Motion of charged particles around a magnetic black hole/topological star with a compact extra dimension
aa r X i v : . [ g r- q c ] F e b Motion of charged particles around a magnetic blackhole/topological star with a compact extra dimension
Yen-Kheng Lim ∗ Department of Physics, Xiamen University Malaysia, 43900 Sepang, Malaysia
February 18, 2021
Abstract
We study the motion of charged particles in a family of five-dimensional solu-tions describing either a black hole or topological star with a fifth compact dimensionstabilised by a magnetic flux. The particle’s trajectory is shown to move along thesurface of a Poincar´e cone. The radial motion shows a rich structure where the ex-istence of various bound, plunging, or escaping trajectories depend on theconstantsof motion. Curves of energy and angular momentum corresponding to sphericalorbits show a swallow-tail structure highly reminiscent to phase transitions of ther-modynamics. When the momentum along the compact direction is varied, the is acritical point beyond which the swallow-tail kink disappears and becomes a smoothcurve.
The study of geodesics and particle motion within a spacetime is an important toolto understand the properties of a gravitational system. For instance, the stability ofcircular orbits is linked to quasinormal modes of the spacetime [1]. From an astrophysicalstandpoint, the size of innermost stable circular orbits of charged particles around a blackhole may reveal features of magnetic fields in its vicinity [2]. In fact, as the direct imagingof a black hole is now a reality [3], one can seriously consider the optical appearance ofthe black hole, which can be calculated by considering null geodesics around it [4–7].In recent decades, theories of gravity with extra dimensions have received attention dueto developments in theoretical physics such as string theory, holographic correspondences,and braneworld scenarios, among many others. A simple and natural candidate of a ∗ Email: [email protected] α, β ) in thenotation of the present paper. If α > β , the spacetime carries a horizon and hencedescribes the black hole with the compact extra dimension stabilised by the magneticcharge.On the other hand, if α < β , it describes a type of soliton star [15]. In this case, ifa 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 fifthdimension is appropriately fixed. In Refs. [13, 14], Bah and Heidmann allow the presenceof 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 magneticfield determines the minimum radial distance where the spacetime caps off. When themagnetic field is zero, this does not happen as the spacetime is simply a direct productbetween a Schwarzschild/Minkowski spacetime and a circle.In this paper, we study the motion of charged particles in this family of solutionsdescribed in the preceding paragraphs. We shall use the terminology magnetic Kaluza–Klein black hole (KKBH) to refer to the case α > β , and magnetic topological star (TS)to refer to the case α < β . Since we always consider a non-zero magnetic field throughoutthe paper, we will often drop the term ‘magnetic’ since it will be understood that it willbe present at all times.A guiding intuition in understanding the physics of this problem is the fact that theparticle is under the influence of two forces. First is the spherically-symmetric grav-itational attraction towards the KKBH/TS, and second is the Lorentz force due to aspherically-symmetric magnetic field. A similar situation occurs for a charged particlearound a magnetically-charged Reissner–Nordstr¨om black hole, which was studied byGrunau and Kagramanova in [16]. There, the authors obtained exact analytical solutionsin terms of Weierstraß functions. Further studies of particles in the Reissner–Nordstr¨omspacetime were subsequently done by other authors in [17–19]. A non-relativistic ana-logue of this problem is the dyon-dyon interaction studied by Schwinger et al. in [20].(See also [26, 27].) In these similar/analogue problems, the motion of the electric chargeis known to move on the surface of a Poincar´e cone [21].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 allthe aforementioned problems is that the equations of motion in the angular coordinates2re the same. They depend only on the electromagnetic properties of the system whichare spherically-symmetric, and independent of the other parameters that distinguishesthe different systems. Relative to a choice of coordinate axes, the opening angle andorientation of the Poincar´e cone depend on the product between the particle charge andthe magnetic field strength, as well as its angular momentum.The radial motion of the particle can be classified into categories depending on whetherit has access to the horizon, may escape to infinity, or bound in a finite domain. Thesedepend 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 theparticle orbits the black hole indefinitely. When a particle’s energy exceeds a certainthreshold, 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 [16,22]as well as rotating black holes [23]. On the other hand, if no horizon is present (the TScase) the particle may have two disconnected bound domains. Furthermore, a bounddomain may still exist whent the energy exceeds the aforementioned threshold.The rest of the paper is organised as follows. In Sec. 2 we describe the spacetime givenby [13, 14] and derive the equations of motion for a charged particle in it. The parameterspace of conserved quantities of the particle, as well as its domains of motion are studiedin Sec. 3. In particular, we obtain the Poincar´e cone in Sec. 3.1 and study the domainsof radial motion in Sec. 3.2. Conclusions and closing remarks are given in Sec. 5. In thispaper, we use geometrical units where the speed of light equals unity and the conventionfor Lorentzian metric signature is ( − , + , + , + , +). The five-dimensional KKBH/TS spacetime is described by the metric [13–15]d s = − U d t + V d w + d r U V + r d θ + r sin θ d φ , (1a) U = 1 − αr , V = 1 − βr , (1b)where α and β are constants and w is the coordinate representing the compact fifthdimension. If α > β , the solution describes a KKBH and has a horizon at r = α . On theother hand, if α < β , the solution describes the TS spacetime where the spacetime capsoff at r = β .In this latter case, the periodicity of w can be appropriately fixed to remove conicalsingularities, as was done in [15], or such that certain orbifold singularities are allowed, aswas done in [13, 14]. Here, we need not choose a particular periodicity for w and mainlyfocus on the motion of particles in the non-compact directions, namely ( r, θ, φ ).3he gauge potential of this solution is given by A = − g cos θ d φ. (2)The Maxwell tensor is then obtained by taking the exterior derivative, F = d A . Forthe potential (2), the Maxwell tensor describes a spherically symmetric inverse-squaremagnetic field, whose strength is parametrised by g . The metric (1) with the gaugepotential (2) satisfies the Einstein–Maxwell equations in five dimensions provided that g = αβ . The motion of a test particle of charge per unit mass e is described by a spacetimecurve x µ ( τ ), where τ is an appropriate affine parameter. Here we shall take our choice ofparametrisation for τ such that g µν ˙ x µ ˙ x ν = − , (3)where over-dots denote derivatives with respect to τ . The motion is governed by theLagrangian L ( x, ˙ x ) = g µν ˙ x µ ˙ x ν + eA µ ˙ x µ . For the spacetime described by (1) and (2), theLagrangian is explicitly L ( x, ˙ x ) = 12 (cid:18) − U ˙ t + V ˙ w + ˙ r U V + r ˙ θ + r sin θ ˙ φ (cid:19) − eg cos θ ˙ φ. (4)Since the magnetic field strength and the particle charge always appear together in theequations of motion, it will be convenient to define q = eg .The conjugate momenta are obtained by p µ = ∂ L ∂ ˙ x µ . Explicitly, they appear as follows: p t = ∂ L ∂ ˙ t = − U ˙ t, (5a) p w = ∂ L ∂ ˙ w = V ˙ w, (5b) p φ = ∂ L ∂ ˙ φ = r sin θ ˙ φ − q cos θ, (5c) p r = ∂ L ∂ ˙ r = ˙ rU V , (5d) p θ = ∂ L ∂ ˙ θ = r ˙ θ. (5e)Since our spacetime has three Killing vectors ∂ t , ∂ w , and ∂ φ , the momenta along thesedirections are constants of motion. We shall denote the constants by p t = − E, p w = P, p φ = L, (6) We normalise our gauge field such that the Einstein–Maxwell action appears as I ∝ R d x √− g ( R − F µν F µν ). w -direction, and the angular momen-tum of the particle, respectively. The evolution of t , w , and φ are determined by the firstintegrals ˙ t = EU , ˙ w = PV , ˙ φ = L + q cos θr sin θ . (7)The equations of motion for r and θ can be obtained from the Euler–Lagrange equations,giving ¨ θ = − r ˙ θr + cos θ ( L + q cos θ ) r sin θ + q ( L + q cos θ ) r sin θ , (8)¨ r = 12 (cid:18) U ′ U + V ′ V (cid:19) ˙ r + rU V ˙ θ − V U ′ E U + U V ′ P V + U V ( L + q cos θ ) r sin θ , (9)where primes denote derivaties with respect to r .We also note that Eq. (3) can be regarded as another equation of first integrals. UsingEq. (7) to express ˙ t , ˙ w , and ˙ φ in terms of the constants of motion, we have˙ r U V + r ˙ θ − E U + P V + ( L + q cos θ ) r sin θ = − . (10)Eqs. (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 algorithmin C.A deeper analytical insight can be found by considering the Hamilton–Jacobi equation H (cid:0) ∂S∂x , x (cid:1) + ∂S∂τ = 0, where H ( p, x ) = g µν ( p µ − eA µ ) ( p ν − eA ν ) is the Hamiltonian ob-tained from the Legendre transform of the Lagrangian. Explicitly, the Hamilton–Jacobiequation for our present context reads12 " − U (cid:18) ∂S∂t (cid:19) + 1 V (cid:18) ∂S∂w (cid:19) + U V (cid:18) ∂S∂r (cid:19) + 1 r (cid:18) ∂S∂θ (cid:19) + 1 r sin θ (cid:18) ∂S∂φ + q cos θ (cid:19) + ∂S∂τ = 0 . (11)For this system, the Hamilton–Jacobi equation is completely separable, giving the firstintegrals ˙ t = EU , ˙ w = PV , ˙ φ = L + q cos θr sin θ , (12a) r ˙ θ = ± s Q + L − ( L + q cos θ ) sin θ , (12b) r ˙ r = ± p r ( V E − U P ) − r U V ( r + L + Q ) , (12c)5here Q is the Carter-like [24] separation constant.We further simplify the equations by introducing a Mino-type parameter [25] definedby d τ d λ = r , and changing variables to x = cos θ . Then the equations of motion nowbecome d t d λ = r EU , (13a)d w d λ = r PV , (13b)d φ d λ = L + qx − x , (13c)d x d λ = ∓ p X ( x ) , (13d)d r d λ = ± p R ( r ) , (13e)where X ( x ) = Q (1 − x ) − ( L + qx ) , (14a) R ( r ) = r (cid:0) V E − U P (cid:1) − r U V (cid:0) Q + L + r (cid:1) . (14b) The polar motion with x = cos θ is governed by X ( x ) in Eq. (14a). Since d x d λ must bereal in Eq. (13d), the particle is allowed to move in the domain where X ( x ) ≥
0. Clearly,we see that no such domain exist if Q + L ≤
0. Therefore the Carter-like constant Q is restricted to Q > − L . When this is satisfied, X ( x ) is non-negative in the domain x − ≤ x ≤ x + , x − ≤ x ≤ x + , (15)where x ± = − qL ± p ( Q + q )( Q + L ) Q + L + q . (16)The domain (15) is non-empty if ( Q + q )( Q + L ) ≤
0. Since we already argued abovethat
Q > − L , we then have Q + q ≥ Q, L ) ∈ (cid:8) Q + L > Q + q ≥ (cid:9) . (17)Eq. (13d) can be integrated explicitly upon a choice of branch and initial conditions.6ere, let us consider two specific choices, x (0) = x + and x (0) = x − . In the former,we take the upper (negative) sign of Eq. (13d), wheras in the latter we take the lower(positive) sign. These choices will give an increasing λ as the particle evolves away fromtheir respective initial conditions. The integration is then Z xx ± d x ′ p X ( x ′ ) = ∓ Z λ d λ ′ x ( λ ) = x + + x − ± x + − x − (cid:16)p Q + L + q λ (cid:17) x ( λ ) = 1 Q + L + q h − qL ± p ( Q + q )( Q + L ) cos (cid:16)p Q + L + q λ (cid:17)i . (18)We can also obtain an analytical solution for φ expressed as a function of x by eliminating λ from Eq. (13c) and (13d), givingd φ d x = ∓ L + qx (1 − x ) p X ( x ) . (19)The integral can be performed with the aid of partial fraction decomposition on the factor1 / (1 − x ). The result is φ ( x ) = (sgn( L − q ) + sgn( L + q )) π ± ( sgn( L − q ) arcsin " ( L − q ) p ( Q + q )( Q + L ) (cid:18)
11 + x − Q + L + q − qL ( L − q ) (cid:19) − sgn( L + q ) arcsin " ( L + q ) p ( Q + q )( Q + L ) (cid:18) − x − Q + L + q + qL ( L + q ) (cid:19) , (20)where the ‘ ± ’ signs are in accordance to the choice of initial conditions of x ( λ ) in Eq. (18).We have also defined the sign function as sgn( x ) which returns ± x ≷ x = 0.It is worth noting that the angular equations of motion are independent of α and β ,and are purely due to the Lorentz interaction between the charge and the spherically-symmetric magnetic field. As alluded to in the Introduction, the equations of motion for x = cos θ and φ are in fact identical to the equations of motion for a charged particlearound a magnetically-charged Reissner–Nordstr¨om black hole [16, 18, 19, 22], as well asthe non-relativistic dyon-dyon interaction [20, 26] where the motion is confined to thePoincar´e cone.We will show that the Poincar´e cone also exists in our context of the KKBH/TSspacetime as well. If we take our coordinates ( r, θ, φ ) to define a naive Euclildean three- Note that in the argument of the cosine function, λ lies outside the square root. { ˆ e r , ˆ e θ , ˆ e φ } as the ortho-normal basis in spherical coordinates, the vector ~J = − q ˆ e r − r sin θ ˙ φ ˆ e θ + r ˙ θ ˆ e φ (21)is conserved throughout the motion, i.e., dd τ ~J = ~
0. In the non-relativistic case, thisquantity is the total angular momentum of the system [20, 26, 27]. This implies that theparticle moves on the surface of a cone which subtends an angle χ such thatcos χ − ~J (cid:12)(cid:12) ~J (cid:12)(cid:12) · ˆ e r = q p q + Q + L . (22)If we further define a Cartesian coordinate system in this naive Euclidean space by x = r sin θ cos φ, x = r sin θ sin φ, x = r cos θ, (23)the angle ψ of the cone’s axis with the x -direction (the axis passing through the northand south poles) is given bycos ψ = ~J (cid:12)(cid:12) ~J (cid:12)(cid:12) · ˆ e = L p q + Q + L , (24)where Eq. (12) was used and ˆ e is the unit vector along x . The motion in the r -direction is governed by the function R ( r ) defined in Eq. (14b), whichwe will rewrite here as R ( r ) = c r + c r + c r + c r + c , (25)where c = − (cid:0) P − E (cid:1) , (26a) c = α + β − βE + αP , (26b) c = − (cid:0) αβ + Q + L (cid:1) , (26c) c = ( α + β ) (cid:0) Q + L (cid:1) , (26d) c = − αβ (cid:0) Q + L (cid:1) . (26e)From Eq. (13e), the requirement that d r d λ be real means the particle can only access thedomains of r where R ( r ) ≥
0. We identify these domains by studying the root structureof R ( r ), which serves as possible boundaries of the domains.8t will be convenient to define K = L + Q , as L and Q always appear in this combi-nation in R ( r ). To aid our discussion below, we introduce the following terminology forthe possible domains of r such that R ( r ) ≥ • Plunging orbits . A finite domain of r which contains the horizon r = α . Particlesin this domain may fall into the horizon. • Escaping orbits . A (semi-)infinite domain of r , where R ( r ) remains positive as r → ∞ . Particles in this domain can escape to infinity. • Bound orbits . A finite domain bounded by two roots of R ( r ). Particles in thisdomain are in stable bound orbits, neither falling into the black hole or escaping toinfinity.Suppose we start with a case where R ( r ) has four real roots. Varying E , P , and K willgenerally vary the positions of each root. A pair of roots will coalesce into a degenerateroot when R ( r ) = R ′ ( r ) = 0. A particle located at this point will will solve the equationsof motion for constant r , which we will call a circular orbit . This condition is satisfiedwhen the energy and angular momentum satisfies E = E = ( r − α ) [2( r − β ) + r (2 r − β ) P ] r ( r − β ) (2 r − α ) , (27a) L + Q = K = r [ α ( r − β ) + r ( α − β ) P ]( r − β ) (2 r − α ) . (27b)For a given α , β , and P , plotting Eq. (27) as a parametric curve in r on the E - K planewill serve as a boundary separating domains for which R ( r ) has various numbers of realor complex roots.We can identify which pair among the four roots are degenerate by evaluating thesecond derivative R ′′ ( r ) (cid:12)(cid:12)(cid:12) E = E ,L = L = − α ( r − β ) ( r − α ) + r ( α − β ) ( r − α + β ) r + 6 αβ ) P ]( r − β ) (2 r − α ) . (28)If the double root of R ( r ) is a local minimum, this corresponds to an unstable circularorbit. On the other hand, if this double root is a local maximum, the circular orbit isstable as a small perturbation puts it in small oscillations about its original radius. The critical circular orbit (CCO) is r CCO such that R ′′ ( r CCO ) = 0. This is the critical radiuswhere a circular orbit changes from being unstable to stable, or vice versa. Indeed, if The term ‘circular’ used here has an interesting roundabout connotation. Usually, constant- r solu-tions of spherically symmetric equations of motion reduces to a circle because the symmetry confines themotion to a plane containing the origin. When spherical symmetry is not present, constant- r trajectoriesmay lie on a sphere, and are typically called spherical orbits [28]. At the same time we have shown in theprevious subsection that the trajectories lie on a Poincar´e cone. The intersection between a cone and asphere whose apex and centres coincide is, again, a circle. = β = 0, and letting α = 2 m , we recover the innermost stable circular orbit (ISCO) ofthe Schwarzschild black hole, r ISCO = 6 m . Looking at large r , we find that, asymptotically, E ∼ P − (cid:0) α + ( α − β ) P (cid:1) r + O (cid:0) r (cid:1) , (29a) K ∼ (cid:0) α + ( α − β ) P (cid:1) r + 14 (3 α + 4 β )( α − β ) P + 34 α + O (cid:0) r (cid:1) . (29b)Next we will explore the E - K space as α varies, starting from α > β (the KKBHcase), and then for α < β (the TS case). α > β (KKBH) We begin with α > β , which correspond to spacetimes with an event horizon. The E - K space can be organised into domains separated by curves E = E and K = K , as definedin Eq. (27). Fig. 1 demonstrates the domains for the concrete example α = 2, β = 1, and P = 0 . α > β , we find that there are two branches of circular orbits. The unstable branchhas R ′′ ( r ) > α < r < r CCO and is the upper branch depicted in Fig. 1.This branch tends to infinity as r → α . On the other hand, the stable branch is the onewith R ′′ < r CCO < r < ∞ . This branch asymptotically approaches E = 1 + P as r → ∞ , in accordance to Eq. (29).The horizontal red line is E = 1 + P . Values of E below this line makes the leadingcoefficient c of R ( r ) negative. We shall call this case ‘A’. Conversely, values of E abovethis line shall be called case ‘B’ for which c is positive.First we look at Case A. As the leading coefficient c is negative, we have R ( r ) < r beyond its largest root. This implies that the particle is unable to escapeto infinity. The parameters giving four real roots of R ( r ) lie in domain A1 in Fig. 1. Thestructure of these four roots can be understood with the aid of the Descartes rule of signs.For K = L + Q satisfying (17), along with α, β ≥
0, we see that c < c >
0, and c ≤ c can be rearranged as c = β (cid:20) αβ (cid:0) P (cid:1) − E (cid:21) . Since E < P and α > β , this term is positive. By the Descartes rule of signs, R ( r )in this case has four positive roots. Since R ( α ) = E α ( α − β ) >
0, the number of rootslocated outside the horizon is either 3 or 1. From these considerations, we conclude that, As we shall see in the following, the reason we use the terminiology ‘critical circular orbit’ ratherthan ‘innermost’ is there may be multiple points staisfying R ′′ ( r ) = 0, and hence ‘innermost’ is no longeraccurate. . . . . .
15 12 16 20 24 28 R ′′ ( r ) < R ′′ ( r ) > r → ∞ r → α / A1A2 B1B2
Figure 1: Parameter space described by angular momentum L and energy E , plottedhere for α = 2, β = 1, and P = 0 .
2. The blue curves corresponds to the circular orbits,where the the second derivatives of R ( r ) on these orbits are indicated. The segment with R ′′ ( r ) < R ′′ ( r ) > r = r CCO . The four domains markedA1, A2, B1, and B2 have functions R ( r ) sketched in Figs. 2a, 2b, 2c, and 2d, respectively.for E < P , there is always a plunging orbit and at most one bound orbit. A sketchof such a situation is shown in Fig. 2a, which occurs when E and K takes values in thesubdomain A1 of Fig. 1. For r > r CCO , the two largest roots will coalesce, shrinking thedomain of bound orbit to a point, giving a stable circular orbit. On the other hand, if r < r
CCO , the second and third largest roots coalesce such that the plunging and bounddomains are only separated by a point, which is the unstable circular orbit. For this case( α > β ), we see that it is indeed appropriate to call r CCO the innermost stable circularorbit.Varying E and K further until the pair of roots become complex, the function R ( r )will appear as sketched in Fig. 2b, where only a plunging orbit can exist. This occurswhen E and K takes values in subdomain A2 of Fig. 1.We now turn to Case B, where E > P . Now c is positive. Whether c is positiveor negative, the Descartes rule of signs tells us that R ( r ) has either three positive rootsor one positive root. Since R ( r ) is positive at the horizon, we conclude that there areno bound orbits in this case. The possible types of motion are either plunging and/orescaping orbits. More specifically, there are two subdomains B1 and B2, where domainB1 corresponds to functions R ( r ) appearing in the form sketched in Fig. 2c, and domainB2 have functions appearing as sketched in Fig. 2d.Finally we look specifically at the circular orbits, which lie on the boundary curvesseparating the domains discussed above, as shown in Fig. 1. In particular, the curve (27)for α < r < r CCO is the upper curve. Particles whose energy and angular momentumtaking values along this upper curve are unstable spherical orbits. The lower curve gives11 lunging bound ry (a) Case A1 plunging ry (b) Case A2 plunging escaping ry (c) Case B1 plunging/escaping ry (d) Case B2 Figure 2: Sketches of y = R ( r ) for various cases of energy and angular momentum,corresponding to domains (a)–(d) of the L - E plane of Fig. 1. The vertical red linesindicate the horizon r = α . The allowed domains of r corresponding to R ( r ) ≥ plunging , escaping , and bound are as defined in the main text.stable circular orbits with radii in the range r CCO < r < ∞ . β < α < β (TS) When β > α , the spacetime describes a TS without the presence of a horizon. Thestructure of the parameter space in this case is richer, as there can be up to three branchesof circular orbits, depending on the value of P . Fig. 3 shows the sequence of the curves(27) for the concrete example α = 1 . β = 2, and increasing values of P .We shall attempt to understand thse structures by looking closely at the circular orbitconditions. The various root configurations shall be given labels similar to the black-holecase, with the main difference being that, instead of a horizon, we have the tip of thespacetime r = β . We note that R ( β ) = − P β ( β − α ) ≤ . (30)In other words, as long as P is non-zero, the spacetime tip is not accessible to the particle.We modify our sketch of R ( r ) for the various cases accordingly to obtain Fig. 4. Thevertical lines, now in green, represent the end-of-spacetime position r = β .Looking Eq. (27b), we should keep in mind that this quantity must be positive due toEq. (17). Since now β > α , this quantity changes sign at β √ α √ α ∓ P √ β − α and 32 α. (31)12 . . . B2 B1 B2A2 A2A1 A3 R ′′ ( r ) < r → ∞ R ′′ ( r ) > r → β R ′′ ( r ) < r → α / r ∗ ← r (a) P = 0 .
1. 0 . . . B1B2 A1A2 A2 A3 R ′′ ( r ) < r ∗ ← r R ′′ ( r ) < r → ∞ R ′′ ( r ) > (b) P = 0 . . . . B2 A1A2 A3 (c) P = 0 .
3. 0 . . . B2A2 A3 R ′′ ( r ) < r → ∞ r ∗ ← r (d) P = 0 . Figure 3: Parameter space for the TS spacetime with α = 1 . β = 2, and various P .The horizontal direction represents L + Q and the vertical direction represents E . Thehorizontal red lines correspond to E = 1 + P .13 ound bound ry (a) Case A1 bound ry (b) Case A2 ry (c) Case A3 bound ry escaping (d) Case B1 escaping ry (e) Case B2 Figure 4: Sketches of y = R ( r ) for various cases of energy and angular momentum,corresponding to cases A1–A3, B1–B2 of the K - E plane of Fig. 3. The vertical greenlines indicate r = β . The allowed domains of r corresponding to R ( r ) ≥ escaping and bound are as definedin the main text. Since there are no horizons in this case, there are no plunging orbits.The root r = β √ α √ α + P √ β − α is beyond r < β , so we need not consider this unphysical location.For the next root, we introduce the notation r ∗ = β √ α √ α − P √ β − α > β. (32)At fixed α and β , either r ∗ is larger or smaller than 3 α/
2, depending on the value of P .We consider each case in turn:If 0 < P < α − β √ α √ β − α , then r ∗ < α . In this case K is positive for β < r < r ∗ and α < r < ∞ . An example of this situation is shown in Fig. 3a. The branch β < r < r ∗ corresponds to the middle branch of Fig. 3a, starting from large positive infinity at r → β ,and going to K = 0 as r → r ∗ . For this branch, we find R ′′ ( r ) < r ∗ < r < α gives negative K so we do not considerit here. Next, increasing r continuously from α , we have a stable branch until r reachesthe critical point r CCO . This point correspond to the sharp cusp in Fig. 3a. After thiscritical point, the remaining branch r CCO < r < ∞ is a branch of stable circular orbits.In summary, for the TS case we have two branches of stable circular orbits, β < r < r ∗ and r CCO < r < ∞ .If α − β √ α √ β − α < P < P crit for some P crit , then r ∗ > α . Here we have a stable branchfrom r ∗ , increasing until a critical point, which we shall denote by r CCO1 . The point14 K ( r CCO1 ) , E ( r CCO1 )) is the upper-right sharp cusp of Fig. 3b. As r increases further, weget the unstable branch which ends at another critical point r CCO2 , where its correspondingpoint is the lower-left cusp in Fig. 3b. After which we reach a stable branch r CCO2 < r < ∞ . As P increases towards P crit , the two critical points r CCO1 and r CCO2 apporach eachother, as can be seen in going from Fig. 3b to Fig. 3c.As P → P crit , the two critical points coalesce. For the example of α = 1 . β = 2, the value is about P crit ≈ . P further beyond that results ina smooth curve separating domains A2 and A3 (see Fig. 3d), and this single branch isstable. This is highly reminiscent to the swallow-tail graphs of thermodynamics depictingphase transitions.Finally, as P increases further towards αβ − α , the single circular orbit branch approachesthe line E = 1 + P , thus shrinking the domain A2. If P continues to increase beyondthat, K in Eq. (27b) becomes negative for any r and no circular orbits can occur. α < β (TS) For β > α , the position r = α is beyond the physical range. Then K only changessign at r ∗ > β . In this case, the situation depicted in Fig. 3a does not exist here. Theremaining sequence of structure as P > P reaches P crit , the unstable branch disappears and the two stable branchmerges into one, shown in Fig. 5d. For the example α = 1, β = 2, the value of the criticalmomentum is about P crit ≈ . In Sec. 3.1, it was shown that the trajectory of the orbits lie on a Poincar´e cone whoseorientation ψ and opening angle χ is given by Eq. (24) and (22), respecively. In thisSection, we shall explore some concrete examples and plot the orbits, thus demonstratingthe geometrical significance of the cone.From Eqs. (24) and (22), we see that the angles ψ and χ are determined by L , Q ,and q . We first construct some examples of circular orbits whose cones lie parallel tothe x -axis. For this we require ψ = 0 or ψ = π . This is achieved when Q = − q . Forconstant r , the requisite values of E and K are determined by Eq. (27a) and (27b). Thenthe appropriate value of L is chosen to satisfy the equation L = K − Q .Starting with q = 0, we have a ‘cone’ with opening angle χ = π , which is simplythe flat x - x plane. When q is increased, the opening angles change in accordance toEq. (22). Some examples are shown in Fig. 6. Next if we wish to consider fixed opening15 . . . . R ′′ ( r ) > r → ∞ R ′′ ( r ) < A2 A1B2 B1 (a) P = 0. 0 . . . . R ′′ ( r ) > r → ∞ R ′′ ( r ) < A2 A3 A1B2 R ′′ ( r ) > r ∗ ← r (b) P = 0 . . . . . A2 A3B2 (c) P = 0 .
05. 0 . . . . R ′′ ( r ) > r → ∞ A2 A3B2 r ∗ ← r (d) P =. Figure 5: Parameter space for the soliton/topological star with α = 1, β = 2, and various P . The horizontal direction represents K = L + Q and the vertical direction represents E . The horizontal red lines correspond to E = 1 + P .16 − − − x − − − x − − − Figure 6: Three spherical orbits of radius r = 15 and momentum P = 0 . α = 2, β = 1. From top to bottom, the values of q are 0,3, and 9. The values of E and K are as calculated from Eq. (27a) and (27b). In eachcase, Q = − q , ensuring the cone is vertical and the orbits are all parallel to the x - x plane. The q = 0 orbit lies on a ‘cone’ of opening angle π , which is simply the x = 0plane. For q = 3 and q = 9, their opening angles are χ = 1 . χ = 0 . − − x − − x − − − (a) Constant r = 15. x − − x − − x − . − . − . − . − . . . . (b) E = √ . K = 16. Figure 7: Orbits around a black hole spacetime of α = 2 and β = 1 of a particle ofcharge q = 3 and momentum P = 0 .
25. From highest to lowest, the angular momentaare L = 2 (blue), L = 3 (green), and L = 4 (orange). For Fig. 7a, the values of E and K are determined from Eq. (27). This time their underlying cones are not plotted to avoidcluttering the figure.angles, Eq. (22) tells us that χ can be fixed if q and K = Q + L are fixed. So varying L will now change the angle of the cone with respect to the x -axis. Some examples of this,with constant and non-constant r are shown in Fig. 7. When α < β we have seen in the previous section that there exist two distinct domains ofbound orbits, which is denoted as Case A1 and sketched in Fig. 4a. Changing the energyand angular momentum may cause two of the roots to coalesce and become complex,leaving a single connected domain of bound orbits, denoted Case A2 and sketched inFig. 4b.
E < √ P ) Let us first consider Case A1 in further detail. In this case, R ( r ) has four real roots,which we denote to have the following order: r ≤ r ≤ r − ≤ r + . (33)In Case A1, we have bound orbits where the particle can exist either in the interval [ r , r ]or [ r − , r + ]. If we place our particle with initial conditions r (0) = r and d r d λ (0) = 0, the18 − − x − − −
20 2 4 6 x − − − − Figure 8: Orbits around a topological star spacetime with α = 1 . β = 2, for a particlewith q = 3, P = 0 . E = √ . K = 6 . Q = 1, and L = √ K − Q . The particlein the inner domain r < r < r is depicted with a blue curve, where r = 2 . r = 2 . r − ≤ r ≤ r + is shown by the bluecurve, where r − = 3 . r + = 8 . r = β = 2.trajectory of the particle is given by the analytical solution r ( λ ) = ( r + − r ) r + ( r − r ) r + sn ( ηλ, p ) r + − r + ( r − r )sn ( ηλ, p ) , (34)where sn ( ψ, k ) is the Jacobi elliptic function of the first kind, and η = 12 p (1 + P − E )( r + − r )( r − − r ) , p = s ( r + − r − )( r − r )( r + − r )( r − − r ) . (35)To describe bound orbits in the outer domain, let us choose the initial conditions r (0) = r − and d r d λ (0) = 0. Then its trajectory will be described by the solution r ( λ ) = ( r + − r ) r − − ( r + − r − ) r sn ( ηλ, p ) r + − r + ( r − − r + )sn ( ηλ, p ) . (36)An example of orbits in these two domains are shown in Fig. 8.In case A2, the function R ( r ) has two real roots, which we denote by r ≤ r . We19 − − x − − −
20 2 4 x − Figure 9: Orbits around a topological star spacetime with α = 1 . β = 2, for a particlewith q = 3, P = 0 . E = √ . K = 7 . Q = −
1, and L = −√ K − Q . The particlemoves in the domain r < r < r where r = 4 . r = 10 . R ( r ) are complex and are given by m ± i n , where m = 2 . n = 0 . r = β = 2.further denote the other pair of complex conjugate roots as m ± i n . Then the function R ( r ) is written as R ( r ) = (1 + P − E )( r − r )( r − r ) (cid:2) ( r − m ) + n (cid:3) . (37)The particle can exist in the domain r ≤ r ≤ r where R ( r ) ≥
0. Choosing initialconditions r (0) = r and the upper sign for Eq. (13e), the analytical solution is r ( λ ) = Br + Ar cot (cid:8) arcsin [sn ( δλ, ρ )] (cid:9) B + A cot (cid:8) arcsin [sn ( δλ, ρ )] (cid:9) , (38)where A = ( m − r ) + n , B = ( m − r ) + n ,δ = p AB (1 + P − E ) , ρ = 12 r − ( A − B ) + ( r − r ) AB . (39)An example of such an orbit is shown in Fig. 920 .2.2 ‘High energy’ bound orbits (
E > √ P ) For particles of ‘high’ energy
E > √ P , bound orbits may still exist for sufficientlysmall P where case B1 exists. In this case, the function R ( r ) has four real roots r ≤ r ≤ r − ≤ r + and is non-negative in the domain r ≤ r ≤ r − . The leading coefficient of R ( r ) is positive. So the function can be written as R ( r ) = (cid:0) E − − P (cid:1) ( r − r )( r − r )( r − − r )( r + − r ) . (40)Choosing initial conditions r (0) = r and the upper sign in Eq. (13e), the analyticalsolution is r ( λ ) = ( r − − r ) r − ( r − − r ) r sn ( ζ λ, b ) r − − r − ( r − − r )sn ( ζ λ, b ) , (41)where ζ = 12 p ( E − P − r + − r )( r − − r ) = i η, b = s ( r − − r )( r + − r )( r + − r )( r − − r )) . (42)An example of such an orbit is shown in Fig. 10. x − . − . − . . . . . . x − . − . − . . . . . x − − Figure 10: Orbits around a topological star spacetime with α = 1 . β = 2, for a particlewith q = 3, P = 0 . E = √ . K = 8 . Q = 1, and L = √ K − Q . The particlemoves in the domain r < r < r − where r = 2 . r − = 2 . r = β = 2. 21 Conclusion
In this paper we have derived and analysed the equations of motion for a charged particlein the magnetic black hole/topological star solution. The angular motion was found tobe similar to analogous systems involving interacting electric and magnetic monopoles.In particular, the electric charge moves along the surface of a Poincar´e cone. The angleand orientation of the cone depends on the charge and the angular momentum, as iswell-known since Poincar´e’s original non-relativistic analysis.On the other hand, the radial motion shows a richer structure of possibilities in thetopological star case. In particular, up to two distinct domains of bound orbits may existfor fixed energy and total angular momentum. On the E - K space, the points repre-senting circular orbits exibit a swallow-tail structure where each branch correspond tostable/unstable circular orbits. When the w -momentum is varied, the swallow-tail kinkdisappears, leaving just a single stable branch. This is highly reminiscent of phase tran-sitions behaviour in thermodynamics, where swallow-tail structures appear in the graphsof intrinsic parameters. Investigating this similarity and the possibility of carrying overthermodynamic concepts into particle mechanics of this spacetime might be a candidateof future study. References [1] V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin,
Geodesicstability, Lyapunov exponents and quasinormal modes , Phys. Rev. D (2009)064016, [ arXiv:0812.1806 ].[2] V. P. Frolov, A. A. Shoom, and C. Tzounis, Spectral line broadening in magnetizedblack holes , JCAP (2014) 059, [ arXiv:1405.0510 ].[3] Event Horizon Telescope
Collaboration, K. Akiyama et al.,
First M87 EventHorizon Telescope Results. I. The Shadow of the Supermassive Black Hole ,Astrophys. J. (2019), no. 1 L1, [ arXiv:1906.11238 ].[4] J. P. Luminet,
Image of a spherical black hole with thin accretion disk , Astron.Astrophys. (1979) 228–235.[5] T. Johannsen, A. E. Broderick, P. M. Plewa, S. Chatzopoulos, S. S. Doeleman,F. Eisenhauer, V. L. Fish, R. Genzel, O. Gerhard, and M. D. Johnson, TestingGeneral Relativity with the Shadow Size of Sgr A* , Phys. Rev. Lett. (2016),no. 3 031101, [ arXiv:1512.02640 ].[6] J.-P. Luminet,
Seeing Black Holes : from the Computer to the Telescope , Universe (2018), no. 8 86, [ arXiv:1804.03909 ].227] S.-W. Wei, Y.-C. Zou, Y.-X. Liu, and R. B. Mann, Curvature radius and Kerr blackhole shadow , JCAP (2019) 030, [ arXiv:1904.07710 ].[8] R. Gregory and R. Laflamme, Black strings and p-branes are unstable , Phys. Rev.Lett. (1993) 2837–2840, [ hep-th/9301052 ].[9] R. Gregory and R. Laflamme, The Instability of charged black strings and p-branes ,Nucl. Phys. B (1994) 399–434, [ hep-th/9404071 ].[10] T. Harmark and N. A. Obers,
Phases of Kaluza-Klein black holes: A Brief review , hep-th/0503020 .[11] B. Kol, The Phase transition between caged black holes and black strings: A Review ,Phys. Rept. (2006) 119–165, [ hep-th/0411240 ].[12] R. Gregory,
The Gregory–Laflamme instability , pp. 29–43. 2012. arXiv:1107.5821 .[13] I. Bah and P. Heidmann,
Topological Stars and Black Holes , arXiv:2011.08851 .[14] I. Bah and P. Heidmann, Topological Stars, Black holes and Generalized ChargedWeyl Solutions , arXiv:2012.13407 .[15] S. Stotyn and R. B. Mann, Magnetic charge can locally stabilize Kaluza–Kleinbubbles , Phys. Lett. B (2011) 269–272, [ arXiv:1105.1854 ].[16] S. Grunau and V. Kagramanova,
Geodesics of electrically and magnetically chargedtest particles in the Reissner-Nordstr¨om space-time: analytical solutions , Phys. Rev.D (2011) 044009, [ arXiv:1011.5399 ].[17] R. M. Gad, Geodesics and geodesic deviation in static charged black holes ,Astrophys Space Sci (2010) 107–114, [ arXiv:0708.2841 ].[18] D. Pugliese, H. Quevedo, and R. Ruffini,
Motion of charged test particles inReissner-Nordstrom spacetime , Phys. Rev. D (2011) 104052, [ arXiv:1103.1807 ].[19] M. Sharif and S. iftikhar, Effects of electromagnetic field on the motion of particlesin dyonic Reissner-Nordstr¨om black hole , Int. J. Mod. Phys D (2017) 1750091.[20] J. S. Schwinger, K. A. Milton, W.-y. Tsai, L. L. DeRaad, Jr., and D. C. Clark, Nonrelativistic Dyon-Dyon Scattering , Annals Phys. (1976) 451.[21] H. Poincar´e,
Remarques sur une exp´erience de M. Birkeland , Compt. Rendus (1896) 530–533. 2322] E. Hackmann, V. Kagramanova, J. Kunz, and C. Lammerzahl,
Analytic solutions ofthe geodesic equation in higher dimensional static spherically symmetricspace-times , Phys. Rev. D (2008) 124018, [ arXiv:0812.2428 ]. [Addendum:Phys.Rev.D 79, 029901 (2009)].[23] D. C. Wilkins, Bound Geodesics in the Kerr Metric , Phys. Rev. D (1972) 814–822.[24] B. Carter, Global structure of the Kerr family of gravitational fields , Phys. Rev. (1968) 1559–1571.[25] Y. Mino,
Perturbative approach to an orbital evolution around a supermassive blackhole , Phys. Rev. D (2003) 084027, [ gr-qc/0302075 ].[26] J. Sivardi´ere, On the classical motion of a charge in the field of a magneticmonopole , Eur. J. Phys. (2000) 183.[27] Y. M. Shnir, Magnetic Monopoles . Text and Monographs in Physics. Springer,Berlin/Heidelberg, 2005.[28] E. Teo,
Spherical orbits around a Kerr black hole , Gen. Rel. Grav. (2021), no. 110, [ arXiv:2007.04022arXiv:2007.04022