial\theta}\right)^{2}$
		$\displaystyle+\frac{1}{r^{2}\sin^{2}\theta}\left(\frac{\partial S}{\partial\phi}+q\cos\theta\right)^{2}\Bigg{]}+\frac{\partial S}{\partial\tau}=0.$		(11)

For this system, the Hamilton–Jacobi equation is completely separable, giving the first integrals

	$\displaystyle\dot{t}$	$\displaystyle=\frac{E}{U},\quad\dot{w}=\frac{P}{V},\quad\dot{\phi}=\frac{L+q\cos\theta}{r^{2}\sin^{2}\theta},$		(12a)
	$\displaystyle r^{2}\dot{\theta}$	$\displaystyle=\pm\sqrt{Q+L^{2}-\frac{\left(L+q\cos\theta\right)^{2}}{\sin^{2}\theta}},$		(12b)
	$\displaystyle r^{2}\dot{r}$	$\displaystyle=\pm\sqrt{r^{4}\left(VE^{2}-UP^{2}\right)-r^{2}UV\left(r^{2}+L^{2}+Q\right)},$		(12c)

where $Q$ is the Carter-like [31] separation constant.

We further simplify the equations by introducing a Mino-type parameter [32] defined by $\frac{\mathrm{d}\tau}{\mathrm{d}\lambda}=r^{2}$, and changing variables to $x=\cos\theta$. Then the equations of motion now become

	$\displaystyle\frac{\mathrm{d}t}{\mathrm{d}\lambda}$	$\displaystyle=\frac{r^{2}E}{U},$		(13a)
	$\displaystyle\frac{\mathrm{d}w}{\mathrm{d}\lambda}$	$\displaystyle=\frac{r^{2}P}{V},$		(13b)
	$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}\lambda}$	$\displaystyle=\frac{L+qx}{1-x^{2}},$		(13c)
	$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}\lambda}$	$\displaystyle=\mp\sqrt{X(x)},$		(13d)
	$\displaystyle\frac{\mathrm{d}r}{\mathrm{d}\lambda}$	$\displaystyle=\pm\sqrt{R(r)},$		(13e)

where

	$\displaystyle X(x)$	$\displaystyle=Q^{2}(1-x^{2})-\left(L+qx\right)^{2},$		(14a)
	$\displaystyle R(r)$	$\displaystyle=r^{4}\left(VE^{2}-UP^{2}\right)-r^{2}UV\left(Q+L^{2}+r^{2}\right).$		(14b)

In this paper we have derived and analysed the equations of motion for a charged particle in the magnetic black hole/topological star solution. The angular motion was found to be similar to analogous systems involving interacting electric and magnetic monopoles. In particular, the electric charge moves along the surface of a Poincaré cone. The angle and orientation of the cone depends on the charge and the angular momentum, as is well-known since Poincaré’s original non-relativistic analysis.

On the other hand, the radial motion shows a richer structure of possibilities in the topological star case. In particular, up to two distinct domains of bound orbits may exist for fixed energy and total angular momentum. On the $E^{2}$-$K$ space, the points representing circular orbits exhibit a swallow-tail structure where each branch correspond to stable/unstable circular orbits. When the $w$-momentum is varied, the swallow-tail kink disappears, leaving just a single stable branch. This is highly reminiscent of phase transitions behaviour in thermodynamics, where swallow-tail structures appear in the graphs of intrinsic parameters. Investigating this similarity and the possibility of carrying over thermodynamic concepts into particle mechanics of this spacetime might be a candidate of future study.