Charged Q-balls and boson stars and dynamics of charged test particles

# Charged Q-balls and boson stars and dynamics of charged test particles

Yves Brihaye    Valeria Diemer 111née Kagramanova    Betti Hartmann Faculté de Sciences, Université de Mons, 7000 Mons, Belgium
Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany
Universidade Federal do Espírito Santo (UFES), Departamento de Física, Vitória (ES), Brazil
July 11, 2019July 11, 2019
July 11, 2019July 11, 2019
###### Abstract

We construct electrically charged -balls and boson stars in a model with a scalar self-interaction potential resulting from gauge mediated supersymmetry breaking. We discuss the properties of these solutions in detail and emphasize the differences to the uncharged case. We observe that -balls can only be constructed up to a maximal value of the charge of the scalar field, while for boson stars the interplay between the attractive gravitational force and the repulsive electromagnetic force determines their behaviour. We find that the vacuum is stable with respect to pair production in the presence of our charged boson stars. We also study the motion of charged, massive test particles in the space-time of boson stars. We find that in contrast to charged black holes the motion of charged test particles in charged boson star space-times is planar, but that the presence of the scalar field plays a crucial rôle for the qualitative features of the trajectories. Applications of this test particle motion can be made in the study of extreme-mass ratio inspirals (EMRIs) as well as astrophysical plasmas relevant e.g. in the formation of accretion discs and polar jets of compact objects.

###### pacs:
04.40.-b, 11.25.Tq

## I Introduction

Non-topological solitons in contrast to topological solitons ms () are not stable due to a topological charge, but due to a globally conserved Noether charge that results from a continuous symmetry existing in the system fls (); lp (). The non-topological soliton most frequently discussed is the -ball coleman () and its gravitating counterpart, the boson star kaup (); misch (); flp (); jetzler (); new1 (); new2 (); Liddle:1993ha (). These are solutions appearing in complex scalar field models that possess a continuous U(1) symmetry. Due to the recent confirmation of the existence of a fundamental scalar field in nature cern_lhc () these models seem to be very appealing. When the U(1) symmetry is global, the Noether charge corresponds to the particle number. When the U(1) symmetry is gauged the product between the Noether charge and the constant defining the coupling between the gauge field and the scalar field can be interpreted as total charge of the soliton.

As is well known, soliton solutions exist in models with a subtle interplay between non-linearity and dispersion and as such specific potentials of the scalar field are necessary to obtain -ball solutions. It was shown in coleman () that a potential of at least 6th order in the scalar field is necessary to obtain solutions and these solutions have been constructed in vw (); kk1 (); kk2 (); bh (). Note that when gravity is added to the system this restriction is not present, however, using the same scalar field potential in the gravitating case leads to a proper flat space-time limit of the solutions kk1 (); kk2 (); bh2 (). Unfortunately, 6th order potentials are non-renormalizable and hence the question arises whether appropriate potentials can be derived from some particle physics models. Indeed, it is well known that in supersymmetric extensions of the Standard Model -ball solutions exist kusenko () and these have been discussed with respect to their astrophysical applications and possible rôle in dark matter models dm (); implications (). In cr (); ct () a potential arising in gauge-mediated supersymmetry breaking has been discussed and the corresponding -ball solutions have been constructed. The gravitating counterparts were discussed in hartmann_riedel2 (); hartmann_riedel ().

Charged -balls and boson stars have also been studied previously jetzler2 (); Arodz:2008nm (); Kleihaus:2009kr (); Pugliese:2013gsa (); Tamaki:2014nua (). In particular, a detailed study in models with a V-shaped potential (in which compact objects arise Arodz:2008jk (); Arodz:2008nm ()) has been presented in Kleihaus:2009kr (). In these latter mentioned models with a V-shaped potential boson stars with a well-defined outer surface can be considered as toy models for or even alternatives to compact objects such as neutron stars or even supermassive black holes schunck_liddle () residing e.g. in the center of galaxies. Hence, the applications of boson stars is two-fold:

• They can be considered as toy models for neutron stars. Neutron stars are very difficult to model due to a lacking general equation of stay governing the matter inside the star. Boson stars of stellar size masses might be interesting in this respect. And then, since very compact objects accrete matter around them, it is interesting to study the accretion of matter to boson stars as well. This can e.g. be done by studying the geodesic motion of test particles in the space-time of boson stars – in analogy to the studies done in black hole space-times tejeda1 (); tejeda2 (). In fact, the assumption made in this so-called ballistic limit is that of non-interacting, free test particles. The dynamics of these type of particles in uncharged boson star space-times has been discussed in Eilers:2013lla (). However, since in astrophysical settings we would expect the accreted matter to consist of a plasma, i.e. charged constituents, one of the aims of this paper is to study the motion of charged particles in boson star space-times.

• While objects with a “hard core” as alternatives to supermassive black holes in the center of our galaxy have been ruled out Broderick:2005xa (), the objects constructed in this paper do not have a well-defined surface outside which the energy density is strictly vanishing. To our knowledge no observation yet clearly rules out a non-compact horizonless object as a possible center of our galaxy. In fact, recent observations of radio waves with wavelength of mm emitted from the galactic center suggest that the size of this object (Sagittarius A) is smaller than the expected apparent event horizon size of the assumed black hole Doeleman:2008qh (). The question that we try to address then is how a possible charge of such an object could influence rotation curves of stars around the galactic center and whether any signal of the charge of either the object or the accreted matter would show up in the radio wave signals.

Our paper is organised as follows: in Section II, we discuss the field theoretical model of electrically charged -balls and boson stars. In Section III we present numerical solutions to the coupled system of ordinary, non-linear differential equations. In particular, we will put emphasize on the rôle of the charge on the stability and existence of the solutions. In Section IV, we discuss the motion of charged, massive test particles in the space-time of electrically charged boson stars and also comment on possible astrophysical applications.

## Ii Charged Q-balls and boson stars

In the following we will discuss the field theoretical model to describe the space-time of charged, non-spinning, non-compact -balls and boson stars in which the test particles will move.

The action of the field theoretical model reads:

 ~S=∫√−gd4x(R16πG+Lm) (1)

where is the Ricci scalar, denotes Newton’s constant and the matter Lagrangian is given by

 Lm=−14FμνFμν−DμΦDμΦ∗−U(|Φ|) (2)

where denotes a complex scalar field and we choose as signature of the metric . is the field strength tensor of a U(1) gauge field and denotes the covariant derivative with gauge coupling . is the scalar field potential that arises in gauge-mediated supersymmetry breaking (GMSB) in Supersymmetric extensions of the Standard Model cr (); ct ():

 U(|Φ|)=m2η2susy(1−exp(−|Φ|2η2susy)) . (3)

In GMSB the supersymmetry (SUSY) breaking happens in a hidden sector and the breaking is mediated via gauge fields to the visible sector of the Standard Model Giudice:1998bp (). These fields are the so-called messenger fields with corresponding messenger mass scale . These are coupled to the gauge multiplets with coupling constant . The soft breaking mass scale is then given by and is on the order of TeV. The lightest supersymmetric particle (LSP) predicted by GMSB is the gravitino with mass . Hence, limits of the parameter are typically given in terms of this mass as follows

 1TeV√m≤√ηsusy≤√g4π√m3/2Mrpl√m , (4)

where is the reduced Planck mass. The recent results from the ATLAS and CMS detectors at LHC seem to indicate that the minimal GMSB is ruled out since it predicts a mass of the Brout-Englert-Higgs (BEH) boson of less than 122 GeV which is clearly in disagreement with the confirmed value Arbey:2011ab (). However, extended GMSB models still allow for the measured value of the BEH boson and new models have been discussed (see e.g. CahillRowley:2012gu (); CahillRowley:2012cb ()). Current bounds on depend on the next-to-lightest supersymmetric particle (NLSP) and the decay channels. Mostly the gravitino is assumed to have masses above roughly 1 keV and not larger than 1 GeV (limits exist also from Cosmology and Astrophysics). The messenger scale could then be between roughly 1 TeV and up to TeV.

The matter Lagrangian (2) is invariant under a local U(1) transformation. As such the locally conserved Noether current , , associated to this symmetry is given by

 jμ=−i(Φ∗DμΦ−Φ(DμΦ)∗)  with   jμ;μ=0 . (5)

The globally conserved Noether charge of the system then reads

 N=−∫√−gj0d3x , (6)

which corresponds to a particle number such that is the total charge. Note that since is positive the sign of will determine the sign of the charge of the -balls and boson stars. Finally, the energy-momentum tensor is given by

 Tμν = gμνLM−2∂LM∂gμν=(FμαFνβgαβ−14gμνFαβFαβ) (7) − 12gμν((DαΦ)∗(DβΦ)+(DβΦ)∗(DαΦ))gαβ+(DμΦ)∗(DνΦ)+(DνΦ)∗(DμΦ)−U(Φ|) .

The coupled system of ordinary differential equations is given by the Einstein equations

 Gμν=8πGTμν (8)

with as in (7) and the matter field equations

 ∂μ(√−gFμν)=√−geΦ∗DνΦ   ,   DμDμΦ−∂U∂|Φ|2Φ=0. (9)

In the following, we want to study non-spinning -balls and boson stars. For the metric we use the following Ansatz in isotropic coordinates

 ds2=−f(r)dt2+l(r)f(r)[dr2+r2dθ2+r2sin2θdφ2] . (10)

For the matter fields we choose

 Φ=ϕ(r)eiωt , Aμdxμ=A(r)dt , (11)

such that our solutions will carry only electric charge and denotes the internal frequency of the scalar field. The concrete expression for the charge of the solution (6) is then

 Q=8π~e∞∫0√l3f2r2(ω−~eA)ϕ2dr (12)

The initially complex scalar field can be chosen to be real. The reason is the U(1) gauge symmetry that allows us to gauge away any non-vanishing phase. We hence apply the following U(1) gauge transformation

 Φ→Φe−iωt  ,  A→A+ω~e . (13)

In order to be able to use dimensionless quantities we introduce the following rescalings

 r→rm  ,  ω→mω  ,  ϕ→ηsusyϕ  ,  A→ηsusyA (14)

and find that the equations depend only on the dimensionless coupling constants

 e=ηsusym~e   ,   α=8πGη2susy=8πη2susyM2pl , (15)

where is the Planck mass. Note that with these rescalings the scalar boson mass becomes equal to unity. The charge of the scalar field will be given in multiples of which can be between unity and . on the other hand is in general very small, but in this paper we will also consider larger values of in order to understand the qualitative behaviour of the boson star solutions.

We can also write

 αe2=8πm2G~e2=2m2Gαfs=274⋅m2G (16)

where in our natural unit system () denotes the fine structure constant with . Since stable boson stars should only exist when the gravitational attraction compensates the electromagnetic repulsion (see e.g. Jetzer:1993nk ()) we would expect that they exist only for , i.e. for .

The Lagrangian gets rescaled like such that the masses we will give in our numerical analysis below are measured in units of which is (with the given values as above) between roughly 1 TeV and TeV. The upper limit here corresponds to the Planck mass which - of course - we would not be able to reach for boson stars. However, these are just rough approximations and we note that boson stars in supersymmetric extensions of the Standard Model can be quite heavy.

Using these rescalings the field equations of motion read

 ϕ′′=−rl′+4l2rlϕ′−(ω−eA)2lf2ϕ+lfϕexp(−ϕ2) , (17)
 A′′=−(f′f+l′2l+2r)A′+2lfe(eA−ω)ϕ2 (18)

for the matter field functions and

 f′′=f′(−l′2l+f′f−2r)+2αl(2(ω−eA)2ϕ2f−1+exp(−ϕ2)) , (19)
 l′′=l′(l′2l−3r)+4αl2((ω−eA)2ϕ2f2−1−exp(−ϕ2)f) (20)

for the metric functions. The prime now and in the following denotes the derivative with respect to . Note that the equations are unchanged under . The coupled set of non-linear ordinary differential equations (17) - (20) has to be solved subject to appropriate boundary conditions. These are given by the requirement of regularity at the origin

 f′|r=0=0  ,  l′|r=0=0  ,  ϕ′|r=0=0  ,  A′(0)=0 (21)

and by the requirement of finite energy, asymptotically flat solutions

 f(r=∞)=1  ,  l(r=∞)=1  ,  ϕ(r=∞)=0  ,  A(r=∞)=−ωe . (22)

For numerical calculations, we have found it more convenient, however, to adapt the boundary conditions. As can be easily verified, the equations depend only on the combination . We take advantage of this and impose instead of . The asymptotic behaviour of the matter fields reads

 A(r→∞)∼μ+Qr  ,  ϕ(r→∞)∼1rexp(−√1−(ω−eμ)2r) , (23)

where is the value of at , which will be computed numerically by the program. Furthermore, we impose an additional condition on in order to avoid the trivial solution . For that we add an equation , hence making a function. By imposing the additional boundary conditions and we find a consistent solution with constant.

The set of coupled, nonlinear ordinary differential equations (8)-(9) has been studied with a V-shaped potential in Kleihaus:2009kr (), however in Schwarzschild-like coordinates. In hartmann_riedel (); Eilers:2013lla () uncharged boson stars with an exponential potential of the form (3) have been considered.

## Iii Soliton solutions

We have solved the set of differential equations numerically using the ODE solver Colsys colsys (). The relevant parameters are and the electric coupling constant . Once choosing these constants, families of solutions labeled by (and hence ) can be constructed.

### iii.1 Charged Q-balls

This corresponds to the case , i.e. to solitons in flat space-time. In this case we have . These solutions are called “-balls” in the literature and we hence refer to them as such here. Let us first discuss the restrictions on the parameter that we have. From vw () we know that in the uncharged case, i.e. for the frequency is bounded from above and below, i.e. . To see how this constraint changes in the presence of an electromagnetic field note that we can rewrite (17) in analogy to the uncharged case as follows

 12ϕ′2+12(ω−eA)2ϕ2−12U(ϕ)=E−2r∫0ϕ′2rdr , (24)

where is an integration constant. This describes the frictional motion of a particle with “coordinate” at “time” in an effective potential of the form

 V(ϕ)=12(ω−eA)2ϕ2−12U(ϕ) . (25)

The requirements then are that and for some vw (). In our case with the potential (3) this leads to the following conditions

 (eμ−ω)2<1  ,  (eμ−ω)2>0 , (26)

where the latter condition is not really an extra condition since all quantities are assumed to be real. Keeping these conditions in mind we have studied the case of fixed and varying in order to understand how the dependence of on changes in the presence of an electromagnetic field. In the case , i.e. for uncharged -balls, the limit corresponds to . This is the so-called thick wall limit. On the other hand, corresponds to , the thin wall limit. In both cases, the mass and the charge of the solutions diverges. This is shown in Fig.1(a), where we give the mass and particle number of the -balls in dependence on for and for . Let us first discuss the case again in detail: in the limit (which corresponds to ) the -balls have and are hence unstable to decay into individual quanta of mass . Increasing the parameter progressively we can construct a family of stable solutions with for (this corresponds to ). This is illustrated in Fig. 1(b). Increasing further the frequency approaches zero, the mass and particle number of the -ball both tend to infinity.

For , we now observe that a different phenomenon exists such that the solutions do not exist on the full interval , but that -balls for exist only on a finite interval of , where both and are finite and depend on . Hence, charged -balls have neither a thin wall limit nor a thick wall limit considering the scalar field only.

This is shown in Fig.2, where we give and the quantity in dependence on (left) as well as and as function of (right). At both and we find that the solutions cease to exist because and at the same time the particle number (and with it ) tend to infinity. This is related to the argument with the potential given above (see (26)), but here the lower bound on is not really a constraint. In fact, the solutions cease to exist because the quantity tends to unity at both and (see the subfigure of Fig.2(b)). Our results seem to indicate that in the limit the electric field tends to zero, while in the limit the electric field spreads over all space. The physical interpretation is the following: remembering that the integral of the scalar field over corresponds to the particle number , our results suggest that a minimal number of needs to be present to allow for charged -balls. Furthermore, if becomes too large (at ) the electric repulsion among the individual particles becomes too strong and the solution spreads over all space.

We find that the minimal value of increases with , while the maximal value of decreases with . This is shown in Fig.3, where we give and in dependence on . increases from zero at , while decreases from infinity. At a sufficiently large value of we find that . This means that for no -balls can be constructed. This is related to the electric repulsion that becomes too strong to support stable configurations.

### iii.2 Charged boson stars

Here, the matter fields are coupled to the metric field, i.e. . In Fig.4 we show the profiles of the functions of a typical charged boson star solution.

First, we have fixed and and varied in order to understand the dependence of the physical quantities on . In the limit , the solutions coincide with the uncharged boson stars constructed in hartmann_riedel (). We were able to construct a branch of charged boson stars for . The numerical results suggest that the branch is not limited by any critical value of as in the -ball case. In the limit , i.e. the thick wall limit the gauge field function becomes a constant function, where the constant depends on the value of . This is shown in Fig.5 (upper figure), where we give the mass and the charge as function of for and , respectively. Both the uncharged and the charged boson stars show a typical spiraling behaviour at the minimal value of , while and both tend to zero at the maximal value of . We find that the maximal value of is independent of (like it is independent of ). However, the minimal value of decreases for increasing . This can be understood when noticing that the gravitational field acts as an attractive force, while the electric field is repulsive in nature. Hence the electric repulsion counterbalances gravity. As such boson stars with larger masses and particle numbers are possible as compared to the uncharged case. The mass of the boson stars in dependence on the particle number shows a spike-like behaviour typical for boson star solutions. On the main branch of solutions boson stars are stable (), while on the second branch (corresponding to the spirals in the --plot) the solutions develop an instability to decay into individual scalar boson (since ).

We also observe that the mass and charge possess several local minima and maxima in dependence on . This is shown in Fig.5 (bottom left). Now, the question is what happens at the minimal value of in this case. This is shown in Fig.5 (bottom right), where we give , and as function of . Clearly, tends to a finite value for . In contrast to the case of -balls, however, this thin wall limit is not regular since at the same time and tend both to zero.

Fixing and increasing , our results hence indicate that charged boson stars exist up to very large values of . This contrasts with the case of charged -balls. This behaviour can be easily understood when realizing that increasing the central density increases the gravitational attraction which compensates the electrostatic repulsion. In particular, the values of the metric functions at , i.e. and tend to zero for becoming large. At the same time, however, the mass and particle number remain finite and the parameter does not approach zero.

Although we only give the data for and here, we have checked that the underlying properties hold for generic values of these parameters.

In Jetzer:1993nk () it was argued that the vacuum becomes unstable to pair production in the presence of a boson star when the charge of a boson star fulfills . Using all the rescalings used in this paper we find that this bound translates into a bound for the (dimensionless) particle number computed in our numerical analysis:

 N≥4πe3ηsusym . (27)

Now, this bound of course depends strongly on the value of . For the upper bound TeV and choosing we find that for values of pair production would occur, while for the lower bound TeV and again we find . As can be clearly seen in Fig.5 the maximal value of is always on the order of and we would hence expect the vacuum to be stable with respect to pair production in the presence of our charged boson stars.

## Iv Motion of a charged test particle in the space-time of a boson star

In the following, we want to study the motion of test particles in the space-time of a charged boson star. The Hamilton-Jacobi equation describing such a motion reads

 −2∂S∂τ=gμν(∂S∂xμ−qAμ)(∂S∂xν−qAν) , (28)

where denotes the charge of the test particle and is the action.

These have a solution of the form Chandrasekhar83 ()

 S=12δτ−Et+Lφ+S(r)+S(θ) . (29)

Here is an affine parameter which corresponds to proper time for massive test particles. The parameter is equal to for a massless and equal to for a massive test particle, respectively. Since no massless, charged particles exist in nature, we will here concentrate on the massive particles and will set in the following.

The constants and are the conserved energy and the angular momentum of a test particle related to the Killing vector and , respectively. Using (29) and the components of the metric tensor (10) we can rewrite the equation (28) as follows

 −δr2l(r)f(r)+l(r)f2(r)r2(E+qA(r))2−r2(∂S(r)∂r)2=(∂S(θ)∂θ)2+L2sin2θ , (30)

where we collected radial coordinate dependent terms on the left and polar coordinate dependent terms on the right hand side. Since the right and left sides of (30) are equal and each of them depends only on one coordinate they have to be equal to a constant (a separation constant). Thus, the expressions for and yield

 (∂S(r)∂r)2=l(r)f2(r)(E+qA(r))2−δl(r)f(r)−Kr2 , (31) (∂S(θ)∂θ)2=K−L2sin2θ . (32)

Now introducing

 R≡(∂S(r)∂r)2   ,   Θ≡(∂S(θ)∂θ)2 (33)

we rewrite the solution (29) in the following way

 S=12δτ−Et+Lφ+∫r√Rdr+∫θ√Θdθ . (34)

Differentiating (34) with respect to , , and , using (31) and (32) and setting the result equal to zero (since the variation of the action with respect to the constants of motion should vanish), we get the following four expressions

 ∫r1r2dr√R=∫θdθ√Θ , (35) τ=∫rl(r)f(r)dr√R , (36) t=∫rl(r)f2(r)(E+qA(r))dr√R , (37) φ=∫θLsin2θdθ√Θ . (38)

From (36) we infer the differential equation for the radial coordinate and the polar coordinate, respectively

 drdτ=f(r)l(r)√R   ,   dθdτ=f(r)r2l(r)√Θ . (39)

To omit the dependence on the radial coordinate in the equation for the polar coordinate we introduce a new affine parameter  Mino03 ()

 dλ=f(r)r2l(r)dτ . (40)

With this the equations (39) simplify to

 dθdλ=√Θ   ,   drdλ=r2√R . (41)

The differential equations for the remaining two coordinates and are now easy to derive. For the azimuthal coordinate and the time coordinate , respectively, we get

 dφdλ=Lsin2θ   ,   dtdλ=r2l(r)f2(r)(E+qA(r)) . (42)

Equations (41) and (42) describe the motion of a charged test particle in the boson star space-time given by the metric (10).

### iv.1 Components of the equation

To visualize the motion we need to integrate the differential equations (41) and (42). This reads:

 λ−λ0=∫rr0drr2√R , (43) λ−λ0=∫θθ0dθ√Θ , (44) φ−φ0=∫θθ0Lsin2θdθ√Θ , (45) t−t0=∫rr0l(r)f2(r)(E+qA(r))dr√R , (46)

where index zero denotes an initial value. Since the coordinate and functions , and (and correspondingly ) are given numerically, only numerical solutions to the radial and time equations (43) and (46) are possible. The integrals in the equations (44) and (45), on the other hand, can be solved analytically.

#### iv.1.1 Polar motion

Let us first consider equation (44) for the polar coordinate . Introducing a new variable we can rewrite the equation as follows

 λ−λ0=∫θθ0dθ√Θ=−∫ξξ0dξ√K−L2−Kξ2 , (47)

where . For physical motion the values of (as well as ) must be real, moreover, the condition must be fulfilled. Introducing

 Θξ=K−L2−Kξ2 (48)

in (47) and studying its zeros given by

 ξ1,2=±√K(K−L2)−K (49)

we can get limits on the values of . In the following we will consider the different possible values:

• Since the square root in (49) must be positive it immediately follows that . Rewriting (49) in the form

 ξ1,2=∓√K−L2K≡∓√1−L2K . (50)

we find that which leads to physical values of . Moreover, will be bounded from above and below such that the motion will be between a minimal and maximal value of which can be between and . If then . This corresponds to , i.e. corresponds to motion in the equatorial plane.

• Introducing where and rewriting (49) as

 ξ1,2=±√K1+L2K1≡∓√1+L2K1 . (51)

we find that the values of in (51) belong to unphysical branches with and for non-vanishing . If then and the result of integration in (47) contains complex values which is unphysical ( must be real).

• This case is not possible for non-vanishing since the values of are complex (because of in (47)) and hence unphysical.

To summarize we note that physical values of the coordinate in (47) are possible provided that

 K>0andK≥L2 . (52)

Integration of (47) for and thus yields:

 λ−λ0=−(−1√Karcsin(−Kξ√K(K−L2)))∣∣ξξ0  . (53)

Inversion of (53) where gives the final solution for the coordinate :

 θ=arccos(√K(K−L2)Ksin(√K(λ′−λ))) , (54)

where is a constant.

#### iv.1.2 Azimuthal motion

We have the following integral for the azimuthal coordinate

 φ−φ0=∫θθ0Lsin2θdθ√Θ=−∫ξξ0L1−ξ2dξ√K−L2−Kξ2 . (55)

The statements made on the values of the separation constant in Sec. IV.1.1 are also true for the equation (55). The integral in (55) can be solved by elementary functions as follows

 φ=φ0+12L|L|arctan2ξ√L2(K−L2−Kξ2)ξ2(L2+K)−K+L2∣∣ξξ0 . (56)

Substituting with given by (54) into (56) we get .

Using the components of the equations for the - and the -coordinate we can give the motion explicitly. This reads

 dθdφ=sinθ√KL2sin2θ−1 , (57)

which can be solved by

 cot2θ=(KL2−1)sin2φ . (58)

The turning points of the motion in -direction are given by (using (39))

 dθdτ=0  ⟹  sin2θ=L2K   . (59)

Hence, the -coordinate fulfills

 arcsin(L√K)≤θ≤π−arcsin(L√K) . (60)

Inserting this result into (58) we find that the turning points in -direction correspond to

 sin2φ=1  ⟹  φ=π2+nπ   ,  n∈Z . (61)

The angular motion of test particles is hence planar, where the inclination of the plane with respect to the equatorial plane is determined by the ratio . For the particle moves in the equatorial plane . Hence, the stronger differs from the stronger the plane in which the particle moves is inclined with respect to the equatorial plane. Note that this is in stark contrast to the motion of charged test particles in the space-time of charged black holes GruKa (). In this latter case, the charged particles move in general on non-planar orbits.

### iv.2 Numerical results

In the following we will describe our results for the motion of charged, massive test particles in charged boson star space-times. We solve the radial differential equation (43) numerically using MATLAB with the recursive adaptive Lobatto quadrature with an absolute error tolerance of . For the polar - and azimuthal -coordinates we use the results (54) and (56). For visualization of the trajectories we use a spherical coordinate system with , and .

We have chosen the space-time of a boson star with , and some particular choices of the parameter . The physical values of the corresponding solutions are given in Table 1. Whenever we give the value of in the following, the corresponding values of mass and charge can be read off from this table.

#### iv.2.1 The effective potential

In order to understand what type of orbits are possible, we define an effective potential

 V±eff=−qA(r)± ⎷f2(r)l(r)(δl(r)f(r)+Kr2) , (62)

such that we can rewrite the equation describing radial motion as follows

 (drdλ)2=r4R≡r4l(r)f2(r)(E−V+eff)(E−V−eff) . (63)

The values of at which mark turning points of the motion. In order for to be real and positive, we have to require the positiveness of the right hand side of (63). Values of for which the right hand side of  (63) is negative are not allowed. In the following, we will refer to these values of as forbidden regions. Note that (63) is completely invariant under . It hence contains a charge conjugation (C) - time reversal (T) symmetry and furthermore a parity (P) symmetry since only the angular momentum of the particle enters. Thus, the equation (63) is invariant under a CPT transformation and particles with negative energy and charge can be interpreted as anti-particles with positive energy and charge . Now, we can distinguish different type of orbits :

• Bound orbits (BOs): these orbits have two turning points corresponding to two values of at which . The motion of the particle varies between a minimal radius and a maximal radius, where both have finite values. This is often also referred to as planetary motion.

• Escape orbits (EOs): these orbits have only one turning point corresponding to one value of at which . The motion of the particle varies between a finite minimal radius and . Hence, the particle approaches the boson star from infinity, scatters of the boson star and moves back to infinity.

In Fig.6 we show the effective potential (62) for a charged test particle with , angular momentum and separation constant . This corresponds to motion in a plane with . The particles move in a boson star space-time with , and different values of the central value of the scalar field . We find that depending on the value of either BOs or EOs exist and that two regions for are possible: and . Now, remembering that our particles are charged, we can interpret particles with and charge as anti-particles with energy and charge .

In Fig. 7 we show the effective potential (62) for varying separation constant for in Fig. 7(a), for in Fig. 7(b) and for in Fig. 7(c). For small values of bound orbits exist, while for increasing only escape orbits are present. Since for a fixed value of the value of determines how strong the plane of motion is inclined with respect to the equatorial plane this suggests that bound orbits are only possible if the inclination is not too big, i.e. the range of does not vary too strongly along the trajectory.

The charge of a test particle also influences the effective potentials. For large positive charges the effective potential will be shifted downwards along the -axis, which means that for positive bound orbits are no longer possible. This is related to the electromagnetic repulsion between the positively charged boson star and the positively charged test particle. If is too big, the electromagnetic repulsion is too strong and the particle gets simply scattered by the boson star. On the other hand, bound orbits exist for negative values of . With the anti-particle interpretation this is also obviously explained. Since a particle with negative energy and large positive is equivalent to an anti-particle with positive energy and large negative charge the electromagnetic interaction between the positively charged boson star and the negatively charged test particle is attractive and bound orbits become possible. We present potentials for large positive charges in Fig. \subrefpotential_massive11_q1 for and in Fig. \subrefpotential_massive11_q3 for . The boson star has in this case. For decreasing negative charges the potential moves upwards along the -axis, which leads to two possible regions for bound orbits with positive energies. In this case for negative energies only escape orbits can be found. Again, the reason is the increased electromagnetic attraction in the case of decreasing negative and the increased repulsion between the particle with negative energy and negative charge which can be interpreted as an anti-particle with positive energy and positive charge . Examples of potentials for negative charges are shown in Fig. 9(a) for and in Fig. \subrefpotential_massive11_q-4 for .

#### iv.2.2 Examples of orbits

In the following, we will present some typical orbits possible in the space-time of boson stars.

In the Fig. 10 we give an example of orbits for which correspond to the effective potential in Fig. 6(a). In Fig. 10(a) and Fig. 10(c) we show bound orbits.

In Fig. 10(c) we give the motion of test particles with smaller energies. In Figs. 10(b) and 10(d) the corresponding - motion is shown. Two escape orbits for larger values of are presented in the Figs. 10(f) and 10(e). We comparing the orbits we find that for larger values of the orbit has a simple parabolic form, whereas the particle with lower energy moves around the boson star, slightly changes its direction before going to infinity again. In Fig. 11 we show bound orbits for small negative charges of the test particle. A bound orbit for is given in Fig. 11(a) and corresponds to the potential in Fig. 9(a). For a corresponding bound orbit is shown in Fig. 11(b) for the effective potential in Fig. 9(b).

All orbits are confined to a plane which is inclined with respect to the equatorial plane. This feature is peculiar to charged boson star space-time and can be observed in all cases considered below. Even a large absolute value of a charge of a test particle does not tear the orbit out of a plane as it is known from the charged particle motion in the Reissner-Nordström space-time GruKa ().

A small increase of in the potentials for and in Figs. 6(a) and 6(b) does not influence the potentials visibly. Hence, the case of maximal mass (and charge) does not differ strongly from that of minimal internal frequency at . This is apparent when comparing Fig. 12 with Fig. 10.

but have closely lying values of turning points as can be seen from the Table 2.

For small value of () bound orbits are possible for very small positive values of the energy as shown in Fig. 13(a) and Fig. 13(c). Figs. 13(b) and 13(d) illustrate the motion of these orbits. The escape shown in Fig. 13(e) and Fig. 13(f) have parabolic form. The corresponding effective potential is shown in Fig. 6(c).

For large values of the scalar field (here ) with effective potential given in Fig. 6(d) the particle trajectories display more structure. This is shown in the Fig. 14. Bound orbits given in Fig. 14(a) and Fig. 14(c) are different than the trajectories for smaller considered before. Here a test particle circles around the boson star a number of times during a single revolution. Also the corresponding escape orbits given in Fig. 14(e) and Fig. 14(f) show this property. For large charges and positive energies of a test particle only escape orbits can be found as we have already argued with the help of the effective potentials in Fig. 8. We illustrate this in Fig. 15(a) where an escape orbit for a test particle with and an escape orbit with in Fig. 15(b).