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Charged particle motion in Kerr-Newmann space-times

Hongxiao Xu∗and Eva Hackmann†

ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany

The motion of charged test-particles in the gravitational ﬁeld of a rotating and electromagnetically

charged black hole as described by the Kerr-Newman metric is considered. We completely classify

the colatitudinal and radial motion on the extended manifold −∞ ≤ r≤ ∞, including orbits crossing

the horizons or r= 0. Analytical solutions of the equations of motion in terms of elliptic functions

are presented which are valid for all types of orbits.

∗hxu@uni-bremen.de

†eva.hackmann@zarm.uni-bremen.de

2

I. INTRODUCTION

The Kerr-Newman solution to the Einstein-Maxwell equations describes the gravitational ﬁeld of a rotating and

electromagnetically charged stationary black hole [1]. It generalizes both the static and charged Reissner-Nordstr¨om

metric [2, 3] as well as the rotating Kerr metric [4]. The latter is of very high importance not only for general relativity

but also from an astrophysical point of view, as many black hole candidates were found in recent years, which are

expected to rotate. Although it is not very likely that they also carry a net charge, some accretion scenarios were

studied which may create such black holes [5, 6].

On way to explore the gravitational ﬁeld of a Kerr-Newman black hole is to consider the geodesic motion of (charged)

test-particles in this space-time. Already shortly after the discovery of this solution many aspects of the geodesic

motion were studied, among others, timelike equatorial and spherical orbits of uncharged particles [7] and the last

stable orbit of charged particles [8] (see also Sharp [9] and references within). Later the motion of charged particles

was studied by Biˇc´ak et al [10, 11] including, besides some general discussion for the radial motion, a discussion of a

number of special cases like motion along the symmetry axis and circular motion of ultrarelativistic particles. Only

recently, Kov´aˇr et al [12] found oﬀ-equatorial circular orbits of charged particles which are unstable outside the outer

horizon. A comprehensive analysis of photon orbits in Kerr-Newman space-time was presented by Calvani and Turolla

[13] including the extended manifold with negative values of the radial coordinate and naked singularities.

Analogously to the uncharged case, the geodesic equation in Kerr-Newman space-time can be separated by introduc-

ing an additional constant of motion (besides the constants associated to the obvious symmetries of the space-time),

the Carter constant [14], which ensures the integrability of the equations of motions. The resulting structure of the

equations is essentially the same as in Schwarzschild space-time, where the equations of motion can be solved ana-

lytically in terms of elliptic functions as ﬁrst demonstrated by Hagihara in 1931 [15]. However, due to the remaining

coupling of radial and colatitudinal equation, the generalization of his method to Kerr(-Newman) space-time was not

straightforward. This issue was solved by Mino [16] by introducing a new time parameter, often called the Mino time,

which completely decouples the equations of motions and enable a straightforward application of elliptic functions.

This was already used to analytically solve the geodesic equation for bound timelike orbits in Kerr space-time by

Fujita and Hikida [17] and for general timelike and lightlike orbits in Kerr space-time in [18].

In this paper, we will discuss the geodesic motion of charged test-particles in Kerr-Newman black hole space-

times. For the sake of completeness, we will include a magnetic charge of the black hole which was not done in

the cited references but has interesting eﬀects on the colatitudinal motion. After introducing the relevant notations

and equations of motion in the next section, we proceed with a complete classiﬁcation of timelike orbits of (charged)

particles in Kerr-Newman space-time. This includes oﬀ-equatorial orbits, trajectories crossing the horizons, and orbits

with negative values of the radial coordinate. In the fourth section, we will present analytical solutions in terms of

elliptic functions dependent on the Mino time for all coordinates. The paper is closed by a summary and conclusion.

II. GEODESICS IN KERR-NEWMAN SPACE-TIME

A. Kerr-Newman space-time

The Kerr-Newman spacetime is a stationary, axisymmetrical, and asymptotically ﬂat solution of the Einstein-

Maxwell equation

Gµν =−2gαβFµα Fνβ −1

4gµν Fαβ Fαβ ,

where Gµν is the Einstein tensor and Fµν the electromagnetic tensor. Throughout, the units are selected such that

c= 1 for the speed of light and G= 1 for the gravitational constant. In Boyer-Lindquist coordinates the metric takes

the form

ds2=ρ2

∆dr2+ρ2dθ2+sin2θ

ρ2(r2+a2)dφ −adt2−∆

ρ2asin2(θ)dφ −dt2(1)

with

ρ2(r, θ) = r2+a2cos2θ , (2)

∆(r) = r2−2Mr +a2+Q2+P2,(3)

and M > 0 the mass, athe speciﬁc angular momentum, Qthe electric, and Pthe magnetic charge of the gravitating

source. (The existence of magnetic charges has not been proven yet but it will still be considered for the sake of

3

completeness.) We restrict ourselves here to the case that two horizons exist, given by the coordinate singularities

∆(r) = 0, r±=M±pM2−a2−Q2−P2. The only genuine singularity is for ρ2= 0, where r= 0 and θ=π/2 is

fulﬁlled simultaneously. This means that a test particle approaching r= 0 from above or below the equatorial plane

does not terminate at r= 0 as it would in Schwarzschild space-time but continues to negative values of r. For large

negative values of rthis can be interpreted as a “negative universe”, see [19].

The Kerr-Newman metric reduces to the Kerr metric for Q=P= 0 describing the exterior of rotating non charged

black holes. It reduces to the Reissner-Nordstr¨om metric for a= 0 which describes the exterior of a non rotating but

charged black hole. In the case P=Q=a= 0 the Kerr-Newman metric is reduced to the Schwarzschild metric.

The electromagnetic potential is given by

A=Aνdxν=Qr

ρ2(dt −asin2θdφ) + 1

ρ2Pcos θadt −(r2+a2)dφ.(4)

from which the electromagnetic tensor can be calculated by F=1

2(∂νAµ−∂µAν)dxµ∧dxν. By the interchanges

Q→P,P→ −Qthe electromagnetic potential ˇ

Aof the dual electromagnetic tensor ˇ

Fcan be obtained.

B. Equations of motion

The equations of motion for a test particle of normalized mass ǫ, electric charge e, and magnetic charge hcan be

obtained by the Hamiltonian

H=1

2gµν (πµ+eAµ+hˇ

Aµ)(πν+eAν+hˇ

Aν) (5)

where πµdescribe the general momenta. By introducing

ˆ

Q=eQ +hP

√e2+h2,ˆ

P=eP −hQ

√e2+h2,ˆe=pe2+h2(6)

the Hamiltonian can be reduced to

H=1

2gµν (πµ+eAµ)(πν+eAν),(7)

where all quantities of the this equation refer to ˆ

Q,ˆ

P, and ˆe. Therefore, the discussion of a test particle without

magnetic charge is suﬃcient.

We can obtain three constants of motion directly, since Hdoes not depend on τ,φ, or t. The ﬁrst, ǫ2=−gνµ ˙xν˙xµis

the normalization condition with ǫ= 1 for timelike and ǫ= 0 for lightlike trajectories. (The dot denotes diﬀerentiation

with respect to an aﬃne parameter τ.) The second and third equation

E=−πt=−gtt ˙

t−gtφ ˙

φ+eAt,(8)

L=πφ=gφt ˙

t+gφφ ˙

φ−eAφ,(9)

describe the conservation of energy Eand angular momentum in zdirection, respectively. A fourth constant of motion

can be obtained by considering the Hamilton-Jacobi equation

−∂τS=1

2gµν (∂µS+eAν)(∂νS+eAµ).(10)

With the ansatz S=1

2ǫτ −Et +Lφ +S1(r) + S2(θ) it can be shown that the Hamilton-Jacobi equation indeed

separates with the Carter constant Kas separation constant, see [14].

With these four constants the equations of motion become

dθ

dγ 2

=¯

K−ǫ¯a2cos2θ−T2(θ)

sin2θ=: Θ(θ),(11)

d¯r

dγ 2

=R2(¯r)−(ǫ¯r2+¯

K)¯

∆(¯r) =: R(¯r),(12)

dφ

dγ =¯aR(¯r)

¯

∆(¯r)−T(θ)

sin2θ,(13)

dt

dγ =(¯r2+ ¯a2)R(¯r)

¯

∆(¯r)−¯aT(θ),(14)

4

where

R(¯r) = (¯r2+ ¯a2)E−¯a¯

L−e¯

Q¯r , (15)

T(θ) = ¯aE sin2θ−¯

L+e¯

Pcos θ . (16)

All quantities with a bar are normalized to M, i.e. x= ¯xM for x=r, a, L, Q, P as well as K=¯

KM2and ¯

∆(¯r) =

¯r2−2¯r+ ¯a2+¯

Q2+¯

P2. Here γis the normalized Mino time [16] given by dγ =Mρ−2dτ with the eigentime τ.

III. CLASSIFICATION OF MOTION

In this section we will classify the types of orbits in terms of colatitude and radial motion. We will analyze which

orbit conﬁguration, i.e. which set of orbit types, may appear for given parameters ¯a, ¯

Q, ¯

P , E , ¯

L, ¯

K, e and which region

in parameter space a given orbit conﬁgurations occupies. Here we assume that ǫ= 1, that is, we restrict ourselves to

test particles with mass, but the discussion for light may be done analogously. The whole analysis will be based on

the conditions Θ(θ)≥0 and R(¯r)≥0 which are necessary for geodesic motion.

For both colatitudinal and radial motion, we will ﬁrst give some general properties as symmetries and notation of

orbit types. We then proceed with the determination of possible orbit conﬁgurations, i.e. sets of orbit types which

are possible for given parameters (for more than one possible orbit type the actual orbit is determined by initial

conditions). Each orbit conﬁguration covers a particular region in the parameter space. Finally, we will analyze how

these regions look like and determine their boundaries in parameter space.

A. Colatitudinal motion

The coordinate θmay only take a speciﬁc value θ0∈[0, π] if Θ(θ0)≥0 is valid. We will analyze in the following

whether this is fulﬁlled for a given parameter set.

1. General properties

First, we notice that Θ does not depend on ¯

Qand that eand ¯

Ponly appear as the product e¯

P. The function Θ

has the following symmetries:

•A change of sign of e¯

Phas the same eﬀect as reﬂecting θat the equatorial plane: Θ|−e¯

P(θ) = Θ|e¯

P(π−θ). In

particular is Θ symmetric with respect to the equatorial plane if e¯

P= 0: Θ|e¯

P=0(θ) = Θ|e¯

P=0(π−θ).

•A simultaneous change of sign of ¯

Land Eresult in a reﬂection at the equatorial plane: Θ|−¯

L,−E(θ) = Θ|¯

L,E (π−θ)

and Θ¯

L=0=E(θ) = Θ¯

L=0=E(π−θ).

Therefore, we assume w.l.o.g. e¯

P≥0 and E≥0 in the following. The condition Θ(θ)≥0 also shows that ¯

K≥0 is

a necessary condition for geodesic motion as all other (positive) terms are subtracted.

The Carter constant also encodes some geometrical information if considered in its alternative form ¯

C=¯

K−(¯aE −

¯

L)2. Because of Θ(π/2) = ¯

K−(¯aE −¯

L)2=¯

Ca particle may only cross or stay in the equatorial plane if ¯

C≥0.

For equatorial orbits even ¯

C= 0 is necessary as dθ

dγ π

2= 0 needs to be fulﬁlled. For ¯

K= 0 geodesic motion is only

possible if ¯a=¯

L= 0 or in the equatorial plane with ¯aE =¯

L.

The function Θ(θ) constains a term which diverges for θ= 0, π given by −(¯

L−e¯

Pcos θ)2

sin2θ. This fact suggests to

distinguish between two cases:

•¯

L6=±e¯

P: In this case Θ(θ)→ −∞ for θ→0, π, that is, the north or south pole will never be reached.

•¯

L=±e¯

P: In this case Θ(θ)→¯

K−¯a2for θ→0 and ¯

L=e¯

Pas well as for θ→πand ¯

L=−e¯

P. Therefore,

a particle with ¯

L=e¯

Pmay reach the north pole and a particle with ¯

L=−e¯

Pthe south pole if in addition

¯

K≥¯a2. For the subcase ¯

L= 0 = e¯

Pboth north and south pole may be reached if ¯

K≥¯a2.

If the parameters are such that the poles θ= 0, π can not be reached it is convenient to consider ν= cos θinstead of

θ. In terms of νthe diﬀerential equation for colatitudinal motion reads

dν

dγ 2

=

4

X

i=0

biνi=: Θν(ν),(17)

5

where b0=¯

K−(¯

L−¯aE)2=¯

C,b1= 2e¯

P(¯

L−¯aE), b2=−¯

K−¯a2−e2¯

P2+ 2¯a2E2−2¯aE ¯

L,b3= 2¯aEe ¯

P, and

b4= ¯a2(1 −E2).

For a given set of parameters of the space-time and the particle diﬀerent types of orbits may be possible. We call

an orbit

•northern or N, if it stays in the northern hemisphere θ < π/2,

•normal or E, if it crosses or stays in the equatorial plane θ=π/2,

•southern or S, if it stays in the southern hemisphere θ > π/2.

Equatorial orbits with θ≡π/2 are a special case of normal orbits. In addition to the above notions we add an index

Nif the north pole θ= 0 and Sif the south pole θ=πis reached, for example NNfor a northern orbit reaching the

north pole.

2. Orbit conﬁgurations

Let us now analyze which orbit conﬁgurations, i.e. which sets of the above introduced orbit types, are possible

for given parameters. We use the necessary condition for colatitudinal motion Θ(θ)≥0 for this, which implies to

analyze the occurrence of real zeros of Θ in [0, π] and the behavior of Θ at the boundaries θ= 0, π giving the sign of

Θ between its zeros. (The orbit a test particle with the given parameters actually follows in a space-time with the

given parameters depends on the initial values.)

(A) Case ¯

L6=±e¯

PHere Θ(θ) = −∞ at θ= 0, π which implies that Θνhas an even number of zeros in (−1,1)

(counted with multiplicity). If Θνhas no real zeros there, no colatitudinal motion is possible, which gives a restriction

to the permitted sets of parameters for geodesic motion. In the case of 2 real zeros there is a single orbit of type N,

E, or S, which is stable at a constant θif the 2 zeros coincide. If all 4 zeros of Θνlie in (−1,1) all combinations of

two orbit types except EE are possible. For two or more coinciding zeros this point is stable if it is a maximum of

Θνand unstable otherwise.

(B) Case ¯

L=±e¯

PHere we have to consider four subcases: (B1) ¯

K < ¯a2: The same orbit types as in (A) are

possible. (B2) ¯

K > ¯a2,¯

L6= 0: In this case Θ has diﬀerent signs at θ= 0, π which implies that Θ has an odd number

of real zeros in (0, π). For 1 real zero there is one orbit of type NNor ENfor ¯

L=e¯

Pand one of type SSor ESfor

¯

L=−e¯

P. If Θ has 3 real zeros in (0, π ) there is one additional orbit not reaching a pole. (B3) ¯

K > ¯a2,¯

L= 0 = e¯

P:

Here Θ >0 at θ= 0, π, i.e. Θ has an even number of zeros in (0, π). Also, for e¯

P= 0 the function Θ is symmetric with

respect to the equatorial plane. For no real zeros there is one orbit of type ENS which reaches both poles, and for 2

real zeros an NNand an SSorbit. More zeros in [0, π ] are not possible. (B4) ¯

K= ¯a2: Here an orbit with constant

θ= 0 (θ=π) is possible for ¯

L=e¯

P(¯

L=−e¯

P). For ¯

L= 0 no other than the two constant orbits are possible but

for ¯

L6= 0 it is Θ → −∞ at the other boundary. In the latter case, if the orbit is stable, there may be one additional

orbit of type E,N, or S.

For an overview of this diﬀerent orbit conﬁguration see table I.

3. Regions of orbit conﬁgurations in parameter space

It is now of interest for which sets of parameters a given orbit conﬁguration changes. As Θ ≥0 is necessary for

geodesic motion, this happens if the behavior of Θ at the boundaries changes, which means a switch from one of the

above cases (A),(B1), ..., (B4) to another, or if the number of real zeros of Θ changes. The latter occurs at that

parameters for which Θ has multiple zeros. With these two conditions the diﬀerent regions of orbit conﬁgurations in

parameter space can be completely determined. The ﬁrst condition was already analyzed above.

For θ∈(0, π) the function Θ has the same zeros as the polynomial Θνand we may use Θνinstead of Θ for all

orbits not reaching θ= 0, π. If ¯

L=±e¯

Pthen ν0=±1 is a zero of Θνbut does not correspond to a turning point of

the colatitudinal motion. If in addition ¯

K= ¯a2then ν0=±1 is a double zero of Θνand θ= 0, π a simple zero of Θ,

which does correspond to a turning point of θ. Keeping this in mind we will also use Θνfor these cases but discuss

the occurence of multiple zeros at θ= 0, π separatly using Θ.

The condition for a double zero ν0is dΘν

dν (ν0) = 0 = Θν(ν0). This can be read as 2 conditions on 2 of the 5

parameters E,¯

L,¯

K,e¯

P, and a. Solving these two conditions for Eand ¯

Ldependent on the position of the double

6

zeros range of θ∈[0, π] types of orbits

2N

2E

2S

4N,N

4N,E

4N,S

4E,S

4S,S

(A) ¯

L6=±e¯

P; (B1) ¯

L=±e¯

P,¯

K < ¯a2

zeros range of θ∈[0, π] types of orbits

1NN

1EN

3NN,N

3NN,E

3NN,S

3EN,S

(B2) ¯

L=e¯

P,¯

L6= 0, ¯

K > ¯a2

zeros range of θ∈[0, π] types of orbits

0EN,S

2NN,SS

(B3) ¯

L= 0 = e¯

P,¯

K > ¯a2

zeros range of θ∈[0, π] types of orbits

2NN

3NN

3EN

4NN,N

4NN,E

4NN,S

(B4) ¯

L=e¯

P,¯

L6= 0, ¯

K= ¯a2

zeros range of θ∈[0, π] types of orbits

4NN,SS

(B4) ¯

L= 0 = e¯

P,¯

K= ¯a2

TABLE I: Overview of diﬀerent orbit conﬁgurations for colatitudinal motion. The vertical bar of the second column

denotes θ=π/2 and the thick lines Θ ≥0, i.e. regions where a motion is possible. Dots represent single zeros and

circles double zeros. If zeros merge, the resulting orbits are stable if a line is reduced to a point and unstable if lines

merge. The conﬁgurations (B2) and (B4) with ¯

L=−e¯

Pare obtained by a reﬂection at the equatorial plane.

zero ν0and the other parameters yields

E1,2=e¯

P

2¯aν0±1

2pν2

0(1 −ν2

0)( ¯

K−¯a2ν2

0)( ¯

K+ ¯a2−2¯a2ν2

0)2

(ν2

0−1)( ¯

K−¯a2ν2

0)¯aν0

,(18)

¯

L1,2=e¯

P(ν2

0+ 1)

2ν0±1

2

ν0(1 −ν2

0)( ¯

K−¯a2)( ¯

K+ ¯a2−2¯a2ν2

0)

pν2

0(1 −ν2

0)( ¯

K−¯a2ν2

0)( ¯

K+ ¯a2−2¯a2ν2

0)2.(19)

The expressions for Eand Ldiverge at ν0= 0 for e¯

P6= 0, at ν0=p¯

K/¯a2, and at ν0=±1 for E, what suggests to

consider the 2 conditions for double zeros directly for these points:

Equatorial orbits For ν0= 0 the 2 conditions imply that either ¯

K= 0 and ¯

L= ¯aE or ¯

K= (¯aE −¯

L)2and e¯

P= 0

are necessary and suﬃcient for the existence of equatorial orbits. The asymptotic behavior of Eand ¯

Lat ν0= 0 also

displays these conditions,

E1,2=e¯

P

2¯aν0∓¯

K+ ¯a2

2¯a√¯

K+O(ν2

0),(20)

¯

L1,2=e¯

P

2ν0±¯

K−¯a2

2√¯

K+O(ν0) = ¯aE1,2±p¯

K+O(ν0).(21)

In the case ¯

K= 0, ¯

L= ¯aE the equatorial orbit is the only possible geodesic orbit (for Θ 6≡ 0) and, thus, stable,

whereas for ¯

K= (¯aE −¯

L)2,e¯

P= 0 the sign of A:= ¯a2(E2−1) −¯

L2has to be considered: The orbit is stable if A≤0

(for Θ 6≡ 0) and unstable if A > 0. In the case A= 0 the zero θ=π/2 is even fourfold. For ¯

K= ¯a2,e¯

P= 0 = ¯

L, and

E2= 1 the function Θ is identical to zero.

Orbits with θ≡0, π Geodesic motion along the axis θ= 0, π is possible only if θ= 0, π is a double zero of Θ. The

2 conditions on double zeros show that ¯

L=e¯

Pand ¯

K= ¯a2are necessary and suﬃcient for θ≡0, and ¯

L=−e¯

P,

¯

K= ¯a2for θ≡π. This can also be seen by considering the asymptotic behaviour of ¯

L1,2and E1,2as ν0approaches

±1: it is given by limν0→1¯

L1,2→e¯

Pand limν0→−1¯

L1,2=−e¯

Pwhereas E1,2diverges for ¯

K6= ¯a2.

7

Let us discuss the stability of the orbits θ≡0, π : The orbits are unstable if ¯a2−(¯aE −¯

L/2)2>0 and stable if

¯a2−(¯aE −¯

L/2)2<0. For ¯a2−(¯aE −¯

L/2)2= 0 the poles θ= 0, π are fourfold zeros and the orbit θ≡0 is stable

if E=¯

L

2a−1, and the orbit θ≡πif E=¯

L

2a+ 1. In the special case of ¯

L= 0 = e¯

Pthe two orbits are unstable if

E2<1 and stable if E2>1. For ¯

L= 0 = e¯

P,¯

K= ¯a2, and E2= 1 again Θ ≡0.

Orbits with θ≡ ±p¯

K/¯a2The singularity ν0=±p¯

K/¯a2is located in (−1,1) and is not equal to zero only if

0<¯

K < ¯a2. Assuming this, orbits with constant cos θ≡ ±p¯

K/¯a2can exist only if ¯a¯

L−(¯a2−¯

K)E=±e¯

P√¯

Kis

fulﬁlled. This can be infered from the asymptotes of ¯

L1,2in terms of E1,2around ν0=±p¯

K/¯a2,

¯

L1,2=¯a2−¯

K

¯aE1,2±e¯

P√¯

K

¯a+O

sν∓r¯

K

¯a2

.(22)

The expressions (18) and (19) depend linearly on e¯

Pand a rescaling of parameters ( ¯

L/¯a,¯

K/¯a2,e¯

P /¯a) removes

the rotation parameter ¯acompletely from the equations. Only the dependence on ¯

Kis not obvious. If we solve the 2

conditions dΘν

dν (ν0) = 0 = Θν(ν0) for Eand ¯

Kinstead of ¯

Lthis yields

E1,2=e¯

P

2¯aν0±1

2p(e¯

P(ν2

0+ 1) −2¯

Lν0)2+ 4ν2

0¯a2(ν2

0−1)2

¯aν0(ν2

0−1) (23)

¯

K1,2= ¯a2−e¯

P(ν2

0+ 1) −2ν0¯

L

2ν2

0(ν2

0−1) (e¯

P(ν2

0+ 1) −2ν0¯

L)±q(e¯

P(ν2

0+ 1) −2¯

Lν0)2+ 4ν2

0¯a2(ν2

0−1)2(24)

Note that E1and ¯

K1behave regular at ν0= 0 and ¯aE1=¯

L+O(ν0), ¯

K1=O(ν2) whereas E2and ¯

K2are ﬁnite

at ν0= 0 only if e¯

P= 0. Then ¯

K→(¯aE −¯

L)2as expected from the analysis of equatorial orbits above. Also, for

ν0→ ±1 the conditions ¯

L=±e¯

P,¯

K→¯a2are recovered.

Triple zeros are also of interest as they correspond to parameters where a stable orbit with constant θbecomes

unstable and vice versa. Therefore, will will study them here. The 3 conditions for such points are 0 = d2Θν

dν2(ν0) =

dΘν

dν (ν0) = Θν(ν0) which we read as 3 conditions on E,¯

L, and e¯

Pyielding

E1,2=±1

2

2ν6

0¯a4+ 6ν2

0¯

K¯a2−(3ν4

0¯a2+¯

K)( ¯

K+ ¯a2)

p(1 −ν2

0)( ¯

K−¯a2ν2

0)(1 −ν2

0)( ¯

K−¯a2ν2

0)¯a,(25)

¯

L1,2=∓1

2p(1 −ν2

0)( ¯

K−¯a2ν2

0)( ¯

K−¯a2)(¯a2ν6

0−3ν4

0¯a2+ 3ν2

0¯

K−¯

K)

(ν2

0−1)2(¯

K−¯a2ν2

0)2,(26)

e¯

P1,2=±ν3

0(¯

K−¯a2)2

p(1 −ν2

0)( ¯

K−¯a2ν2

0)(ν2

0−1)( ¯

K−¯a2ν2

0).(27)

In particular, triple zeros are also double zeros and the asymptotic behavior of ¯

Las a function of Eand ¯

Kat the

singularities ν0=±p¯

K/¯a2is determined by (22). The points ν0=±1 were studied above but we note here that

θ= 0, π is a double zero of Θ if ν=±1 is a triple zeros of Θν.

Besides the equatorial orbits and orbits with constant θ= 0, π discussed above, fourfold or even higher order zeros

are only possible for parameters corresponding to Θ ≡0, which are ¯a=e¯

P=¯

L=¯

K= 0 with arbitrary Eor E=±1,

¯

K= ¯a2, and ¯

L=e¯

P= 0.

The results of this section are visualized in ﬁgures 1 to 4. As the ﬁve dimensional parameter space can not be

completely pictured we have to ﬁx at least two parameters. For this we choose ¯a, which can be removed from equations

(18), (19), (23), and (24) by a rescaling of parameters, and e¯

P, which enters only linearly in (18) an (19). As a three

dimensional plot of E,¯

L, and ¯

Kis still confusing we present two dimensional plots of ¯

Lover Eand ¯

Kover E. Special

cases to be discussed are then e¯

P= 0, ¯

L=±e¯

P,¯

K= ¯a2, and ¯

K= 0. For the latter geodesic motion is only possible

on a stable orbit of constant θ(if not Θ ≡0), more precise on the equator for ¯

L= ¯aE or on e¯

Pcos θ=¯

Lfor ¯a= 0.

In each plot, we use the following conventions:

•Solid lines indicate double zeros of Θ which correspond to stable or unstable orbits of constant θ. We use red

lines for orbits with constant θ < π

2, blue for constant θ > π

2, and green for equatorial orbits.

•Dashed lines denote orbits with turning points at the equatorial plane and are given by ¯

L= ¯aE ±√¯

K. They

also mark the transition from Nor Sto Eorbits and are asymptotically approached by solid lines corresponding

to orbits with constant θnear the equatorial plane.

8

¯

K= 3, e¯

P= 11.25 ¯

K= 0.5, e¯

P= 0.3

FIG. 1: Orbit conﬁgurations for the colatitudinal motion with ¯a= 0.5, ¯

K > ¯a2, and e¯

P > 0. For a general

description see the text. The orbit conﬁgurations on the solid lines contain an orbit with constant θ. They are

unstable if marked by the red solid line starting at the dot and approaching the dashed line, and stable

otherwise.The dash dotted line at ¯

L=e¯

P(¯

L=−e¯

P) denotes an orbit crossing the north (south) pole and

corresponds to the orbit conﬁgurations (B2). The other regions correspond to the conﬁgurations (A). Small plots on

the top are enlarged details of the lower plot.

•Dash dotted lines correspond to orbits which cross a pole. They only appear in ¯

Lover Eplots for ¯

K≥¯a2

and are located at ¯

L=±e¯

P. In this case they are asymptotes to the solid lines corresponding to orbits with

constant θnear a pole.

•Dotted lines are asymptotes to solid lines. They do NOT separate diﬀerent orbit conﬁgurations. For ¯

Lover E

plots they only appear for ¯

K < ¯a2and are approached for ν0→ ±√¯

K/¯a. If ¯

Kis plotted versus Ethey are

approached for ν0→ ±1.

•Single dots mark triple zeros which separate stable from unstable orbits.

•The labels N,E, and Sindicate the orbit conﬁgurations summarized in table I. Solid, dashed, and dash dotted

lines indicate transitions from one orbit conﬁguration to another. Here dash dotted lines are special in the sense

that on both sides there are always the same orbit conﬁgurations but only the line itself corresponds to another

conﬁguration. Regions where a colatitudinal motion is forbidden are marked with ‘none‘.

Note that we plot usually only positive values of Eas negative values can be obtained by E→ −E,¯

L→ −¯

Land a

reﬂection at the equatorial plane (e.g. an Norbit becomes an Sorbit).

9

(a) ¯

K= ¯a2(b) ¯

K= 0.2

FIG. 2: Orbit conﬁgurations for the colatitudinal motion with ¯a= 0.5, ¯

K≤¯a2, and e¯

P= 0.3. For a general

description see the text. The orange solid lines in (a) correspond to orbits of constant θ= 0, π, which are unstable

for energies less than the one marked by the dots. These orbits correspond to the conﬁgurations (B4), other regions

(in both plots) to the conﬁgurations (A) or (B1). In both plots, the orbits labeled by the red solid line starting at

the dot and approaching the dashed line are unstable. All other orbits on solid lines are stable. Small plots on the

top are enlarged details of the lower plot.

B. Radial motion

The discussion of the radial motion will be analogous to that of the θ-motion. An orbit can only take a speciﬁc

value in [−∞,∞] if the radicand of the right hand side of eq. (12) is larger than or equal to zero at that point, i.e. if

R(¯r) = ((¯r2+ ¯a2)E−¯a¯

L−e¯

Q¯r)2−(ǫ¯r2+¯

K)(¯r2−2¯r+ ¯a2+¯

Q2+¯

P2)≥0.

We will analyze for which values of the parameters E,¯

L,¯

K,e,¯

Pand ¯

Qthis inequality is satisﬁed.

1. General properties

The three parameters e,¯

Q, and ¯

Pappear only in the two combinations e¯

Q=: Qand ¯

Q2+¯

P2=: P2which may be

considered instead. The function Rhas the following symmetries:

•It depends quadratically on ¯

P(and P): R|−¯

P=R|¯

P(R|−P =R|P).

•A simultaneous change of sign of ¯

Qand eresults in the same motion: R|−¯

Q,−e=R|¯

Q,e.

•Also, a simultaneous change of sign of ¯

L,Eand e(or Q) results in the same motion: R|−E,−¯

L,−e=R|E, ¯

L,e

(R|−E,−¯

L,−Q =R|E, ¯

L,Q).

10

(a) ¯

K= 0.5 (b) ¯

K= 0.2

FIG. 3: Orbit conﬁgurations for the colatitudinal motion with ¯a= 0.5 and e¯

P= 0. For a general description see the

text. The green solid lines correspond to equatorial orbits which are unstable from dot to larger ¯

Land else stable.

The black solid line marks two stable orbits of constant θ6=π

2which are symmetric with respect to the equatorial

plane. The dash dotted line in (a) marks the case (B3) with an EN S orbit between the green solid lines and an

NNSSconﬁguration else.

Thus it suﬃces to consider ¯

P≥0, ¯

Q≥0, and e≥0 (or, equivalently, P ≥ 0 and Q ≥ 0). Note that ¯

K≥0 was a

necessary condition for the colatitudinal motion to be possible at all and, therefore, this condition remains valid.

The sign of ¯

C=¯

K−(¯aE −¯

L)2again encodes some geometrical information. At ¯r= 0 the polynomial R(0) =

−(¯

Q2+¯

P2)(¯

L−¯aE)2−(¯a2+¯

Q2+¯

P2)¯

Ccan only be positive if ¯

C≤0. Since ¯

C≥0 needs to be satisﬁed for an

orbit to reach the equatorial plane this implies that ¯

C= 0 is necessary for an orbit to hit the ring singularity. For

¯

Q2+¯

P26= 0 also ¯

L−¯aE = 0 is neccessary to hit the singularity and, thus, also ¯

K= 0.

The zeros of the parabola ¯

∆(¯r) are the horizons ¯r±= 1 ±p1−¯a2+¯

Q2+¯

P2with ¯

∆(¯r)<0 in between. Therefore,

the polynomial R(¯r) is always postive for ¯r∈[¯r−,¯r+]. This implies that there can not be any turning points or

spherical orbits between the horizons. There is a turning point at a horizon if R(¯r±) = 0 (see eq. (15)).

For a given set of parameters of the space-time and the particle diﬀerent types of orbits may be possible, for which

we use the terminologies

•transit or T, if ¯rstarts at ±∞ and ends at ∓∞,

•ﬂyby or F, if ¯rstarts and ends at +∞or −∞,

•bound or B, if ¯rremains in a ﬁnite interval [¯rmin,¯rmax].

We add an index +, 0, or −to a ﬂyby or bound orbit if it stays at ¯r > 0, crosses ¯r= 0, or stays at ¯r < 0. Also, a

superscript ∗will be added if the orbit crosses the horizons, i.e. contains the interval [¯r−,¯r+]. For example, the orbit

F∗

+comes from inﬁnity, crosses the two horizons, turns at some 0 <¯r < ¯r−, and goes back to inﬁnity. If more than

one orbit is possible for a given set of parameters, the actual orbit of the test particle is determined by the initial

conditions.

2. Orbit conﬁgurations

We will now analyze which sets of the above introduced orbit types are possible for given parameters. Geodesic

motion is only possible in regions of R(¯r)≥0 and, therefore, the possible orbit conﬁgurations are fully determined

by the number of real zeros of the polynomial Rand its sign at ±∞. The latter is determined by the sign of E2−1

(or the lower order coeﬃcients if E2= 1, what we will consider separately). This suggest to introduce the following

classes of orbit conﬁgurations.

11

(a) ¯

L= 0.5 (b) ¯

L=e¯

P

FIG. 4: Orbit conﬁgurations for the colatitudinal motion with ¯a= 0.5 and e¯

P= 0.3. For a general description see

the text. For both plots, solid lines with dashed asymptotes correspond to unstable and all other to stable orbits. In

(b) the regions above the solid orange line at ¯

K= ¯a2correspond to the conﬁgurations (B2) and below to (B1). The

line itself corresponds to (B4) and covers all possible conﬁgurations (leftmost corresponds to topmost in table I)

with the exception of the NNNconﬁguration which is only possible for large values of e¯

P.

(I) Case E2>1 Here R(¯r)→ ∞ for ¯r→ ±∞ and Rmay have none, two, or four real zeros. For no zeros there

is a transit orbit, for two zeros there are two ﬂyby orbits, and for four zeros there are two ﬂyby and a bound orbit.

(II) Case E2<1 For such energies a test particle can not reach ±∞,R(¯r)→ −∞ for ¯r→ ±∞. As R(¯r)>0

between the horizons it has at least 2 real zeros and there is always one bound orbit crossing the horizons. If Rhas

four real zeros there is an additional bound orbit.

(III) Case E2= 1 Here the behaviour of Rat inﬁnity depends on the sign of 1 −e¯

Q. For 1 −e¯

Q > 0 it is

R(¯r)→ ±∞ for ¯r→ ±∞ and the other way around for 1 −e¯

Q < 0. In both cases Rhas one or three real zeros. For

one real zero there is a ﬂyby orbit which crosses the horizons and for three real zeros there is a ﬂyby and a bound

orbit. The ﬂyby orbit reaches +∞for 1 −e¯

Q > 0 and −∞ for 1 −e¯

Q < 0. If also 1 −e¯

Q= 0 one has to consider the

sign of the second order coeﬃcient and so on.

For an overview of this diﬀerent orbit conﬁguration for the radial motion see table II.

3. Regions of orbit conﬁgurations in parameter space

After considering the possible orbit conﬁgurations for the radial motion we will now ﬁnd the sets of parameters for

which a given orbit conﬁguration changes to another. The orbit conﬁgurations are fully determined by the signs of

R(±∞) as categorized above and the number of real zeros of R, which changes if two zeros of Rmerge. The latter

occurs if the two conditions on double zeros R(¯r0) = 0 and dR

d¯r(¯r0) = 0 are fulﬁlled. Read as two conditions on Eand

12

real zeros range of ¯rtypes of orbits

0 T

2F∗

0,F+

2F0,F∗

+

2F−,F∗

+

2F−,F∗

0

4F∗

0,B+,F+

4F0,B∗

+,F+

4F−,B∗

+,F+

4F−,B∗

0,F+

4F0,B+,F∗

+

4F−,B+,F∗

+

4F−,B0,F∗

+

4F−,B−,F∗

+

4F−,B−,F∗

0

(I) E2>1

real zeros range of ¯rtypes of orbits

1F∗

0

3F−,B∗

0

3F−,B∗

+

3F0,B∗

+

3F∗

0,B+

(III) E2= 1, e¯

Q > 1

real zeros range of ¯rtypes of orbits

2B∗

+

2B∗

0

4B∗

+,B+

4B∗

0,B+

4B+,B∗

+

4B0,B∗

+

4B−,B∗

+

4B−,B∗

0

(II) E2<1

real zeros range of ¯rtypes of orbits

1F∗

+

1F∗

0

3B∗

+,F+

3B∗

0,F+

3B+,F∗

+

3B0,F∗

+

3B−,F∗

+

3B−,F∗

0

(III) E2= 1, e¯

Q < 1

TABLE II: Overview of diﬀerent orbit conﬁgurations for radial motion. The vertical bars of the second column mark

¯r= 0, ¯r= ¯r−, and ¯r= ¯r+(from left to right). The dots represent the real zeros of R(turning points) and the thick

lines R≥0, i.e. regions where a motion is possible. If zeros merge, the resulting orbits are stable if a line is reduced

to a point and unstable if lines merge.

¯

Lthis implies

E1,2=e¯

Q

2¯r0±1

2p¯

∆(¯r0)(¯r2

0+¯

K)(¯r0¯

∆(¯r0) + (¯r0−1)( ¯

K+ ¯r2

0))2

¯

∆(¯r0)(¯r2

0+¯

K)¯r0

,(28)

¯

L1,2=e¯

Q(¯a2−¯r2

0)

2¯a¯r0±1

2p¯

∆(¯r0)(¯r2

0+¯

K)(¯r0¯

∆(¯r0) + (¯r0−1)( ¯

K+ ¯r2

0))2

¯

∆(¯r0)(¯r2

0+¯

K)(¯r0¯

∆(¯r0) + (¯r0−1)( ¯

K+ ¯r2

0))¯r0¯a

×(¯r0¯

∆(¯r0)(¯a2−¯

K) + ( ¯

K+ ¯r2

0)(¯r2

0−¯r0(¯

P2+¯

Q2)−¯a2)) .(29)

These expressions diverge at ¯r0= 0 and at the horizons ¯r0= ¯r±. We consider these points separately, see below. In

addition, ¯

Ldiverges also at ¯r0=±∞,¯

L1,2→±sign(¯r0)−e¯

Q

2¯a¯r0+O(1) for ¯r0→ ±∞. This implies that for e¯

Q > 1

other orbit conﬁgurations than for e¯

Q < 1 may appear. At the limits ¯r0→ ±∞ the energy Eremains ﬁnite,

E1,2→ ±sign(¯r0) there.

Orbits at ¯r= 0 As the ring singularity is located at ¯r= 0, θ=π

2only those orbits which are not equatorial do

not terminate at ¯r= 0. From the discussion of the colatitudinal motion equatorial orbits occur for ¯

K= 0, ¯aE =¯

Lor

¯

K= (¯aE −¯

L)2,e¯

P= 0. If we exclude these parameters ¯r= 0 is a multiple zero if and only if e¯

Q6= 0, ¯

K=e2¯

Q2¯

∆(0),

and ¯aE −¯

L=e¯

Q¯

∆(0)

¯a. This can also be infered from the asymptotic behaviour of E1,2and ¯

L1,2at ¯r0= 0, which is

13

given by

E1,2=e¯

Q¯

∆(0) ±p¯

K¯

∆(0)

2¯

∆(0)¯r0∓¯

∆(0)(1 + e2¯

Q2)−e2¯

Q2

2e¯

Q¯

∆(0) +O(¯r0),(30)

¯

L1,2= ¯aE1,2±pK¯

∆(0)

¯a+O(¯r0).(31)

Here the equation for E2implies two facts: First, at ¯

K=e2¯

Q2¯

∆(0) the regions of orbit conﬁgurations essentially

change because the sign of the r−1

0term changes and E2switches between ±∞. Second, for ¯

K=e2¯

Q2¯

∆(0) the

expressions for E2,L2, and e¯

Qare the same as (35)-(37) with ¯r0= 0, which correspond to triple zeros.

Orbits at the horizons It was noted above that a horizon is a turning point of the radial motion if R(¯r±) = 0. This

implies that the horizons can not be multiple zeros because of dR

d¯r(¯r±) = −2(¯r±−1)(¯r2

±+¯

K)6= 0 for R(r±) = 0. (This

is valid for massive test particles only. For light rays a horizon is a multiple zero if ¯

K= 0 and ¯a¯

L=E(¯r2

±+ ¯a2)−e¯

Q¯r±

but the corresponding orbit is always unstable.) The asymptotic behaviour of E1,2and ¯

L1,2near the horizons is given

by

lim

¯r0→¯r+

E1,2→ ±√2

4

√¯r+−1q¯r2

++K

¯r+√¯r0−¯r+

+O(1) ,(32)

lim

¯r0→¯r−

E1,2→ ∓√2

4

√¯r−−1q¯r2

−+K

¯r−√¯r0−¯r−

+O(1) ,(33)

lim

¯r0→¯r±

¯

L1,2→E1,2

¯a(2¯r±−¯

P2−¯

Q2) + O(1) .(34)

Now let us turn back to the general case. The expressions (28) and (29) depend linearly on Q=eQ and the

parameter ¯acan be removed by a rescaling of the other parameters ( ¯

L

¯a,Q

¯a,P

¯a2,¯

K

¯a2) and the radial coordinate ( ¯r

¯a).

(Note that also distances measured in units of Mhave to be rescaled). The dependence on ¯

Kand P2=¯

Q2+¯

P2is less

obvious. This can be studied by solving (28) and (29) for these parameters but the expressions are quite cumbersome

and we do not give them here. However, in Figure 9 orbit conﬁgurations for varying ¯

Kare shown and all possible

orbit types already appear in Figures 5 to 8.

Let us now analyze where triple zeros occur as they mark transitions from stable to unstable orbits. Solving the

three conditions d2R

d¯r2(¯r0) = dR

d¯r(¯r0) = R(¯r0) = 0 for E,¯

L, and Q=e¯

Qyields

E1,2=±1

2

¯

K¯

∆2(¯r0) + (¯r2

0+¯

K)(3¯r2

0−2¯r0+¯

K)¯

∆(¯r0)−(¯r2

0+¯

K)2(¯r0−1)2

(¯r2

0+¯

K)¯

∆(¯r0)p(¯r2

0+¯

K)¯

∆(¯r0),(35)

L1,2=∓1

2p(¯r2

0+¯

K)¯

∆(¯r0)

(¯r2

0+¯

K)2¯

∆2(¯r0)¯a¯

K(3¯r2

0−¯a2+ 2 ¯

K)¯

∆2(¯r0) + (¯r2

0+¯

K)(¯r4

0−¯

K(¯r2

0−2¯r0+ ¯a2)−¯a2(3¯r2

0−2¯r0)) ¯

∆(¯r0)

+ (¯r2

0+¯

K)2(¯a2−¯r2

0)(¯r0−1)2,(36)

Q1,2=∓¯r3

0¯

∆2(¯r0)−(¯r2

0+¯

K)(2¯r3

0−¯r2

0+¯

K)¯

∆(¯r0) + (¯r2

0+¯

K)2¯r0(¯r0−1)2

(¯r2

0+¯

K)¯

∆(¯r0)p(¯r2

0+¯

K)¯

∆(¯r0).(37)

Here again, the horizons ¯r±are singularities and in between triple zeros are not possible. Fourfold zeros may only

occur for ¯r < 0, given by E1,2=±2¯r−1

2√¯r(¯r−1) ,¯

L1,2=±¯r2−2¯rP2−¯a2

2¯a√¯r( ¯r−1) ,Q1,2=±¯r

√¯r(¯r−1) , and ¯

K=¯r(¯

∆(0)−¯r)

¯r−1, or at the

ring singularity if E=±1

√¯

∆(0) ,¯

K= 0, Q=1

E, and ¯

L= ¯aE.

The regions of diﬀerent orbit conﬁgurations are visualized in Figures 5-8. As the parameter space is six dimensional,

we have to ﬁx at least three parameters for plotting. We always choose to ﬁx ¯aand Q=e¯

Qfor the reasons outlined

above. Here Q=±1 and Q=±p¯

K/ ¯

∆(0) will separate quite diﬀerent plot structures: at Q=±1 the behaviour of

¯

L1,2at inﬁnity changes and at Q=±p¯

K/ ¯

∆(0) the behaviour of Eat ¯r= 0. We will restrict here to positive values

of Qbut allow all values of ¯

Land E. The values e¯

Q < 0 are recovered by (E, ¯

L)→(−E, −¯

L) as noted in the section

on general properties. Therefore, we distinguish between four diﬀerent regions: (i) Q<min{1,q¯

K

¯

∆(0) },(ii) q¯

K

¯

∆(0) <

Q<1, (iii) 1<Q<q¯

K

¯

∆(0) , and (iv) max{1,q¯

K

¯

∆(0) }<Q. Note that for e¯

Q= 0 the polynomial Ris unchanged

by the transformation (E, ¯

L)→(−E, −¯

L) and, therefore, regions of orbit conﬁgurations diﬀer only slightly from this

14

(a) e¯

Q= 0.3, e¯

Q < 1<q¯

K

¯

∆(0)

(b) e¯

Q= 0.9, e¯

Q < 1<q¯

K

¯

∆(0)

FIG. 5: Orbit conﬁgurations for the radial motion with ¯a= 0.6, ¯

K= 1, and P2= 0.4 for case (i). For a general

description of colours and linestyles see the text. The spherical orbits marked by the dark blue (green) solid lines

starting at the dots and approaching the light blue dashed (the black dash dotted) lines are stable. In (b) also the

orbits on the red solid line between the two dots are stable. All other spherical orbits are unstable. Note that for the

small e¯

Qof (a) regions of orbit conﬁgurations are only slightly deformed by the transformation E→ −E,¯

L→ −¯

L

but remain unchanged otherwise. Small plots on the top are enlarged details of the lower plot.

symmetry for small Qin region (i). However, if Qis larger than (37) with ¯r0=1

2(¯

∆(0) −¯

K−p(¯

∆(0) −¯

K)2+ 4 ¯

K)

(local minimum) the structure changes as an additional pair of triple zeros appears, see Figure 5b.

For better comparison, we always plot ¯

Lover Eand indicate the dependence on ¯

Kand P=¯

P2+¯

Q2by slowly

varying them in a plot series. In each plot, we use the following conventions:

•Solid lines indicate double zeros of Rwhich correspond to stable or unstable spherical orbits of constant ¯r. We

use red lines for orbits with constant ¯r < 0, blue for constant 0 <¯r < ¯r−, and green for constant ¯r > ¯r+.

•Dashed lines denote orbits with turning points at ¯r= 0 and are given by ¯

L±= ¯aE ±√¯

K¯

∆(0)

¯a. They mark

15

FIG. 6: Orbit conﬁgurations for the radial motion with ¯a= 0.6, ¯

K= 0.5, P2= 0.4, and e¯

Q= 0.9, q¯

K

¯

∆(0) < e ¯

Q < 1.

For a general description of colours and linestyles see the text. The spherical orbits on solid lines starting at a dot

and approaching the black dash dotted lines or the light blue dashed lines are stable (two green lines, a dark blue

line, and a red line). All other spherical orbits are unstable. In the detailed plot on the upper right the solid blue

line approaches the light blue dashed line so close from below that they are hard to distinguish; the regions

indicated there are meant to be between them. The same holds for the lower right plot with the red solid line

approaching from above.

transitions between orbits with diﬀerent indices. Red or blue solid lines corresponding to orbits with constant

¯rnear ¯r= 0 asymptotically approach ¯

L±.

•The dash dotted line marks E2= 1. Between E=±1 no orbit can reach inﬁnity. In addition, E=±1 is

asymptotically approached by red or green solid lines with ¯r→ ±∞

•Dotted lines are asymptotes to solid lines for ¯r→¯r±. They do NOT separate diﬀerent orbit conﬁgurations.

•Single dots mark triple zeros which separate stable from unstable orbits.

•The labels T,F, and Bindicated the orbit conﬁgurations summarized in table II. Solid, dashed, and dash

dotted lines indicate transitions from one orbit conﬁguration to another.

IV. ANALYTICAL SOLUTIONS

We will now solve the equations of motion (11) - (14) with the initial conditions

θ(γ0) = θ0,¯r(γ0) = ¯r0, φ(γ0) = φ0, t(γ0) = t0,(38)

In addition the initial direction, i.e. the sign of dx

dγ (γ0) for x=θ, ¯r, has to be speciﬁed. We denote this by σx=

sgn dx

dγ (γ0).

16

FIG. 7: Orbit conﬁgurations for the radial motion with ¯a= 0.6, P2= 0.4, ¯

K= 1, and e¯

Q= 1.1, 1 < e ¯

Q < q¯

K

¯

∆(0) .

For a general description of colours and linestyles see the text. The spherical orbits on solid lines starting at a dot

and approaching the black dash dotted lines or the light blue dashed lines are stable (two dark blue lines, a green

line, and a red line). All other spherical orbits are unstable. In the detailed plot on the upper right one of the solid

red lines approaches the light blue dashed line so close from below that they are hard to distinguish; the regions

indicated there are meant to be between them.

A. θ-motion

The equation of motion (11) needs to be solved. We ﬁrst concentrate on the case in which the poles are not reached,

i.e. θ(γ)∈(0, π). Then it is convenient to substitute ν= cos(θ), compare (17), and to solve the equivalent equation

of motion

dν

dγ 2

=

4

X

i=0

biνi= Θν(ν) (39)

with ν(γ0) = ν0:= cos(θ0) and sgn dν

dγ (γ0)=σν:= −σθ. If Θν(ν) has a zero of multiplicity two or more the

solution can be solved by elementary functions. In general Θν(ν) is a polynomial of order four with simple zeroes

only and can be solved with the following procedure that uses the Weierstrass elliptic function ℘(see [18, 20]. We

transform the equation to the Weierstrass form by the substitution ν= ( 4

a3ξ−a2

3a3)−1+νΘ, where νΘis an arbitrary

zero of Θνand ai=1

(4−i)!

d(4−i)Θν

dν(4−i)(νΘ). This leads to

dξ

dγ 2

= 4ξ3−gθ2ξ−gθ3(40)

17

(a) e¯

Q= 2 (b) e¯

Q= 10

FIG. 8: Orbit conﬁgurations for the radial motion with ¯a= 0.6, ¯

K= 1, P2= 0.4, and 1 <q¯

K

¯

∆(0) < e ¯

Q. For a

general description of colours and linestyles see the text. The spherical orbits on the blue (green) solid lines starting

at a dot and approaching the light blue dashed (black dash dotted) lines are stable. In addition, the spherical orbits

on the lower red solid line are stable but all others are unstable. Small plots on the top are enlarged details of the

lower plot.

with

gθ2=1

12a2

2−1

4a1a3, gθ3=−1

216a3

2+1

48a1a2a3−1

16a0a2

3.(41)

The initial conditions are ξ(γ0) = ξ0:= 1

4(a3

ν0−νΘ+a2

3) and sgn dξ

dγ (γ0)=σξ:= −sgn(a3)σν. The solution of eq. (40)

can now be expressed in terms of the Weierstrass elliptic ℘function,

ξ(γ) = ℘(γ−γθ,in;gθ2;gθ3),(42)

where γθ,in is a constant such that ℘(γ0−γθ,in) = ξ0and sgn(℘′(γ0−γθ ,in)) = σξ. This is e.g. fulﬁlled by γθ,in =

γ0−σξRξ0

∞

dτ

√4τ3−gθ2τ−gθ3

(with the principal branch of the square root). The solution for θis then given by

θ(γ) = arccos a3

4℘(γ−γθ,in;gθ2;gθ3)−a2

3

+νΘ.(43)

Let us now consider orbits which reach the poles. First, let θ(γ) be in the open interval (0, π) for γ∈(γ1, γ2) but on the

endpoints γ1,2the orbit may reach the poles, θ(γ1,2)∈ {0, π}. Then the solution on (γ1, γ2), given by eq. (43), is also

valid on the complete interval [γ1, γ2] because the right hand side of eq. (43) is continuous on the whole closed interval

18

¯

K= 0 ¯

K= 0.02 ¯

K= 0.0684

¯

K= 0.08 ¯

K= 0.6¯

K= 1.0

FIG. 9: Orbit conﬁgurations for the radial motion with ¯a= 0.6, P2= 0.4, e¯

Q= 0.3 and varying ¯

K. For a general

description of colours and linestyles see the text. Note that at ¯

K= 0.0684 the plots change from case (ii) to (i).

P2= 0.1P2= 0.195 P2= 0.2

FIG. 10: Orbit conﬁgurations for the radial motion with ¯a= 0.6, ¯

K= 0.8, e¯

Q= 1.2 and varying P2. For a general

description of colours and linestyles see the text. Note that at P2≈0.1956 the plots change from case (iii) to (iv).

19

P2= 0.5P2= 0.627 P2= 0.63

FIG. 11: Orbit conﬁgurations for the radial motion with ¯a= 0.6, ¯

K= 0.8, e¯

Q= 0.9 and varying P2. For a general

description of colours and linestyles see the text. Note that at P2≈0.6277 the plots change from case (i) to (ii).

with limits θ(γ1,2) as γapproaches γ1,2. In general let γi,i≥1, be the parameters with θ(γi)∈ {0, π}and γi< γi+1 .

Deﬁne θi:= θ|[γi−1,γi]and solve the diﬀerential equation in each interval with the condition θi(γi−1) = θi−1(γi−1),

sgn dθi

dγ (γi−1)=−sgn dθi−1

dγ (γi−1). The switch in sign of dθi

dγ in γicanonically identiﬁes θon [0, π].

B. ¯r-motion

The procedure to solve the equation of motion for the ¯r, see (12)

d¯r

dγ 2

=R2(¯r)−(ǫ¯r2+¯

K)¯

∆(¯r) = R(¯r) (44)

is analogous to that in the previous section. Again, the right hand side is a polynomial of fourth order. If it has a

zero of multiplicity two or more the diﬀerential equation can be solved in terms of elementary functions. The general

case can be solved with the substitution ¯r=c34ξ−c2

3−1+ ¯rR, where ¯rRis a zero of Rand ci=1

(4−i)!

d(4−i)R

d¯r(4−i)(¯rR).

This leads to

¯r(γ) = c3

4℘(γ−γ¯r,in;g¯r2;g¯r3)−c2

3

+ ¯rR(45)

with g¯r2,g¯r3given as in (41) with ai=ciand g¯ri=gθi. The parameter γ¯r,in only depends on the initial

conditions, ℘(γ0−γ¯r,in ;g¯r2;g¯r3) = 1

4c3

¯r0−¯rR+c2

3and sgn(℘′(γ0−γ¯r,in;g¯r2;g¯r3)) = −sgn(c3)σ¯r, e.g. γ¯r,in =

γ0+ sgn(c3)σ¯rRξ0

∞

dτ

√4τ3−g¯r2τ−g¯r3

.

C. φ-motion

The equation describing the φ-motion, see (13), can be rewritten as

φ(γ) = φ0+Zγ

γ0

¯aR(¯r)

¯

∆(¯r)dγ −Zγ

γ0

T(θ)

sin2θdγ , (46)

where the right hand side is separated in a part which only depends on ¯rand one that only depends on θ. We will

now treat both integrals separately.

20

The θ-dependent integral

Let us start with the integral

Iφθ(γ) = Zγ

γ0

T(θ)

sin2θdγ. (47)

If we insert the expression for θgiven by eq. (43), which we write symbolically as θ=θ(℘(γ−γθ,in)), we get

Iφθ(γ) = Zγ

γ0

T(θ(℘(γ−γθ,in))

sin2θ(℘(γ−γθ,in)) dγ =Zγ

γ0

Rφθ(℘(γ−γθ,in))dγ (48)

with Rφ(θ(℘(γ−γθ,in)) a rational function of ℘(γ−γθ,in). A partial fraction decomposition then yields

Iφθ(γ) = Zγ

γ0

αθ+αθ1

℘(γ−γθ,in)−βθ1

+αθ2

℘(γ−γθ,in)−βθ2

dγ (49)

with

αθ= ¯aE +e¯

P νΘ−¯

L

1−ν2

Θ

, αθ1,2=1

8

(e¯

P±¯

L)a4

(νΘ±1)2, βθ1,2=−1

12

3a4±a3−νΘa3

νΘ±1.(50)

The integral over each summand in eq. (49) can be expressed in terms of the Weierstrass ζ=ζ(·;gθ2;gθ3) and

σ=σ(·;gθ2;gθ3) function (see [20, 21])

Iφθ(γ) =αθ(γ−γ0) + αθ1

℘′(v1)ln σ(γ−v1)

σ(γ0−v1)−ln σ(γ+v1)

σ(γ0+v1)+ 2(γ−γ0)ζ(v1)

+αθ2

℘′(v2)ln σ(γ−v2)

σ(γ0−v2)−ln σ(γ+v2)

σ(γ0+v2)+ 2(γ−γ0)ζ(v2)(51)

where v1, v2need to chosen such that ℘(v1+γθ ,in) = βθ1, and ℘(v2+γθ,in) = βθ2.

The ¯r-dependent integral

The procedure to solve the ¯r-dependent integral

Iφ¯r(γ) = Zγ

γ0

¯aR(¯r)

¯

∆(¯r)dγ

is analogous to the previous section. We substitute ¯r= ¯r(℘(γ−γ¯r,in)) of eq. (45) and obtain a rational function as

integrand, Rφ¯rof ℘(γ−γ¯r,in), which we decompose into partial fractions

Iφ¯r(γ) = Zγ

γ0

Rφ¯r(℘(γ−γ¯r,in))dγ (52)

=Zγ

γ0

α¯r+α¯r1

℘(γ)−β¯r1

+α¯r2

℘(γ)−β¯r2

dγ . (53)

Here the constants are given by

α¯r= ¯aR(¯rR)

¯

∆(¯rR), β¯r1,2=c3

12 +±p−c2

4(−1 + ¯a2+¯

Q2+¯

P2)−c4¯r+c4

4¯

∆(¯r)(54)

and α¯r1,2the coeﬃcients of the partial fraction decomposition. The solution has the the form of eq. (51) with

αθ1,2=α¯r1,2,βθ1,2=β¯r1,2and v1, v2such that ℘(v1+γ¯r,in) = β¯r1and ℘(v2+γ¯r,in ) = β¯r2. Also the ℘,ζ, and σ

functions refer to the parameters g¯r2,g¯r3.

21

D. t-motion

The solution to the diﬀerential equation describing the t-motion, see (14), can be found in a similar way as for the

φmotion. An equivalent formulation of the diﬀerential equation is

t(γ) = t0+Zγ

γ0

(¯r2+ ¯a2)R(¯r)

¯

∆(¯r)dγ −Zγ

γ0

¯aT(θ)dγ , (55)

where the right hand side is a again separated in an ¯rand a θdependent part. We treat these integrals separately.

The θ-dependent integral

We solve the integral

Itθ(γ) := Zγ

γ0

¯aT(θ)dγ

with the same ansatz as in the foregoing section: First, we substitute θ=θ(℘(γ−γθ,in )) of eq. (43) and then decompose

the integrand Rtφwhich is a rational function of ℘(γ−γθ,in ) in partial fractions,

Itθ(γ) = Zγ

γ0

Rtθ(℘(γ−γθ,in))dγ

=Zγ

γ0

αθ+αθ1

℘(γ)−βθ

+αθ2

(℘(γ)−βθ)2dγ , (56)

where

αθ=−¯a¯

L−¯a2Eν2

Θ+ ¯a2E+ ¯ae ¯

P νΘ, βθ=1

12a3,

αθ1=−1

4¯aa4(2¯aEνΘ−e¯

P), αθ2=−1

16¯a2Ea2

4.

(57)

The solution written in terms of ζ=ζ(·;gθ2;gθ3) and σ=σ(·;gθ2;gθ3) is given by

Itθ(γ) = αθ(γ−γ0) + αθ1

℘′(v)ln σ(γ−v)

σ(γ0−v)−ln σ(γ+v)

σ(γ0+v)+ 2(γ−γ0)ζ(v)

+αθ2

℘′2(v)−ζ(γ−v) + ζ(γ0−v)−ζ(γ+v) + ζ(γ+v)−℘(v)(γ−γ0)

−℘′′(v)

2℘′(v)ln σ(γ−v)

σ(γ0−v)−ln σ(γ+v)

σ(γ0+v)+ 2(γ−γ0)ζ(v) (58)

where vsuch that ℘(v+γθ,in) = βθ.

The ¯r-dependent integral

Now the Integral

It¯r:= Zγ

γ0

(¯r2+ ¯a2)R(¯r)

¯

∆(¯r)dγ (59)

will be solved. The substitution ¯r= ¯r(℘(γ−γ¯r,in)) of eq. (45) again leads to a rational function Rt¯rof ℘(γ−γ¯r,in ),

which becomes after a partial fraction decomposition

It¯r=Zγ

γ0

Rφ¯r(℘(γ−γ¯r,in))dγ

=Zγ

γ0

α¯r+α¯r1

℘(γ)−β¯r1

+α¯r2

℘(γ)−β¯r2

dγ . (60)

Here α¯r,β¯r1,2are deﬁned as in eq. (54), and α¯r1,2are the coeﬃcients of the partial fraction decomposition. The

solution has the the form of eq. (51) with αθ1,2=α¯r1,2,βθ1,2=β¯r1,2and v1, v2such that ℘(v1+γ¯r ,in) = β¯r1and

℘(v2+γ¯r,in) = β¯r2. The ℘,ζ, and σfunctions here refer to g¯r2and g¯r3.

22

FIG. 12: Charged particle orbit in Kerr-Newman space-time. Here a B+orbit in the northern hemisphere (Norbit)

with ¯a= 0.6, ¯

P= 0.47, ¯

Q= 0.16, e= 158.11, E= 1.98, ¯

L=−62.62, ¯

K= 33.33 is shown. Left: three-dimensional

plot. The gray spheres correspond to the horizons. Right: projection on (x, z) plane. The dotted lines correspond to

extremal θvalues.

-1 012345

-1

1

2

3

4

5

6

FIG. 13: Charged particle orbit in Kerr-Newman space-time. Here a F∗

+orbit in the northern hemisphere (Norbit)

with ¯a= 0.6, ¯

P= 0.47, ¯

Q= 0.16, e= 158.11, E=−11.83, ¯

L=−5.18, ¯

K= 33.33 is shown. Left: three-dimensional

plot. The gray spheres correspond to the horizons. Middle: projection on (x, z) plane. The dotted lines correspond

to extremal θvalues. Right: projection on (x, y) plane. Dotted lines correspond to the horizons. Note that φ(γ)

diverges at the horizons at some ﬁnite γ0. In this plot we stopped at some γ0−∆γ(∆γsmall) and continued with

γ0+ ∆γon the other side of the horizon.

E. Examples

The analytical solutions to the equations of motions are given by (43), (45), (51), and (58), respectively with the

appropriate constants (54). Here, we use these results to exemplify the orbit structure in Kerr-Newman space-time,

see Figs. 12 and 13.

23

V. SUMMARY AND CONCLUSIONS

In this paper we discussed the motion of charged particles in the gravitational ﬁeld of Kerr-Newman space-times

describing stationary rotating black holes with electric and magnetic charge. We demonstrated that it is suﬃcient to

consider test-particles with electric charge only as an additional magnetic charge would only lead to reparametrization.

After that we classiﬁed the orbits in radial and colatitudinal direction. For both a large variety of orbit conﬁgurations

was identiﬁed as summarized in tables I and II. In particular, we also identiﬁed orbits crossing the horizons or r= 0.

These conﬁgurations were then assigned to regions in the parameter space pictured in several ﬁgures. The boundaries

of these regions are amongst others given by parameter combinations which represent orbits of constant ror θ, which

were discussed in detail. For all orbit conﬁgurations analytical solutions to the equations of motion were presented in

terms of elliptic functions dependent on the Mino time.

For the sake of completeness we considered here a black hole endowed with magnetic charge although such was not

found until now. This has a big impact on the motion in θdirection in contrast to electric charge, which does not

inﬂuence the colatitudinal motion at all. Not only deviates the motion from the symmetry to the equatorial plane for

a nonvanishing magnetic charge but also additional types of orbits appear. For example, stable oﬀ-equatorial circular

orbits outside the horizon do exist in this case, which are not possible else [12]. They are given by the intersection

of a red solid line with E2<1 as in Fig. 1 and a green solid line with E2<1 as in Fig. 5. (E.g. ¯a= 0.5, Q= 0.3,

P= 0.6, e¯

P≈2.1, ¯

K= 1, E≈0.9, and ¯

L≈2.4 results in an orbit with constant ¯r= 3 and co s θ≈0.8.) On the

contrary, the magnetic charge does not inﬂuence the radial motion as is appears only in the combination ¯

P2+¯

Q2.

Still, to our knowledge the discussion of the radial motion in this paper is the most complete so far for Kerr-Newman

space-times.

The analytical solution presented here are largely based on the 19th century mathematics of elliptic functions

already used by Hagihara [15] to solve the geodesic equation in Schwarzschild space-time. However, a key ingredient

here is the introduction of the Mino time which decouples the radial and colatitudinal equations of motion. We

presented the results here in terms of Weierstrass elliptic functions which may be rewritten in terms of Jacobian

elliptic functions. The advantage of our presentation is that one formula is valid for all orbit conﬁgurations.

ACKNOWLEDGMENT

We would like to thank Claus L¨ammerzahl for suggesting this research topic and for helpful discussions. E.H.

acknowledges ﬁnancial support from the German Research Foundation DFG and support from the DFG Research

Training Group 1620 Models of Gravity.

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