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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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