Charged objects in magnetized Schwarzschild spacetime: mapping parameter space and measuring magnetic fields with gravitational wave frequencies

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Charged objects in magnetized Schwarzschild

spacetime

mapping parameter space and measuring magnetic fields with

gravitational wave frequencies

Julia Schuring

Report Bachelor Project Physics and Astronomy

Studentnumber 11215453

Faculty Faculteit der B`etawetenschappen, VU

Faculteit der Natuurwetenschappen, Wiskunde en Informatica, UvA Second examiner prof. dr. R. Fleischer

Daily supervisor dr. G. Koekoek (Maastricht University) Conducted between 03-02-2020 & 07-07-2020

Size course 15 EC

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Abstract

It is well known in astrophysics that compact objects in the universe can be ac-companied by magnetic fields although the origin, details and strengths of such fields are still subject of ongoing research. In this thesis the motion of a charged object orbiting a static, axially symmetric dipole magnetized Schwarzschild black hole (BH) is theoretically examined. As opposed to the usual method in which an effective potential is calculated, in this thesis this is done by the geodesic deviation method. The already well known result is found that the innermost stable circular orbit (ISCO) can shift from 6GM_c2 to ∼ 3

GM

c2 for large

enough magnetic fields. In addition, the geodesic deviation method allows for a complete mapping of the parameter space which is presented in this thesis. An interesting new result that is found is the existence of ’forbidden annuli’, a region in between the event horizon (2GM_c2 ) and the photon sphere (3

GM

c2 ) in which

cir-cular orbits are not allowed to exist even though at smaller and bigger radii they are. These forbidden regions only occur when the charge and magnetic field are of sufficiently high enough strength. We present formulas that give the location of the shifted ISCO and outer radius of the forbidden annulus as a function of the charge, mass and strength of the magnetic dipole field, and show that each can constitute a ’barrier’ where the inspiraling mass will produce a surge in gravitational waves. Lastly, an expression for the gravitational wave frequency produced at each of the barriers is given and is proposed as a method to measure the existence and strength of magnetic fields by gravitational wave observations.

Samenvatting

In 1916 publiceerde Albert Einstein zijn meetkundige theorie van de zwaartekracht. Hij beargumenteerde dat zwaartekracht niet een kracht is, maar een effect van de tijdruimte zelf. Massa’s die in het heelal aanwezig zijn kunnen de structuur van ruimtetijd krommen, zoals wanneer je een bowlingbal op een trampoline legt, het springdoek vervormt. Een extreem voorbeeld van zo’n massa is een zwart gat: een astronomisch object dat de ruimtetijd zo erg kromt dat op een bepaalde afstand zelfs licht niet meer kan ontsnappen. In het onderzoek dat is gedaan in deze bachelorscriptie is er gekeken naar de aanwezigheid van sterke magnetische velden rondom zwarte gaten, en is onderzocht wat voor invloed deze velden hebben op de beweging en stabiliteit van geladen massa’s die in cirkelvormige banen om zo’n zwart gat heen bewegen. In zo’n situatie zal een bewegend geladen object in een magnetisch veld niet alleen de zwaartekracht voelen, maar ook de Lorentzkracht. De Lorentzkracht zorgt ervoor dat stabiele cirkelvormige banen dichterbij het zwarte gat mogelijk zijn dan wanneer er geen zwaartekracht aanwezig is. Bovendien ontstaan er dichtbij het zwarte gat rin-gen waar instabiele cirkelvormige banen verboden zijn, terwijl daar omheen de banen wel toegestaan zijn. Bij beide effecten vinden we een mogelijke barri`ere die ervoor kan zorgen dat een object, dat langzaam naar het zwarte gaat toebe-weegt in cirkelvormige banen, opeens exponentieel uit zijn baan wordt gehaald. Dit zal een signaal genereren in de data van gravitatiegolven. Hiermee denken we een manier te hebben gevonden om de sterkte en aanwezigheid van sterke magnetische velden rondom zwarte gaten aan te tonen.

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1 Introduction

The nature, state, and future of objects in the universe, from elementary objects to the most massive compact objects, are determined by forces that can be bro-ken down to the four fundamental interactions: the gravitational, EM, strong, and weak interactions. On large scales — like the distances that we come across in astrophysics and cosmology — the effects of gravity and the electromagnetic (EM) force dominate the motion of objects. It is believed that many compact objects, like neutron stars and black holes (BHs), are surrounded by a magnetic field. In many astrophysical systems the interplay between these two fields can be observed from planet Earth, leading to the field of multi-messenger astron-omy.

Recent research shows that there is very little known about the exact prop-erties of such fields around compact objects and due to the complexity of the problem methodology for empirical observation is limited. How strong these magnetic fields can get is still subject of ongoing research. The presence of such a field is indicated by observed effects around compact objects, like relativistic jets, synchrotron radiation and humps in emission lines of quasars [1, 2]. The common idea is that processes in the accretion disk could create such fields. Ac-cretion disks are formed by accumulated material that end up in orbit around massive stellar objects (see figure 1a for a visual illustration). Eatough et al. for instance found that a strong magnetic field of ∼ 102 _{Gauss in the}

accre-tion disk could clarify the multi-frequency radio emission of a pulsar close to Sgr A*_{, the super massive BH in the center of our Galaxy [3]. The theory of}

general-relativistic magnetohydrodynamic (GRMHD) accretion disks could ex-plain the amplification of magnetic fields up to ∼ 108 _{Gauss [4, 5]. This model}

is a dynamical general relativistic approximation and describes the plasma in the accretion disk as a fluid. GRMHD simulations have been investigated and it is found that nonlinear properties of the disk, which allow for shocks and turbulence, can create instabilities that amplify already present EM fields (see for example the article by Porth et al. [6]). Because of the dynamic flow of the plasma, the magnetic fields that are created are also varying in magnitude. A more extreme situation is the one where the central object is taken to be a quasar. A quasar is an active centre of a galaxy in which a super massive BH (∼ 106_{− 10}10_M

) is embedded. It is believed that quasars are the result of

the accretion disks around the BHs. In recent study it was found that in the spectrum of quasar PG0043+039 certain emission lines appear — thought to be cyclotron lines [2]. A model that included magnetized plasma with a magnetic field strength of ∼ 108_{Gauss in the inner magnetic accretion regions could}

the-oretically explain the line emitting regions of the BH. Whether fields exist with higher magnitudes of order is a question of ongoing research.

The aim of this thesis is to contribute to this open question. In this thesis, the theoretical aspects of a magnetized Schwarzschild BH on a charged orbiting object are investigated, leading to a prediction on how the magnetic fields can

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be measured by the means of gravitational wave (GW) observations.

The report is built up as follows. The theoretical background of the research is discussed in section 2.1. The effect of Schwarzschild spacetime on the magnetic field is examined. In section 2.2 the method of the geodesic deviation will be explained that is used to approach the problem. A short introduction on GW cosmology and detectors will be given in section 2.3. In section 3 the zeroth and first order deviation equations of the equation of motion are calculated to make statements about the existence and stability of circular orbits in the equatorial plane, and in section 4 this is made numerical by mapping the parameter spaces for a dipole field. The theoretical implications of this section will be investigated in 5 and applied to an existing binary system with magnetic field. A method is proposed to calculate the presence of strong magnetic fields in the universe with the help of future GW observations. Points of discussion and possible next steps for future research are given in section 6, and an overall conclusion is made in section 7. Throughout this thesis a geometrized unit system (c = G = 1) is used unless explicitly stated otherwise.

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Figure 1: In this thesis the motion of a charged object in circular motion around a magnetized Schwarzschild black hole is investigated. The magnetic field that we consider is a static axially symmetric dipole field. In figure (b) the situation is sketched. In reality the magnetic field lines will also be curved due to the central mass. The common idea in literature is that magnetic fields around compact objects are created by processes in the accretion disk. Accretion disks are structures in the equatorial plane of compact objects formed by accumulated material that ends up in orbit around the masses. The value of the innermost stable circular orbit (ISCO) determines the inner radius of the disk, as seen in figure (a).

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2 Theory

2.1 Static, axisymmetric magnetic field in Schwarz-schild

spacetime

In this section the theory behind the study is being discussed. Explained is how the metric tensor looks like in Schwarzschild spacetime and how it transforms the EM field. To use this approximation, where one only considers that the curved spacetime interacts with the magnetic field but not the other way around, a condition on the magnetic field is given that will be checked trough out thethesis. Lastly, current and future GW detectors will be discussed.

2.1.1 Linearization of Schwarzschild spacetime

The theory of classical mechanics describes the motion of everyday of objects, and it has been the main theory of mechanics for the longest time of history because it describes the physics these objects very well. In this theory, motions are described by the laws of Newton. Nevertheless, as found by 20th _century

physicists, the validation of classical mechanics is restricted to non-relativistic velocities (which is at velocity scales v c) and to weak gravitational fields (which is at length scales r GM_c2 ). The theory of general relativity generalizes

the theory of Newtonian space and time. One of the implications is that gravity is no longer a force but an effect of curved spacetime which makes it possible to study the motion of objects that move relativistically and/or in a strong gravi-tational field near compact objects like BHs and neutron stars.

The theory of general relativity describes how the geometry of curved space-time, and the mass-energy in it, are related. This relation is summarized by the famous Einstein field equations (EFEs):

Rµν−

1

2gµνR + Λgµν= 8πG

c4 Tµν. (1)

Here, Rµν is the Ricci curvature tensor, R the scalar curvature, Tµν the

energy-momentum tensor, and Λ is the cosmological constant which comprises the energy density of space. In general relativity, the structure of spacetime is captured by a metric tensor gµν that describes the local geometry of spacetime.

The EFEs are ten nonlinear partial differential equations which can be solved for the ten metric coefficients. Consequently, there is not one general solution to the EFE — there are many possible metrics that solve the equations. The EFEs are nonlinear, which means that the partial derivatives in it appear to higher powers than the first, and that the behaviour they describe is usually very complicated and might possibly become chaotic [7, 8]. By using different techniques and assumptions it is possible to find analytical solutions by approximation. The technique that is often used is the linearization of the EFE, where an unknown metric is approximated as one known solution to the EFE gK

µν plus a small

perturbation δgµν:

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One of the assumptions that can then be made is that we’re looking at spacetime in vacuum, so when the energy-momentum tensor Tµν is zero. It has been well

known since 1916 that the perturbed part, δgµν, corresponds to GWs that travel

with the speed of light. How these GWs are detected is discussed in section 2.3 . Most massive objects in the universe have an angular momentum ~J and can best be described using the Kerr-metric. Due to the rotation of the object, spherical symmetry is destroyed, which makes the algebra of the Kerr-metric calculations very complex [9]. In this thesis a non-rotating, spherically symmet-ric uncharged BH is considered, for which the Schwarzschild-metsymmet-ric is sufficient. In the case of Schwarzschild spacetime spherical coordinates (t, r, θ, ϕ) can be used due to spherical symmetry. The covariant Schwarzschild-metric is given by

gµν = diagonal(−(1 − 2M/r), (1 − 2M/r)−1, r2, r2sin2θ) (3)

In general relatively the motion of a test particle in curved spacetime is described by the forceless geodesic equation of motion

d2xµ dτ2 + Γ µ λν dxλ dτ dxν dτ = 0 (4)

where Γµ_λν(u) is the affine connection field, Γµ_λν(u) with coefficients that are called the Christoffel symbols. The expressions for the symbols in Schwarzschild spacetime are written out in A.2 and will the used throughout the thesis. 2.1.2 The transformed magnetic field

The theory of electrodynamics is described by the Maxwell equations and can neatly be summarized by

∂µFµν= −Jν (5)

where Jµ is the source of the electromagnetic field and Fµν is the contravariant Maxwell field tensor given by

Fµν = gµκgνλ(∂κAλ− ∂λAκ) (6)

with Aµthe electromagnetic four-vector potential. In spherical coordinates the

field tensor takes the form of [10]

Fµν =        0 Er Eθ/r Eϕ/(r sin θ) −Er 0 −Bθ(r)/r Bθ/(r sin θ) −Eθ/r Bϕ/r 0 −Br/(r2sin θ)

−Eϕ/(r sin θ) −Bθ/(r sin θ) Br/(r2sin θ) 0

       . (7)

The covariant field tensor follows by contraction with the corresponding space-time metric gµν.

The curvature of spacetime transforms the dynamics of electromagnetism. To extend the Maxwell equations to general spacetime, equation (5) can be modified

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using the principle of general covariance which states that in curved spacetime the partial derivative ∂µ should be substituted with the covariant derivative

Dµ, which is given by the partial derivative plus a correction with the affine

connection Γν_µλ in which the curvature of spacetime is embedded [11]. Since we are not interested in how the magnetic field was created, but only in how the field is transformed by the gravitational field, we use the so called sourceless Maxwell equations. This means that the current density Jν _{is equal to zero.}

So, in general spacetime the sourceless Maxwell equations look like ∂µFµν+ Γµ_µλFλν+ ΓνµλF

µλ_{= 0.} ₍₈₎

As we know from non-relativistic electrodynamics, a moving charged object in a magnetic field feels the attraction or repulsion of the Lorentz interaction. In general relativity, the Lorentz interaction is given by fµ_{= qF}µ

ν dx

ν

dτ [11]. So, by

adding this force to equation (4), the equation of motion of a charged object that is both affected by a magnetic and strong gravitational field is then given by the equation of motion

d2xµ dτ2 + Γ µ λν dxλ dτ dxν dτ = q mF µ ν dxν dτ (9)

where q and m represent the charge and mass of the object respectively. In this thesis, a static, axisymmetric magnetic field is considered, so the fol-lowing four statements are true:

1. The magnetic field is static, so there is no time dependence 2. We only consider the magnetic part of the field, so ~E = 0

3. The magnetic field is axisymmetric, this means that components of the magnetic field are independent of ϕ

4. The field is smooth, so there will be no discontinuities in the magnetic field lines

By using these statements and the symmetries of the situation investigated in this thesis is possible to find expressions for the field tensor Fµν and the magnetic

vector potential Aµ. This has been computed before (for complete derivation

see for example [12, 13]) and will not be recalculated here. This results in the field tensor Fµν for an axisymmetric, static magnetic field:

Fµν =        0 0 0 0 0 0 0 r sin θBθ 0 0 0 −r2_{sin θB} r 0 −r sin θBθ r2sin θBr 0        . (10)

and a magnetic vector potential which only has an angular component Aϕthat

is dependent on r and θ [1]: Aϕ= alr2sin2θ C 3/2 l (cos θ) Q (0,2) l ( r M − 1) (11)

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where al represents a constant that ensures that the vector potential reduces

to the flat-space solution for large values of r [14]. C_l3/2 and Q(0,2)_l are the Gegenbauer polynomials and Jacobi functions of the second kind respectively and describe the behaviour of the vector potential for the lth multipole of the field. The expressions for the first few multipoles are given in A.1. Magnetic fields in nature are mathematically described by a multipole expansion. This expansion consists of multiple terms (the multipoles) which together make up the physical vector potential. By specifying the multipole the explicit expression for (11) can be found. To describe the complete, exact solution for Aϕ, an

infinite sum of multipole terms is generally needed. However, it is sufficient to only consider the first few terms because the weight of the terms decreases for larger values of l.

2.1.3 Schwarzschild approximation

In section 2.1.1 it was stated that the Schwarzschild-metric is a solution to the sourceless EFE. But, in our case, there is a source term: Tµν is nonzero because

of the magnetic field. The T00 _{component in this case is:}

T00= B

2

2c2_µ 0

Therefore, the magnetic field is also interacting with spacetime and should be included in the metric. But maybe we can assume that this contribution is so small, that we can use the source-free solution of the Schwarzschild metric. So, the magnetic field in the problem should be big enough to have a detectable in-fluence on the charged objects, but not too big to have an effect on the spacetime metric, as that would complicate the problem. Summarizing, for Schwarzschild spacetime to hold in the situation where a BH is magnetized, the following condition should be true:

δgµν gSµν

where gS

µν is now the known Schwarzschild-metric. This condition has been

numerically calculated by Gal’tsov and Petukhov, among others, who compared the relative contribution to the metric of the magnetic field with the mass M of the BH, finding [15]:

B ∼ 1019M

M Gauss. (12)

This condition will be checked throughout the thesis for consistency of the method.

2.2 Geodesic deviation method

Solving the equation of motion given by equation (4) is, in general, a very com-plicated task. By the use of symmetries, assumptions and simplifying methods it is possible to find approximate analytical solutions. One of these methods is the world-line deviation method, developed by Kerner et al. in 2001 that will

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be used throughout this thesis [16, 17].

In the world-line deviation procedure, successive approximations are used where one begins with a known analytical solution to the equation of motion and adds higher order deviations to it to improve accuracy. By doing this, the coordinates xµ become, for the first order only, linear in some expansion parameter σ that is small, unitless, and invariant. The physical interpretation of σ is related to the eccentricity of the orbit as laid out in the work by Koekoek and Van Holten but the technical details are not important for the remainder of this thesis [17]. Subsequently, higher order terms in σ can be computed, by which even non-linear effects are included in the approximation. Because of the small nature of σ, while improving the accuracy of the overall solution, every higher order itself adds less precision than the previous order.

The exact solution xµ _{will thus be an infinite sum of terms with orders in σ:}

xµ= xµ₀+ σnµ+σ

2

2 m

µ_{+ ...} ₍₁₃₎

The corrections nµ, mµ, etc. can be found separately by substituting this sum into the equation of motion. Note that the Christoffel symbols Γµ_λν and the field tensor Fµν are also dependent on the coordinates, so they modify as well.

For purposes of this thesis that we will explain in the upcoming section, only the zeroth and first order deviations will be calculated, so let us only consider the first deviation equation for now. For the computation of higher orders the exact same procedure can be used, and those results can be found in the work by Koekoek and Van Holten.

From direct Taylor expansion up to first order it can be found that Γµ_λν(xµ₀+ σxµ) = Γµ_λν(xµ₀) + σ(nκ∂κΓ µ λν(x µ 0)) (14) F_λνµ(xµ₀+ σxµ) = F_λνµ(xµ₀) + σ(nκ∂κF µ λν(x µ 0)) (15)

By substituting the perturbed expressions, collecting all the terms with equal orders of σ, and throwing out higher orders that are nonlinear, the seperate deviation equations are obtained:

0th order: x¨µ₀+ (Γµ_λνx˙λ₀x˙ν₀− q mF µ νx˙ ν 0) = 0 1st order: n¨µ+ 2(Γµ_λνx˙λ₀− q mF µ ν) ˙n ν_{+ (∂} κΓµ_λνx˙λ0x˙ ν 0− q m∂κF µ νx˙ ν 0)n κ_{= 0} (16) These equations will be referred to as the ’geodesic deviation equations’. When a situation is specified where the spacetime metric and EM fields are captured, it is relatively simple to calculate the motion of an object following this geodesic.

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2.3 GW detectors

General relativity tells us that massive objects that are orbiting each other in-duce a time-varying quadrupole moment that generates gravitational radiation in the form of GWs. These GWs, as discussed earlier, are small distortions of spacetime that travel with the speed of light. GW cosmology is interesting because it offers a complete new way to look at the universe. Before, astrophysi-cists were dependent on EM radiation only. The usage of this form of radiation to study the universe is limited due to its strong interaction with matter. The gravitational interaction on the other hand is weak, and the first detection of GWs in 2017 opened up a complete new field of multi-messenger astronomy. GWs are detected through the effects of strain h, a dimensionless quantity that is proportional to the ratio of the change in distance to the distance between two point masses (∆L_L ) and time interval to time between two events(∆t_t ). These tiny deviations in distances and time intervals arise at a frequency that is equal to the GW frequency which itself is proportional to the angular frequency of the rotating massive objects. Both the strain and frequency of GWs can be measured with highly sensitive detectors on Earth. For this thesis we are only interested in the GW frequency, as will be explained in section 5.

At this moment two kinds of detectors are in use: ground-based interferometers like the LIGO and Virgo detectors, and pulsar timing array (PTA) experiments like the IPTA collaboration. For 2034 another interferometer is planned to be in operation, which is the space-based LISA detector. Laser interferometers like LIGO and LISA measure tiny fluctuations in the distances between two point masses. Ground-based interferometers are sensitive to frequencies in the range of 102_-103_{Hz while the LISA mission will be able to measure the frequency band}

of 10-4-10-1Hz [18, 19]. Another way to detect GWs is by the means of PTAs which measure the time distortion that a GW produces. A radio pulsar is a fast rotating neutron star that emits EM radiation in the form of pulses with highly regular time intervals. Due to this rotational stability they are used as very precise astronomical clocks and can even reach the accuracy of atomic clocks. A passing GW affects the regular train of pulsed radiation, causing a fluctua-tion in the observed pulsar frequency [20]. By analyzing the distorted pulsar signal with radio telescopes, the GW and its corresponding characteristics can be traced back. PTAs can detect GW frequencies in the order of 10-9_-10-6_Hz

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3 The existence and stability of circular orbits

in the equatorial plane

The trajectory of objects orbiting a compact object surrounded by a magnetic field is described by the equation of motion, as given in section 2.1.2. The equation can normally not be solved analytically due to the complexity of the problem. Yet, as described in section 2.2, it is possible to approximate the motion by first finding an exact solution to the zeroth order equation, and then adding higher order deviations to this solution to approach the general solution. The equation of motion (9) will be solved for the orbital motion of an object in Schwarzschild spacetime that is close to a circular orbit. This is relevant due the fact that closed orbits in Schwarzschild spacetime become more circular over time due to the emission of gravitational radiation. So, for the zeroth order solution we assume circular orbits in the equatorial plane as reference orbits. For now we are only interested in the first order deviation because it determines whether the circular orbits are stable in time or not.

In the next two sections, these zeroth and first order solutions to the geodesic equations will be calculated.

3.1 Circular orbits in the equatorial plane

3.1.1 Zeroth order equation of motion

Whether circular orbits are allowed in the equatorial plane with the axially symmetric magnetic field included can be found by solving the zeroth order equation of motion. In section 2.2 we found that this equation is given by

¨ xµ₀+ (Γµ_λνx˙λ₀x˙ν₀− q mF µ νx˙ ν 0) = 0 (17)

where the dots denote the derivative to proper time τ . Writing this out for each component and using the symmetry of the Schwarzschild Christoffel symbols (see A.2) gives

¨ xt₀+ 2Γt_trx˙t₀x˙r₀= 0 (18a) ¨ xr₀+ Γr_tt( ˙xt₀)2+ Γr_rr( ˙xr₀)2+ Γr_θθ( ˙xθ₀)2+ Γr_ϕϕ( ˙xϕ₀)2− q mF r ϕx˙ ϕ 0 = 0 (18b) ¨ xθ₀+ 2Γθ_rθx˙r₀x˙θ₀+ Γθ_ϕϕ( ˙xϕ₀)2− q mF θ ϕx˙ ϕ 0 = 0 (18c) ¨ xϕ₀ + 2Γϕ_rϕx˙r₀x˙₀ϕ+ 2Γϕ_θϕx˙θ₀x˙ϕ₀ − q mF ϕ rx˙ r 0− q mF ϕ θx˙ θ 0= 0 (18d)

We can use the mathematical and physical properties of circular orbits to solve these differential equations. A circular orbit is characterized by two variables: the distance to the barycenter r and the angular velocity uϕ_{. Because these}

variables are both constant in a circular orbit, both the total energy and the angular momentum of the object are conserved. The four-velocity (the deriva-tive of xµ _{with respect to proper time τ ) of a object in a circular orbit is of}

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the form uµ = ( ˙xt, ˙xr, ˙xθ, ˙xϕ) = (ut, 0, 0, uϕ), because r is constant and the polar angle can be chosen to be θ = π/2 (which is true anyhow in the equatorial plane). Filling this in and writing out the expressions for the Christoffel symbols gives us the four conditions for circular orbits to exist in an axially symmetric magnetic field in Schwarzschild spacetime:

˙ ut= 0 (19a) M r2 1 − 2M r (ut)2− r1 − 2M r (uϕ)2− q mF r ϕu ϕ_{= 0} _(19b) −q mF θ ϕu ϕ_{= 0} _(19c) ˙ uϕ= 0 (19d) In the second and third condition, the magnetic field comes into effect. We can check for what magnetic fields these conditions are satisfied. Since we assume that neither q, m, and uϕ _{are zero (the latter because that would mean radial}

infall of the object), condition (19c) tells us that Fθ

ϕ should be zero for circular

orbits to exist. This component of the field tensor is given by

Fθ_ϕ= gθθ∂θAϕ= −sin θBr (20)

In the equatorial plane, the radial component of the transformed magnetic field must thus be zero, and the magnetic field should be perpendicular to the plane for circular orbits to exist. This result supports the fourth statement that we made in section 2.1.2, because if the axisymmetric magnetic field would not be perpendicular in the equatorial plane, the direction of the field in the radial direction would be opposite just above and under the plane. This would mean a discontinuity in the field lines. So, when is this the case?

Let us look at the complete expression for Fθ

ϕ by using the expression for Aϕ

that we found in section 2.1.2: Fθ_ϕ= Q(0,2)_l r

M − 1

(2 sin θ cos θ C_l3/2(cos θ) + sin2θ ∂θC 3/2

l (cos θ)) (21)

which simplifies in the equatorial plane (θ = π/2) to the condition Fθ_ϕ= Q(0,2)_l r M − 1 (∂θC 3/2 l (cos θ)) = 0.

The Jacobi functions of the second kind Q0,2_l are generally not zero, but the Gegenbauer polynomials C_l3/2, given by equation (61) in A.1, can be. For even values of l, the polynomials consist of even powers of x and a constant term, while for odd values they are composed only by odd powers of x. When x = cos θ and the derivative to θ is taken, we obtain ∂θcosnθ = −n sin θ cosn−1θ

terms and for odd values of l also a constant term with λ. This means that for cos(θ = π₂) = 0, all terms cancel but the constant term. So, condition (19c) is always satisfied for even multipoles, while for odd multipoles the zero term

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should come from the Jacobi polynomials of the second kind. To summarize, only the even multipoles 2l will ensure the possibility of circular orbits in the equatorial plane.

There is another condition that applies to the four-velocity uµ namely the nor-malization:

gµνdxµdxν= −1.

This gives us a fifth condition for circular orbits to exist in the equatorial plane: −1 −2M r (ut)2+1 −2M r −1 (ur)2+ r2(uθ)2+ r2sin2θ(uϕ)2= −1 using ur_{, u}θ_{= 0 and solving for u}t_gives:

ut= s

1 + r2_(uϕ₎2

1 −2M_r . (22)

Substituting this into equation (19b), we obtain a quadratic equation for the angular velocity uϕ_{, which can be solved using the quadratic formula:}

r1 − 3M r (uϕ)2+ q mF r ϕu ϕ₋M r2 = 0 ⇒ uϕ= −_mqFrϕ± q (_mqFr ϕ)2+4Mr (1 − 3M r ) 2r(1 −3M_r ) . (23)

Here, q and m are the charge and the mass of the object respectively. M is the mass of the massive object, and r is the constant radius of the circular. Frϕ

contains the polar direction of the magnetic field, and is given by Fr_ϕ= grrFrϕ= 1 − 2M r r sin θBθ(r), (24)

but it can also be written in terms of the magnetic potential Aϕthat we

deter-mined in section 2.1.2: Frϕ= 1 − 2M r ∂rAϕ. (25)

The expression for Aϕand thus Frϕdepends on the specification of the multipole

l.

3.1.2 Angular velocity

The angular velocity that meets the conditions for circular orbits is given by equation (23). When the magnetic field is turned to zero, or when the object has no charge, these two different values reduce to the angular velocity of a geodesic orbit: uϕ_geo= ± s M r3_{(1 −} 3M r ) (26)

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With the magnetic field included, it seems to be the case that for the same _Mr and |qµ_m| combination there are two ”accepted” possibilities for the magnitude of the angular velocity that give a circular orbit. This splitting result reminds of the quantum mechanical Zeeman effect.

In atomic physics the Zeeman effect denotes the splitting of one spectral line (in our case: one value of angular velocity) into several components (two different values of angular velocity within same orbit) when an external magnetic field is switched on. But this is how far the comparison goes. In the quantum mechan-ical effect the energy of the object is shifted due to the coupling of the magnetic dipole moment associated with the spin of the electron and the dipole moment of the external magnetic field [22]. This is obviously not what happens in the case of a magnetized BH as we’re not dealing with spin here. It’s more as if the angular momentum of the charged object is coupling with the dipole moment of the magnetic field. But what really happens, is that the Lorentz force and its effect are dependent on the sign of the angular velocity of the object. As we see in equation 26, in the case where there is no magnetic field, there are two possible orbits per fixed radius _Mr to the BH. A object can both move with an angular velocity uϕgeo clockwise and counterclockwise. Here, the

mag-nitude of uϕgeois the same but opposite in direction. When there is an external

magnetic field, this opposite sign has an impact on the effect of the Lorentz force (fr_{= qF}r

ϕuϕ). With a classical point of view, it can be seen as if all the

forces need to be balanced for circular orbits to exist. When the Lorentz force is aligned with the gravitational output, the object needs a higher angular velocity to prevent plunging into the BH compared to the case where the magnetic force and gravitational force are opposite in direction. This is why one magnitude of the angular velocity should be bigger and opposite in sign than the other at the same distance to the BH. From now on, we thus need to check both the plus and minus sign of the angular velocity for the same orbit.

In figure 2 the top view on the situation is sketched. Without a magnetic field, the possible orbits are given by the magnitude of uϕ_geo from equation (26) and can have two directions at a fixed position R. With a magnetic field present, there are two magnitudes |uϕ₁| (purple orbits) and |uϕ₂| (turquoise orbits) possi-ble. These orbits can also have two directions, which brings us to four possible orbits per fixed radius R. Note that the magnetic field itself does not change the magnitude of the angular velocity of the object. The Lorentz force is perpen-dicular to the angular direction of motion, and thus does no work on the object. It can therefor only change the direction of motion and the total energy of the system is preserved as we would expect in the case of circular orbits. Equation (23) simply tells us what angular velocity the object should have to move in a circular orbit.

Whether the overall sign of uϕ _{is positive or negative, so does the object move}

clockwise or counterclockwise, depends on the sign of the charge q, the direc-tion of the magnetic field, the region we are looking, and the sign of the root. Combining both the magnitude and the direction of uϕ _{results in four possible}

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other higher because of different influences of the Lorentz force. For the same magnitude two orbits can exist: one moving clockwise, and the other moving counterclockwise.

sign _mqFr

ϕ region sign root magnitude direction

+ r > 3M + lower counterclockwise − higher clockwise 2M < r < 3M + lower counterclockwise − higher counterclockwise − r > 3M + higher counterclockwise − lower clockwise 2M < r < 3M + higher clockwise − lower clockwise

Table 1: Possible equatorial circular orbits of a charged object in a magnetic field in Schwarzschild spacetime. There are four orbits possible for a fixed radius

r

M to the BH: two orbits have a higher magnitude of angular velocity u ϕ _but

are opposite in sign. The other two are lower in magnitude and also equal but opposite in sign. Which orbit is relevant depends on the initial values of the charge q, the direction of the magnetic field, the region and the direction of the geodesic orbit. This result shows the effect of the Lorentz force on the system.

Figure 2: Top view of the binary system where a mass m orbits another mass M . The possible circular orbits with and without field are sketched. Without a magnetic field, the possible orbits are given by the magnitude of uϕgeo from

equation (26) and can have two directions at a fixed position R. With a magnetic field present, there are two magnitudes |uϕ₁| (purple orbits) and |uϕ₂| (turquoise orbits) possible. These orbits can also have two directions, which brings us to four possible orbits per fixed radius R.

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3.2 Stability of the circular orbits in the equatorial plane

3.2.1 First order deviation

Stable orbits are orbits that objects will come back to when slightly perturbed. We define the equatorial circular orbits to lie in the ˆr,ˆϕ-plane, at the constant polar angle θ = π/2 with constant radius. This means that any deflection from this plane of movement could result in instability of the orbit. The first order deviations nµ can be defined as the distance between the equatorial circular reference orbit and a position slightly outside the orbit, and how this distance evolves in time determines the stability of the orbit. When it becomes larger for bigger values of proper time τ , the orbit is unstable because then the object is moving away from the orbit. When it goes to zero in time, the orbit is stable because the object gets pushed back. So, to say something about the stability of the equatorial circular orbits found in the previous section, the first order geodesic deviation equation needs to be solved:

¨ nµ+ 2(Γµ_λνx˙λ₀− q mF µ ν) ˙n ν_{+ (∂} κΓµ_λνx˙λ0x˙ ν 0− q m∂κF µ νx˙ ν 0)n κ_{= 0} ₍₂₇₎

We can again write this out for every separate coordinate to obtain four coupled differential equations ¨ nt+ 2Γt_trut ˙nr= 0 (28a) ¨ nr+ 2Γr_ttut˙nt+ 2(Γr_ϕϕuϕ− q mF r ϕ) ˙n ϕ₊_∂ rΓrtt(u t₎2_{+ ∂} rΓrϕϕ(u ϕ₎2 − q m∂rF r ϕ nr− q m∂θF r ϕu ϕ_nθ_{= 0} (28b) ¨ nθ− q mF θ ϕ˙n ϕ₋ q m∂rF θ ϕu ϕ_nr₊_∂ θΓθϕϕ(u ϕ₎2₋ q m∂θF θ ϕu ϕ_nθ_{= 0} _(28c) ¨ nϕ+ (2Γϕ_ϕruϕ− q mF ϕ r) ˙n r₋ q mF ϕ θ˙n θ_{= 0} _(28d)

In the following subsection we will use these to investigate the stability of circular orbits in the equatorial plane.

3.2.2 Realness and stability

The condition that has to be met for equatorial circular orbits to exist in the first place, is that the angular velocity uϕ has to be real for these orbits to be physical. This means that the expression under the root in equation (23) has to be positive. This is always true for values of r > 3M . But for 2M < r < 3M , so between the Schwarzschild radius and the photonsphere of the BH, this is not always the case. In this region, the following condition must be true for equatorial circular orbits to exist:

q mF r ϕ 2 ≥ −4M (1 − 3M r ) r . (29)

Evidently, within this region not all orbits can exist but only specific combina-tions of the charge q and mass m of the object, the magnitude of the magnetic

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field and the strength of the gravitational field. The multipole of the magnetic field needs to be specified to investigate this specific relation.

When the first condition for realness is met, it is meaningful to calculate whether the orbits are stable by solving equation 28 and checking whether these devia-tions increase or decrease in time. In section 3.1.1 it was found that Fθ

ϕis equal

to zero for circular orbits, so these terms cancel. With this, equation (28b) decouples from the others and becomes a second order differential equation for nθ_: ¨ nθ+∂θΓθϕϕ(u ϕ₎2₋ q m∂θF θ ϕu ϕ_nθ_{= 0} ₍₃₀₎

which is the equation of motion for a simple harmonic oscillator. When we assume that the diversion is at a maximum nθ

0 at τ = 0, and when we use that

for a circular orbit in the equatorial plane ∂θΓθϕϕ= 1 and ∂θFθϕ= 1

r2∂θ2Aϕ, the

solution for nθ _{can be found:}

nθ(τ ) = nθ₀cos(ωθτ ) ⇒ ωθ= r (uϕ₎2₋ q mr2∂ 2 θAϕuϕ (31)

The expression for nθ _{can be rewritten using Euler’s formula:}

nθ(τ ) =n θ 0 2 eiωθτ_{+ e}−iωθτ_. ₍₃₂₎

When the frequency ωθis real, nθ(τ ) is a cosine and the object oscillates around

the equatorial plane in time. We’ll call this a semi-stable orbit, since the object does not get totally pushed back into the plane but stays close. When the frequency is imaginary, so ωθ→ ˜ωθ= iωθ, this cancels out the i in the exponents,

resulting into the expression for nθ(τ ): nθ(τ ) =n θ 0 2 e−ωθτ_{+ e}ωθτ_. ₍₃₃₎

In time, the first exponents grows to zero but the second exponent grows expo-nentially. This means that the distance to the equatorial plane becomes bigger in time and that the resulting orbit is unstable.

Thus, the radial stability depends on whether ωθis real or imaginary, so whether

the expression under the root is positive or negative. This depends on the sign of the charge q and the angular velocity uϕof the object, and on the expression for the magnetic four-potential Aϕ, so on the magnetic field. For semi-stable

orbits in the polar direction the following condition has to be true: (uϕ)2− q

mr2∂ 2

θAϕuϕ ≥ 0. (34)

For the other three equations in 28, things are not as easy because the four-positions of the coordinates are coupled. To solve the problem, a trick can be

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used where the coupled t and ϕ equations are integrated and substituted into to the equation for r to obtain the solutions to the first order deviation equation, for details see the article by A. N. Aliev and D.V. Gal’tsov [23]. Found is that all three deviations in t, r and ϕ are dependent on the radial frequency ωr:

ω_r2=3 −8M r (uϕ)2−2M r3 1 − M r (ut)2 − q m[(∂rF r ϕ− Γ ϕ rϕF r ϕ− Γ r ϕϕF ϕ r)u ϕ₊ q mF r ϕF ϕ r] (35)

The same argument for polar stability can be made for radial stability: when (ωr)2 ≥ 0, the circular orbit is radially semi-stable but when ωr is imaginary

the deviations grow in time and the circular orbit will be unstable. What the conditions are for stability depends on the radius of the circular orbit (_Mr), on the charge q and m of the object, and on the magnetic field. The latter needs to be specified to find explicit conditions and regions for stable circular orbits.

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4 Mapping parameter space for a dipole field

As described before, magnetic fields in nature are mathematically described by a multipole expansion. The first term dominates the expansion and is thereby often a good approximation to the exact potential [22]. Because there are no magnetic monopoles in nature, the first and simplest term of the magnetic field is given by the dipole field, when l = 0. Let us look at the existence and stability of circular orbits in the equatorial plane in such a field.

Recall that for equatorial, circular orbits, the magnetic four-potential is given by (11). When we fill this in for l = 0, we obtain the following expression for a magnetic dipole field in Schwarzschild spacetime, see A.3 [24]:

Aϕ=

µ sin2θ

r h(r) (36)

where µ is the magnetic dipole moment and h(r) is a function that is only dependent on M and r: h(r) = − 3r 3 8M3 ln(1 −2M r ) + 2M r + 2M2 r2 . (37)

We can also specify the relationship between the magnetic dipole moment µ and the strength of the magnetic field. The magnitude of the magnetic field in the equatorial plane is given by the polar direction of the field, which depends on the distance r and the dipole moment:

Bθ(r) =

−µ

r3(h − r h

0_). ₍₃₈₎

This relation will be useful later when we compare magnitudes of the mag-netic dipole moments with the magmag-netic fields strengths around compact objects found in literature.

4.1 Realness of the angular velocity

For equatorial circular orbits to exist, the combination of the charge q and mass m of the charged object, and the magnetic and gravitational field need to fulfill the condition (29). Now that the multipole is specified, we can substitute the expression for the dipole field in Fr

ϕ: Fr_ϕ= (1 −2M r )∂rAϕ (39) =−µ r2 1 −2M r (h(r) − r h0(r)) (40) The realness condition (29) becomes:

qµ m

2

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where we will call R the realness function: R = s −4M r3_{(1 −} 3M r ) (1 −2M_r )2_{(h − r h}0₎2 (42)

This means that for some combination of values of q, µ, m, and _Mr, circular orbits can exist and for some other combination it can not. Circular orbits can always exist for r > 3M but in between the event horizon (r = 2M ) and the photon sphere (r = 3M ), the angular velocity can become imaginary.

An overview of the conditions for realness of the angular velocity uϕ is given in table 2. When the value of qµ_m is equal to this realness function, the root in expression (23) is zero, and the splitting of two different values for uϕ as discussed in section 3.1.2 disappears. This means that with that combination of the charged object, the magnetic field and that distance to the BH, there is just one possible value for the magnitude of the angular velocity for a circular orbit to exist.

To see whether the found circular orbits are physically real, we need to check the stability of the orbits. This will be done in the upcoming subsection.

region condition uϕ _orbit

r > 3M real exists 2M < r < 3M qµ_m 2 ≥ R2 _real _exists qµ m 2 < R2 _imaginary _non-existent

Table 2: Conditions for realness of the angular velocity uϕ_{. Outside the photon}

sphere (r > 3M ) the angular velocity for circular orbits is real, but in between the event horizon (r = 2M ) and the photon sphere, the value of qµ_m

2

needs to fulfill the realness condition. The function R is given by equation (42).

4.2 Stability

Found was that circular orbits are allowed close to the BH when we consider charged objects in a magnetic dipole field without a radial component in the equatorial plane. But this does not automatically mean that such orbits are likely to be physical. As we have seen in section 3.2, it might be the case that when the object gets a little push or pull by some external forces, like when other objects fly by, the balance between the angular velocity, the magnetic field and gravity can no longer hold the object into place. This can cause the object to leave its circular orbit to, for example, fly away into space or to get gobbled up by the BH. The circular orbit thus needs to be stable as well to be physical. In the previous section we saw that the stability of the circular orbits depend on the two deviation frequencies ωθ and ωr. When one or both

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of these values become imaginary, the perturbations nµgrow exponentially. By taking the combination of the charge q, mass m, and magnetic dipole moment µ as one variable qµ_m, and plotting this against the variable _Mr, a parameter space diagram can be made to check for what combinations the circular orbits are stable. We thus need to check the reality conditions for the two deviation frequencies.

The lowest value of r

M where stable circular orbits can exist is called the

in-nermost stable circular orbit (ISCO). In the sourceless case, so when there are no magnetic effects, this value is found to be r = 6M [25]. In the upcom-ing sections, the ISCO will be calculated for the magnetized situation. Note that, because there are two frequencies that can cause deviations, there are two seperate ISCOs, namely for both the polar (ISCOθ) and radial direction

(ISCOr). When not otherwise specified, we will mean by ISCO the last radius

where both the frequencies are real for circular orbits. In section 4.3.2 the re-sults will be combined to form an overall conclusion of the stability regions and the ISCO in the case of a magnetized BH.

4.2.1 Stability in the polar direction

For semi-stability in the polar direction of the orbits, equation (34) needs to be true. For a dipole field in the equatorial plane the second derivative to θ of the magnetic vector potential Aϕcan be calculated, which then gives:

(uϕ)2+2µqh mr3u

ϕ_{≥ 0.} ₍₄₃₎

When this condition is met, the polar frequency ωθ is real and circular orbits

are semi-stable in the polar direction.

As we saw earlier, for various radii _Mr there are four orbits possible. Two pairs of orbits with the same magnitude in angular velocity, but opposite in sign. We thus need to include both the expression for the uϕ_{with the plus sign and the one}

with the minus sign in front of the root. Because there are two options here, two expressions for ω2

θ will be used, one with the uϕ plus-root-definition (ωθ2)+, and

one with the negative-root-definition (ω2

θ)−. The realness of the polar frequency

depends on three variables: the combination qµ_m, the distance to the BH _Mr and the mass of the BH M , which will be treated as a free parameter. We can now look at the parameter space of these variables, to see where ω_θ2 is positive or negative. This will be done in section 4.3 where we use the worked example of M = 10 to get a visual idea of the stability regions.

4.2.2 Stability in the radial direction

For the stability in the radial direction, the frequency ωrgiven by equation (35)

is a little bit more complicated. But in principle the same can be done as for ω2

θ. Once again we need to take in mind that the root in the expression for the

angular velocity uϕcan both be positive (ω2_r)+and negative (ω2_r)−. We need to check the parameter space of ωr2to see for what combinations of

qµ m and

r M the

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frequency ωr is real or imaginary. Again, when ωr is imaginary the orbits are

unstable in the radial direction, and when ωr is real the orbits are semi-stable

in this direction.

4.3 Worked example M = 10

It is now useful to fill in some values for the variables involved to get a visual idea of the regions of existence for the angular velocity uϕ _{and stability for the}

each direction. We take the mass of the center BH M as a free parameter, for the case when M = 10, and plot the parameter space for the combinations of

qµ m and

r

M. Note that we are still using geometrized units here, so the numerical

results found have no physical meaning in the real world. 4.3.1 Parameter space diagrams

The parameter space for the angular velocity uϕ _{is illustrated in figure 3 and}

4. We took qµ_m to be in the range of ±250 and looked at the region close to the BH (2M < r < 5M ). In the first figure the realness in the case of the negative root in equation 23 is represented for both positive and negative combinations of qµ_m. When the angular velocity is real, the sign of it is also indicated. In The second figure the same is done for the positive root. It can be seen that in the 2M < r < 3M region the angular velocity for circular orbits can both be positive of negative when the sign of the root is fixed. To find not just the sign but also the magnitude of the angular velocity uϕone can fill in equation (23). The red regions indicate that there are no circular orbits possible in that region of parameter space.

When plotting the realness function R from equation (42) for the 2M < r < 3M region, the parabolic shape of the function gets exposed. Remember that for circular orbits to be possible, the combination |qµ_m| needs to be bigger than the realness function. So, it can be seen that between the event horizon (2M ) and the photon sphere (3M ) the realness of uϕ_{depends on the value of |}qµ

m|.

Some-thing interesting happens here. There seem to be two regions where uϕ_{can be}

real: close to r = 2M and close to r = 3M . In between these regions, uϕ _is

imaginary and no circular orbits can be possible. The possible implications of this ’forbidden annulus’ will be examined in section 5. Furthermore, when |qµ_m| is equal or bigger than the maximum value of the realness function, the realness condition is always met. Figure 5 shows that this value depends on the mass of the BH. For M = 10 the |qµ_m|min is 191.706, for M = 20 the |qµ_m|min is 766.824,

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Figure 3: The parameter space for the realness of the angular velocity uϕwith the minus root. We took qµ_m to be in the range of ±250 and looked at the re-gion close to the BH (2M < r < 5M ). It can be seen that uϕ is imaginary in a region with specific combinations be-tween qµ

m and r

M. For circular orbits

to be real, the angular velocity is neg-ative in the r > 3M region, and can be either positive or negative in the 2M < r < 3M region, depending on the sign of qµ_m.

Figure 4: The parameter space for the re-alness of the angular velocity uϕ with the plus root. We took qµ_m to be in the range of ±250 and looked at the region close to the BH (2M < r < 5M ). It can be seen that uϕ is imaginary in a region with spe-cific combinations between qµ_m and r

M. For

circular orbits to be real, the angular ve-locity is positive in the r > 3M region, and can be either positive or negative in the 2M < r < 3M region, depending on the sign of qµ_m.

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Figure 5: The realness function from equation (42) plotted for the region in between the event horizon (r = 2M ) and the photon sphere (r = 3M ). Increasing the mass of the BH M will increase the maximum of the function, with the radius at which this maximum appears does not change. When the value of |qµ_m| is bigger than this maximum, the angular velocity uϕ _{is real and circular orbits can exist. Within a}

specific range of values |qµ_m|, there are circular orbits possible with real angular velocity uϕ_{close to r = 2M and r = 3M , but in between the angular velocity will be imaginary.}

In figures 6a, 6b, 6c, and 6d the parameter space of ω2

θ is plotted. Once again,

we took qµ_m to be in the range of ±250 and looked at the region close to the BH (2M < r < 5M ). In figure 6a it can be seen that for the combination of (ω2

θ)−

(so the negative root in the definition for the angular velocity uϕ) and positive values of qµ_m, the stability in the 2M < r < 3M regions first increases, until it reaches a minimum value for the ISCOθ. When qµ_m is further increased, the

ISCOθ shifts towards higher values.

In figure 6b the regions of stability for the combination of (ω2_θ)− with negative values of qµ_m gives a different result. The ISCOθ converges to a certain value.

In the region outside the photon sphere (r > 3M ) the orbits are always stable, because here the angular velocity (see figure 3) and qµ_m have the same sign, so condition (43) is always met.

Notice how the plots of figure 6a and 6d, and 6b and 6c look alike. The only difference is that they are mirrored in the y-axis, which indicates a certain sym-metry in the problem. Apparently, changing both the sign of the root in uϕ

and the sign of either the charge q or magnetic dipole field µ gives you the same regions of stability. The difference lies in the value of the angular velocity of these orbits, which is the splitting effect as discussed in section 3.1.2.

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(a) (b)

(c) (d) .

Figure 6: The parameter space diagrams for the polar stability close to the central BH (2M < r < 5M ). The red region indicates where the angular velocity uϕ is imaginary. It is shown that when the root in uϕ and qµ_m are opposite in sign (figures (a) and (d)), the stability in the 2M < r < 3M regions first increases when qµ_m in increased, until it reaches a minimum value for the ISCOθ. For higher values of qµ_m the ISCOθ shifts towards higher values. When

the signs are equal (figure (b) and (c)), the ISCOθconverges to a certain value.

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The symmetry that we saw for the parameter space of ω2_θalso appears in the ω_r2 parameter spaces. In figures 7a-7d the parameter spaces diagram are presented for the case M = 10.

Let us first look at the case where the root and qµ_m are opposite in sign. Outside the photon sphere r > 3M there are two possibilities for radially stable circular orbits. Either when qµ_m is relatively small (0 < qµ_m < ∼ 2.045, see figure 7a and 7c) or relatively large (qµ_m > ∼ 2120, see figure 7b and 7d). In figure 7a and 7c the ratio qµ_m is taken to be small. For these small values, the ISCOr

starts at _Mr = 6 and shifts to higher values until ω2

r becomes negative for all

values. When qµ_m is big enough (starting from qµ_m ∼ 321) the stability region starts around r = 2M as illustrated in figure 7b and 7d. When qµ_m is further increased, the maximum value of _Mr where radially stable orbits can exist shifts to higher values.

Now consider the situation where the sign of the root and the sign of qµ_m are equal, so figure 8a and 8b. We considered _Mr from 2 to 7, and qµ_m from 0 to ±200. Whenqµ_m is bigger than zero, the ISCOrshifts from ISCOr= 6M towards

ISCOr = 3.02M . For r > 6M the circular orbits are radially stable

indepen-dently of the value of qµ_m. In the 2M < r < 3.02M region no stable orbits are possible because when qµ_m > 0 in this region, ω2

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(a) (b)

(c) (d) .

Figure 7: The parameter space diagrams for the radial stability when the sign of the root in uϕ and qµ_m are opposite. In the left two figures the distance r is taken from 6M to 100M , while two images to the right are relatively close to the BH (2M < r < 10M ). It can be seen that when qµ_m is small, the ISCOr

starts at r = 6M when qµ_m = 0 and shifts to higher values when qµ_m is increased until ω2

r becomes negative for all values. When qµ

m is large, starting from from qµ

m ∼ 321, radially stable orbits can exist close to the BH, and when qµ

m is further

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(a) (b) .

Figure 8: The parameter space diagrams for the radial stability when the sign of the root in uϕ _and qµ

m are equal. We took qµ

m to be in the range (0, ±250)

and _Mr between 2M and 7M . For r > 6M the circular orbits are radially stable independently of the value of qµ_m. When qµ_m is bigger than zero, the ISCOrshifts

from ISCOr= 6M towards ISCOr= 3.02M . In the 2M < r < 3M region ωr2is

either imaginary (because uϕ_{is imaginary) or negative.}

4.3.2 Comparing the stability regions

The previous stability results for the polar and radial direction can be combined to say something general about the possible regions of stability around black magnetized BHs. In figures 9a-9d the combined parameter spaces of the absolute values |qµ_m| and the distance r

M to the central BH are illustrated. Figure 9a and

figure 9b represent the situations where the sign of the root in uϕ _and qµ m are

opposite, while 9c and 9d represent the situations where the signs are equal. When either the charge q of the smaller object is zero, or when the magnetic dipole moment µ is zero, it can be seen that the ISCO has a radius of r = 6M , just as expected for the case where no magnetic forces play a role.

To obtain stable circular orbits for the situation where qµ_m and the root are opposite in sign, you need a small value of qµ_m and large radii, as shown in figure 9a. Figure 9b shows that when qµ_m is too big, the seperate directions of r and θ can be stable, but the stability regions do not overlap.

In the case of equal signs of qµ_m and the root, stable circular orbits can exist for all values when r > 6M . When the value of qµ_m is increased, the ISCO shifts from 6M towards the photon sphere 3.02M , as shown in 9d. This shifted ISCO reminds us of the effect of the Lorentz interaction, as explained in section 3.1.2. In figure 9c it is shown that closer towards the event horizon the circular orbits are stable in the polar direction, but not in the radial direction.

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(a) Parameter space for the existence and stability of circular orbits in the equatorial plane of a BH (M = 10) in a magnetic dipole field when the sign of qµ_m and the root of uϕ _{are opposite.}

Stable circular orbits are possible for (r > 6M ) for relatively small values of

qµ m.

(b) Parameter space for the existence and stability of circular orbits in the equatorial plane of a BH (M = 10) in a magnetic dipole field when the sign of qµ_m and the root of uϕ _{are opposite.}

Whenqµ_m is too big, the seperate direc-tions of r and θ can be stable, but the stability regions do not overlap.

(c) Parameter space for the existence and stability of circular orbits in the equatorial plane of a BH (M = 10) in a magnetic dipole field when the sign of qµ_m and the root of uϕ are equal. It is shown that closer towards the event horizon the circular orbits are stable in the polar direction, but not in the radial direction.

(d) Parameter space for the existence and stability of circular orbits in the equatorial plane of a BH (M = 10) in a magnetic dipole field when the sign of qµ_m and the root of uϕ are equal. When the value ofqµ_m is increased, the ISCO shifts from 6M towards the pho-ton sphere 3M .

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5 Applications to GW physics

In the previous sections we illustrated the parameter space for stable and unsta-ble, equatorial, circular orbits in a magnetic dipole field around a BH and found that some interesting effects occur for particular combinations of the charge-to-mass ratio _mq of the orbiting object and the magnetic dipole moment µ. We find that the ISCO is dependent on the variables µ, q, m, and M , in contrast to the non-magnetized situation where the ISCO is only M dependent. From the worked example we find that RISCOcan even shift from 6M up to ∼ 3M .

Also, in the 2M < r < 3M region, there are annuli where unstable circular orbits cannot exist while for smaller and bigger radii they can when |qµ_m| is of sufficiently high enough strength.

We defined the ISCO to be the last radius at which circular orbits are stable in both the polar and radial direction. At RISCO, the circular orbits are starting

to become unstable, which implies that a small push will make the deviations from the circular orbit to exponentially grow in time. Such a push could be the result of the emission of GWs, as we will see in 5.1. At the forbidden annulus, the angular velocity itself becomes imaginary. This radius indicates the last circular orbit (LCO). We expect that at one of the RISCO and Rannulus radii

circular orbits will cease to exist. This means that an object that is gradually circling towards a BH (for example, an object that is losing angular momentum and energy due to the emission of radiation) would suddenly have to change its manner of motion. The outer radius of the forbidden annulus and the shifted ISCO each thus constitute a ’barrier’ where we can expect a surge in the output of radiation that can be detected on Earth. Here, we will look at the GW out-put of binary systems. Which of these barriers will result in a surge in the GW data is determined by the parameters of the system at hand. In each case, the radius of the barrier responsible for the surge is directly related to the strength of the magnetic field and can thus be used as a means to measure magnetic field strengths by GWs.

In this section, we will calculate expressions for the GW frequency at the mo-ment of the peak, for each of the two barriers. We will calculate this explicitly for an existing binary system. In this chapter natural units will be restored to obtain numerical results.

5.1 GW physics

Throughout this thesis we have assumed that the radius and orbital frequency of the examined circular orbits are constant in time. In other words, the as-sumption was made that for such orbits the energy and angular momentum are conserved. But, as one might remember from their electrodynamics classes, accelerated charged objects send out EM radiation and thus lose energy and an-gular momentum over time. Furthermore, the theory of general relativity states that a system with a varying second mass moment creates GWs that carry away energy and angular momentum from the radiating system. This causes the

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ra-dius r to decrease over time, which influences the value of the angular velocity, as can be observed in equation (23). This means that our previous assumption of perfectly circular orbits is not valid anymore. However, assuming this loss is a slow process, the rate of inspiraling will be much smaller than the orbital ve-locity (dr_dt ωr) and the previous found results are still a good approximation. In GW literature, such slowly varying orbits are therefore called quasi-circular orbits. How good this approximation is depends on the rate of inspiraling and changing angular frequency compared to the object its tangential velocity and angular frequency itself. To be more precise, the following two conditions should be met for the quasi-circular approximation to be valid:

(1) ˙r ωGWr

(2) ˙ωGW ωGW2

(44) As long as the two conditions are satisfied, the quasi-circular approximation can be used that assumes a circular orbit with slowly varying radius. This condi-tions can be checked using the Newtonian approximated as is laid out in A.4 and will be calculated for the example in section 5.4.

We found that between the radius of the ISCO, RISCO, and the outer radius

of the forbidden annulus, Rannulus, circular orbits can exist in the zeroth order

equation of motion. However, from the first order deviations nµ it can be seen that the circular orbits start becoming unstable at the exact radius of the ISCO. Any small push or pull will cause the object to leave its orbit for a bit. Mag-giore proposes that such a push can emanate from the emission of GWs [26]. If so, when the ISCO is reached, circular orbits will cease to exist and this is the moment where the plunge starts and where the mass m will exponentially fall into the central mass M . We expect that this sudden change in orbit at RISCO will lead to a spike or chirp in the GW frequency output. However, if

the exponent is small, the orbiting object will continue to approximately move in a quasi-circular orbit towards the outer radius of the forbidden annulus, be-cause the deviation from quasi-circular motion will then only grow very slowly in time. This condition can be made precise by demanding that the time needed for the deviation to build up be much smaller than the time needed to overlap the distance from ISCO to the outer radius of the forbidden annulus. This must be true for the deviation in θ and r-directions. The frequency ωθof the former,

however, will be a real number until the outer radius of the forbidden annulus is reached, as evidenced by (31); thus, there will be no exponentially growing deviation in polar direction before the outer radius of the forbidden annulus is reached. For the deviation in radial direction, the condition that the deviation is not given enough time to build up, is given by

|ωr| uϕ (45)

This condition needs to be checked for every case to be investigated. If this con-dition is met, this means that the orbiting mass will continue its quasi-circular motion even after the ISCO is crossed, and thus that the end of quasi-circular

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motion will take place at the outer radius of the forbidden annulus. If this con-dition is not met, the quasi-circular motion will already cease when the mass crosses the ISCO. Thus, equation (45) determines whether the surge in GW output will take place at the barrier of the ISCO, or at the barrier at the for-bidden annulus.

We are now interested in the angular GW frequency that such an inspiraling system will produce at either of the two barriers. The GW frequency fGW is

twice the orbital frequency uϕ_obs divided by 2π of the orbiting object as seen from an observer on Earth that measures time t:

fGW = 2uϕ_obs 2π = 1 π dxϕ dt

Until now we have used the proper time coordinate τ , which describes the coordinate system of the moving object itself. To calculate the GWs and its frequency, therelationship dτ_dt = utcan be used to find the angular velocity uϕ_obs:

fGW =

1 π

uϕ ut

which gives by using the expressions of uϕ _{and u}t _{in equations (23) and (22)}

fGW = 1 π v u u u u t 1 − 2GM rc2 2r(1−3GM rc2 ) D 2 +r c 2 (46)

where D is the numerator of uϕ_:

D = −q mF r ϕ± r q mF r ϕ 2 +4GM r 1 − 3GM rc2 (47) As promised, this expression for the GW frequency will be specialised to each of the two barriers, resulting in a series of steps that will allow a measured GW frequency to be assigned to the value of the magnetic dipole moment µ. Before doing so, the charge of the orbiting object needs to be specified.

5.2 Charge of the orbiting BH

Assuming that the orbiting object around the central mass is a BH, we must find the charge-to-mass ratio for such a compact star if we want to make statements about the exact value of the magnetic dipole moment µ. We will be using two ratios that are found in existing literature.

In their 1978 article, Bally and Harrison derived that all macroscopic bodies in the universe, BHs included, are positively charged with a charge-to-mass ratio of approximately 100CM−1, or 5.0310−29Ckg-1 _{[27]. To compare: the}

charge-to-mass ratio of an electron is -1.76×1011_Ckg-1_{. Other theoretical research shows}

Charged objects in magnetized Schwarzschild spacetime: mapping parameter space and measuring magnetic fields with gravitational wave frequencies