Stationary Fields in Black-Hole Space-Times

DOCTORAL THESIS

Pavel C´ıˇzekˇ

Stationary ﬁelds in black-hole space-times

Institute of Theoretical Physics

Supervisor of the doctoral thesis: doc. RNDr. Oldˇrich Semer´ak,DSc. Study programme: Theoretical physics, astronomy and astrophysics Study branch: Physics

Prague 2017

Title: Stationary ﬁelds in black-hole space-times

Author: Pavel C´ıˇzekˇ

Institute: Institute of Theoretical Physics

Supervisor: doc. RNDr. Oldˇrich Semer´ak,DSc., Institute of Theoretical Physics

Abstract:

Motivated by modelling of astrophysical black holes surrounded by accretion structures, as well as by theoretical interest, we study two methods how to obtain, within stationary and axisymmetric solutions of general relativity, a metric describing the black hole encircled by a thin ring or a disc. The first is a suitable perturbation of a Schwarzschild black hole. Starting from the seminal paper by Will [1974], we showed that it is possible to express the Green functions of the problem in a closed form, which can then be employed to obtain, e.g., a reasonable linear perturbation for a black hole surrounded by a thin finite disc. In the second part we tackle the same problem using the Belinski-Zakharov generating algorithm, showing/confirming that in a stationary case its outcome is unphysical, yet at least obtaining a modest new result for the (static) “superposition” of a Schwarzschild black hole with the Bach–Weyl ring.

Keywords: general relativity, black hole, perturbations, Belinski–Zakharov method

I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act.

In ...... date ...... signature of the author

i Acknowledgement

First of all my parents and wife have my deepest gratitude for supporting me during studies. I am grateful for my supervisor’s his patience, many clever remarks and point- ing out direction of research. And also for his time and helpfulness during quite hectic ﬁnishing of this thesis. People from Institute of Theoretical Physics of Friedrich Schiller University of Jena have my thanks for many insights during my stay there. Especially I would like to mention Marcus Ansorg (in memoriam) for arrangement of this visit as well as for his comments related to coordinate systems. Although not used explicitly in this thesis they allowed me to see problems presented here from diﬀerent perspective. There were many other people that helped me in along the way. However if I try to give their exhaustive list I would probably fail due to the sheer number of them. Nevertheless they all have my gratitude.

ii Contents

List of Figures 3

List of Abbreviations 4

List of Symbols 5

Introduction 9 Plan of the thesis ...... 11

1 Circular space-times 13 1.1 Metric ...... 13 1.2 Energy-momentum tensor ...... 14 1.2.1 Single-component perfect ﬂuid ...... 15 1.2.2 Two-component dust source ...... 16 1.2.3 Relation between one-component and two-component interpretations ...... 16 1.2.4 Linear velocity ...... 17 1.3 Einstein ﬁeld equations ...... 17 1.3.1 Equations for metric functions ...... 18 1.3.2 Boundary conditions ...... 19 1.3.3 Ernst equation ...... 21 1.4 Some important solutions ...... 21 1.4.1 Schwarzschild black-hole ...... 21 1.4.2 Chazy–Curzon metric ...... 23 1.4.3 Bach–Weyl ring ...... 24 1.4.4 Kerr black-hole ...... 25 1.5 Associated linear problem ...... 26 1.5.1 Linear problem associated with the Ernst equation . . . . 26 1.5.2 Belinskii–Zakharov method ...... 27

2 Perturbative approach 30 2.1 Expansion in perturbing mass ...... 30 2.2 Multipole expansion: Spherical harmonics ...... 32 2.2.1 Methods of solving linear PDE ...... 32 2.2.2 Choice of background metric ...... 33 2.2.3 Metric function (ζ)...... 35 2.3 Boundary conditions ...... 36 2.4 Green functions ...... 38 2.4.1 Solving the expanded ﬁeld equations ...... 38 2.4.2 Green functions ...... 39 2.4.3 Angular velocity of the horizon ...... 40 2.4.4 Linear perturbation due to a thin ring ...... 41 2.4.5 Convergence ...... 42 2.4.6 Closed form of Green functions ...... 46 2.5 Disc solution ...... 51 2.5.1 Gravitational potential ...... 51

1 2.5.2 Dragging ...... 52 2.5.3 Physical properties ...... 54

3 Belinskii–Zakharov method 58 3.1 Unknown generating function ...... 58 3.2 Two–soliton solutions ...... 64 3.2.1 Generated metric ...... 64 3.2.2 Asymptotic behaviour at spatial inﬁnity ...... 65 3.2.3 Axis ...... 67 3.2.4 Equatorial plane ...... 67 3.2.5 Castro & Letelier’s solution revisited ...... 68 3.3 Avoiding discontinuity ...... 68 3.3.1 Diagonal matrix Γkl(ρ, z)...... 68 3.3.2 Continuity of physical metric ...... 69 3.3.3 Normal derivatives across equatorial plane ...... 74 3.4 Example ...... 76

Conclusion 78

A Metric function transformations 80

B Christoﬀel symbols 81 B.1 Weyl–Lewis–Papapetrou form of metric ...... 81 B.2 Carter–Thorne–Bardeen form of metric ...... 82

C Properties of Legendre / Gegenbauer functions 83 C.1 Definition of Gegenbauer functions ...... 83 C.2 Basic properties ...... 84 C.2.1 Values at special points ...... 84 C.2.2 Orthogonality ...... 84 C.2.3 Asymptotic behaviour ...... 85 C.2.4 Recurrence relation ...... 86 C.2.5 Sum of double product ...... 86 C.3 Miscilaneous relations ...... 89 C.3.1 Relation between integration and differentiation ...... 89 C.3.2 Product simplification ...... 89 C.3.3 Generalized Neumann integral ...... 89 C.4 Triple product ...... 91 C.4.1 CCC ...... 91 C.4.2 DCC ...... 98 C.4.3 DDC, DDD ...... 100 C.5 Quadruple product ...... 101 C.5.1 CCCC ...... 101 C.5.2 DCCC ...... 101 C.5.3 DDCC, DDDC, DDDD ...... 102

Bibliography 103

Attachments 106

2 List of Figures

1.1 Metric functions of Schwarzschild black-hole ...... 22 1.2 Orbital velocities of Schwarzschild BH ...... 23 1.3 Metric functions of Bach-Weyl ring...... 25

2.1 Oblate spheroidal and ﬂat-ring coordinate systems ...... 34 2.2 Linear perturbation of Schwarzschild BH by ring — truncated series 42 2.3 Linear perturbation of Schwarzschild BH by ring — full calculation 46 2.4 Example of disc parameters ...... 56

3.1 Integration path of F ...... 60 3.2 Example of F of BZM...... 63 3.3 Metric functions of Bach–Weyl ring surrounding black-hole . . . . 70 3.4 Example of stationary metric generated from Bach-Weyl ring . . . 76 3.5 Example of stationary metric generated from Bach-Weyl ring — detail ...... 77

3 List of Abbreviations

BH Black hole. BZM Belinski–Zakharov method. CTB metric Carter–Thorne–Bardeen form of metric (1.6) EFE Einstein field equations. GF Green function. l.h.s. Left hand side (of equation). ODE Ordinary differential equations. PDE Partial differential equation. QED What was to be demonstrated (Quod erat demonstrandum). r.h.s. Right hand side (of equation). WLOG Without loss of generality. WLP metric Weyl–Lewis–Papapetrou form of metric (1.5) ZAMO Zero angular momentum observer.

4 List of Symbols

Coordinates

µ x Generic spacetime coordinates. (x0 being time-like) t, φ, ρ, z Cylindric spacetime coordinates. t, φ, θ, r Spherical spacetime coordinates. ζ = ρ + iz Complex spatial coordinate used in associated linear problems.

General Symbols

= χg One-form. ◦µ ◦ χµ (µ) One-form, expressed using local ◦ tetrad. µ = gχµ Vector. ◦ ◦χ (µ) Vector, expressed using local tetrad. ◦ 4 µ µ µ = µ Einstein’s summation rule. ◦ • µ=1 ◦ •

εµ1...µn P Levi-Civita symbol. Totally antisymmetric,

ε1...n = +1. µ µ ...µ µ µ ...µ +... ◦ 1 2 n −◦ 2 1 n [µ1...µn] = n! Antisymmetric part of tensor. ◦ µ µ ...µ + µ µ ...µ +... = ◦ 1 2 n ◦ 2 1 n Symmetric part of tensor. ◦(µ1...µn) n! Γα = gαµ g 1 g Christoﬀel’s symbol. βγ µ(β,γ) − 2 αβ,µ i Imaginary unit, i2 = 1. Complex conjugate of− . ◦ ◦ 1 ∂k k k = k! ∂λ◦k Coeﬃcient of λ in Taylor expansion {◦} λ=0 of . 1 j = k ◦ δ = Kronecker’s delta symbol. jk 0 j = k Lin[... ]6 (Suitable) linear combination of given object. O (f) Function behaves (up to multiplication constant) like f in given limit. Z Integers. + Z0 Non-negative integers. N = Z+ Natural numbers (positive integers). R Real numbers. C Complex numbers. Real part. < Imaginary part. =

5 (Diﬀerential) Operators

Dot product. As in Euclidean cylindric / · spherical coordinates. Tr Trace of matrix. det Determinant of matrix. d Total diﬀerential. ∂ µ ,µ = ∂x◦µ Partial derivative by x . ◦ Covariant derivative by xµ. ◦;µ ξ Lie derivative (along vector ﬁeld ξ). L ∂ ∂ = , Gradient. As in Euclidean cylindric / spherical ∇ ∂ρ ∂z ∂ 1 ∂ coordinates. = ∂r , r ∂θ 1 ∂ ρ ∂ z = + ◦ + ◦ Divergence. As in Euclidean cylindric / ∇· ρ ◦ρ ∂ρ ∂z 2 ∂ r cot θ spherical coordinates. = r r + ∂r◦ + r θ 1 ∂ θ◦ ◦ + r ∂θ◦ = Laplace operator. 4 ∇ · ∇ Special functions

Pn(x) n-th Legendre polynomial. Qn(x) Legendre function of second kind. (λ) Cn (x) n-th Gegenbauer polynomial. (λ) Dn (x) Gegenbauer function of second kind, see (C.3). a, b F ; x Gauss hypergeometric function. 2 1 c ∞ x x Γ(x) = t e− dx (Complete) Gamma function. 0 R∞ x x Γ(x, z) = t e− dx Incomplete gamma function. z z R x x γ(x, z) = t e− dx Incomplete gamma function. 0 R1 K(k) = dx Complete elliptic integral of the ﬁrst √(1 x2)(1 k2x2) 0 − − kind. R1 1 k2x2 E(k) = −1 x2 dx Complete elliptic integral of the sec- 0 − q ond kind. R1 Π(α, k) = dx Complete elliptic integral of the third (1 αx2√(1 x2)(1 k2x2) 0 − − − kind. R 1 when x > 0 H(x) = Heaviside (step) function. 0 when x < 0 x sin(t) Si (x) = t dt Sine integral. 0 R

6 Scalars κ Surface gravity. s Line element (interval). g = det gµν Determinant of metric tensor. ν (νk) Gravitational potential (metric function). And its expansion into λ-series. ω (ωk) Dragging (metric function). And its expansion into λ-series.

ζ (ζk) Metric function. And its expansion into λ- series. k Metric function (in section 1.1) or argument of Elliptic integral (in the rest of the thesis). A Metric function. B Metric function. dφ Ω = dt Angular velocity. ΩH Angular velocity of the (black-hole) horizon. 2ν v = ρBe− (Ω ω) Linear velocity (with respect to − ZAMO). µ R = Rµ Scalar curvature. µ T = Tµ Trace of energy-momentum tensor. Rest mass-energy density. p Pressure. σ Surface rest mass-energy density (pre- ν λ cisely σe − denotes surface rest mass-energy density). ν λ P Azimuthal pressure (precisely P e − denotes the azimuthal pressure). σ Surface rest mass-energy density of ± streams going in positive / negative direction. (with respect to φ) δ Dirac’s delta-distribution. Ernst potential. aE Angular momentum. b Radius of the ring. V Newtonian gravitational potential. m Perturbation or ring mass. N Number of solitons. Also used as normalization constant in section 3.1. Total mass of the space-time. M D, H Mass of the disc and black-hole. M M Total angular momentum of the J space-time. , Angular momentum of the black-hole. JD JH

7 Vector (ﬁelds)

µ ∂xµ η = ∂t Time Killing vector field. µ ∂xµ ξ = ∂φ Axial Killing vector field. µ µ µ χ = η + ΩHξ Killing field normal to black-hole horizon. uµ Fluid velocity (one stream). uµ Fluid velocity of streams going in pos- ± itive / negative direction. (with respect to φ)

Tensors

gµν Metric tensor (with signature -+++). gab t φ part of metric. α α α − Rβγδ = 2Γβ[δ,γ] + 2Γβ[δΓγ] Riemann tensor. α Rµν = Rµαν Ricci tensor. G = R 1 Rg Einstin’s tensor. µν µν − 2 µν Tµν = ( + p)uµuν + pgµν Energy-momentum tensor. (Deﬁnition corresponds to perfect ﬂuid.)

Other symbols

λ Expansion parameter (proportional to disc / ring mass) in chapter 2. λ Spectral parameter of inverse scattering methods. K iζ = K+−iζ In the case of associated linear prob- q lem of Ernst equation (chapter 1). = α z + In the case of Belinski-Zakharov (α z)2 + −ρ2 method (chapters 1 and 3). Ψ− Matrix solution of the associated lin- p ear problems, , Integration path (or it’s part), see ◦ C C section 3.1. ◦

8 Introduction

Objects generating a very strong gravity (neutron stars, black-holes) play an interesting role in the most energetic astrophysical sources like active galactic nuclei, X-ray binaries or gamma-ray bursts. In these systems, they strongly interact with matter and fields (such an interaction actually must happen in order that their systems be observable). Should the surrounding be also taken into account when trying to analytically model their gravitational fields, it becomes in itself a challenge. Namely, such strong sources have to be treated within general relativity rather than by Newtonian physics — and general relativity is non-linear, so it is typically very difficult to find the gravitational field (space-time) of a compound system. Models used in astrophysics usually neglect the gravitational effect of the surroundings, and in most cases the strong-gravity object really dominates the field by orders of magnitude. However, this dominance may not be true for some components of the field (simply because of symmetries of the system) and mainly for higher derivatives of the potential/metric. These higher derivatives (space- time curvature) are in turn crucial for stability of motion of the surrounding matter, so, because of its own self-gravity (non-linear effect of sources on themselves), the actual surrounding matter can assume quite a different configuration than the test, non-gravitating one. In some very symmetric situations, (some of) the Einstein field equations reduce to a linear problem and the exact solution is almost as easy as in the Newtonian case. This is for example the case for static and axially symmetric configurations. However, for the modelling of astrophysical systems, this class of space-times is too restricted, because it does not involve rotation — a typical ingredient of celestial systems (often even necessary as a support against gravity). It is thus desirable to proceed to stationary (but non-static) settings where at least a stationary rotation is “permitted”. Stationary and axially symmetric models can indeed well approximate most of the astrophysical systems. The step from static to stationary space-times is quite non-trivial, as indicated by the history of exact solutions of Einstein’s equations itself. The famous Schwarzschild solution Schwarzschild [1916], historically the first solution of general relativity, was found about one month after publication of this theory by Einstein. It describes any spherically symmetric vacuum region of space-time. However, it turned out to be a pioneering solution, because it can describe a black-hole — a hardly imaginable, intrinsically dynamical object separated from the outer world by a horizon. It required some 50 years for this theoretical con- struction to be accepted by the (astro)physical community. And about the same time was needed for a generalization of the Schwarzschild solution to a rotating one: the latter was found by Kerr in 1963 Kerr [1963]. It is the same year when quasars were discovered, today “canonically” explained in terms of power driven by supermassive Kerr-type black-holes. Shortly after the Schwarzschild solution, Weyl Weyl [1917] showed that in fact all static and axially symmetric (electro-)vacuum space-times can be described by a simple (“Weyl”) metric containing just two unknown functions, provided that one works in a suitable coordinates which were later also given his name. Still more importantly, one of those unknown functions i) has quite a clear physical

9 meaning, namely it is a counter-part of the Newtonian gravitational potential, and, even ii) like in the Newtonian theory, it is given by Laplace (Poisson) equation, which is linear, so a superposition of partial solutions can be applied. The second metric function can be found by a line integral if the potential is already known. The Weyl metrics will be important as a starting point of this thesis. (See chapter 1.3.) Several metrics belonging to the Weyl class were found soon using the Newto- nian analogy — for example the Bach–Weyl solution Bach and Weyl [1922] for a circular thin ring (which will be important below), or, later, the Morgan–Morgan solution Morgan and Morgan [1969] for families of circular thin discs. However, it also turned out that a simple transfer of results from the Newtonian realm may not always yield an expected outcome (see section (1.4.2)). When speaking about stationary and axially symmetric space-times, it is almost automatically supposed that besides the “independence on time and azimuth” they also have another important geometrical property — their meridional plane, locally spanned by directions orthogonal to both basic symmetries, is integrable, i.e. existing as a global surface. This property is standardly called “orthogonal transitivity” (see section 1.1) and the metrics without this property are only very little known. Physically, the orthogonal transitivity means that the source elements have to move only in the directions of the time and axial symmetry (circular motion, hence “circular” space-times), namely they must not perform a solenoidal motion (although a combination of such motions could keep the time and axial symmetries). An important development in the history of the above circular space-times was the reformulation of the problem in terms of a complex potential by ErnstErnst [1968]. This enabled to encode the problem in a very compact form and stimu- lated further study in the field. This lead to discoveries of new classes of solutions, e.g. the Tomimatsu–Sato ones Tomimatsu and Sato [1973] which provide a certain generalization of the Kerr solution. Even more importantly, the Ernst formulation lead to the proof that the stationary axisymmetric problem is completely integrable and boosted the effort to find transformations under which the equation remains satisfied, i.e. of the “generation techniques” by which one can transform one its solution into another one of the same symmetry class. These typically start from a linear problem associated with the original field equations, which is more promising for solution than the original non-linear formulation. In this thesis, we will specifically mention some results of the so-called inverse-scattering methods, see e.g. Neugebauer and Meinel [2003] andBelinskii and Verdaguer [2004]. Both of the above mentioned generating methods can be used to generate new solutions by inserting “solitons” into the known metrics. Usually such attempts result in a metric with unwanted singularities, but one remarkable result was achieved in this way — the metric for a rigidly rotating disc of dust (see Neugebauer and Meinel [1994] and Neugebauer and Meinel [1995]). In chapter 3, we will employ the inverse-scattering method discovered by Belinskii & Zakharov (surveyed in Belinskii and Verdaguer [2004]) with the aim to obtain a metric for a black-hole surrounded by a ring or disc. Many papers were interested in BZM applied on Weyl space-times — mentioning few of them Chaudhuri and Das [1997], Zellerin and Semerák[2000] and Semerák[2002] (this one also discussed

10 non-physical properties of obtained solution similarly to the examined by section (3.3)). Inspired by article de Castro and Letelier [2011] Bach–Weyl ring is going to be used as seed metric. Leaving aside numerical approaches, there is still another option how to tackle Einstein (or in fact any difficult) equations: perturbation techniques. These are not appropriate for any purposes, but can at least provide solutions which are “close” to some known exact solutions. Clearly our problem of a heavy and compact centre surrounded by a light source is a very good candidate for such an approach, the more that the centre is a black-hole whose metric is known and simple (at least in the Schwarzschild case). In particular, we will follow the results by Will [1974] who proposed a method how to perturb a Schwarzschild metric by a slowly rotating and light thin circular ring. Also mentioned should be the famous Teukolsky equation Teukolsky [1973] which “masters” (even non- stationary) perturbations of a Kerr black-hole due to physical fields of different kinds (different spins). Recently the primary target of approximation techniques has been non-stationary strong-field systems, in particular collisions of compact objects as a source of gravitational waves. There, their results are being matched to those of numerical treatment. Actually, numerical methods can tackle almost any situation (without particular symmetries) and in some cases factually represent the only option. On the other hand, analytical results still have their importance, because they allow to study whole families of solutions, dependence of their properties on parameters, stability, etc., better than numerical “shooting”.

Plan of the thesis

This thesis consists of three main chapters. The first chapter is of a purely review type and introduces basic concepts necessary to study stationary and axisymmetric black-holes perturbed by some surrounding matter — the metric of circular space-times (section 1.1), a suitable energy-momentum tensor (1.2), the form of the Einstein equations (1.3), an overview of several basic solutions of the given class (1.4), and some information on two basic generation techniques (1.5). The second chapter deals with the application of a perturbative approach to circular space-times. An expansion analogous to the one suggested by Will [1974] is introduced, which leads to the set of linear differential equations whose fundamental solution can be found. The complete solution can however be achieved only for sufficiently simple background metric; regarding the astrophysical motivation, it is natural to choose the Schwarzschild black-hole as the background metric (section 2.2.2), plus appropriate boundary conditions (section 2.3). Will found the Green functions for the two crucial metric functions of this problem (section 2.4.2). Our own contribution centers on finding the closed form of these Green functions (see 2.4.6), which is much more suited for further work (e.g. for treating extended sources, where the Green functions have to be integrated) and for numerical evaluation. In section 2.5, we employ the above closed-form Green functions to derive the first-order perturbation of the Schwarzschild black-hole due to a simple type of finite thin disc, including its interpretation and several illustrations. The correspondence between the ring of matter as a source and the Green functions is examined in section 2.4.4 and the basic physical properties of

11 the perturbed-solution disc are checked in section 2.5.3. In the third chapter we mention some results we obtained using generation techniques, namely the Belinskii–Zakharov inverse-scattering method. In particular, the spectral function F is calculated in section 3.1, which opens the possibility to generate new metrics. We specifically consider a two–soliton solution. In section 3.2 we derive expressions for the generated metric and discuss its asymptotic behaviour. Unfortunately, it turns out that without special conditions the metric is not continuous across the equatorial plane (section 3.3). The solution is regular in a static case, but in the stationary one only the continuity of the metric itself can be ensured (normal derivatives of the metric functions have jumps in the equatorial plane, which indicates the presence of a kind of supporting surface). An example of the generated stationary metric, together with its problems just mentioned, is given in section 3.4. Many “technical” parts of calculations are shifted to (three) appendices. Ap- pendix A just shows transformation between two metric forms used in this thesis and appendix B brings expressions for the corresponding connection coefficients (Christoffel symbols). Most important is the last appendix C where the properties of orthogonal polynomials are examined, crucial for putting the Green functions of our perturbation problem to the closed form; they include some theorems which may not be well known. In particular, the Gegenbauer functions (those of the first-kind being commonly known as Gegenbauer polynomials) and the corresponding, ultraspherical differential equation (C.1) are discussed, because they are important for the more involved, “dragging”part of the problem. Originally the matter presented in this appendix was supposed to become a separate, purely mathematical paper (supporting the paper C´ıˇzekandˇ Semerák[2017]), but I have been unable to finish it in time. However, the whole thesis can be understood even without reading this long appendix, assuming the reader is familiar with the Gegenbauer polynomials and trusts the crucial relation (C.104) (it is of course possible to check the final results ex post, without going into details of their derivation).

12 1. Circular space-times

1.1 Metric

In this part we summarize basic properties of space-times which are stationary, axially symmetric and orthogonally transitive. Of many general-relativity text- books which cover this subject, we follow here Stephani et al. [2003], chapter 19. Stationary axisymmetric space-times are characterized by the existence ot two µ µ commuting Killing vector fields, one of them (η ) time-like (η ηµ < 0) and the µ µ other one (ξ ) space-like (ξ ξµ > 0). Usually it is also required that the integral lines of ηµ be closed in analogy with Euclidean (Minkowski) space. This assumption does not affect Einstein’s equations which only determine local curvature (not global topology), but it is important for the interpretation of results. Namely, without it the plane-wave solutions are included as well. (These correspond to different boundary conditions and some quantities — for example total angular momentum — loose their meaning.) Commutativity of the vector fields ηµ and ξµ implies that they are surface- forming. Metric can be expected simple if the corresponding integral surfaces are covered by coordinates (t and φ) directly given by parameters of the Killing symmetries, ηα = ∂xµ/∂t, ξα = ∂xµ/∂φ, with t being time and φ azimutal angle.1 In addition, stationary axisymmetric space-times are almost “by default” assumed to satisfy another important property called orthogonal transitivity: the bivector orthogonal to both ηµ and ξµ (spanning the meridional plane) must be surface-forming as well. The necessary and sufficient condition for this property reads (Kundt and Trümper [1966])

µ µ η Rµ[αηβξγ] = 0 = ξ Rµ[αξβηγ] (1.1) and through the Einstein equations it can also be rewritten in terms of the energy- momentum tensor, µ µ η Tµ[αηβξγ] = 0 = ξ Tµ[αξβηγ]. (1.2) Simply speaking, one requires invariance under the transformation (t t, φ φ), which physically means that the source elements have to move only→ in − Killing→ directions,− not within the meridional plane. Hence the term circular space-times. Thanks to orthogonal transitivity, the remaining two coordinates (x2 and x3) can be chosen to cover the meridional planes (locally orthogonal to both Killing vector ﬁelds) and the metric takes form

gtt gtφ 0 0 gtφ gφφ 0 0 gµν =   , (1.3) 0 0 g22 g23  0 0 g g   23 33 1   g = 2η = 0 0 = g ηα + g ηα + g ηα = g . Lη µν (µ;ν) ⇒ µν,α µα ,ν να ,µ µν,t Similarly for the axial symmetry and φ.

13 where all the components (gtt, gtφ, gφφ, g22, g23 and g33) depend only on the coordinates x2 and x3. The remaining freedom in the choice of x2 and x3 can be fixed in various ways, usually according to additional features of the given specific solution. One of generic options follows from the fact that all two-dimensional Riemannian manifolds are locally conformally flat and can thus be covered by “Cartesian”- type coordinates isotropically. This type of coordinates is called Weyl and the metric assumes in them the form

gtt gtφ 0 0 g g 0 0 g = tφ φφ , (1.4) µν  0 0 e2k 0   0 0 0 e2k     where k is a metric function parametrizing g22 = g33. Another ambiguity arises in expressing the (t, φ) part of the metric tensor, the two mostly used “extreme” cases being the Weyl–Lewis–Papapetrou (WLP) form

2 2ν 2 2 2 2ν 2 2ζ 2ν 2 2 ds = e (dt + Adφ) + ρ B e− dφ + e − dρ + dz = − 2ν 2 2 2 2 2ν 2 2ζ 2ν 2 2 2 = e (dt + Adφ) + r sin θB e− dφ + e − dr + r dθ (1.5) − and the “Carter–Thorne–Bardeen” (CTB) form2

2 2ν 2 2 2 2ν 2 2ζ 2ν 2 2 ds = e dt + ρ B e− (dφ ωdt) + e − dρ + dz = − − 2ν 2 2 2 2 2ν 2 2ζ 2ν 2 2 2 = e dt + r sin θB e− (dφ ωdt) + e − dr + r dθ , (1.6) − − where cylindrical coordinates (ρ, z) or spherical coordinates (r, θ) are employed instead of generic x2 and x3.3 The metrics contain 4 unknown functions (as in the original case (1.4)) — ν, ω (or A), ζ and B. Transformations between the two sets of metric functions can be found in appendix A and the corresponding Christoﬀel symbols are given in appendix B. In both cases, the function ν can be interpreted as gravitational potential, ω represents angular velocity of azmuthal rotational dragging, ζ describes scaling of the space-like 2-metric and B is a defect of circumference with respect to the radius ρ. In a static limit (ω = 0) both metrics reduce to the Weyl form

2 2ν 2 2 2 2ν 2 2ζ 2ν 2 2 ds = e dt + ρ B e− dφ + e − dρ + dz = − 2ν 2 2 2 2 2ν 2 2ζ 2ν 2 2 2 = e dt + r sin θB e− dφ + e − dr + r dθ . (1.8) − 1.2 Energy-momentum tensor

The sources orbiting around some centre can be kept on their orbits due to hoop stresses and by the centrifugal eﬀect. The source we consider in this thesis is

2This was used, for example, by Bardeen and Wagoner [1971] and Will [1974], the articles followed closely in perturbation part of this thesis. 3The coordinates are connected by relations z r = ρ2 + z2, cos θ = , ρ = r sin θ, z = r cos θ. (1.7) ρ2 + z2 p p 14 a circular thin disc (or infinitesimal ring) which bears neither electric charge nor current. Following astrophysical motivation, we rather adhere to the second limit possibility and interpret the source as an orbiting particles or fluid. Two basic pictures are now possible: a single-component fluid, having certain surface density, orbiting with some suitable orbital velocity and in general generating some (suitable) pressure in the azimuthal direction; or a combination of two non- interacting (presureless), dust components with suitable surface densities and orbiting on counter-rotating circular geodesics. Below we give some details on these two interpretations. Note also that for an infinitesimally thin disc encircling symmetrically a black- hole, the energy-momentum tensor vanishes everywhere but “across” the disc, hence it can be included as a part of boundary conditions for vacuum field equations (see section 1.3.2). These are naturally chosen so that the space-time is reflection symmetric with respect to the equatorial plane (fixed by the disc), and this symmetry is also expected to hold for all the metric functions. µ Besides the Tν itself, it is useful to define a surface energy-momentum tensor µ Sν by

+ 0+ ∞ µ µ 2ν 2ζ µ µ µ T = S e − δ(z), T g dz = T g dz S (1.9) ν ν ⇒ ν zz ν zz ≡ ν Z 0Z −∞ − in order to separate the distributional part of source and work with ﬁnite quantities.

1.2.1 Single-component perfect ﬂuid Energy-momentum tensor of a perfect ﬂuid reads

Tµν = ( + p)uµuν + pgµν, (1.10) where denotes the rest mass-energy density, p is pressure and uµ is the 4-velocity of fluid elements (assumed to be a time-like vector and subject to normalization µ u uµ = 1). In the dust limit, p = 0. To satisfy the assumptions of stationarity, axial symmetry− and circularity, the fluid velocity is restricted to the t and φ directions, uµ = ut, uφ, 0, 0 = ut (1, Ω, 0, 0) = utηµ + uφξµ, (1.11) φ t Ω = dφ/dt = u /u denoting the angular velocity of the fluid. Clearly u[αξβηγ] = 0 and the property of orthogonal transitivity holds. 2 Provided that Sφ St + 4St Sφ 0, the energy-momentum tensor can be φ − t φ t ≥ diagonalized to

T µν = uµuν + pwµwν,Sµν = σuµuν + P wµwν, (1.12) where uµ denotes total velocity of fluid, wµ stands for spatial vector orthogonal to fluid velocity ( still belonging to the t–φ surface), and p are mass-energy density of the fluid and azimuthal pressure and σ and P describes surface mass-energy

15 density and azimuthal pressure4. Vector ﬁeld wµ is given by u u wµ = φ , t , 0, 0 , w = ρB uφ, ut, 0, 0 = utρB ( Ω, 1, 0, 0) . (1.14) ρB −ρB µ − − 1.2.2 Two-component dust source Another possibility how to interpret the disc is a combination of two non-interacting, dust components orbiting on counter-rotating circular geodesics. In this case the energy-momentum tensor reads

µν µ ν µ ν µν µ ν µ ν T = +u+u+ + u u ,S = σ+u+u+ + σ u u , (1.15) − − − − − − where uµ denote orbital velocities of the “co-rotating” and “counter-rotating” stream,5± are the mass-energy densities of the streams and σ are the corre- ± ± sponding surface mass-energy densities as introduced above. Because the dust streams are not interacting with each other (and themselves), µ ν their orbital velocities must obey geodesic equation u ;νu = 0. Clearly only the radial component (µ = ρ) is non-trivial, ± ±

ν ρ ν ρ ν µ ρ t 2 ρ ρ 2 ρ 0 = u u ;ν = u u ,ν + u u Γνµ = u Γtt + 2Ω Γtφ + Ω Γφφ ± ± ± ± ± ± ± ± ± 1 t 2 ρρ 2 = u g gtt,ρ + 2gtφ,ρΩ +gφφ,ρ Ω , −2 ± ± ± ⇒ g g 2 g Ω = tφ,ρ tφ,ρ tt,ρ , (1.16) ± −g ± g − g φφ,ρ s φφ,ρ φφ,ρ where Ω = dφ/dt = uφ /ut denotes angular velocity of the stream. ± ± ± ± 1.2.3 Relation between one-component and two-component interpretations Because the energy-momentum tensor must come out the same in both cases, equations (1.15) and (1.12) can be used to express relations between single- and two-component quantities,6

2 2 2 2 µ 2 σ P = σ+ + σ , σ + P = σ+ + σ + 2σ+σ (u+u µ) , − − − − − µ 2 µ 2 σ = σ+ (u+uµ) + σ (u uµ) . (1.17) − − 4The relationship between quantities is given by

2ν 2ζ 2ν 2ζ = σe − δ(z),P = pe − δ(z). (1.13)

ν ζ ν ζ To be precise σe − and P e − denotes surface mass-energy density and azimuthal pressure. 5The quotation marks indicate that in general one has to be careful what these terms mean: usually they are understood in the sense of the φ coordinate, but actually it is more appropriate to take them “with respect to the geometry”, i.e. with respect to zero-angular-momentum orbits. 6Many such relations can be found; we only list here the simplest ones, following from the µν µν µν comparison of S gµν , S Sµν and S uµuν .

16 1.2.4 Linear velocity For the interpretation purposes, it is also useful to deﬁne the linear velocity with respect to zero-angular-momentum observer (ZAMO) which orbits, in the Killing directions, with Ω = ω, 2ν v ρBe− (Ω ω) . (1.18) ≡ − In terms of this quantity, the ﬂuid “bulk” four-velocity uµ can be written

2ν ν ν 2 e e e Ω ut = − , uµ = − , − , 0, 0 . (1.19) 1 v2 ⇒ √1 v2 √1 v2 − − − It is also useful to endow the ZAMO family with a tetrad adapted to the global coordinates,

ν ν ν e ν µ ν µ e = e− dt + ωe− dω, e = dφ, e = e − dρ, e = e − dz. (1.20) (t) (φ) ρB (ρ) z

In this frame the perfect-ﬂuid energy-momentum tensor (1.10) has only ﬁve non- zero components,

+ pv2 p + v2 T (t)(t) = ,T (φ)(φ) = , 1 v2 1 v2 ( +−p)v − T (t)(φ) = ,T (ρ)(ρ) = T (z)(z) = p. (1.21) 1 v2 − Analogous relations hold for the corresponding surface energy-momentum tensor. Linear velocity can be used in the relations between single- and two-component interpretations. Projections of the velocities are given by

α vv 1 α v v u uα = ± − , u wα = ± − , ± √1 v2 1 v2 ± √1 v2 1 v2 − − ± − − ± α β v+v 1 gαβu+u = −p− , p (1.22) − 2 2 1 v+ 1 v − − − where v denote linearp velocitiesp corresponding to the two dust streams. Equa- ± tions (1.17) then read

2 2 2 2 2 (1 v+v ) − σ P = σ+ + σ , σ + P = σ+ + σ + 2σ+σ −2 2 , − − − − (1 v+) (1 v ) − 2 2 − − (vv+ 1) (vv 1) σ σ σ − . = + 2 − 2 + 2 − 2 (1.23) (1 v ) (1 v+) − (1 v ) (1 v ) − − − − − 1.3 Einstein ﬁeld equations

Einstein ﬁeld equations link the geometry with the energy-momentum distribution. We assume the reader is familiar with this central point of general relativity and only focus on their form speciﬁc to circular space-times and to our particular black-hole–disc problem.

17 1.3.1 Equations for metric functions For the metric (1.6) and for the perfect-ﬂuid energy-momentum tensor (1.10) the Einstein equations read (see, e.g., Bardeen and Wagoner [1971])

2 1 2 4ν 2ζ 2ν 1 + v (B ν) ρ2B e− ω ω = 4πBe − ( + p) + 2p ,(1.24) ∇ · ∇ − 2 ∇ · ∇ 1 v2 − 2 3 4ν 2 2ζ 4ν v ρ B e− ω = 16πρB e − ( + p) , (1.25) ∇ · ∇ − 1 v2 2ζ 2ν − (ρ B) = 16πρBe − p, (1.26) ∇ · ∇ − and B 1 B B B B 0 = ,ρ + ζ ,z ζ ,ρρ ,ρ + ,zz B ρ ,ρ − B ,z − 2B − ρB 2B − 2 2 1 2 2 4ν 2 2 (ν ) + (ν ) + ρ B e− (ω ) (ω ) , (1.27) − ,ρ ,z 4 ,ρ − ,z B,ρ 1 B,z B,ρz B,z 0 = + ζ + ζ B ρ ,z B ,ρ − B − B − 1 2 2 4ν 2ν ν + ρ B e− ω ω . (1.28) − ,ρ ,z 2 ,ρ ,z The latter group of equations can be used to determine the function ζ, assuming that B, ν and ω are already known. Condition for integrability of (1.27) and (1.28) reads

2ρ (ρB ν + Bν ρB ν ) RHS 0 = ,ρ ,z ,z − ,z ,ρ (1.24) 2 2 2 2 B 2ρBB,ρ + ρ (B,ρ) + (B,z) − ρ (ρB ω + Bω ρB ω ) RHS ,ρ ,z ,z − ,z ,ρ (1.25) + 2 2 2 2 − 2 B 2ρBB,ρ + ρ (B,ρ) + (B,z) RHS , +Terms proportional to (powers and derivatives of) (1.26) (1.29) where RHS... denotes right-hand sides of the corresponding equations. (In the vacuum case, = p = 0 and they all vanish.) The condition is fulﬁlled if vacuum equations (1.24), (1.25) and (1.26) hold. Supplemented with appropriate boundary conditions, ζ can be expressed in the form of a line integral, see (2.35) and (2.34). Moreover, assuming dust source (p = 0), (1.26) for B can be satisﬁed in many ways.7 Physically the choice of B corresponds to a coordinate transformation, the relation between coordinates corresponding to some generic B and the ones corresponding to B = 1 (denotedρ, ˜ z˜) reading

ρ˜ = ρB, z˜ = ρB dρ (ρB) dz., (1.30) ± ,z − ,ρ Z where the sign can be chosen arbitrarily. Integrability condition forz ˜ holds as a consequence of (1.26). The simplest solution B = 1 is usually being employed, but it is not ideal in the presence of a black-hole, as we will see in section 1.4.1.

7 1 Except the choice B = ρ− which leads to plane-wave solutions rather than to axially symmetric ones.

18 In the static case (ω = 0), equations (1.24)–(1.28) become much simpler. Equation (1.25) is satisﬁed trivially and (1.26)’s left-hand side reduces to a two- dimensional Laplace operator, with known solutions. With B known, equation (1.24) reduces to a linear partial diﬀerential equation (in the vacuum case and for B = 1, it becomes a Laplace equation in three dimensions), so superposition of solutions can be applied. The knowledge of B and ν is then used to solve (1.27) and (1.28), see (2.35) and (2.34).

1.3.2 Boundary conditions Assuming the existence of a thin equatorial disc, there are four different areas to treat specifically: spatial infinity, symmetry axis, black-hole horizon and equatorial plane.

Spatial infinity We assume the space-time is asymptotically flat, actually, the black-hole–disc configuration is supposed to be the only source present, we do not consider a cosmological “background”, we do not take into account cosmological constant or gravitational radiation. So the space-time should approach a flat one towards spatial infinity, which for our metric components implies

g 1, g 0, g ρ2, g 1. (1.31) tt → − tφ → φφ → zz → In terms of individual metric functions of the (1.5) metric, this means

ν 0,A 0,B 1, ζ 0, (1.32) → → → → while the functions parametrizing the (1.6) form should behave like

ν 0, ω 0,B 1, ζ 0 (1.33) → → → → (all the limits represent ρ2 + z2 ). → ∞ p Axis To avoid a conical singularity, the z = const surfaces have to be locally ﬂat on the axis, which is ensured by

ρ 0+ ζ ln B. (1.34) → ⇒ → Also, derivative by ρ of all the metric coeﬃcients has to vanish at the axis, otherwise string-like singular source would appear there. This requirement implies

ρ = 0 ν = B = ω = ζ = 0. (1.35) ⇒ ,ρ ,ρ ,ρ ,ρ Black-hole horizon The black-hole horizon has a number of special properties valid for any circular space-time. In particular, the dragging angular velocity (ωH) and the surface

19 gravity κ have to be constant over the horizon. The latter is deﬁned by (see Wald [1984] (12.5.14))

2 1 α;β 4ν 2ζ 1 4ν 2 2 κ = χ χ = e − ν ν e− ρ B ω ω , (1.36) −2 α;β ∇ · ∇ − 4 ∇ · ∇ α α α where χ = η + ωHξ stands for a Killing field normal to the horizon. Both of these quantities (ωH and κ) thus should be finite. Since there should be no matter present at the very (stationary) horizon, equation (1.36) can be simplified there, using Einstein equation (1.24), to

6ν 2ζ 2 e − 2ν κ = B e− . (1.37) 4B ∇ · ∇ Also the horizon, in order not to be degenerate, should have ﬁnite (and non– vanishing) azimuthal and latitudinal circumferences. These conditions are en- 8 2ν 2ζ 2ν sured by non–degeneracy of Bρe− and e − .

Equatorial plane Boundary conditions at the equatorial plane in both vacuum and disc regions can be found using (1.24), (1.25) and (1.26). No additional conditions are necessary for ζ because its value is fixed on the axis and equations (1.27) and (1.28) are of the first order. Because the source is located in the equatorial plane only and it is singular (δ-like) in the z direction, there must be discontinuity in the first derivatives of the metric functions.9 As already mentioned above, we also require reflection symmetry with respect to the equatorial plane. This leads to ∂f ∂f = , (1.38) ∂z z=0 − ∂z z=0+ − 2 ∂ f ∂f ∂f ∂f = δ(z) = 2 δ(z), (1.39) ∂z2 ∂z − ∂z ∂z z=0 z=0+ z=0 z=0+ − where f stands for an arbitrary function. It also confirms the assumption that we can treat the vacuum case in the same way –without the δ-like source, both one- sided derivatives have to vanish. Because of the discontinuity on the equatorial plane, the directions of derivatives must be carefully distinguished. Due to the reflection symmetry it is sufficient to know the solution in the “upper” half-space z 0. Unless stated otherwise, the first derivatives in the equatorial plane are ≥ understood as ∂/∂z z=0+. Finally, in the dust| case the boundary condition reads 1 + v2 ν = 2πσ ,B = 0, ,z|z=0+ 1 v2 ,z|z=0 − 8Meaning these expressions are finite and not equal to zero there. 9 Even more exotic boundary conditions could be considered, including discontinuity of the metric itself. However, our astrophysically motivated thin disc is just an idealization of an accretion disc with small but finite thickness, in which case the metric is continuous (including its first and second derivatives). Denoting l the characteristic thickness of the disc, the first 1 2 (second) derivatives of metric behave like l− (l− ) as l 0+. With discontinuous metric we would encounter more strange objects (generating products→ and derivatives of δ distribution, in particular) and the fiedl equations would be even problematic to interpret.

20 Ω ω ve2ν ω = 8πσ − = 8πσ . (1.40) ,z|z=0+ − 1 v2 − ρB(1 v2) − − 1.3.3 Ernst equation One reformulation of the ﬁeld equations for the (electro-)vacuum case should be mentioned — the Ernst equation (see Ernst [1968]). It can be derived using the WLP form of the metric (1.5), usually choosing B = 1. The WLP-metric analogue of equation (1.25) reads

1 4ν 1 4ν ρ− e A ρ− e A ,ρ ,ρ + ,z ,z = 0 (1.41) and can be regarded as an integrability condition for the twist potential , A = ρe4νA , = ρe4νA . (1.42) A,ρ ,z A,z − ,ρ Introducing the (complex) Ernst potential

= e2ν + i , (1.43) E A equations (1.24) and (1.25) can be represented by the Ernst equation

= , (1.44)

1.4 Some important solutions

1.4.1 Schwarzschild black-hole Leaving aside “trivial” Minkowski metric, the simplest solution of Einstein equations is the Schwarzschild metric (Schwarzschild [1916]). It describes any vacuum spherically symmetric (region of) space-time without the cosmological constant (Birkhoff’s theorem: Birkhoff [1923] and Jebsen [1921]). In the extreme case when there is no Tµν anywhere except the very centre (“point-like source”), the solution represents an isolated static and spherically symmetric black-hole. This will below be the object of our perturbation effort. The best known form of the metric 2M dR2 ds2 = 1 dt2 + + R2dϑ2 + R2 sin2 ϑdφ2 (1.46) − − R 1 2M − R employs Schwarzschild coordinates, in particular, we denote by R the Schwarzschild radius (different from the “isotropic” radius r in (1.8), see below); the single metric parameter M represents mass of the black-hole. Such a metric, however, does

21 1 16

14 0.8 12

10 0.6

0.4 6

4 0.2 2

0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 r r

Figure 1.1: Metric functions for Schwarzschild black-hole in isotropic coordinates and mass M = 1). Left: Square of the lapse function ( gtt). Right: Defect of g gθθ gφφ − radius ( rr = r2 = r2 sin2 θ ).

not ﬁt in the form(1.4) and we will rather use the metric written in isotropic coordinates,

2r M 2 M 4 ds2 = − dt2 + 1 + dr2 + r2dθ2 + r2 sin2 θdφ2 . (1.47) − 2r + M 2r The metric functions of (1.5), (1.6) and (1.8) appear in these coordinates as

2r M M 2 M 2 ν = ln − , ω = A = 0,B = 1 , ζ = ln 1 . (1.48) 2r + M − 4r2 − 4r2 See ﬁgure 1.1 for illustration of behaviour of the metric coeﬃcients. In this form, all the metric functions are independent on both angular coordinates and the M black-hole horizon is represented as a sphere with radius r = /2. The only drawback is the complexity of the metric function B. On the other hand, with the simplest possible choice B = 1, the Schwarzschild metric reads

2 2 2 d+ 2M 2 2 d+ + 2M 2 d+ 4M 2 2 ds = − dt + ρ dφ + 2− 2 dρ + dz , (1.49) −d+ + 2M d+ 2M d+ d − − − where d = ρ2 + (z + M)2 ρ2 + (z M)2 (1.50) ± ± − and the metric functionsp are given by p

2 1 d+ 2M 1 (d+ + 2M) ν = ln − , ζ = ln 2 2 ,B = 1. (1.51) 2 d+ + 2M 2 d+ d − − Such form is however less suitable for perturbation, because it leads to a more complicated form of the ﬁeld equations. The function ν assumes a simple form when expressed in terms of inﬁnite

22 4

0.34 0.32 0.3 3 0.28 0.26 0.24 2 0.22 0.2 0.18 0.16 1 0.14 0.12 0.1

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 r r

Figure 1.2: Orbital velocities of circular geodesics about a Schwarzschild black- hole (of mass M = 1). No such orbits exist in the dotted part (it would require superluminal speed). Left: Angular velocity Ω. Right: Linear velocity (with respect to ZAMO, i.e. static observer) v.

10 series of Legendre polynomials Pn(x),

2n+1 ∞ 1 M ν = P (cos θ). (1.53) − 2n + 1 r 2n n=0 X This however does not help in perturbative approach much, because multiplication of Legendre polynomials leads to Clebsh–Gordan coeﬃcients and lengthy sums of terms. Finding of the ring- or disc-like solutions requires orbital velocity of circular orbits in the equatorial plane. Angular velocity is given by (1.16),

g 8√Mr3 Ω = tt,r = , (1.54) −g (2r + M)3 r φφ,r and the corresponding linear velocity is

2ν 2√rM v = rBe− Ω = . (1.55) 2r M − See ﬁgure 1.2 for illustration.

1.4.2 Chazy–Curzon metric To demonstrate pitfalls of a straightforward Newtonian analogy, it is useful to check Chazy–Curzon metric (Chazy [1924] and Curzon [1924]). Newtonian gravitational potential of a point mass reads V = M/r. Because ν in (1.8) denotes the corresponding quantity (also given by Laplace− equation), it could have been

10 We follow notation of chapter 14 of DLMF. Legendre polynomial can be, for example, expressed by formula (14.7.13) there:

1 dn P (x) = (x2 1)n. (1.52) n 2nn! dxn −

23 expected that the ﬁeld of a “point mass” in general relativity would have very similar properties. However, there also appears the second metric function ζ which often deforms (literarily) the picture considerably. Actually, in this case the full metric reads

2 2 2 2M 2 2M 2 2 2 M sin θ 2 2 2 ds = e− r dt + e r r sin θdφ + e− r2 dr + r dθ (1.56) − and the solution as a whole is even not spherically symmetric (ζ function does not possess this symmetry). The “correct” solution of the spherically symmetric problem is the Schwarzschild solution recalled in previous section.

1.4.3 Bach–Weyl ring Another example of distorted Newtonian analogy is the Bach–Weyl ring (see Bach and Weyl [1922]). It describes the ﬁeld generated by a thin (singular) circular ring in the equatorial plane. In the Newtonian case the gravitational potential reads

2π m 1 V (ρ, z) = dφ = −2π ρ2 + z2 + b2 2ρb sin φ Z0 − p 4ρb 2mK (ρ+b)2+z2 = , (1.57) − π (ρq+ b)2 + z2 where m denotes the ring mass,pb is its radius and K (x) stands for the complete elliptic integral of the ﬁrst kind. The corresponding relativistic picture is, again, distorted by the second metric function ζ: it is described by the metric (1.8) with 2m ν = K(k) , −π (ρ + b)2 + z2 B = 1, p m2 (b2 ρ2 z2)(2 k2) ζ = − − − E(k)2 + 4K (k)E(k) 4π2b2 (ρ2 + z2 + b2)(1 k2) − − (3b2 + ρ2 + z2)(2 k2) − K(k)2 , (1.58) − (ρ2 + z2 + b2) where E (k) stands for the complete elliptic integral of the second kind and

4ρb k = (1.59) s(ρ + b)2 + z2 is the argument of the elliptic functions. The metric functions are plotted in ﬁgure 1.3. Distortion due to the metric function ζ is clearly visible. (See Basovn´ıkand Semer´ak[2016] for a detailed study of this solution and a comparison with other singular-ring ones.) The Bach–Weyl ring plays an important role in this thesis, in particular, it represents the source corresponding to the Green functions of our perturbation procedure.

24 2 2

z 1 z 1

0 1 2 3 4 5 6 0 1 2 3 4 5 6 rho rho

±1 ±1

±2 ±2

Figure 1.3: Metric functions for the Bach–Weyl ring with mass m = 1 and radius b = 3. Left: ν. Displayed levels are 0.7, 0.6,..., 0.2. Right: ζ. Displayed levels are 1.0, 0.9,..., 0.1. − − − − − − 1.4.4 Kerr black-hole Another example of a physically signiﬁcant black-hole solution is the Kerr metric (Kerr [1963]), in the Boyer–Lindquist coordinates (generalization of the Schwarzschild ones) having the form

2MR 2MaR sin2 ϑ 2 ds2 = 1 dt + dφ + − − R2 + a2 cos2 ϑ R2 2MR + a2 cos2 ϑ − (R2 + a2 cos2 ϑ)(R2 2MR + a2) sin2 ϑ + − dφ2 + (1.60) R2 2MR + a2 cos2 ϑ − dR2 +(R2 2MR + a2 cos2 ϑ) dϑ2 + . − R2 2MR + a2 − where R denotes radius, M is mass of the black-hole and a is its angular momentum. Indeed, the metric is stationary (not static), so it allows rotation. The metric can be written in the WLP form (1.5) using the coordinate transforma- tion11 ρ = √R2 2MR + a2 sin ϑ, z = (R M) cos ϑ. (1.61) − − The metric functions come out as 1 R2 2MR + a2 cos2 ϑ ν = ln − , 2 R2 + a2 cos2 2MaR sin2 ϑ A = , R2 2MR + a2 cos2 ϑ B = 1, − 1 (R2 2MR + a2 cos2 ϑ)2 ζ = ln − , (1.62) 2 [(R M)2 + (a2 M 2) cos2 ϑ](R2 + a2 cos2) − − where R and ϑ are now considered as functions of ρ and z given by implicit relations (1.61). The Kerr space-time can be generated by thin discs — an infinite differentially (counter-)rotating one Biˇcákand Ledvinka [1993] or a rigidly rotating finite one Neugebauer and Meinel [2003], but in contrast to the Schwarzschild black-hole no real interior solution has been found so far. 11Inverse transformation is not mentioned here because it is quite cumbersome.

25 1.5 Associated linear problem

For a circular space-time, Einstein’s equations (namely the Ernst equation) are known to be completely integrable. However, it is not easy to solve them directly from scratch, so in more generic cases “generating techniques” are being employed. These are mathematical procedures which transform some known, “seed” solution of the problem into another one of the same (circular) class. Ex- amples of such techniques are Bäcklund transformations and inverse-scattering methods (see [Stephani et al., 2003, chapter 10] for quick overview). Most of the (infinite families of) metrics obtained by generating techniques still remain to be interpreted, and it is very probable that almost none of them has a reasonable physical sense. In this thesis we nevertheless include a section on a possible usage of the inverse-scattering method. Generating techniques are mostly not applied directly to the field equations, but rather to an associated linear problem. There are two basic ways how to find this. The one involving Ernst equation (1.44) (e.g. Neugebauer and Meinel [1994], Neugebauer and Meinel [1995] who found the solution for rigidly rotating disc in this manner) and the Belinskii–Zakharov method (great review can be found in Belinskii and Verdaguer [2004]).

1.5.1 Linear problem associated with the Ernst equation Let Ψ be a 2 2 matrix depending on some (“spectral”) parameter λ and on spatial coordinates ×ρ and z. It is useful to introduce a complex coordinate ζ = ρ+iz and consider the matrix Ψ as the function of λ, ζ and ζ. Let the matrix Ψ solve the pair of linear diﬀerential equations (originally, including even Maxwell equations, given by Neugebauer and Kramer [1983])

∂Ψ 0 0 = B + λ B Ψ, (1.63) ∂ζ 0 0 A A ∂Ψ 0 1 0 = A + A Ψ, (1.64) ∂ζ 0 λ 0 B B where the spectral parameter is usually expressed (in terms of still another parameter K) as K iζ λ = − (1.65) sK + iζ and and are complex functions of physical coordinates (ρ, z) alone (they do not dependA B on K or λ). If these functions are related to the Ernst potential by

= E,ζ , = E ,ζ , (1.66) A 2 B 2

26 to overcome this diﬃculty using suitable normalization and a special behaviour of spectral parameter on the axis (λ = 1). A considerable amount of work on this formulation has been done by the Jena GR group (see e.g. Neugebauer and Meinel [2003] for a survey).

1.5.2 Belinskii–Zakharov method The Belinskii–Zakharov inverse-scattering method, originaly published in Belin- skii and Zakharov [1978], assumes the metric has the form

2 a b 2 2 ds = gabdx dx + f(dρ + dz ), (1.67) where gab denotes its Killing (t, φ) part. They, as well as f, depend on ρ and z only. In addition, the determinant of the 2-metric is assumed to amount to 2 det gab = ρ . Such a condition is satisﬁed by metrics (1.5) and (1.6), if B = 1 is chosen.− Introducing 2 2 matrices × 1 1 U ρg g− ,V ρg g− , (1.68) ≡ ,ρ ≡ ,z 1 where g− denotes the inverse of g and multiplication is the ordinary matrix multiplication, the Einstein equations (1.24) and (1.25) take the form

U,ρ + V,z = 0. (1.69)

The remaining two equations — (1.27) and (1.28) — can be written as f 1 1 ,ρ = + Tr U 2 V 2 , (1.70) f −ρ 4ρ − f 1 ,z = Tr (UV ) . (1.71) f 2ρ Again, (1.69) represents the integrability condition of these two equations. There- fore, f can be obtained by quadrature provided that U and V are known. Equation (1.69) seems simple at first glance. However, the matrices U and ∂M 1 ∂ ln det M V are not independent. For example, using the relation Tr ∂x M − = ∂x , one can find that ∂ U ∂ V Tr = Tr . (1.72) ∂z ρ ∂ρ ρ In order to solve the equation (1.69), Belinskii and Zakharov introduced an additional (spectral) parameter λ and differential operators

∂ 2λ2 ∂ ∂ 2λρ ∂ Dˆ = , Dˆ = + (1.73) 1 ∂z − λ2 + ρ2 ∂λ 2 ∂ρ λ2 + ρ2 ∂λ and a generating matrix Ψ (again 2 2) which satisﬁes the linear system × ρV λU ρU + λV Dˆ Ψ = − Ψ, Dˆ Ψ = Ψ. (1.74) 1 λ2 + ρ2 2 λ2 + ρ2 The normalization of Ψ is ﬁxed by boundary condition

Ψ(ρ, z, λ = 0) = g(ρ, z) (1.75)

27 which is in agreement with (1.68). Knowing the solution of (1.74) (let us denote it by Ψ0 and the corresponding original/seed 2-metric by g0), the new solution of the system (1.74) can be found by prescription

Ψ(ρ, z, λ) = χ(ρ, z, λ)Ψ0(ρ, z, λ), g(ρ, z) = χ(ρ, z, 0)g0(ρ, z), (1.76) where χ is the so-called dressing matrix. It has to satisfy ρV λU ρV λU ρU + λV ρU + λV Dˆ χ = − χ χ 0 − 0 , Dˆ χ = χ χ 0 0 , (1.77) 1 λ2 + ρ2 − λ2 + ρ2 2 λ2 + ρ2 − λ2 + ρ2 where U0 and V0 are defined by (1.68) with the seed metric g0 plugged in. There are several restrictions which the matrix χ has to satisfy (see [Belinskii and Verdaguer, 2004, ch. 1.3] for details). In particular, its determinant has to be det χ = 1 in order that the generated metric have correct determinant (det g = ρ2). However, it may be useful to omit this condition in the procedure and normalize− the obtained metric only later (we will denote the renormalized quantities by the superscript ph (meaning physical). For the so-called soliton solutions when the dressing matrix contains only simple poles in the λ plane, it can be expressed in algebraic terms. It can finally be used to find a new metric,

N (k) (l) Π(k)(l)mc md gab = (g0)ab (g0)ac(g0)bd , (1.78) − µ(k)µ(l) k,lX=1 N N 2 ρ µ(k) f = C f k=1 det Γ , (1.79) N 0 N Q (k)(l) 2 2 (µ(k) + ρ ) k=1 Q where

µ = α z + (α z)2 + ρ2, (1.80) (k) k − k − (k) 1 m(k) C ρ, z, µ − , a = b Ψ( p (k)) ab (1.81) m(k)m(l)(g ) a b 0 ab , Γ(k)(l) = 2 (1.82) ρ + µ(k)µ(l) N 1 Π = Γ − Γ Π = δ . (1.83) (k)(l) (k)(l) ⇔ (k)(m) (m)(l) (k)(l) m=1 X The indices a, b, c, d t, φ and k, l, m 1, 2,...,N , where N denotes the number of “solitons”.∈ The { parentheses} are∈ used { to distinguish} between the soliton and coordinate indices. (Einstein’s summation rule still applies, i.e. when the same index appears twice it is summed over. The constants α(k) correspond to (k) the position of the solitons on the axis and the constants Ca encode physical quantities like black-hole mass, angular momentum, etc. (See section (3.2) for a discussion of the two-soliton solution.) Both these sets of constants can be chosen arbitrarily, as well as the integration constant CN which ﬁxes scaling of the ρ and z coordinates.

28 Concerning the physical parameters of the solution, it can be shown that

( 1)N ρ2N det g = det g (1.84) −N 0 2 µ(k) k=1 Q and so [Belinskii and Verdaguer, 2004, (1.110)] ρ gph = g, (1.85) det g | | N p N+1 µ(k) f ph = Cphf k=1 det Γ . (1.86) N 0 N k 1 (k)(l) N2 Q− ρ 2 µ(k) µ(l) k=2 l=1 − Q Q

29 2. Perturbative approach

The pair of equations (1.24) and (1.25) is, due to their non-linearity, hard to solve. A usual way how to proceed is the perturbation one: start from some known exact solution, make slight perturbation of some appropriate quantities (“X + δX”), substitute the decomposition into the equations, subtract the X- part which is satisfied exactly and linearize the rest in δX. For a sufficiently simple original, exact “background”, one can hope the linear equations could be solved better than the original, non-linear ones. However, general relativity is non-linear and to include its effects (namely, the “back reaction” of the source on itself, i.e. “self-gravitation”) properly, it is desirable to extend the perturbation order beyond the first, linear one. One thus arrives at the solution of the field equations in terms of series expansions of the relevant quantities in terms of some small parameter(s). In particular, this chapter tackles, in a perturbative way, the problem of a slowly rotating and light annular thin disc encircling a black-hole. The system is studied within the class of circular space-times and the target of perturbation is a Schwarzschild black-hole. In order that the disc’s mass and angular-momentum effect can be considered as a perturbation, one should restrict to m 1 1 1, (2.1) M 1 v2 1 2M − − r where m denotes the disc rest mass, M is the black-hole mass, v stands for a (maximal) linear velocity of the dust and r stands for radius (of the inner rim) of the disc. Physically, this assumption of course means weak self-gravitation of the disc (non-linear effect of the dust to its own gravitational field) and should ensure that the perturbation series may be cut after just several terms. (For astrophysical purposes, this should certainly be enough, because it is likely that a bigger error arises as a consequence of the model idealizations anyway.) Perturbations of black-hole space-times are a “classical” GR problem, recently mainly studied in a non-stationary regime in connection with collisions of compact objects and a consequent gravitational emission. Much weaker interest has been focused on stationary perturbations, so one has to mainly search in older literature. Actually, the methods derived by Bardeen and Wagoner [1971] and Will [1974] can still be taken as a starting point. In this chapter we follow closely our recent paper C´ıˇzekandˇ Semerák[2017] (conference proceedings C´ıˇzekandˇ Semerák[2009], C´ıˇzek[2011]ˇ and C´ıˇzekandˇ Semerák[2012] can be mentioned for some preliminary results).

2.1 Expansion in perturbing mass

Let the expansion parameter be denoted by λ. Provided that the series (2.3) converges, λ need not be speciﬁed further, important is that the r.h. sides of the ﬁeld equations (1.24)–(1.28) are proportional to it, see (2.4) and (2.5). One suitable suggestion could be m 1 λ = . (2.2) M 1 v2 − 30 The metric functions are expanded as

∞ ∞ ∞ k k k ν = νkλ , ω = ωkλ , ζ = ζkλ , (2.3) Xk=0 Xk=0 Xk=0 where νk, ωk and λk are functions of ρ and z. Clearly ν0, ω0, ζ0 and B are metric function describing the background (vacuum) solution. Let us remind that the metric function B need not be expanded, because in our case (thin disc or ring) the equation (1.26) can be solved exactly — it is a homogeneous linear partial diﬀerential equation (PDE). The surface rest mass density of our source is given by ν 3ν 2ζ σ = λe τ, = λτe − δ(z), (2.4) where τ describes the disc mass distribution and the term eν ensures that the total rest mass of the disc, given by

ρ+ 2ζ 3ν = √gφφgρρgzzdφdρdz = ρBe − dφdρdz = 2πλ τρBdρ, (2.5) M discZZ discZZ ρZ − depends on λ linearly; ﬁnally, ρ and ρ+ are inner and outer radii of the disc. − Equations ((1.24) and (1.25)) for ν and ω lead to the following equations for individual powers of λ:

2 ∞ 2 ρ2B 4ν ν 1 + v B ν e− ω ω πBτδ z e , ( k) k l j j l = 4 ( ) 2 (2.6) ∇ · ∇ − 2 − − ∇ · ∇ 1 v j,l=0 k 1 jX+l k − − ≤ k ν 2 3 4ν 2 ve− ρ B e− ωk j = 16πρB τδ(z) , (2.7) j − 2 ∇ · ∇ − 1 v k 1 j=0 − − X where 1 ∂kf f = (2.8) { }k k! ∂λk λ=0 is the λk-term arising from the Taylor expansion of f and k 1. (In particular, the k = 0 case corresponds to vanishing r.h. sides of (2.6)≥ and (2.7), i.e. to ν and ω given by vacuum equations (1.24) and (1.25) and thus reducing to a given background metric. Finally, the linear velocity v is given by the metric functions and their derivatives. The above perturbation equations can be solved iteratively. If νj and ωj are known for all 0 j < k, equations (2.6) and (2.7) can be written as ≤

2 2 4ν0 2 4ν0 (B ν ) 2ρ B e− ν ω ω ρ2B e− ω ω = ∇ · ∇ k − k∇ 0 · ∇ 0 − ∇ k · ∇ 0 BR = k , (2.9) ρ2 + z2 2 3 4ν0 2 3 4ν0 ρ B e− ω 4 ρ B e− ν ω = ∇ · ∇ k − ∇ · k∇ 0 M 4ρ2S = k (2.10) B(ρ2 + z2)3

31 where Rk and Sk stand for the already known “source terms”,

k 1 ρ2 z2 ρ B2 ∞ ρ B2 − + 2 4ν 2 4ν0 R = e− ω ω + e− ω ω k  k l j j l j k j B 2 − − ∇ · ∇ 2 ∇ · ∇ − j,l=0 j=1  jX+l

2.2 Multipole expansion: Spherical harmonics

2.2.1 Methods of solving linear PDE Linear PDE are usually being solved by expansion (with respect to some set of functions of coordinates) or by using a Green function. The latter case can be applied to linear diﬀerential equations only. Assume the equation takes the form Ofˆ = R(x1, x2, . . . xN ), (2.13) where Oˆ denotes some linear diﬀerential operator, R some source term and x1, . . . xN are coordinates. The boundary conditions have the form

Dfˆ = 0, (2.14) ∂Σ

ˆ where D stands for some other linear diﬀerential operator and ∂Σ is the border of the domain Σ. Green function is deﬁned then as a particular solution of (2.13) with a δ-like r.h. side and satisfyingG the boundary conditions (2.14), i.e. Oˆ (x1, . . . , xN , y1, . . . , yN ) = δ(y1 x1)δ(y2 x2) . . . δ(yN xN ), (2.15) G − − − Dˆ = 0. (2.16) G ∂Σ

(Despite its name, the Green function is deﬁned using distributions. The term fundamental solution, used in a more general sense, may be more appropriate.) Equation (2.13) is solved by

f(x1, . . . , xN ) = R(y1, . . . , yN ) (x1, . . . , xN , y1, . . . , yN )dy1 ... dyN . (2.17) G ZZΣ

32 This clearly satisfies the boundary conditions (2.14). In the case of a compact source (thin disc in our case), the integral on the r.h. side of (2.17) simplifies considerably. The problem is then to find the Green function itself, i.e. a solution of the corresponding differential equation with the special r.h. side and given boundary conditions. Fundamental systems of ordinary differential equations (ODEs) can be found straightforwardly, but for PDEs no generic direct procedure is known. We are going to find the Green functions of (2.9) and (2.10) using series expansion and, ex post, summation of the obtained series (see section 2.4.6). Series expansion, used in the paper Will [1974] we start from, reduces PDE to a set of ODEs. In a general case, these equations are not independent and their solution is not simple. However, they can decouple in some cases if using an expansion into eigenfunctions of a suitable part of the original differential equation. An important attribute of the series solution is its convergence speed. Gener- ally the series converge exponentially fast, but one has to be much more careful in the case of singular sources (spatially less than three-dimensional). In our case of the thin disc (spatially a two-dimensional source) the convergence is still power-law and breaks down just at some particular points (analogue of the Gibbs phenomenon occurs due to the discontinuities). The convergence of the multipole expansion towards the exact Green function is illustrated, for a linear perturbation by a massive thin ring, in figures 2.2 and 2.3. Convergence problems also appear in numerical simulations using spectral methods (see, for example, the survey Grandclément and Novak [2009]). They can sometimes be eliminated by a clever choice of coordinates. For example, Ansorg et al. [2003] obtained better behaviour in the oblate spheroidal coordinates

ρ = a cosh u cos v, z = a sinh u sin v, (2.18) where a is an arbitrary ﬁxed parameter describing the outer rim of the disc. Also toroidal coordinates, employed in Ansorg et al. [2009], can be used. For our purpose, even more suitable (for obtaining smooth boundary conditions) seem to be the ﬂat-ring coordinates given by

a sin v√cos2 u + k2 sin2 u a cos v 1 k2 sin2 u ρ = , z = , (2.19) 2 2 2 2 2 2 − 2 2 cos u + k sin u sin v cos u +pk sin u sin v where a and k are arbitrary fixed parameters describing the inner and outer rims of the disc (see Kuchel et al. [1987]). Unfortunately, these do not fit well into the series expansion. Both coordinate systems mentioned are sketched in figure 2.1.

2.2.2 Choice of background metric As already mentioned, we will start from a static background metric in order to decouple the set of equations (2.9) and (2.10). Among the black-hole solutions, the simplest is the Schwarzschild metric which, in the limit case (vacuum everywhere), describes an isolated, spherically symmetric black-hole (section 1.4.1). Although the Schwarzschild metric can be written in canonical Weyl coordinates ρ, z and the ﬁeld equations look quite simple in them, such a possibility naturally involves the B = 1 choice. This means that the black-hole horizon degenerates to a ﬁnite segment of the symmetry axis (ρ = 0, z ( M, +M)), ∈ − 33 4 4

3 3

z 2 z 2

1 1

0 1 2 3 4 5 6 0 1 2 3 4 5 6 rho rho

Figure 2.1: Contours of constant u and v in coordinate systems adapted to discs. Left: Oblate spheroidal coordinates with a = 2. Right: Flat-ring coordinates with a = 2 and k = 0.5. which is not an ideal option for treating the respective boundary conditions and for physical interpretation. In isotropic coordinates r, θ, we saw the horizon is a sphere located at radius r = M/2. This is given by the choice M 2 B = 1 , (2.20) − 4r2 where M denotes the black-hole mass. Since the function B is not subject to perturbation, the horizon remains on r = M/2 even after perturbation. The remaining background-metric functions read 2r M M 2 ν = ln − , ω = 0, ζ = ln 1 . (2.21) 0 2r + M 0 0 − 4r2 Substituting into the equations (2.9) and (2.10), we have

1 ∂ ∂ν 4r2 M 2 ∂ ∂ν BR (4r2 M 2) k + − sin θ k = k , (2.22) r2 ∂r − ∂r 4r4 sin θ ∂θ ∂θ r2 1 ∂ (2r + M)7 ∂ω (2r + M)7 ∂ ∂ω M 4S k + sin3 θ k = k . (2.23) r2 ∂r 26r2(2r M) ∂r 26r6(2r M) sin3 θ ∂θ ∂r r4B − − The dependence of the l.h. sides on θ is quite simple and Legendre polynomi- (3/2) als Pn(cos θ) and Gegenbauer polynomials Cn (cos θ) are eigenfunctions of the angular part of those diﬀerential operators, namely 1 d dP (cos θ) sin θ n = n(n + 1)P (cos θ), (2.24) sin θ dθ dθ − n (3/2) 1 d 3 dCn (cos θ) (3/2) 3 sin θ = n(n + 3)Cn (cos θ), (2.25) sin θ dθ " dθ # − where n(n + 1) and n(n + 3) are the respective eigen-values. It is known that analytic− functions can− be expanded in the basis of these polynomials. Deﬁning

∞ ∞ (3/2) νk(x, θ) = νkn(x)Pn(cos θ), ωk(x, θ) = ωkn(x)Cn (cos θ), (2.26) n=0 n=0 X X 34 where x a is dimensionless radius given by r M 2 x 1 + , (2.27) ≡ M 4r2 equations (2.22) and (2.23) read

∞ d dν (x2 1) kj j(j + 1)ν P (cos θ) = R , (2.28) dx − dx − kj j k j=0 X ∞ d dω (x2 1) (x+1)4 kj j(j+3)(x+1)4ω C(3/2)(cos θ) = S , (2.29) − dx dx − kj j k j=0 X where r is now considered a function of x. (3/2) Using the orthogonality of polynomials Pj(cos θ) and Cj (cos θ) (section C.2.2), the equations can be separated to yield d dν (x2 1) kj j(j + 1)ν = R , (2.30) dx − dx − kj kj d dω (x2 1) (x + 1)4 kj j(j + 3)(x + 1)4ω = S , (2.31) − dx dx − kj kj where Rkj and Skj are coeﬃcients of expansion of the r.h. sides of (2.28) and (2.29), i.e.

1 2j + 1 R = R (x, cos θ)P (cos θ)d(cos θ), (2.32) kj 2 k j Z1 − 1 2j + 3 S = S (x, cos θ)C(3/2)(cos θ) sin2 θd(cos θ). (2.33) kj 2(j + 1)(j + 2) k j Z1 − 2.2.3 Metric function (ζ) As already mentioned in section 1.3.1, the metric function ζ can be obtained by quadrature. Boundary condition on the axis (1.34) suggests a suitable integration path — along the axis to a given radius and then along the contour of constant radius (in our case all this path is through vacuum). Rewriting (1.27) and (1.28) using (2.20) gives

M 2 1 2 3 4ν r 1 + 2Bν ν ω ω sin θB e− 4r2 ,r ,θ − 2 ,r ,θ ζ,θ = 2 + (2.34) M 2 2 2 1 + 4r2 + B cot θ 2 2 2 M 2 2 2 ν,θ 2 ω,θ 2 2 4 4ν r cot θ B ν + frac14 ω r sin θB e− r4 − ,r − r2 ,r − r2 + 2 h M 2 2 2 i 1 + 4r2 + B cot θ and the ζ function itself is obtained by prescription

ζ = ln B + ζ,θdθ. (2.35) Z0

35 For ν and ω known, the calculation of ζ is just a “technical” issue and will not be investigated in detail. Exact expressions for ζk are anyway quite complicated and not very useful, while the integral (2.35) usually requires numerical evaluation. Relation (2.34) simpliﬁes considerably on the horizon

ζ,θ r= M = 2δν,θ r= M , (2.36) | 2 | 2

∞ j where δζ = νjλ , i.e. perturbation of gravitational potential. The fact that ra- j=1 dial derivativesP of dragging (see page 51) vanishes there was used as well. Finally ζ itself reads

M M ζ = ln B + 2 δν 2 δν r= M , . (2.37) r= 2 r= 2 2 | | − | θ=0 The B function is here kept as it describes diverging part of ζ.

2.3 Boundary conditions

Spatial inﬁnity Boundary conditions at spatial inﬁnity are given by (1.33). When expanded according to (2.3) and (2.26), they read

ν 0, ω 0, (2.38) kj → kj → as x . → ∞ Axis Boundary conditions on the symmetry axis (1.34) hold as a consequence of the dependence of angular part on cos θ. In our spherical-type coordinates, vanishing of the ρ-derivative at the axis is equivalent to vanishing of the derivative with respect to θ, ∂ 1 ∂ = 0. (2.39) ∂ρ r ∂θ ρ=0 θ=0,π

The expected dependence of the angular part on cos θ implies

∂f(cos θ) = f 0(cos θ) sin θ = 0, (2.40) ∂θ − |θ=0,π θ=0,π

where f(x) is an arbitrary function and f 0(x) is its derivative with respect to the argument. In the last equality it has been assumed that f 0(1) is bounded. In the case of polynomials (whether Legendre or Gegenbauer) such assumption is satisﬁed.

36 Black-hole horizon Unperturbed metric has the horizon located at sphere r = M/2 corresponding to the singularity of metric function B, which is not subjected to the perturbation scheme. It can be expected that coordinates of horizon stay the same1. Using orthogonality of Legendre (C.14) and Gegenbauer polynomials (C.15) the boundary conditions on horizon can be separated. Constancy of dragging reads

ωkj r= M = 0, (2.41) | 2 where j > 1 (ωk0 can be arbitrary). Condition of non–degenerate azimuthal circumference reads

P∞ j νj λ ν − j=1 Bρe− M = 2M sin θe (2.42) r= 2 r= M 2

implying ﬁniteness of νj on the horizon so νjk must not diverge there. Physically meaningful latitudinal circumference is ensured by this as well. Given (2.37) it can be concluded that

M M ζ ν = ln(4) + δν 2 δν r= M , . (2.43) r= 2 r= 2 2 − | | − | θ=0 Finiteness of perturbation of gravitational potential δν on the horizon ﬁnishes sketch of the proof of non–degeneracy. Constancy surface gravity, due to non-linearity of the equations (1.36) and (1.37) is not analysed here explicitly.

Equatorial plane There are two ways how to treat a thin source located in the equatorial plane. 1. Consider it as a limit case of source that is spatially extended (see sections 1.2 and 1.3.1). The reflection symmetry is of course assumed. 2. Include it by prescribing jumps of the normal derivative of the metric functions there (see section 1.3.2). If the source is absent, the normal derivatives vanish — see (1.40). We will use the first view for a calculation of the metric and the second one for the analysis of properties of the solution. Actually, the r.h. sides of (2.6) and (2.7) involve the equatorial boundary conditions automatically since the source terms are δ-like and they do show reflection symmetry. In terms of the orthogonal polynomials, the necessary (and sufficient) condition for reflection symmetry is, regarding their parity (C.12), that the odd expansion coefficients has to vanish,

νk(2j+1) = 0, ωk(2j+1) = 0, (2.44)

+ where j Z0 . ∈ 1To prove that it is necessary to calculate all orders of expansions. Being it not done in this thesis the position of horizon remains assumption. The results from low order of mass expansion (see Green functions (2.81) and (2.96)) indicates that nothing dramatic takes place and the assumption holds.

37 2.4 Green functions

2.4.1 Solving the expanded ﬁeld equations Fundamental solution Fundamental solutions of (2.30) and (2.31) are already presented in the paper Will [1974]. They can be written in the form

νjk = Lin [Pk(x), Qk(x)] , (2.45)

ωjk = Lin [Fk(x),Gk(x)] , (2.46) where Lin[] means a suitable linear combination, Pk,Qk denote Legendre functions of the ﬁrst (second) kind and k, k + 3 1 + x Fk(x) = 2F1 − ; , (2.47) 4 2 ∞ dx0 Gk(x) = Fk(x) ; (2.48) 4 2 (1 + x0) [Fk(x0)] Zx a, b F ; x denotes the Gauss hypergeometric function, 2 1 c a, b ∞ Γ(a + k)Γ(b + k)Γ(c)xk F ; x = , (2.49) 2 1 c Γ(a)Γ(b)Γ(c + k)k! Xk=0 where Γ(x) is the Gamma function

∞ x 1 t Γ(x) = t − e− dt. (2.50)

Z0 + Note that k Z0 ensures that the Legendre function of the first kind Pk and Fk(x) are polynomials∈ of degree k. Asymptotic (x ) behaviour of the fundamental solutions is given by → ∞ k k 1 k k 3 P (x) x , Q (x) x− − ,F (x) x ,G (x) x− − , (2.51) k ≈ k ≈ k ≈ k ≈ and the behaviour at the horizon (x 1+) by → 1 P (x) = 1, Q (x) ln(x 1),F (x) = δ ,G (x) const. = 0, (2.52) k k ≈ 2 − k k0 k ≈ 6 where δkj is the usual Kronecker symbol. See [DLMF, ch. 14.8] or section C.2.3 for the properties of Legendre functions. The behaviour of Fk and Gk follows from the definition of the Gauss hypergeometric function ([DLMF, (15.4.24)]) and of the function Gk. In the last case we have actually used an alternative definition k 3 ( 1)k(k + 2)!(k + 3)! 1 + x − − k, k + 3 2 G (x) = − F ; , (2.53) k 48(2k + 3)! 2 2 1 2k + 4 1 + x which can be obtained thanks to [DLMF, (15.10.6)] and regarding that only one fundamental solution (unique up to a multiplicative constant) vanishes when x (the multiplicative constant can be fixed by comparing the leading-order terms→ ∞ of the hypergeometric function and of (2.48).

38 Source terms

Let us denote by ν(x, x0, θ, θ0) and ω(x, x0, θ, θ0) the Green functions of equations (2.22) and (2.23),G respectively, andG by

R = S = δ(x0 x)δ(cos θ0 cos θ), (2.54) k k − − the corresponding source terms, with (x0, θ0) representing the position of the source. (Note that diﬀerential operators on the l.h. sides of the perturbation equations — and thus also the Green functions — do not depend on the perturbation order k, whereas the source terms do depend on it.) Together with (2.32) and (2.33) we have

1 2j + 1 R = δ(x0 x)δ(cos θ0 cos θ)P (cos θ)d(cos θ) = kj 2 − − j Z1 − 2j + 1 = P (cos θ0)δ(x0 x), (2.55) 2 j − 1 2j + 3 (3/2) 2 S = δ(x0 x)δ(cos θ0 cos θ)C (cos θ) sin θd(cos θ) = kj 2(j + 1)(j + 2) − − j Z1 − 2j + 3 (3/2) 2 = C (cos θ0) sin θ0δ(x0 x), (2.56) 2(j + 1)(j + 2) j − where up to now [Will, 1974, (50)] has been followed (except for his expansion in horizon angular velocity).

2.4.2 Green functions The Green functions can be expanded into spherical harmonics analogously to (2.26),

∞ (x, x0, θ, θ0) = (x, x0, θ0)P (cos θ), (2.57) Gν Gνn n n=0 X ∞ (3/2) (x, x0, θ, θ0) = (x, x0, θ0)C (cos θ). (2.58) Gω Gωn n n=0 X They have to obey (2.30) and (2.31) with the r.h. sides (2.55) and (2.56). The latter, however, are δ-like, so νn and ωn are, up to a scaling factor, Green’s functions of (2.30) and (2.31).G Since theG source vanishes almost everywhere (except the point x = x0 where it is singular), the solution of (2.30) and (2.31) has the form

Aνj(x0, θ0)Pj(x) + Bν>j(x0, θ0)Qj(x) when x x0 ≥ Aωj j ω>j j ≥ where A and B are suitable functions of the source position (x0). The overlap ◦◦◦ ◦◦◦ of intervals in (2.59) and (2.60) is intentional: the equations are of the second

39 order and the Green functions should be continuous (discontinuity is only in the first derivatives), so both definitions have to coincide at x = x0. Green functions has to also obey boundary conditions. Those at spatial infinity, combined with (2.51), give

Aν>j = Aω>j = 0, (2.61)

+ where j Z0 . On the horizon, uniformity of the angular velocity requires ωkj = 0 (see (2.41))∈ for j > 0 and (2.52) yields

Bω

Bν

+ for all j Z0 . Finally, one has to take care of the ﬁrst derivatives in order to get ∈ δ(x x0). Analogously to (1.39), it can be shown that − ∂2f ∂f ∂f = δ(x x0). (2.64) ∂x2 ∂x − ∂x − x=x0 x=x0+ x=x0 − The above requirements ﬁx the unknown functions A and B and the ◦◦◦ ◦◦◦ expanded Green functions read 2j + 1 (x, x0, θ0) = P (cos θ0) sin θ0P (x )Q (x ), (2.65) Gνj − 2 j j < j > 2j + 3 (3/2) 2 (x, x0, θ0) = C (cos θ0) sin θ0F (x )G (x ), (2.66) Gωj −2(j + 1)(j + 2) j j < j > where x< = min(x, x0), x> = max(x, x0). (2.67)

Note that ωj has one degree of freedom related to Bω<0 on top of (2.66) — see discussionG in the next section.

2.4.3 Angular velocity of the horizon Special attention deserves the term in (2.66) which arises from freedom in the choice of Bω<0 and which has been omitted. Consider ﬁrst the function

3 (3/2) 2 1 2 1 B C (cos θ0) sin θ0F (x0)G (x) = B sin θ0 . (2.68) − ω<0 4 0 0 0 − ω<0 4 (x + 1)3

It is smooth everywhere (including spatial inﬁnity, horizon and ring radius) and solves the homogeneous equation (2.31) (with j = 0). Hence,

sin∈ θ0 ˜ω0(x, x0, θ0) = ω ( , 0, θ0) ω< (2.69) G G 0 § § − B 0 ( + ) 4 § ∞ 3

40 is also a Green function of (2.31), just with a diﬀerent value of angular momentum of the horizon. A solution with general r.h. side can be obtained by

ω˜ = S (x0, θ0) ˜ (x, x0, θ0)dx0d(cos θ0) = k0 k0 Gω0 ZZ = S (x0, θ0) (x, x0, θ0)dx0d(cos θ0) k0 Gω0 − ZZ 2 sin θ0 S (x0, θ0)B dx0d(cos θ0) = − k0 ω<0 4(x + 1)3 ZZ Jk = S (x0, θ0) (x, x0, θ0)dx0d(cos θ0) + , (2.70) k0 Gω0 (x + 1)3 ZZ where the constants Jk are proportional to Bω<0 and to the mass distribution. These constants clearly adjust the horizon angular velocity and can be interpreted as ﬁxing the own angular momentum of the black-hole (note in passing that the part of the Green function corresponding to Bω<0 is smooth across the source). More precisely, the angular velocity of the horizon is determined by

∞ 2 k sin θ0 Jk ω ω λ , ω S x0, θ0 x0 θ0 . H = Hk Hk = k0( ) 3 d d + (2.71) 4(x0 + 1) 8 Xk=0 Z Our choice differs from the original paper Will [1974]. Will used additional expansion parameter β = kωH (besides the “mass” perturbation denoted there by γ) rather than considering the constants Jk (he did not mention the freedom in the horizon angular momentum). In any case, the introduction of constants Jk serves the same purpose and, at least by my opinion, is simpler to work with. (Note that interaction of angular momentum of the disc and of the black-hole may however be present in higher orders of the mass perturbation (2.3), so there one would have to be more careful. In order to fix the ambiguity in the black-hole angular momentum, we put Jk = 0 in the rest of this thesis. Effectively it means that the black-hole angular momentum is kept zero in perturbation, so the black-hole is just being “carried along” by dragging induced by the external source. (We will confirm this explicitely after deriving a specific solution for a simple finite thin disc.)

2.4.4 Linear perturbation due to a thin ring Green functions of the equations (2.28) and (2.29) have a clear physical interpretation — they describe linear perturbation of due to a circular thin (singular) ring of matter. We assume this ring is located in the equatorial plane, θ0 = π/2, on some radius r = r0 (a ring placed oﬀ the equatorial plane would require some kind of supporting struts in order to remain stationary, otherwise it would be pulled towards this plane). Two further parameters are suﬃcient to describe it — its mass m and its linear orbital velocity v (we note that the total mass of the system will be denoted by ). The velocity should equal a local Keplerian orbital velocity, which for a linearM perturbation of a Schwarzschild black-hole is given by (1.55) (namely, a correction due to self-gravity of the ring is of the second or higher order in λ). The ring mass density is described by m = δ(ρ ρ0)δ(z), (2.72) 2πρ −

41 4 4

3 3

z 2 z 2

1 1

0 1 2 3 4 5 6 0 1 2 3 4 5 6 rho rho

Figure 2.2: Linear perturbation of a Schwarzschild black-hole (having mass M = 1) by a ring of matter (located at r = 3). Metric functions are calculated using series truncated to 30 orders of the “multipole” expansion. Left: Perturbation of gravitational potential (ν1). Right: Perturbation of dragging (ω1). so one can conclude that π π ν (x, θ) x, x0, θ, , ω (x, θ) x, x0, θ, (2.73) 1 ∝ Gν 2 1 ∝ Gω 2 (in the second equation we have already set J1 = 0). The multiplicative constant can be ﬁxed by combining (2.72), (1.55), (2.4), (2.11) and (2.12):

2 2 2 2mM x0(x0 + 1) (x0) 1 π ν (x, θ) = − x, x0, θ, , (2.74) 1 (r )3(x 2) Gν 2 0 0 −p 4 2 4m√Mr0(2r0 + M)(x0 + 1) (x0 1) π ω x, θ x, x0, θ, , 1( ) = 4 − ω (2.75) − (r0) (x0 2) G 2 − where r0 is now considered to be a function of x0, and λ = 1 (more precisely, mλ being mass of the ring).

2.4.5 Convergence Important aspect of every perturbation scheme is its convergence. Indeed, in our case the series (2.57) and (2.58) do not behave very well and their truncation may be an issue. This is apparent in ﬁgure 2.2 where ν1 and ω1 for the ring perturbation are calculated using the ﬁrst 30 terms of the multipole expansion. As expected, convergence is the worst close to the radius of the source and, besides the very source (in the equatorial plane), it shows itself particularly at the axis. In tables 2.1 and 2.2 the convergence is illustrated in a numerical detail. In order to estimate the convergence speed, we combine (2.65) and the asymptotic behaviour from section C.2.3, to obtain a lower bound on the convergence speed of (2.57), 2j + 1 = P (cos θ)P (x )Q (x )P (0) (2.76) |Gνj| 2 | j j < j > j | ≈ 1 +j 2 2 1 x< + x< 1 1 − f(x, x0, θ) + O , (2.77) ≈ j x + x2 1 j > p > − ! p 42 N Exact θ x 10 20 30 40 50 (see (2.81)) 2,00 -0,19807 -0,19813 -0,19813 -0,19813 -0,19813 -0,19813 2,50 -0,22468 -0,22664 -0,22680 -0,22682 -0,22682 -0,22682 2,90 -0,26750 -0,28772 -0,29629 -0,30058 -0,30293 -0,30636 π 2 3,00 -0,28370 -0,31884 -0,34032 -0,35583 -0,36797 3,10 -0,25857 -0,27846 -0,28700 -0,29134 -0,29374 -0,29738−∞ 3,50 -0,19552 -0,19818 -0,19849 -0,19853 -0,19854 -0,19854 4,00 -0,15445 -0,15478 -0,15479 -0,15479 -0,15479 -0,15479 2,00 -0,17340 -0,17344 -0,17344 -0,17344 -0,17344 -0,17344 2,50 -0,16688 -0,16747 -0,16754 -0,16755 -0,16755 -0,16755 2,90 -0,15563 -0,15601 -0,15780 -0,15881 -0,15894 -0,15858 π 3 3,00 -0,15196 -0,15038 -0,15329 -0,15671 -0,15813 -0,15585 3,10 -0,15012 -0,15044 -0,15221 -0,15323 -0,15337 -0,15299 3,50 -0,14007 -0,14071 -0,14085 -0,14086 -0,14086 -0,14086 4,00 -0,12584 -0,12599 -0,12600 -0,12600 -0,12600 -0,12600 2,00 -0,16020 -0,16016 -0,16016 -0,16016 -0,16016 -0,16016 2,50 -0,15061 -0,15003 -0,15008 -0,15008 -0,15008 -0,15008 2,90 -0,14321 -0,13879 -0,14019 -0,14108 -0,14074 -0,14062 π 4 3,00 -0,14178 -0,13419 -0,13694 -0,14025 -0,13889 -0,13816 3,10 -0,13822 -0,13388 -0,13526 -0,13617 -0,13582 -0,13570 3,50 -0,12655 -0,12582 -0,12591 -0,12592 -0,12592 -0,12592 4,00 -0,11458 -0,11444 -0,11445 -0,11445 -0,11445 -0,11445 2,00 -0,15110 -0,15113 -0,15113 -0,15113 -0,15113 -0,15113 2,50 -0,13920 -0,13971 -0,13975 -0,13974 -0,13974 -0,13974 2,90 -0,12758 -0,12952 -0,13141 -0,13019 -0,13032 -0,13046 π 6 3,00 -0,12402 -0,12610 -0,13108 -0,12705 -0,12729 -0,12817 3,10 -0,12310 -0,12499 -0,12687 -0,12564 -0,12577 -0,12591 3,50 -0,11656 -0,11716 -0,11722 -0,11721 -0,11721 -0,11721 4,00 -0,10712 -0,10725 -0,10725 -0,10725 -0,10725 -0,10725 2,00 -0,14420 -0,14434 -0,14434 -0,14434 -0,14434 -0,14434 2,50 -0,13010 -0,13270 -0,13242 -0,13246 -0,13245 -0,13245 2,90 -0,10950 -0,13056 -0,11932 -0,12593 -0,12186 -0,12343 0 3,00 -0,10054 -0,13646 -0,10871 -0,13221 -0,11144 -0,12127 3,10 -0,10552 -0,12621 -0,11502 -0,12169 -0,11754 -0,11915 3,50 -0,10822 -0,11153 -0,11104 -0,11112 -0,11111 -0,11111 4,00 -0,10153 -0,10208 -0,10206 -0,10206 -0,10206 -0,10206

Table 2.1: Convergence of the Green function for gravitational potential . The Gν situation corresponds to a ring located in the equatorial plane on radius x0 = 3. N denotes the number of terms in the truncated expansion.

43 N Exact θ x 10 20 30 40 50 (see (2.96)) 2,00 -0,43591 -0,43605 -0,43605 -0,43605 -0,43605 -0,43605 2,50 -0,50673 -0,51175 -0,51218 -0,51222 -0,51223 -0,51223 2,90 -0,63084 -0,68904 -0,71424 -0,72701 -0,73402 -0,74433 π 2 3,00 -0,67968 -0,78323 -0,84794 -0,89513 -0,93229 3,10 -0,59006 -0,64596 -0,67049 -0,68309 -0,69010 -0,70079−∞ 3,50 -0,37462 -0,38087 -0,38162 -0,38173 -0,38174 -0,38175 4,00 -0,24485 -0,24550 -0,24551 -0,24551 -0,24551 -0,24551 2,00 -0,37082 -0,37087 -0,37087 -0,37087 -0,37087 -0,37087 2,50 -0,33726 -0,33758 -0,33779 -0,33781 -0,33782 -0,33781 2,90 -0,29896 -0,29065 -0,29373 -0,29732 -0,29866 -0,29767 π 3 3,00 -0,29039 -0,27174 -0,27375 -0,28423 -0,29251 -0,28643 3,10 -0,27625 -0,26820 -0,27111 -0,27462 -0,27598 -0,27498 3,50 -0,22901 -0,22911 -0,22943 -0,22948 -0,22948 -0,22948 4,00 -0,17923 -0,17936 -0,17938 -0,17938 -0,17938 -0,17938 2,00 -0,33785 -0,33783 -0,33783 -0,33783 -0,33783 -0,33783 2,50 -0,29105 -0,29114 -0,29139 -0,29139 -0,29139 -0,29139 2,90 -0,24557 -0,24287 -0,25224 -0,25304 -0,25057 -0,25097 π 4 3,00 -0,23163 -0,22347 -0,24540 -0,25018 -0,23813 -0,24098 3,10 -0,22598 -0,22333 -0,23241 -0,23322 -0,23075 -0,23116 3,50 -0,19376 -0,19385 -0,19426 -0,19426 -0,19425 -0,19425 4,00 -0,15499 -0,15498 -0,15500 -0,15500 -0,15500 -0,15500 2,00 -0,31638 -0,31633 -0,31633 -0,31633 -0,31633 -0,31633 2,50 -0,26684 -0,26508 -0,26545 -0,26543 -0,26542 -0,26543 2,90 -0,23803 -0,21449 -0,22923 -0,22750 -0,22750 -0,22613 π 6 3,00 -0,23586 -0,19140 -0,22626 -0,22328 -0,20593 -0,21686 3,10 -0,21926 -0,19660 -0,21091 -0,20924 -0,20616 -0,20787 3,50 -0,17658 -0,17434 -0,17494 -0,17490 -0,17489 -0,17490 4,00 -0,14080 -0,14058 -0,14060 -0,14060 -0,14060 -0,14060 2,00 -0,29876 -0,30073 -0,30070 -0,30070 -0,30070 -0,30070 2,50 -0,20835 -0,25531 -0,24665 -0,24810 -0,24787 -0,24790 2,90 0,04811 -0,45000 -0,00854 -0,37008 -0,08545 -0,20972 0 3,00 0,19023 -0,72430 0,42764 -0,91957 0,59767 -0,20094 3,10 0,05373 -0,42505 0,00482 -0,35188 -0,06733 -0,19249 3,50 -0,11763 -0,17360 -0,15918 -0,16257 -0,16180 -0,16194 4,00 -0,12397 -0,13108 -0,13062 -0,13065 -0,13064 -0,13064

Table 2.2: Convergence of the Green function for the dragging angular velocity . The situation corresponds to a ring located in the equatorial plane on radius Gω x0 = 3. N denotes the number of terms in the truncated expansion.

44 where f(x, x0, θ) denotes some suitable function independent of j. Except at the radius of the source (x = x0), the series converge very well (exponentially fast). 6 The problem occurs at the radius of the source (x = x0 x< = x>). The exponential convergence disappears (the base of the exponential⇒ part at (2.77) becomes 1) and only 1/j remains, which is not sufficient to ensure convergence. A bit more precise (and lengthy) analysis shows that the actual divergence is only bound to the axis. Off the axis, the Abel–Dirichlet criterion reveals that the series (2.57) converges conditionally (due to alternating signs of its terms). This behaviour is known from spectral methods in general as already mentioned (see Grandclément and Novak [2009]). In passing, one might think this behaviour is simply due to the singularity of the ring. It is not so. In the thin-disc case2 (i.e. with a distribution of matter described by regular Rk)

xR+ 2j + 1 ν < R (x )P (min(x, x ))Q (max(x, x )) | j| 2 kj R j R j R × xRZ − P (cos θ)P (0)dx < × j j R| 1 +j 2 2 1 x<0 + (x0)< 1 1 < − f(x, xR , xR+, θ) + O , (2.78) j2 2 − j x>0 + (x0)> 1! p − p where Rkj are given by (2.32), xR denotes the inner/outer rim of the disc, and ± x<0 and x>0 are deﬁned by

x when x xR+ xR when x xR x0 = , x0 = − − (2.79) < x otherwise ≤ > x otherwise ≤ R+ and f(x, xR , xR+, θ) denotes suitable j-independent function. Again, conver- − gence is exponentially fast outside the radii of the disc. Within these radii the convergence is only like 1/j2, i.e. a power-law behaviour like for the ring. In

2 Let us sketch the proof using νj assuming x > xR+. Other cases are analogous (up to possible split of integration interval into two). Asymptotic relation (C.16), (C.18) and (C.19) can be used to give bound to ν in the form | j| 1 1 1 π 1 νj 4 cos + j θ cos j + 1 | | → √ √ 2 2 2 2 j+ 2 × jπ sin θ x 1 x + √x2 1 − − x R+ 1 +j 2 1 2 Rkj(xR) xR + x 1 dxR < 4 2 R × xR 1 − xRZ − q − p 2 xR++√x 1 R+ − f 0(x, xR , xR+, θ) max( Rkj ) 4 1 +j dt < − | | x2 1 t 2 < j+ 1 R j x + √x2 1 2 − t q Z 2 − xR +√xR 1 − −− 1 +j 2 2 xR+ + x 1 1 R+ − < f(x, xR , xR+, θ) j2  x+√qx2 1  − −   2 where t = xR + xR 1, f(x, xR , xR+, θ) and f 0(x, xR , xR+, θ) are suitable j-independent function and j is assumed− to be suﬃciently− large. − p 45 4 4

3 3

z 2 z 2

1 1

0 1 2 3 4 5 6 0 1 2 3 4 5 6 rho rho

Figure 2.3: Linear perturbation of a Schwarzschild black-hole (having mass M = 1) by a ring of matter (located at r = 3). Left: Perturbation of gravitational potential (ν1). Right: Perturbation of dragging (ω1). contrast to the ring perturbation, the (power-law) convergence is absolute here, but, on the other hand, it spans over a region with ﬁnite volume (compared to just a shell r = r0 for ring). A similar analysis can be performed for linear perturbation of dragging ω1. It is quite a lengthy technical exercise, with the same results — exponential 1 2 convergence outside the radii of the source and a power-law one (like j− or j− ) where the matter is present (in the equatorial plane). The calculation can also be challenging from the numerical point of view. Qj(x) and Gj(x), when expressed in terms of elementary functions, can be split into two terms, which are orders of magnitude larger than the ﬁnal result and almost precisely cancel each other. This circumstance can be healed either by using many digits of precision or by a suitable representation of the problematic functions. Many rapidly converging recipes are known for Legendre functions of the second kind and (2.53) helps with the evaluation of Gj(x).

2.4.6 Closed form of Green functions Surprisingly, (2.57) and (2.57) can be expressed in a closed form. The main arguments and proofs are given in appendix C, (C.104). (Although crucial for our results, this technical part would not look “organic” if included in this chapter.) As an advertisement, we present here a counter-part of ﬁgure 2.2 computed using the closed-form Green functions — see ﬁgure 2.3. Improvement of convergence is clearly seen in smoothness of the contours. Formulas used to produce these plots are given in the rest of this section.

46 Green function for gravitational potential

Green function for gravitational potential ( ν), as given by (2.57) and (2.65), already has the form (C.104), so3 G

1 ∞ = (2j + 1)P (x )Q (x )P (cos θ)P (cos θ0) = (2.80) Gν −2 j < j > j j j=0 X cos(θ+θ0) dξ = , 2 2 2 − π x + (x0) + ξ 1 2ξxx0 [cos (θ θ0) ξ][ξ cos (θ + θ0)] cos(Zθ θ ) − − − − − − 0 p p where the hypergeometric function is expressed in terms of elementary functions and simpliﬁcation was done using assumption x, x0 > 1. The notation of x<> is in accordance with (2.67) and in the integral it has been dropped due to its symmetry (with respect to the exchange of x<>). Further calculation reveals K(k) ν = , (2.81) G − 2 π xx cos θ cos θ )2 (x2 1)((x )2 1) sin θ sin θ 0 − 0 − − 0 − − 0 r where p

2 2 2 4 (x 1)((x0) 1) sin θ sin θ0 k = − − 2 . (2.82) (xx cos θ cospθ )2 (x2 1)((x )2 1) sin θ sin θ 0 − 0 − − 0 − − 0 These expressions simplify considerablyp when B = 1. Let us denote the corresponding coordinates, respectively spherical and cylindrical, by (t, φ, r,˜ θ˜) and (t, φ, ρ,˜ z˜). In accordance with (1.30), the transformation reads M 2 M 2 ρ˜ = ρ 1 , z˜ = z 1 + , (2.83) − 4(ρ2 + z2) 4(ρ2 + z2) and the Green function (2.81) is given by MK(k) ν = . (2.84) G −π (˜ρ +ρ ˜ )2 + (˜z z˜ )2 0 − 0 Analogously, (2.82) takes the formp 4˜ρρ˜ k2 = 0 . (2.85) (˜ρ +ρ ˜ )2 + (˜z z˜ )2 0 − 0 In fact, such a simple form is not surprising. The result is actually well known from the Newtonian theory. Given B = 1 and static background (ω0 = 0), equation (2.6) becomes a Laplace equation. Its Green function for point charge is well known (1/(4π ~r ~r0 )). Therefore, the Green function adapted to axially symmetric source (i.e.| ring)− | reads

1 d~r0 1 ρ˜0 0 = | | = dφ = ν 2 2 2 G −4π ~r ~r0 −4π ρ˜ + (˜ρ0) + (˜z z˜0) 2˜ρρ˜0 cos φ ringZ | − | Z − −

3 p Assuming cos (θ θ0) cos (θ + θ ). Otherwise integration bounds have to be interchanged. − ≥ 0

47 ρ˜ = 0 K(k) . (2.86) −π (˜ρ +ρ ˜ )2 + (˜z z˜ )2 0 − 0 It follows thatp M M d~r0 ν = ν0 = | | . (2.87) G ρ˜0 G − ρ˜0 ~r ~r0 ringZ | − | The difference in normalization is due to the slightly different normalization factor at the delta functions on the r.h. side of the field equation.

Green function for dragging At first glance, no such simple way exists in the case of dragging, since the corresponding fundamental system is given in terms of the Gauss hypergeometric functions. Curiously enough, they can be expressed using the Gegenbauer function according to 12( 1)k F (x) = − Oˆ C(3/2)(x), (2.88) k k(k + 1)2(k + 2)2(k + 3) x k ( 1)k G (x) = − Oˆ D(3/2)(x), (2.89) k 12 x k where d2 d d Oˆ (x 1) (x 1) = (x 1)2 (2.90) x ≡ − dx2 − dx − dx + and k 1 for the purpose of (2.88) (k Z0 in the case of (2.89)). The formula (2.88)≥ can be proven by a straightforward∈ calculation. The formula (2.89) is a bit more tricky. First, it solves the homogenous equation (2.31). It is also clear that it vanishes at infinity. Therefore, (2.89) has to be proportional to the Gk(x) given by (2.48) (the second solution Fk(x) does not vanish asymptotically). Finally, the multiplicative constant can be fixed by comparing the leading terms at x . Combining this knowledge with (2.58) and (2.66) brings the desired alternative→ ∞ formulation of the Green function for dragging,

∞ 2j + 3 2 (3/2) (3/2) = sin θ0F (x )G (x )C (cos θ)C (cos θ0) = Gω − 2(j + 1)(j + 2) j < j > j j j=0 X 3 d ∞ j + = ∆ Oˆ (x 1)2 2 D(3/2)(x ) − x> dx < − j(j + 1)3(j + 2)3(j + 3) j > × < j=1 X d (3/2) (3/2) (3/2) Cj (x<)Cj (cos θ)Cj (cos θ0) = ×dx< 3 ∞ d 2 j + 2 = ∆˜ Oˆ (x + 1)− − x> dx < (j + 1)3(j + 2)3 × < j=0 X x< 2 (3/2) (3/2) (3/2) (3/2) (ζ 1)D (x )C (ζ)C (cos θ)C (cos θ0)dζ = × − j > j j j Z1 x< 2 ˜ ˆ d ζ 1 = ∆ Ox> − 2 − dx< 2(x< + 1) × Z1

48 ∞ 2j + 3 (3/2) (3/2) (3/2) (3/2) D (x )C (ζ)C (cos θ)C (cos θ0)dζ = × (j + 1)3(j + 2)3 j > j j j j=0 X x< d ζ2 1 = ∆˜ Oˆ x> 2 2 − 2 2 − dx< 2π(x< + 1) (x> 1) sin θ sin θ0 × Z1 −

cos(θ θ0) − [cos(θ θ ) ξ][ξ cos(θ + θ )] − 0 − − 0 dξdζ, (2.91) 2 2 2 × x> pζξ + (x> ζξ) (1 ζ )(1 ξ ) cos(Zθ+θ0) − − − − − p where we have used the relations (C.54) and (C.54),

3 (3/2) (3/2) 1 F x G x θ θ0 , ∆ 0( <) 0( >)C0 (cos )C0 (cos ) = 3 (2.92) ≡ −4 −4(x> + 1) x< d 3 ˜ ˆ ζ2 (3/2) x ∆ ∆ + Ox> 2 ( 1)D0 ( >) ≡ dx< 16(x< + 1) − × Z1 (3/2) (3/2) (3/2) 2 ζ θ θ0 ζ , C0 ( )C0 (cos )C0 (cos )d = 3 3 (2.93) × −(x + 1) (x0 + 1) and it has been assumed that cos(θ θ0) cos(θ + θ0). If the latter is not true, these two terms has to be interchanged.− ≥ (Because of the symmetry of the final result (2.96), we do not repeat the proof for the second case.) Note also that symmetry with respect to the exchange x x0 allowed us to drop the x ↔ <> variables as in the case of ν. Above, we deliberatelyG exchanged the order of integration, derivation and summation. When dealing with infinite series, this could be an issue, but luckily, thanks to the fast enough convergence, one can do so. Actually, the convergence is exponential almost everywhere — up to an arbitrarily small neighbourhood of the radius r = r0 and Foubini’s theorem allows us to do so (up to an infinitely small vicinity of the mentioned radius). Let us remind that an example of the Green function for dragging has been given in figure 2.3. When treating more than one-dimensional sources, the Green functions have to be integrated over their volume. The first integration of (2.91) brings logarithms or elliptic functions. Both of them represent quite a serious problem for a ˆ subsequent quadrature. Applying the operator Ox> first and integrating over ζ later avoids such difficulties, because

x< d (ζ2 1) Oˆ (2.94) x> 2 2 − 2 2 dx< 2π(x< + 1) (x> 1) sin θ sin θ0 × Z1 − dζ 5 = k(x0, θ0) I˜k(x0, ξ), × x ζξ + (x ζξ)2 (1 ζ2)(1 ξ2) P > − > − − − − Xk=0 h p i where the functions and I˜ are deﬁned by Pk k 2 (x0, θ0) , 0 3 − 3 2 2 P ≡ π(x + 1) (x0 + 1) sin θ sin θ0

49 (x 1)(x0 1) (x0, θ0) , 1 −3 − 3 − 2 2 P ≡ π(x + 1) (x0 + 1) sin θ sin θ0 2xx0 + (x 1)(x0 1) (x0, θ0) , 2 3 − 3 2− 2 P ≡ π(x + 1) (x0 + 1) sin θ sin θ0 2 (x0, θ0) , 3 3 − 3 2 2 P ≡ π(x + 1) (x0 + 1) sin θ sin θ0 (x0 1)(x 1)(x0 + x) (x0, θ0) , 4 −3 − 3 2 2 P ≡ π(x + 1) (x0 + 1) sin θ sin θ0 (x 1)(x0 1) 5(x0, θ0) − − 2 2 , P ≡ 2π(x + 1)(x0 + 1) sin θ sin θ0

I˜ (x0, ξ) 1, 0 ≡ 1 I˜ (x0, ξ) , 1 ≡ ξ + 1 1 I˜2(x0, ξ) , ≡ x2 + (x )2 + ξ2 1 2xx ξ 0 − − 0 ξ I˜3(x0, ξ) p , ≡ x2 + (x )2 + ξ2 1 2xx ξ 0 − − 0 1 I˜4(x0, ξ) p , ≡ (ξ + 1) x2 + (x )2 + ξ2 1 2xx ξ 0 − − 0 1 I˜5(x0, ξ) p . (2.95) ≡ (x2 + (x )2 + ξ2 1 2xx ξ)3/2 0 − − 0 Finally, it can be concluded that

cos(θ θ ) − 0 ω (x, θ, x0, θ0) = ∆(˜ x, x0) [cos(θ θ ) ξ][ξ cos(θ + θ )] (2.96) G − − 0 − − 0 × Z cos(θ+θ0) p 5 5 (x0, θ0)I˜ (x0, ξ)dξ = ∆(x,˜ x ) (x0, θ0)I (x0, θ0), × Pk k 0 − Pk k Xk=0 Xk=0 where

π 2 2 I (x0, θ0) = sin θ sin θ0, 0 2 I (x0, θ0) = π(1 cos θ )(1 cos θ0 ), 1 − | | − | | a31 a42 a43 I (x0, θ0) = a K(k) a E(k) + a 1 + Π , k , 2 a − 41 − 42 41 a −a r 42 31 31 a41 a31 a42 I (x0, θ0) = (s + s + 2a )K (k) + (s 3s )E (k) + 3 4 a − 21 43 21 a 43 − 21 r 42 41 a2 a2 + 2s2 2s s a + 21 − 43 21 − 21 43 Π 43 , k , a −a 31 31 2a41 a31 a43 I4(x0, θ0) = K(k) + Π , k √a a −a + 1 −a − 31 42 1 31 a + 1 (a + 1)a 3 Π 1 43 , k , − a + 1 −(a + 1)a 1 4 31 50 √a31a42 2 I5(x0, θ0) = 2 (4 2k )K (k) 4E (k) , (2.97) a21 − − with

2 2 2 2 a xx0 + (x 1)((x ) 1), a xx0 (x 1)((x ) 1), 1 ≡ − 0 − 2 ≡ − − 0 − a cos(θ θ0), a cos(θ + θ0) (2.98) 3 ≡ −p 4 ≡ p and a a a a a , s a + a , k 21 43 . (2.99) rs ≡ s − r rs ≡ s r ≡ a a r 31 42 Note that there is no ambiguity in the notation: the moduli deﬁned by the above equation and by (2.82) coincide.

Normal derivatives of metric function on the horizon

Let us return to already proposed vanishing normal derivative of the dragging at the horizon. Given knowledge of Green function we can, ex post, justify such statement. Assuming f(x) being general function of dimensionless radius (2.27) its normal derivative at the horizon reads ∂x 1 M f,r M = f(x),x = f,x = 0. (2.100) r= 2 2 | ∂r r= M M − 4r r= M 2 2

Therefore long as f,x does not diverge at the horizon definition of x ensures vanishing normal derivative there. GF near the horizon are described by polynomial for each multipole moment. Their derivatives are clearly finite — they are polynomials of x as well. So GF has to have finite derivative (with respect to x) at the horizon and subsequently the x derivatives of perturbations of ν and ω must be finite there. In the case of dragging additional the term proportional to Jk has not been examined yet, see (2.70). Luckily is does not spoil anything (its r-derivative vanishes at the horizon as well).

2.5 Disc solution

The Green functions can be used to calculate a general solution of (2.9) and (2.10). The integrals contain non-trivial combination of elliptic functions, so they are in general not easily calculated, but there exists a special case when the metric functions can be expressed using complete elliptic integrals and elementary functions — the case of a thin disc with constant Newtonian density.

2.5.1 Gravitational potential There is no evident way how to construct gravitational potential of a thin disc using Green function (2.81). However, one can proceed in analogy with the Newtonian case (see (2.86)). The Newtonian gravitational potential of a disc with constant surface mass density was found by Lass and Blitzer [1983]. Using

51 their notation (i.e. V (a) denoting gravitational potential of a disc lying in the equatorial plane and extending between origin and the radius x = a), it reads

2 2 V (a) = π z˜ H (˜ra ρ˜) R z˜ + (˜ρ +r ã) E(k(a)) + 2 2 R| | − − z˜ + (˜ρ +r ã) n r˜ ρ˜ 4˜r ρ˜ + (˜r2 ρ˜2)K (k(a))p +z ˜2 a − Π a , k(a) = (2.101) a − r˜ +ρ ˜ (˜r +ρ ˜)2 a a = πM x cos θ H(a2 cos2 θ x2 sin2 θ) R | | − − − M 2 2 2 2 R a31(a)a42(a)E (k(a))+(a cos θ x sin θ)K (k(a)) + − a31(a)a42(a)" − − p(a a x2 cos2 θ)x2 cos2 θ 4 (a2 1)(x2 1) sin θ + 41 32 Π , k(a) , 2 − 2 2 2 − 2 − 2 a cos θ x sin θ a42a31 x cos θ − − p − !# 2 wherer ã = M√a 1 denotes the radius of the ring in coordinatesρ ˜ andz ˜ which correspond to−B = 1 (see (2.83) for transformation), is an arbitrary parameter scaling the mass density (it is proportional to the surfaceR mass density in the Newtonian case) and H (x) stands for the Heaviside function. Further, the symbols ar(x0), ars(x0) and k(x0) are defined analogously to (2.98) and (2.99) (with θ0 = π/2),

2 2 2 2 a (x0) = xx0 + (x 1)((x ) 1), a (x0) = xx0 (x 1)((x ) 1), 1 − 0 − 2 − − 0 − a (x0) = sin θ, a (x0) = sin θ, a (x0) = a (x0) a (x0), 3 p 4 − rs s −p r 2 2 2 a31(x0) 4 sin θ (x 1)((x0) 1) k (x0) = = = 2 − − a42(x0) x2 + (x )2 1 sin θ + 2 sin θ (x2 1)((x )2 1) 0 − − p − 0 − 4˜r ρ˜ a . p = 2 2 (2.102) (˜ra +ρ ˜) +z ˜ Note that due to the existence of the inner and outer rims of the disc, it is necessary to distinguish them. Because of that and similarity of the expressions, it is useful to indicate the dependence on the disc radius x0 explicitly and to substitute there the specific values later accordingly. Constancy of the surface mass density (in the Newtonian case) makes it easy to get the potential of a disc with inner rim as well — it is sufficient to subtract the potential of two discs extending from the origin with the same surface mass density ( parameter), R ν1(x, θ) = V (xR+) V (xR ), (2.103) − − where xR once again denote the radii of the outer/inner rims of the disc. ± 2.5.2 Dragging No such simple Newtonian analogy exists for dragging (of course), so one has to directly integrate the appropriate Green function over the source. Assuming S = = const (this assumption will be clarified later), it reads 1 S xR+ π ω = x, θ, x0, dx0 (2.104) 1 SG 2 xRZ − 52 The main point leading to evaluation of the integral above is to closely examine the integral form of the Green function (2.91), to take into account that the Green function is symmetric with respect to exchange x x0 (see (2.96) and related equations), and to realize that the last expression↔ in (2.91) does even make sense if x> < x< (as long as x<> > 1, i.e. we are not interested in metric below the black-hole horizon). Altogether, this allows to exchange x<> by x, x0 without changing the result (in other words, the symmetry ensures that both substitutions (x<, x>) (x, x0) and (x<, x>) (x0, x) lead to the same result), which leads to the following:→ →

xR+ x0 d ζ2 1 ω = ∆˜ Oˆ 1 x 2 −2 2 S  − dx0 2π(x0 + 1) (x 1) sin θ× xRZ Z1 − −  sin θ  sin2 θ ξ2 − dξdζ dx0 = × x ζξ + (xp ζξ)2 (1 ζ2)(1 ξ2)  sinZ θ − − − − − −  xR+ px0 ζ2 1  = ∆d˜ x0 Oˆ x 2 −2 2 S − S 2π(x0 + 1) (x 1) sin θ × xRZ Z1 − − sin θ sin2 θ ξ2 − dξdζ × x ζξ + (xp ζξ)2 (1 ζ2)(1 ξ2) sinZ θ − − − − − − (xR xR+)(xRp + xR+ + 2) − − = −3 2 2 S (x + 1) (xR + 1) (xR+ + 1) − − sin θ 7 2 2 sin θ ξ Qj(xR+)I˜j(xR+, ξ) Qj(xR )I˜j(xR , ξ) dξ = −S − − − − sinZ θ q j=0 − X h i (xR xR+)(xR + xR+ + 2) − − = −3 2 2 S (x + 1) (xR + 1) (xR+ + 1) − − 7

[Qj(xR+)Ij(xR+) Qj(xR )Ij(xR )] , (2.105) −S − − − j=0 X where 2 1 (1 x)(x0 1) Q (x0) = ,Q (x0) = , 0 2 3 2 1 2 − −3 2 π sin θ(x + 1) (x0 + 1) 4π sin θ(x + 1) (x0 + 1) x0(1 2x) 1 Q (x0) = ,Q (x0) = , 2 2 − 3 2 3 2 3 2 π sin θ(x + 1) (x0 + 1) π sin θ(x + 1) (x0 + 1) 2 (x + x0)(x 1)(x0 1) Q (x0) = ,Q (x0) = 0, 4 2 − 3 − 2 5 4π sin θ(x + 1) (x0 + 1) (1 x) (1 x)(x0 x) Q (x0) = − ,Q (x0) = − − , (2.106) 6 4π sin2 θ(x + 1)3 7 4π sin2 θ(x + 1)3

I˜0,..., I˜5 are deﬁned by (2.95), 1 I˜ (x0, ξ) = , 6 ξ 1 − 53 1 I˜7(x0, ξ) = , (2.107) (ξ 1) x2 + (x )2 + ξ2 1 2xx ξ − 0 − − 0 p and Ik(x0) correspond to (2.97) with θ0 = π/2, i.e.

π 2 I (x0) = sin θ, 0 2 I (x0) = π(1 cos θ ), 1 − | | a31(x0) I2(x0) = a41(x0)K (k(x0)) a42(x0)E (k(x0)) + sa42(x0) {− −

a41(x0) 2 sin θ + 2xx0 Π , k(x0) , a (x ) −a (x ) 31 0 31 0 a41(x0) a31(x0) a42(x0) I (x0) = [a (x0) 3a (x0)] K (k(x0)) 6xx0 E(k(x0)) + 3 4 a (x ) 2 − 1 − a (x ) s 42 0 41 0 2 2 2 2 a21(x0) a43(x0) + 8x (x0) 2 sin θ + − Π , k(x0) , a (x ) −a (x ) 31 0 31 0 2a41(x0) a31(x0) a43(x0) I4(x0) = K(k(x0)) + Π , k(x0) a x a x −a1(x ) + 1 −a31(x ) − 31( 0) 42( 0) 0 0 a3(x0) + 1 (a1(x0) + 1)a43(x0) p Π , k(x0) , − a (x ) + 1 −(a (x ) + 1)a (x ) 1 0 4 0 31 0 a31(x0)a42(x0) 2 I (x0) = 4 2k (x0) K(k(x0)) 4E (k(x0)) , 5 4(x2 1)((x )2 1) − − p − 0 − I (x0) = π( cos θ 1), 6 | | − 2a41(x0) a31(x0) 2 sin θ I7(x0) = K(k(x0)) + Π , k(x0) + a (x )a (x ) −a1(x0) 1 −a31(x0) 31 0 42 0 − 1 sin θ 2 sin θ[a1(x0) 1] +p − Π − , k(x0) . (2.108) a (x ) 1 a (x )(sin θ + 1) 1 0 − 31 0 2.5.3 Physical properties Physical properties of the obtained disc are discussed in more detail in paper C´ıˇzekˇ and Semer´ak[2017]. Using the normal one-sided derivatives of the perturbation (2.103) and (2.105) at the equatorial plane, one has

M 2 2πRM 1 + v2 ν = 2πR 1 + = 0 , (2.109) 1,z|z=0+ 4r2 r v2 0 S x 1 (2r M)2 ω = − = 8SM 2r − (2.110) 1,z|z=0+ −2r (x + 1)3 − (2r + M)6 on the surface of the disc (elsewhere normal derivatives vanish due to the reﬂection symmetry). Unperturbed orbital velocity v0 is given by (1.55). Let us stress that all the relations given below are only valid on the surface of the disc, otherwise mass densities are zero. Considering a two-component dust disc, i.e. energy-momentum tensor in the form (1.15), normal derivatives in the equatorial plane are linked to the dust

54 densities by relations (1.40)

2 2 2 1 + v0 4r + M ν1,z = 2π(σ+ + σ ) = (σ+ + σ ) , (2.111) |z=0+ − 1 v2 − 4r2 8rM + M 2 − 0 − Ω0 ω1,z = 8π(σ+ σ ) = |z=0+ − − − 1 v2 − 0 8r√Mr (2r M)2 = 8π(σ+ σ ) − , (2.112) − − − 4r2 8Mr + M 2 (2r + M)3 − where Ω0 denotes the angular velocity corresponding to v0. Combining these two relations yields the surface mass densities

2 2 3 4r 8Mr + M (Mr) 2 σ R S . = − 2 3 (2.113) ± 8r " ± 2π(2r + M) # For the single-component interpretation, the parameters are obtained using (1.17),

2 2 σ+ + σ σ+ + σ 4σ+σ v0 σ = + − + − + − , (2.114) 2 s 2 (1 v2)2 − 0 2 2 σ+ + σ σ+ + σ 4σ+σ v0 P = − + − + − , (2.115) − 2 s 2 (1 v2)2 − 0 σv2 P v2 = 0 − . (2.116) σ P v2 − 0 The sign of v depends on ratio of the surface mass-densities σ : for σ+ > σ it ± − is positive, otherwise negative. An example of the disc parameters is given in ﬁgure 2.4. The rim(s) are not present there explicitly, because introducing them would only restrict the domain of the functions accordingly. Also, only the part where the circular orbital velocity makes sense (0 v 1) is plotted. Note that for the lower-radius-case the two-component interpretation≤ ≤ has no sense since the obtained mass density is negative for one of the streams (whereas the single-component interpretation is still possible). Total mass and angular momentum can be obtained using the asymptotic behaviour of the metric or by Komar’s integrals. Both results coincide so only the latter option is discussed here. The total mass and angular momentum are given by

1 α;β = lim η dSαβ = (2.117) π Σ M 8 →∞ IΣ

1 α;β α β α = η dSαβ + (2Tβ η T η )dΣα = H + D, 8π horizon − M M I discZ

1 α;β = lim ξ dSαβ = (2.118) Σ J −16 →∞ IΣ 1 1 = ξα;βdΣ + (2T αξβ T ξα)dΣ = + , −16 αβ 2 β − α JH JD horizonI discZ

55 0.1 0.1

0.08 0.08

0.06 0.06

0.04 0.04

0.02 0.02

0 0 2 4r 6 8 10 2 4r 6 8 10 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 2 4r 6 8 10 2 4r 6 8 10

Figure 2.4: Example of parameters of the disc obtained using the closed-form formulas. The conﬁguration is described by R = 0.1, S = 60 and M = 1. Left charts describe parameters of the single-component interpretation, right ones correspond to two-component interpretation (solid lines indicate the positive- direction stream). Top Left: Two-stream surface mass densities. Bottom Left: Two-stream orbital velocities. Top Right: One-stream surface mass density (solid line) and azimuthal pressure (dashed line). Bottom Right: Bulk velocity of the matter.

56 where S is a sphere tending to spatial inﬁnity and H, D, H and D denote contributions of the black hole and of the disc.M Generally,M J each ofJ them is a function of the expansion parameter λ as well. Let us also remind that ηα and ξα stand for the Killing vector ﬁelds. First, since we are interested in a thin disc, its contribution is better expressed α using the surface energy-momentum tensor (1.9) instead of Tβ ,

= 2π (Sφ St)ρBdρ = 2π (σ + P + 2ΩSt )ρBdρ, (2.119) MD φ − t φ discZ discZ v = 2π St ρBdρ = 2π (σ + P ) ρBdρ. (2.120) JD φ 1 v2 discZ discZ − Considering the relation between the surface mass density and normal derivative across the disc (1.40), we have 1 = ν ρBdρ + O , (2.121) MD ,z|z=0+ λ2 discZ

3 3 4ν 1 = ρ B e− ω dρ + O , (2.122) JD ,z|z=0+ λ2 discZ where staticity of the background solution has been used. Finally, employing the knowledge of the linear perturbation of metric, it can be concluded that

= ν ρBdρ + O λ2 , (2.123) MD ,z|z=0+ Z disc 3 3 4ν 2 = ρ B e− ω dρ + O λ . (2.124) JD ,z|z=0+ Z disc Hence, the parameters of the disc, assuming linear order of precision, read

rR+ M 2 M 2 = 2πR 1 + 1 rdr = MD1 4r2 − 4r2 rRZ − 4 2 2 M 1 1 = πRλ rR+ rR 2 2 , (2.125) − − − 16 rR − rR+ − rR+ M 2 M 2 = S 1 dr = JD1 8 − 4r2 rRZ − M 2 M 2 1 1 = S rR+ rR , (2.126) 8 − − − 4 rR − rR+ − (2.127) where rR denotes inner/outer radius of the disc. ± It is easy to show that the black-hole contributions are described by = M + O λ2 , (2.128) MH = O λ2 (2.129) JH (see the paper C´ıˇzekandˇ Semer´ak[2017] for details).

57 3. Belinskii–Zakharov method

3.1 Unknown generating function

In order to use BZM technique to generate solution of Einstein ﬁeld equations one has to solve corresponding system of linear diﬀerential equations (1.74). Paper by de Castro and Letelier [2011] is going to be followed in this chapter. Assuming staticity of seed metric (1.8) the 2-metric g0 is diagonal. The same behaviour can 1 be expected by generating matrix Ψ0. Its non-zero components are expressed by

2 2 2F (Ψ ) = (ρ λ 2λz)e− , (3.1) 0 φφ − − (Ψ ) = e2F , (3.2) 0 tt − where F (ρ, z, λ) denotes function satisfying diﬀerential equations ∂F ∂F ∂F ∂ν ρ λ + 2λ = ρ , (3.3) ∂ρ − ∂z ∂λ ∂ρ ∂F ∂F ∂ν ρ + λ = ρ . (3.4) ∂z ∂ρ ∂z

Reader can check that introduction of term ρ2 λ2 2λz instead of general function is in accordance with (1.74). − − The seed metric is usually chosen to be Minkowski one. In this case BH at the axis of symmetry are generated. This section employs Bach–Weyl ring (see section 1.4.3) instead. Its gravitational potential ν can be expressed in the terms of Legendre functions as ∞ ν = m blP (0)ν (3.5) − l l Xl=0 where l+1 z 2 2 2 νl = ρ + z − Pl . (3.6) ρ2 + z2 ! Due to the linearity of (3.3) and (3.4) one canp search their solution in form

∞ F = m blP (0)F (3.7) − l l Xl=0 where Fl satisﬁes (3.3) and (3.4) with νl on the r.h.s. and b denotes radius of the ring. As the original paper pointed operators in equations (3.3) and (3.4) com- mutes with partial derivative with respect to z. Moreover the functions νl can be expressed by ( 1)l ∂lν ν = − 0 , (3.8) l l! ∂zl 1Also with F diﬀering by factor 2 from de Castro and Letelier [2011], in accordance with ν in the metric (1.8).

58 therefore analogous expression for general Fl can expected

( 1)l ∂lF F = − 0 ,. (3.9) l l! ∂zl where2 z + r F = . (3.10) 0 (z + r + λ)r At the end we can express F as inﬁnite sum

∞ ( 1)l ∂l z + r F = m blP − . (3.11) − l l! ∂zl (z + r + λ)r Xl=0 Castro & Letelier ended simplifying F there and begun calculating solution numerically using truncated series (3.11). However there is a way to calculate F with Bach–Weyl ring as seed solution in the closed form. At ﬁrst let us deﬁne generating functions of components of (3.7) as

2 2 ∞ ( 1)lξl ∂l z ξ + ρ + (z ξ) − (ξ) = l! ∂zl F0(ρ, z, λ) = − − F l=0 z ξ + ρ2 + (z pξ)2 + λ ρ2 + (z ξ)2 P − − − h ρ2 zλp+ ξλ λ ρ2 + (zi pξ)2 = − − − , (3.12) (ρ2 λ2 2zλ + 2ξλ) ρ2 + (z ξ)2 − − p − ∞ l l m (ξ) = m b Pl(0)ξ = . p (3.13) 2 G − l=0 − 1 + (bξ) P In the ﬁrst equation the fact that thep operator is just a shift operator3 have been used. The second case is well known a generating function of Legendre polynomials (f.e. see [DLMF, (18.12.4)]). Moreover orthogonality relation

2π ikφ ilφ k l dt e e− dφ = i t t− = 2πδ , (3.14) − t kl Z0 I C where t = eiφ and is a complex integration path — circle in positive direction around origin (seeC ﬁgure 3.1). All together lead to

∞ ( 1)l ∂l z + r F = m blP − = − l l! ∂zl (z + r + λ)r Xl=0 ∞ l l l ilφ l ilφ l ( 1) ∂ z + r = b P e− N − e N − = l l! ∂zl (z + r + λ)r l=0 X 2π ∞ ∞ k k m l ilφ l ikφ k ( 1) ∂ z + r b e− N − e N φ = Pl − k d = −8π ( )( k! ∂z (z + r + λ)r) Z0 Xl=0 Xk=0 2Keep in mind that r = ρ2 + z2, therefore it must be taken into account when calculating derivatives. p 3In the original article it was created in such way.

59 Figure 3.1: Integration path and singularities employed during calculation of F . Singularities and cuts are given by bold lines (circles). Exact position of tsg with respect to integration path is a priori not known (whether it is inside/outside of the contour). C

2π 1 e iφ i 1 dt = − (Neiφ)dφ = (t) , (3.15) 2π G N F −2π G t F t Z0 IC where N is suitable normalization constant4 and t = Neiφ. In these steps we have deliberately changed order of summation and integration. It could be proven such steps are justified from mathematical point of view5 however it is easier to omit such technicalities and check that the final result (3.27). Before continuing further let us spend some time on singularities of generating functions and . In the latter case the situation is simple. There is a complex cut betweenF t = ibG and t = ib. It is convenient to choose it in such way, that it lies on the imaginary axis. − The former case is a bit more complicated. ρ2 + (z t)2 introduces complex cut between t = z + iρ and infinity and another between−t = z iρ and infinity. p − 4It is necessary to have N > b for (3.13) to converge. 5 The idea of the formal proof is based on Fubini’s theorem and uniqueness of function of complex variables. Radius of convergence (centered around ξ = 0) of (3.13) is b and in the case of (3.12) it reads min ρ2 + z2, t (based on analysis of singularities given below). | sg| 2 2 Assuming domain where b <p ρ + z andb < tsg (i.e. the point (ρ, z) is far enough) power series converges exponentially there and Fubini’s| theorem| can be used to exchange summation p and integration order (as well as splitting of infinite series into two distinct ones in (3.15)). Uniqueness of functions of complex variable can be used to extend solution from domain given above. The restrictions on convergence are, in fact, given by introducing expansions. Therefore they pose (up to the cut discussed later in this chapter) only technical difficulties coming from chosen procedure.

60 Assuming that z and ρ are large enough6 (N 2 < ρ2 + z2) the integration path avoids them. Another singularity is given by z t+λ+ ρ2 + (z t)2 = 0. When λ = 0 then ρ = 0 and t = z and so if z >− N it lies outside integration− path. p When λ = 0 there is (possible — the condition| | is necessary but not sufficient) divergence6 at the point ρ2 2λz λ2 t = − − . (3.16) sg − 2λ It describes “technical” singularity introduced artificially during calculation. λ is going to be such that this singularity does not contribute, see (3.19). Finally t exist in denominator alone so t = 0 is singularity of the integrand of (3.15) as well. Illustration of their positions is given by figure 3.1. Integrand in (3.15) is meromorphic (up to the cuts), therefore integration path can be deformed to 0 = + [ sg]( sg is present in the integration ↓ − ↑ pathC when t is insideC circleC ∪C) without∪C ∪C changing∪C C the result. By defining sg C 1 i (t− ) (t) I = G F dt, (3.17) X −2π t ZCX where X +, , , , sg we can write ∈ { − ↑ ↓ } F = I+ + I + I + I [+Isg] . (3.18) − ↑ ↓ First one 2π iφ i 1 tsg + e iφ Isg = lim − F ie dφ = 0+ π t eiφ t eiφ → 2 G sg + sg + Z0 2π 1 (tsg− ) iφ iφ = G lim tsg + e e dφ = 2πt 0+ F sg → Z 0 2π mλsgn (ρ2 2λz λ2) 1 = − − lim (λ λ ) − + O (1) eiφdφ = 2 2 2 2 2 0+ λeiφ π (ρ 2λz λ ) + 4b λ → − | | 2 − − Z0 pm sgn (ρ2 2λz λ2) = − − ( λ λ). (3.19) (ρ2 2λz λ2)2 + 4b2λ2 | | − − − where itp is assumed λ R. Given λ = µ(k) defined by (1.80) stronger restriction holds — λ 0 — and∈I must vanish. Analogously ≥ sg π i ib + eiφ 1 I = lim − F ± ± dφ = ↑↓ 0+ π ib eiφ ib eiφ → 2 ( + ) G + Z0 ± π ( ib) 1 = F ± lim ± dφ = πb 0+ ib eiφ ∓ 2 → G + Z0 π ( ib) dφ = F ± lim = πb 0+ 2 ∓ 2 → b Z 1 + 2 0 [ (ib+eiφ)] r ± 6The expansion (3.5) converges when r > b, i.e. it is well behaved at infinity.

61 π ( ib) bdφ = F ± lim = πb 0+ iφ 2 ∓ 2 → 2ibe + O ( ) Z0 − ( ib) p2 2i = F ± lim √ − + O √ = 0, (3.20) ∓ 2π 0+ √ 2ib → − where + corresponds to and to . In both cases integrals do not contribute. The last one read ↑ − ↓ ib 1 ± ± 1 ± i (t) (t− ) b (ibu ) 1 I = lim − F G dt = lim F ± du = ± 0 π t 0 π ibu ibu → 2 → 2 G ibZ Z1 ± ± ∓ ± ∓ +1 b (ibu) m = lim ± F + O () − du = 0 π ibu 2 → 2 b Z1 1 + ibu − ± +1 q bm (ibu) b2u2 2ib + 2 = lim ∓ F + O () − ± du = 0 π ibu b2 u2 ibu 2 → 2 s (1 ) 2 + Z1 − ± − +1 bm (ibu) u 2iub = lim ∓ F + O () | | 1 du = 0 π ibu 2 u2 u2 O 2 → 2 √1 u s− ± (1 ) + ( ) Z1 − − − +1 +1 bm (ibu) iu m (ibu) = ∓ F ± du = − F du (3.21) 2π ibu √1 u2 2π √1 u2 Z1 − Z1 − − − where t = ibu. u2( 1 ) = iu + O () have been used to simplify result. Lengthy calculation reveals− ± ± p +1 (ibu) k F du = [(tsg(b ρ) zb) C1 + 1] K (k) + X0(tsg) √1 u2 2√ρb − − Z1 − − √ρb(1 k2) C C2 + − 1 C + 2ρ b2 + t2 Π 2 , k (3.22), k C 2 sg −C2 3 3 where K (() k), E (() k) and Π (n, k) denotes complete elliptic integrals of the ﬁrst, second and third kind,

4ρb k = , (3.23) s(ρ + b)2 + z2 sgn (λ) (z t )2 + ρ2 C = − sg , (3.24) 1 C − p 2 C = t2 (b ρ) 2zbt + b(z2 bρ + ρ2), (3.25) 2 sg − − sg − C = t2 z + t (ρ2 + z2 b2) + b2z, (3.26) 3 − sg sg − and X0(tsg) is suitable function of tsg alone. Finally m √ρb(1 k2) C C2 F = ν(ρ, z) − 1 C + 2ρ b2 + t2 Π 2 , k + − π k C 2 sg −C2 3 3 62 ±0.5

±1

±1.5

±2

±2.5

±3 3 2 1 3 2.5 z 0 2 ±1 1.5 ±2 1 rho 0.5 ±3 0

Figure 3.2: F (ρ, z). This particular example was calculated using parameters m = π, b = 1 and tsg = 1. Choice of X(tsg) is given by (3.28).

k + [(t (b ρ) zb) C 1] K (k) + X(t ) , (3.27) 2√ρb sg − − 1 − sg where X(tsg) stands arbitrary piecewise continuous function of tsg. It is convenient to choose π X(t ) = 2 sgn (λz) 1 + sg 2 2 − tsg + b − n h +sgnp ρ2 + z2 b2 2zt + (ρ2 + z2 b2)2 + 4b2z2 + − − sg − + sgn ρ2 + z2 b2 2zt p (ρ2 + z2 b2)2 + 4b2z2 (3.28) − − sg − − p io in accordance with (3.22). It fixes position of the cut (it extends in the z = 0 plane from ring to infinity) and ensures there is only one (see figure 3.2). Check that solution (3.27) satisfies (3.3) and (3.4) can be performed by straightforward calculation. Also F ν when λ 0 (i.e. tsg ) as required by boundary condition (1.75). Because→ C →1 , C t→2 (b ∞ ρ), C zt2 1 tsg(b ρ) 2 sg 3 sg ≈ − ≈ − ≈ − the coefficients in (3.27) behaves like7 C ρ b 2 − , C3 → z [t (b ρ) zb] C 1 0, sg − − 1 − → 7 On the axis (ρ = 0) the situation is a bit different. Limits of mentioned expressions behaves differently — C 2 0, C3 → π [tsg(b ρ) zb] C1 1 K(k) , { − − − } → −2 ρ2 + z2 2 2 √ρb(1 k ) C1 2 2 C2 pπ − C2 + 2ρ b + t Π , k , k C sg −C2 → 2 2 3 3 4 ρ + z 1 1 p . (3.29) 2 2 → ρ2 + z2 tsg + b q p This ensure validity on one half-axis. However such “wrong” limit affects only infinitesimally small part of the domain and it will not spoil the solution as whole.

63 2 √ρb(1 k ) C1 2 2 − C2 + 2ρ b + tsg 0, k C3 → 1 0 (3.30) 2 2 tsg + b → proving above mentioned limit ofF . p

General Weyl metric

It should be noted that procedure used earlier to ﬁnd spectral function F can be used more generally. Given Weyl metric its gravitational potential can be expanded into multipole moments analogously to (3.5) (νl being deﬁned by (3.6))

∞ ν = βlνl (3.31) Xl=0 using suitable coeﬃcients βl. With their knowledge generating function can be, possibly, found8 ∞ (ξ) = β ξl. (3.32) G l Xl=0 At the end, if we are lucky enough, spectral function F is given by complex integral (3.15).

3.2 Two–soliton solutions

3.2.1 Generated metric The focus is given to two–soliton solutions due to their simplicity. General formulas for new metric have been already introduced in section 1.5.2 — unphysical metric is described by (1.78) and (1.79), physical one is given by (1.85) and (1.86). (Semi-)Explicit expressions for unphysical metric components. (1.80)–(1.82) with adopted assumptions given by this chapter reads

(k) (k) 2F (λ ,ρ,z) m (ρ, z) = C e− k , (3.33) t − t C(k) (k) φ 2F (λk,ρ,z) mφ (ρ, z) = e , (3.34) λk(ρ, z) 2 (k) (l) ρ 2[F (λ ,ρ,z)+F (λ ,ρ,z) ν(ρ,z)] Γ (ρ, z) = C C e k l − (3.35) kl φ φ λ (ρ, z)λ (ρ, z) − k l (k) (l) 2[ F (λ ,ρ,z) F (λ ,ρ,z)+ν(ρ,z)] 2 C C e − k − l / ρ + λ (ρ, z)λ (ρ, z) ., − t t k l o 1 Πkl(ρ, z) = [Γkl(ρ, z)]− and (1.78) takes form

2 C(k)C(l)Π (ρ, z) 2ν(ρ,z) 4ν(ρ,z) t t kl 2[F (λk,ρ,z)+F (λl,ρ,z)] gtt = e e e− , (3.36) − − λk(ρ, z)λl(ρ, z) k,lX=1 8The problem of ﬁnding is much easier since it is not function of ρ and z. G

64 (k) (l) 2 Π (ρ, z) C C 2 kl t φ 2[F (λl,ρ,z) F (λk,ρ,z)] gtφ = ρ e − , (3.37) − λk(ρ, z)λl(ρ, z) λl(ρ, z) k,lX=1 2 2 2ν(ρ,z) 4 4ν(ρ,z) Πkl(ρ, z) gφφ = ρ e− ρ e− − λk(ρ, z)λl(ρ, z) × k,lX=1 (k) (l) Cφ Cφ e2[F (λk,ρ,z)+F (λl,ρ,z)]. (3.38) ×λk(ρ, z)λl(ρ, z) The last unknown function of the solution, f, formula (1.79) reads

2 2 2ζ 2ν 2 λk(ρ, z) f C e − ρ ρ, z , = 2 2 2 det Γkl( ) (3.39) ρ + λk(ρ, z)! kY=1 where ζ and ν denote metric function of seed solution (see (1.58)). Also physical solution can be written using algebraic expression of λ. Nor- malization needed to get physical 2–metric can be obtained from (1.84) ρ6 . det(g) = 2 (3.40) −[λ1(ρ, z)λ2(ρ, z)] Moreover (1.86)

3 ph ph 2ζ 2ν (λ1(ρ, z)λ2(ρ, z)) det Γkl(ρ, z) f = C e − (3.41) 2 [λ (ρ, z) λ (ρ, z)]2 ρ2 2 − 1 gives the remaining physical metric function.

3.2.2 Asymptotic behaviour at spatial inﬁnity General analysis of asymptotic behaviour is already given by Belinskii and Verda- guer [2004]. For detailed analysis refer to this book. Let us begin with asymptotic behaviour (as r ) of spectral function F because the other functions are linked to it. Fixing→ piecewise ∞ continuous part by (3.28) leads to m[1 sgn (λ) sgn (cos θ)] m[1 + sgn (λ) cos θ] 1 F (ρ, z, tsg) − + O . → − 2 t2 + b2 − 2r r2 sg (3.42) Other functions follows. Fromp (1.84) we obtain

2 2 2 4 θ 2 1 r θ − α O det(g) = sin tan 1 k + 2 (3.43) − 2 " − r r # Xk=1 and assuming9 α = α = α 1 − 2 (1) (2) (1) (2) α Ct C + C Ct ph φ φ 1 1 gtt = 1 + 2 m + + O , (3.44) −  C(1)C(2) C(1)C(2)  r r2 t φ − φ t   9Such assumption is physical corresponding to position of solitons aligned with the ring.

65 α gph = 2 (3.45) tφ C(1)C(2) C(1)C(2) × t φ − φ t (1) (2) 4 (1) (2) (1) (2) (1) (2) 4 Ct Ct ∆ + Cφ Cφ Cφ Cφ Ct Ct ∆ 1 ± + − ± cos θ + O , × ∆2 ∆2 r " ± ± # (1) (2) (1) (2) α Ct Cφ + Cφ Ct gph = sin2 θr2 + sin2 θ 2m + 2 r + O (1) , (3.46) φφ  C(1)C(2) C(1)C(2)  t φ − φ t   where ∆ describes discontinuity of F (ρ, z, tsg). Exact choice of ∆ corresponds ± ± to the sign of z, ∆ to θ < π/2 and ∆+ to θ > π/2, where −

∆+ = 1, 1 when ρ < b ∆ = 2m . (3.47) − (e √α2+b2 when ρ > b In the limit r the branch ρ < b of ∆ can be ignored. → ∞ (k) (k)− Rescaling of parameters Ct and Cφ does not change the result, therefore we can choose their scaling arbitrarily — see (3.36), (3.37) and (3.38). It is (1) (2) (1) (2) convenient to put α = Cφ Ct Ct Cφ . Generated solution asymptotically approaches Kerr–NUT solution.− Comparing ﬁrst non-trivial order of expansion in 1/r one can see mass, angular momentum and NUT parameter are given by

M = m C(1)C(2) C(1)C(2), (3.48) Kerr − t φ − φ t (1) (2) 4 (1) (2) Ct Ct ∆ + Cφ Cφ a = ± , (3.49) Kerr ∆2 ± (1) (2) 4 (1) (2) Ct Ct ∆ Cφ Cφ b = ± − . (3.50) Kerr ∆2 ±

The expressions for aKerr and bKerr are ﬂawed. They (generally) change value when crossing equatorial plane. There are two ways to solve this. One is to choose F in such way that there is no discontinuity. It requires no, or at least two seed ring solutions compensating each other. This option is not going to be discussed10 here. Also special choice of constants that ensuring absence of discontinuity in the equatorial plane can be attempted. And, again, there are two possibilities — with vanishing C(k) and without. Former one (as it will be shown later) leads (1) (2) (1) (2) to static metric. Ct Ct = Cφ Cφ = 0 and so aKerr = bKerr = 0. Detailed

10There are several reasons why it is so. Mentioning most important ones (some topics are going to be discussed later in this thesis): The nature of discontinuities in the ﬁrst derivatives gives no physically reasonable solu- • tion. Also it does not change with introduction of second ring. Compensation requires the second ring to have negative mass. Preliminary calculations • showed this behaviour does not vanish after application of BZM. Aim of this section is completing some missing points of paper de Castro and Letelier • [2011]. They are interested in single ring seed metric only.

66 discussion is given by section 3.3.1. The latter approach focuses more closely on (3.45). There is no way (up to the static case) to ensure that aKerr and bKerr does not simultaneously change value in the equatorial plane. However the term proportional to bKerr is multiplied by cos θ therefore it does vanish in equatorial plane (at least in the leading order of expansion). bKerr can be discontinuous there and “only” discontinuity of normal derivatives appears. Focusing solely on aKerr from

(1) (2) 4 (1) (2) (1) (2) 4 (1) (2) Ct Ct ∆+ + Cφ Cφ Ct Ct ∆ + Cφ Cφ − 2 = 2 (3.51) ∆+ ∆ − introduces condition 4m (1) (2) (2) (1) C C C C e √α2+b2 = 0. (3.52) φ φ − t t It also ensures continuity of metric in the whole equatorial plane (see chapter

3.3.2 and equation (3.98)). Inserting it into (3.50) gives bKerr + = bKerr , | − |− where bKerr denotes NUT parameter of solution with z > 0 and z < 0 So something like|± NUT parameter is present. However it is symmetric with respect to the equatorial plane, i.e. it contains cos θ term instead of ordinary cosine. We can conclude that asymptotic structure| | of generated metric closely related to the Kerr–NUT solution (up to the possible discontinuity). There are two possible approaches to make the discontinuity disappear. First one leads to static solution and the second one generates solution symmetric with respect to the equatorial plane (at least asymptotically). In the latter case there is no way to avoid NUT-like term.

3.2.3 Axis

On the axis m F (0, z, tsg) = , (3.53) 2 2 − max(z, tsg) + b where λ 0 has been assumed. Moreoverp ≥ m ν(0, z) = . (3.54) −√z2 + b2 and 2(t z) + O (ρ2) when z < t 2 2 sg sg λ = tsg z + (tsg z) + ρ = ρ2− 4 (3.55) + O (ρ ) when z > tsg − − ( 2(z tsg) q − Careful in the vicinity of axis has to be used as λ can vanish there and limit approach must be employed. Due to the the technical aspects of the discussion and appearance physically severe problems in the equatorial plane it is present here.

3.2.4 Equatorial plane In the equatorial plane there is, generally, a discontinuity of generated metric due to the discontinuity of spectral function F (ρ, z, λ). In some special cases it can be avoided. Complexity of this analysis requires standalone section (3.3.2) dedicated to it.

67 3.2.5 Castro & Letelier’s solution revisited At the first glance solution (3.27) and the one given by de Castro and Letelier [2011] cannot be the same. However two different expansions are used in that paper, one in the area r2 ρ2 + z2 < b2 (denoted “in”, representing Taylor expansion of F from r =≡ 0) and another one when r2 > b2 (denoted “out”, representing Taylor expansion of F with origin at r = ). Such expansions requires r = 0 and r = being regular points of ν and∞ F . The latter is problematic. ∞ As seen in figure 3.2 there is cut in equatorial plane extending from ring towards infinity so r = is not a regular point. No Taylor expansion of (3.27) exist there. Keeping in∞ mind, that solution (3.27) is given up to the function of tsg and b it is possible to choose different function X(tsg) to obtain F with cut extending between r = 0 and ring. In fact arbitrary curve connecting edge of domain of F (i.e. (ρ, z) < 0; + ) ( ; + )) and ring ((ρ, z) = (b, 0)) can be chosen and obtain function∈ with∞ × cut−∞ extending∞ along this curve. In the original paper there has to be non-trivial behaviour along the surface11 r = b. Apart from such complication, one can simply follow the procedure there in order to obtain new metric (as in this chapter). Only thing not seen there is discontinuity of the metric (or, in case of lucky choice of constants, jump in the derivatives) itself.

3.3 Avoiding discontinuity

3.3.1 Diagonal matrix Γkl(ρ, z) It is possible to get rid of discontinuity in the t–φ part of the matrix by suitable (k) 12 (2) choice of C ensuring Γkl(ρ, z) is diagonal. Such assumption requires Ct = (1) Cφ = 0. Following (3.35) Γkl(ρ, z) can be expressed as 2 2ν(ρ,z) 4F (λ1,ρ,z) (1) e − Ct ρ, z , Γ11( ) = 2 2 (3.56) − ρ + λ1(ρ, zh) i Γ12(ρ, z) = Γ21(ρ, z) = 0, (3.57) 2 2 4F (λ2,ρ,z) 2ν(ρ,z) (2) ρ e − Cφ ρ, z . Γ22( ) = 2 2 2 (3.58) λ2(ρ, z)[ρ + λ2(hρ, z)]i Corresponding non–normalized metric takes form 2 2 2 ρ 2ν(ρ,z) 2ν(ρ,z) ρ +λ1(ρ,z) 2ν(ρ,z) g = e + e 2 = e , (3.59) tt λ1(ρ,z) 2 − λ1(ρ, z)

gtφ = = 0, (3.60) 2 2 2 ρ 2 2ν(ρ,z) 2 2ν(ρ,z) ρ +λ2(ρ,z) 2 2ν(ρ,z) g = ρ e− ρ e− 2 = ρ e− (3.61) φφ λ2(ρ,z) 2 − − λ2(ρ, z) 11“In” and “out” solutions are well adopted in their domains. Considering their convergence and behaviour they does not exhibit cut there. Therefore the cut has to be present at the radius of the ring. Exact analysis is not given in the original paper nor here (as previous argument makes it partly obsolete). 12Up to the possible exchange of indices 1 and 2.

68 and using (3.40) physical metric is calcualted

ph 2ν(ρ,z) λ2(ρ,z) 2ν0(ρ,z) gtt = e = e , (3.62) − λ1(ρ,z) ph gtφ = = 0, (3.63)

ph 2 2ν(ρ,z) λ1(ρ,z) 2 2ν0(ρ,z) g = ρ e− = ρ e− , (3.64) φφ λ2(ρ,z) where 2 2 1 α2 z + (α2 z) + ρ ν0 = ν(ρ, z) + ln − − . (3.65) 2 2 2 "α1 z + (α1 z) + ρ # − p − denotes generated gravitational potential. Thisp is a simple superposition of Bach– Weyl ring solution with Schwarzschild black-hole. Such metric is well known and does not require further discussion here. The unknown part (at least based on my knowledge) is expression of the remaining metric function (f or ζ) in closed form. (3.41) gives

3 ph ph λ1λ2 ln f = 2ζ0 2ν0 = ln C + ln f + ln + − 2 λ λ (ρ2 + λ2)(ρ2 + λ2) | 2 − 1| 1 2 + 2 [F (λ , ρ, z) F (λ , ρ, z)] . (3.66) 2 − 1 ph Up to the symmetric case (where α1 = α2) the metric function f obtained by BZM exhibit discontinuity. It has physical− reason — situation, where ring is not aligned with BH must contain some kind of singularity giving additional z–force keeping the ring and BH at their locations. 2 2 Assuming α1 = α2 = α and introducing d = (α z) + ρ metric func- ± tions have form − ± p 2mK(k) 1 d+ + d 2α ν0 = + ln − − (3.67) −π (ρ + b)2 + z2 2 d+ + d + 2α − 2 2 2 2 2 2 m (b r )(2 k ) 2 (3b + r )(2 k ) 2 ζ0 = p − − E(k) +4K (k)E(k) − K(k) + 4π2b2 (r2 + b2)(1 k2) − (r2 + b2) 2− 2 1 (d+ + d ) 4α + ln − − + 2 [F (λ2, ρ, z) F (λ1, ρ, z)] , (3.68) 2 4d+d − − ph where we have chosen constant C2 in such way, that no conical singularity on the axis appears, i.e. that ζ0 = 0. (3.69) |ρ=0 Figure 3.3 gives an example of such generated metric.

3.3.2 Continuity of physical metric After special case leading to the static metric, let us focus on the singularities in the equatorial plane much more rigorously. It is clear (from physical reasons already given) there has to be singularity when the BH is not aligned with ring, i.e. α1 = α2 = α. For convenience in this chapter one-sided limits at the equatorial plane− are denoted by f f . Also usage of ∆ deﬁned by (3.47) z=0+ ± simpliﬁes discussion. ≡ |

69 2 2

z 1 z 1

0 1 2 3 4 5 6 1 2 3 4 5 6 rho rho

±1 ±1

±2 ±2

Figure 3.3: Metric functions of Bach–Weyl ring (mass m = 1 and radius b = 3) surrounding black-hole (mass M = 1). Left: ν. Displayed levels are 2, 1.8, , 0.2. Right: ζ. Displayed levels are 0.1, 0.2,..., 0.8. In the small −area− at the··· inner− rim of ring the values of ζ are even negative.

Continuity of f ph across equatorial plane Formulas from chapter 3.2.1 contains only continuous terms up spectral function F (ρ, z, tsg) that is also fully responsible for discontinuity of metric. F (ρ, z, tsg) deﬁned by (3.27) and (3.28) is continuous everywhere up to the equatorial plane where ρ > b. There it jumps by constant (with respect to ρ and z) value, i.e. 0 when ρ < b F λ, ρ, z F λ, ρ, z m sgn(λ) . ( ) z=0+ ( ) z=0 = ρ > b (3.70) | − | − 2 2 when (√tsg+b Given α = α = α the dependence of diﬀerence of one-sided limits only on 1 − 2 squares of tsg can be used to simplify previous expression 0 when ρ < b F µ , ρ, z F µ , ρ, z ( (1,2) ) z=0+ ( (1,2) ) z=0 = m = − 2 2 when ρ > b − √α +b 1 1 = ln ∆ ln ∆+, (3.71) 2 − − 2 or, equivalently, 1 F µ , ρ, z F µ , ρ, z, . ( (1,2) ) z=0 = ( (1,2) ) ln ∆ (3.72) ± − 2 ±

Expressions for generated metric does not contain spectral function F itself, but rather F (ρ, z) e2F (λk,ρ,z). Its one-sided limits reads k ≡

Fk Fk z=0 = . (3.73) | ± ∆ ± Using fact, that µ(k) is continuous across z = 0 plane, we can express behaviour of (3.33) and (3.34) near the equatorial plane by

(k) C(k)F (k) Ct (k) φ k 1 mt = ∆ , mφ = ∆− . (3.74) z=0 − Fk ± z=0 µ(k) ± ± ±

Subsequently (from (3.35)) limits of matrix Γkl have form 2 2 Γkl(ρ, z) z=0 = Xkl∆ + Ykl∆− , (3.75) | ± ± ± 70 where Xkl and Ykl does not depend on side of limit. They are deﬁned by

e2ν C(k)C(l) ρ2F F C(k)C(l) X t t ,Y k l φ φ . kl = 2 kl = 2ν 2 (3.76) −FkFl ρ + µ(k)µ(l) e µ(k)µ(l) ρ + µ(k)µ(l)

2 Using relation µ(1)µ(2) = ρ several identities for Xkl and Ykl can be found, notably

(1) 2 (2) 2 (Ct ) (Ct ) 2 2 2 2 2 2 F1 µ F2 µ (α + ρ )X X (1) (2) = 1 = 11 22 , (3.77) (1) (2) 2 2 2 Ct Ct ρ X12 F1F2µ(1)µ(2) h (1) 2 (2)i2 (Cφ ) (Cφ ) 2 2 2 2 2 2 F1 µ F2 µ (α + ρ )Y Y (1) (2) = 1 = 11 22 . (3.78) (1) (2) 2 ρ2Y 2 Cφ Cφ 12 F1F2µ(1)µ(2) ph First let us focus on analysis of f , because it depends only on Γkl itself and not its inversion. (3.41) simpliﬁes to

ph 4 ph 2ζ 2ν C2 ρ f e − ρ, z . z=0 = 2 det Γkl( ) z=0 (3.79) ± 4α | ±

Ignoring singular case 13 α = 0 the metric function f ph is continuous if (and only if) det Γkl(ρ, z) does not jump in the equatorial plane. (3.75) is used to express determinant

2 4 4 ρ, z D D D − , det Γkl( ) z=0 = Γ11Γ22 Γ12 z=0 = 4∆ + 0 + 4∆ (3.80) | ± − ± ± − ± where α2 D = X X X2 = X X , 4 11 22 − 12 − ρ2 11 22 D0 = X11Y22 + Y11X22 2X12Y12, − 2 2 α D 4 = Y11Y22 Y12 = Y11Y22. (3.81) − − − ρ2

There is no jump in determinant of Γkl as long as

det Γkl(ρ, z) z=0+ = det Γkl(ρ, z) z=0 | | − 4 4 D 4 − ∆+ ∆ D4 4 4 = 0. (3.82) ⇔ − − − ∆+∆ − 4 4 To ensure that either ∆+ = ∆ (therefore no jump in spectral function F is present — this is the case when −ρ < b) or14 (assuming ρ > b)

2 2 (1) (2) 2 8M D 4 Y11Y22 F1 F2 Cφ Cφ √α2+b2 4 4 e = ∆+∆ = − = = . (3.83) − D X X e4ν (1) (2) 4 11 22 " Ct Ct #

13Physically it correspond to extreme BH. 14 (1) (2) (1) (2) (1) (2) Take note that Ct Ct = 0. Ct Ct = 0 implies Cφ Cφ = 0 and it leads to diagonal case, which was analyzed in previous6 section.

71 Straightforward calculation reveals F (λ , ρ, z) + F (λ , ρ, z) = ν F F = e2ν. (3.84) 1 2 ⇒ 1 2 So (3.83) simpliﬁes to (1) (2) 4m Cφ Cφ e √α2+b2 = . (3.85) C(1)C(2) t t This is a bit surprising result, because D4 and D 4 both depend on ρ and − z in non-trivial way. However at the end spatial dependence of the functions disappears and only constants (with respect to ρ and z) remains.

ph Continuity of gtt across equatorial plane Combining (1.85) and (1.84) gives

ph gab = ρ gab z=0 , (3.86) z=0 ± | ± ± where a, b (t, φ) and the in front of r.h.s. is arbitrary ﬁxed (and does not depend on∈ direction of limit).± The physical metric is continuous as long as its non–normalized component is. (3.36) reads 2 (k) (l) 2ν 4ν Πkl(ρ, z) Ct Ct gtt z=0 = e e . (3.87) | ± − − " µ(k)µ(l) Fk(ρ, z)Fl(ρ, z)# k,l=1 z=0 X ± Due to the continuity of seed metric focus will be given to the bracket. To ph analyze f matrix Πkl (deﬁned as inverse of Γkl) is required. General formula for inversion of 2 2 matrix is quite simple × 1 Γ11 Γ12 − 1 Γ22 Γ12 Πkl = = − (3.88) Γ12 Γ22 det Γ Γ12 Γ11 − so 2 (k) (l) 4 Πkl(ρ, z) Ct Ct Att∆ + Btt = ± (3.89) " µ(k)µ(l) Fk(ρ, z)Fl(ρ, z)# det Γ z=0 k,l=1 z=0 | ± X ± where (employing relations from previous section)

2 2 (1) (2) (1) (2) X22 Ct X11 Ct X12Ct Ct Att = 2 + = 2 2 2 2 2 µ(1) F1 − ρ F1F2 µ(2) F1 X2 X X α2 + ρ2 = 4 12 4 22 11 = 0, (3.90) e2ν − e2ν ρ2 2 2 (1) (2) (1) (2) Y22 Ct Y11 Ct Y12Ct Ct Btt = 2 + = 2 2 2 2 2 µ(1) F1 − ρ F1F2 µ(2) F1 Y X 2 α2 + ρ2 Y X Y X = 4 12 12 22 11 + 11 22 . (3.91) e2ν − e2ν µ µ p (1) (2) Att = 0 implies continuity of the numerator of (3.89) across z = 0 plane. Moreover assuming (3.85) holds also denominator does not jump there and so continuity of gtt across equatorial plane does not introduce any additional conditions.

72 ph Continuity of gtφ across equatorial plane Due to (3.86) it is suﬃcient to prove continuity of non–normalized metric function gtφ. Given (3.37) gtφ reads

2 C(k)C(l)F 2 2 2 2 Πkl(ρ, z) t φ l ρ Atφ∆ + Btφ∆− ± ± gtφ z=0 = ρ = , (3.92) | ± − " µ(k)µ(l) µ(l)Fk # − det Γ z=0 k,l=1 z=0 | ± X ± where (1) (1) (1) (2) (2) (1) (2) (2) X22C C X C C F2 C C F1 X11C C A t φ 12 t φ t φ t φ tφ = 3 2 + + 3 = µ(1) − ρ µ(2)F1 µ(1)F2 ! µ(2) αe2νC(2)C(1) C(2)C(1) C(1)C(2) = t t t φ t φ , (3.93) 4 2 2 2 2 2ρ α + ρ " µ(1)F2 − µ(2)F1 # (1) (1) (1) (2) (2) (1) (2) (2) Y22Cp C Y C C F2 C C F1 Y11C C B t φ 12 t φ t φ t φ tφ = 3 2 + + 3 = µ(1) − ρ µ(2)F1 µ(1)F2 ! µ(2) (1) (2) (2) (1) 2 (1) (2) 2 C C α C C F1 C C F2 = φ φ t φ t φ . (3.94) 2ρ4e2ν α2 + ρ2 " µ(1) − µ(2) # p Assuming (3.85), i.e. continuity of det Γ across equatorial plane gtφ does not jump there as long as

2 2 2 2 Atφ∆+ + Btφ∆+− = Atφ∆ + Btφ∆− − − 2 2 Btφ ∆+ ∆ Atφ 2 2 = 0. (3.95) ⇔ − − − ∆+∆ − Therefore either F (λ, ρ, z) is continuous there (ρ < b) or15 (assuming ρ > b)

(1) (2) 2 2 (1) (2) 4m Btφ Cφ Cφ F1 F2 Cφ Cφ √α2+b2 2 2 e = ∆+∆ = = = . (3.96) − A − (2) (1) e4ν (2) (1) tφ Ct Ct Ct Ct This condition is compatible with (3.85) — it only ﬁxes sign for the term inside absolute value.

ph Continuity of gφφ across equatorial plane ph gφφ has to be continuous across equatorial plane as simple consequence of previous 2 calculations. We know that determinant of t–φ part of metric is gphgph gph = tt φφ− tφ ρ2. Therefore − 2 ph 2 gtφ ρ gph = − . (3.97) φφ ph gtt ph And because r.h.s. contains only continuous expressions (given (3.96) holds) gφφ must have the same property.

15 The special case when Atφ = Btφ = 0 is omitted here because it leads to the already discussed static case.

73 Conclusion of continuity conditions The suﬃcient (and necessary) condition to generate continuous metric is given by (3.96). It can be rewritten to contain static case as well

4m (1) (2) (2) (1) C C C C e √α2+b2 = 0. (3.98) φ φ − t t Clearly it coincides with asymptotic condition (3.52). It should not be surprising because the asymptotic condition is just a special case of general one (3.98). Discussion performed in this chapter showed that (3.52) is not only necessary, but also suﬃcient to prevent metric discontinuity.

3.3.3 Normal derivatives across equatorial plane ∂f ph/∂z z=0 ± 16 Starting with spectral function reveals that, surprisingly enough , its ﬁrst derivatives are continuous across z = 0 plane and reads

dF (λ1,2, ρ, z) m sgn (λ) K(k(ρ, 0)) E(k(ρ, 0)) F,z = + . (3.99) ≡ dz − π α2 ρ2 ρ + b ρ b z=0 2 + ± −

It follows that p

2F (µ(k)(ρ,z),ρ,z) ∂Fk de 2Fk = = 2 Fk z=0 F,z = F,z (3.100) ∂z dz | ± ∆ z=0 z=0 ± ± ± and

(k) ∂m (k) t 2Ct (k) = F,z∆ = 2mt F,z, (3.101) ∂z − Fk ± − z=0 ± (k) (k) ∂m C(k)F m φ φ k 1 1 (k) φ = µ 2F,z + ∆− = 2mφ F,z + .(3.102) ∂z (k) √α2+ρ2 ± α2 + ρ2 z=0 ± p Derivatives of Γkl can be obtained by straightforward calculation

∂Γkl 2 2 = X ∆ + Y ∆− , (3.103) ∂z kl,z kl,z z=0 ± ± ± where 2 µ µ X X F (k) (l) , kl,z = kl 4 ,z + 2 (3.104) "− α2 + ρ2 ρ + µ(k)µ(l) # p2 ρ2 + 2µ µ Y Y F (k) (l) kl,z = kl 4 ,z + 2 (3.105) " α2 + ρ2 ρ + µ(k)µ(l) #

16 It is not as surprising considering thatp (3.3) and (3.4) impairs two conditions on first derivatives of F that are continuous. Supplemented with with boundary condition (1.75) it is sufficient to ensure continuity of first derivatives.

74 and (dis)continuity of the determinant reads

∂ det Γ 4 4 = D4z∆ + D0z + D 4z∆− , (3.106) ∂z − z=0 ± ± ± where 2 D4z = X11,zX22 + X11X22,z 2X12X12,z = D4 8F,z + (3.107) − "− α2 + ρ2 # D = X Y + X Y + Y X + Y X 2X Y 2X Y = 0z 11,z 22 11 22,z 11,z 22 11 22,z − 12,zp12 − 12 12,z 4D = 0 , (3.108) α2 + ρ2 p 3 D 4z = Y11,zY22 + Y11Y22,z 2Y12Y12,z = D 4 8F,z + . (3.109) − − − " α2 + ρ2 #

At the end using (3.41) derivative of the metric functionp is obtained

ph 2ζ 2ν 3 3 ∂f ∂e − ∂ µ(1)µ(2) ∂ det Γ = + 2 + = ∂z z=0 ∂z z=0 ∂z (µ(1) µ(2)) ! ∂z z=0 ± ± − z=0 ± ± ρ 6 ∂ det Γ = + = 2 2 2 −α α + ρ ∂z z=0 ± ρ6 p 4 4 = + D4z∆ + D0z + D 4z∆− . (3.110) −α2 α2 + ρ2 ± − ±

It does not jump atp the equatorial plane if

4 D 4z 4 D 4z − − D4z∆+ + D0z + 4 = D4z∆ + D0z + 4 ∆+ − ∆ − 4 4 D 4z − (∆+ ∆ ) D4z 4 4 = 0. (3.111) ⇔ − − − ∆+∆ − There are, once again, two possibilities how to satisfy this condition — either 2 2 ∆+ = ∆ or − 8F + 3 ,z 2 2 4 4 D 4z D 4 √α +ρ − − ∆+∆ = = 2 = − D4z D4 8F,z + − √α2+ρ2 16F α2 + ρ2 + 1 = ∆4 ∆4 1 + ,z , (3.112) + 2 2 − " 8F,z α + ρ + 2# − p Where (3.96) is assumed to hold17. Therefore generatedp metric must have discontinuity in ﬁrst derivatives in the equatorial plane (assuming we have got rid of discontinuity of metric itself).

17It does not make sense to ensure continuity of the ﬁrst derivatives when the metric itself is discontinuous.

75 2

2 z 1

z 1

0 1 2 3 4 5 rho 0 1 2 3 4 5 6 rho

±1 ±1

±2 ±2

Figure 3.4: Metric functions of generated metric using ring at radius b = 3 with (k) (l) mass m = 2 and BH determined by α = 1. Constants Ct and Cφ are chose in such way that metric is continuous and total angular momentum is aKerr = 0.1 . Left: Gravitational potential ν0. Right: Dragging ω0.

Continuity of derivatives Omitting static case it was shown, that it is impossible to obtain metric that has continuous derivatives across equatorial plane. Because of the behaviour of f ph function and modified NUT-like term at the infinity it is clear that the generated solution has severe physical problems. Therefore there is no need to express (and analyze) derivatives of 2-metric g in great detail. Generally speaking the gravitational potential will have continuous derivatives across equatorial plane. However dragging ω and subsequently f ph (as already proven) does not. This leads to conclusion that even potential interpretation as BH surrounded by infinite disc (with given inner radius) fails. It would require really exotic matter that will only create dragging without gravitational potential. Similar result of discussion is given by Semerák[2002].

3.4 Example

Example of generated metric is present at ﬁgure 3.4. Seed metric contains ring of mass m = 2 at radius b = 3. Position of the black-hole is determined by α = 1 and the constants takes value

C(1) = 0.00738,C(1) = 1.00068, (3.113) t φ − C(2) = 1.00000,C(2) = 0.92557. (3.114) t φ − Such choice is consistent with continuity condition (3.98) (and the degree of freedom involving scaling of all C’s, which does not affect generated metric, is (2) fixed by choice Ct = 1). The only left degree of freedom is fixed by choice of total angular momentum of generated spacetime. Asymptotic physical quantities describing generated spacetime, calculated using (3.48), (3.49) and (3.50), are

MKerr = 3.00137, (3.115)

aKerr = 0.10000, (3.116)

76 2 10

6 1 4

0 1 2 3 4 5 6 0 1 2 3 4 5 6 rho rho ±2

±4 ±1 ±6

±8

±2 ±10

Figure 3.5: Detailed behaviour of metric function near axis. Solid line represents metric function at z = 0, doted denotes z = 0.5 and z = 2. Scaling is chosen to exhibit problematic behaviour of the metric. Left: Gravitational potential (up to the expononet) e2ν. Right: Dragging ω.

b = 0.85241. (3.117) Kerr −

What can be seen is that metric potential ν0 is continuous even with its first derivatives across equatorial plane. However the discontinuity of first derivatives of dragging ω0 at radii r > b. Moreover the behaviour at the axis (and its vicinity) is less regular then ph required. One of the metric functions (gφφ) vanishes near axis (ρ = 0) and ph 18 another one (gtφ ) diverges at the same place (see figure 3.5).

18Divergence of both functions at the same place could be expected because the determinant of t φ part of metric is ρ2, therefore it is regular. − − 77 Conclusion

Inspired by accretion structures likely to exist around astrophysical black-holes, we explored some possibilities of finding stationary and axisymmetric solutions of the Einstein field equations generated by a black-hole encircled, symmetrically, by a thin disc or ring (spatially a two- or one- dimensional source). Two main approaches have been followed — perturbation of a background solution (a Schwarzschild black-hole) with appropriate boundary conditions, and generating a black-hole in some suitable non-vacuum “seed” space-time (e.g. that of a disc or a ring) using the Belinskii–Zakharov inverse-scattering transformation. In the first approach (perturbations), we followed closely the procedure due to Will [1974] who studied a perturbation by a slowly rotating light singular ring. Our contribution consists in putting the Green functions of this perturbation problem into a closed form (involving elliptic integrals), which is more suitable for further work than the original Will’s multipole expansion. In particular, we illustrated on an example that one can, in case of some simple mass distributions, integrate the Green functions and obtain a linear perturbation of the metric for a black-hole encircled by a thin disc (which astrophysically is more realistic than a ring). The results are just being published in C´ıˇzekandˇ Semerák[2017] and we have summarized them in chapter 2 of this thesis. To mention just one specific point, we managed to also abandon — besides the multipole expansion of the Green functions — the “rotational” expansion, namely what in the Will’s version is represented as expansion with respect to the angular velocity of the horizon. This has been possible due to a freedom in the black-hole angular momentum. We fixed the latter in such a way that the angular momentum stays zero even after the perturbation. This does not mean that the black-hole does not rotate (with respect to the asymptotic inertial frame) — its angular velocity is non-zero as it is dragged along by rotation of the surrounding disc. The second part of the thesis has been devoted to the Belinskii–Zakharov generating method. Inspired by the paper de Castro and Letelier [2011], we specifically checked whether the black-hole cannot be “implemented” in a suitable seed metric (namely that of the envisaged external source, e.g. a ring) by a two-soliton version of the inverse scattering procedure. Similarly as already experienced in the literature, we found that one can, in this way, only generate physically reasonable static space-times; in the more desired, stationary case, there is always present an unphysical discontinuity (corresponding to a supporting surface) in the plane of the source. Physically, one can imagine that the inverse-scattering procedure does introduce the black-hole, but “does not care” about adjusting the seed source to such a configuration that it could — without any struts — stay stationary in the new situation. To again mention one slight “achievement”, we have at least found in this way, in a closed form, the second metric function of the corresponding Weyl metric for a Schwarzschild black-hole encircled by a Bach–Weyl ring. Previously this has been only known for the Schwarzschild metric and for the Bach–Weyl ring alone, but, due to the non-linearity in this second function, for the “superposed” space-time it has always been obtained only numerically.

78 Our main future plan is to study the black-hole–disc solution found in the second chapter in more detail and compare it with the results of numerical treatment of similar conﬁgurations which have appeared in the literature.

79 A. Metric function transformations

In this appendix transformations between used metric forms are presented. For this purpose let us denote quantities related to Weyl–Lewis–Papapetrou form of metric (1.5) by subscript and quantities related to Carter–Thorne–Bardeen ◦WLP form of metric (1.6) CTB. The coordinates◦ (t, φ, ρ and z) are in both cases the same. They are introduced before split of the metric forms is done. The diﬀerence is in the interpretation of t–φ part of metric implying only metric functions ν, ω, B and ζ change. Comparing expressions of line element ds2 in (1.5) and (1.6) lead to

2νWLP 2νCTB 2 2 2νCTB 2 e = e + ρ BCTBe− ωCTB, (A.1) 2ν − − 2 2 2ν e WLP A = ρ B e− CTB ω , (A.2) − WLP − CTB CTB 2νWLP 2 2 2 2νWLP 2 2 2νCTB e AWLP + ρ BWLPe− = ρ BCTBe− , (A.3) − 2ζ 2ν 2ζ 2ν e WLP− WLP = e CTB− CTB . (A.4)

Transformation of metric functions follow. To get Carter–Thorne–Bardeen form (1.6) relations

1 e4νWLP A2 ν = ν ln 1 WLP , (A.5) CTB WLP − 2 − ρ2B2 WLP 2νWLP e AWLP ωCTB = , (A.6) ρ2B2 e 2νWLP e2νWLP A2 WLP − − WLP BCTB = BWLP, (A.7) 1 e4νWLP A2 ζ = ζ ln 1 WLP (A.8) CTB WLP − 2 − ρ2B2 WLP are to be applied. Analogously formulas

1 2 2 4ν 2 ν = ν + ln 1 ρ B e− CTB ω , (A.9) WLP CTB 2 − CTB CTB 2 2 4νCTB ρ BCTBe− ωCTB AWLP = , (A.10) 1 ρ2B2 e 4νCTB ω2 − CTB − CTB BWLP = BCTB, (A.11)

1 2 2 4ν 2 ζ = ζ + ln 1 ρ B e− CTB ω (A.12) WLP CTB 2 − CTB CTB give metric functions of Weyl–Lewis–Papapetrou form (1.5).

80 B. Christoﬀel symbols

B.1 Weyl–Lewis–Papapetrou form of metric

Given metric in the form (1.5) non-zero Christoﬀel symbols read AA Γt = ν + ,ρ e4ν, (B.1) tρ ,ρ 2ρ2B2 AA Γt = ν + ,z e4ν, (B.2) tz ,z 2ρ2B2 1 A B A2A Γt = 2Aν + A A ,ρ + e4ν ,ρ , (B.3) φρ ,ρ 2 ,ρ − ρ − B 2ρ2B2 1 B A2A Γt = 2Aν + A A ,z + e4ν ,z , (B.4) φz ,z 2 ,z − B 2ρ2B2 A Γφ = ,ρ e4ν, (B.5) tρ −2ρ2B2 A Γφ = ,z e4ν, (B.6) tz −2ρ2B2 B 1 AA Γφ = ,ρ + ν ,ρ e4ν, (B.7) φρ B ρ − ,ρ − 2ρ2B2 B AA Γφ = ,z ν ,z e4ν, (B.8) φz B − ,z − 2ρ2B2 ρ 4ν 2λ Γtt = ν,ρe − , (B.9) ρ 4ν 2λ Γtφ = [Aν,ρ + A,ρ] e − , (B.10)

ρ 2 4ν 2λ B,ρ 1 2 2 2λ Γ = A ν + AA e − + ν ρ B e− , (B.11) φφ ,ρ ,ρ ,ρ − B − ρ ρ Γρρ = λ,ρ ν,ρ, (B.12) ρ − Γρz = λ,z ν,z, (B.13) ρ − Γzz = ν,ρ λ,ρ, (B.14) z −4ν 2λ Γtt = ν,ze − , (B.15) z 4ν 2λ Γtφ = [Aν,z + A,z] e − , (B.16)

z 2 4ν 2λ B,z 2 2 2λ Γ = A ν + AA e − + ν ρ B e− , (B.17) φφ ,z ,z ,z − B Γz = ν λ , (B.18) ρρ ,z − ,z Γz = λ ν , (B.19) ρz ,ρ − ,ρ Γz = λ ν . (B.20) zz ,z − ,z

81 B.2 Carter–Thorne–Bardeen form of metric

Assuming the metric takes form (1.6) non-vanishing connection coeﬃcients are expressed by

t 1 2 2 4ν Γ = ν ρ B e− ωω , (B.21) tρ ,ρ − 2 ,ρ t 1 2 2 4ν Γ = ν ρ B e− ωω , (B.22) tz ,z − 2 ,z t 1 2 2 4ν Γ = ρ B e− ω , (B.23) φρ 2 ,ρ t 1 2 2 4ν Γ = ρ B e− ω , (B.24) φz 2 ,z φ 1 B,ρ ω 1 2 2 4ν 2 Γ = 2ων ω ω ρ B e− ω ω , (B.25) tρ ,ρ − 2 ,ρ − B − ρ − 2 ,ρ

φ 1 B,z 1 2 2 4ν 2 Γ = 2ων ω ω ρ B e− ω ω , (B.26) tz ,z − 2 ,z − B − 2 ,z φ B,ρ 1 1 2 2 4ν Γ = + ν + ρ B e− ωω , (B.27) φρ B ρ − ,ρ 2 ,ρ

φ B,z 1 2 2 4ν Γ = ν + ρ B e− ωω , (B.28) φz B − ,z 2 ,z ρ B,ρ 1 ω,ρ 2 2 2λ 2 4ν 2λ Γ = ν ρ B e− ω + ν e − , (B.29) tt ,ρ − B − ρ − ω ,ρ ρ B,ρ 1 ω,ρ 2 2 2λ Γ = ν + + + ρ B e− ω, (B.30) tφ − ,ρ B ρ 2ω ρ B,ρ 1 2 2 2λ Γ = ν ρ B e− , (B.31) φφ ,ρ − B − ρ Γρ = λ ν , (B.32) ρρ ,ρ − ,ρ Γρ = λ ν , (B.33) ρz ,z − ,z Γρ = λ + ν , (B.34) zz − ,ρ ,ρ z B,z ω,z 2 2 2λ 2 4ν 2λ Γ = ν ρ B e− ω + ν e − , (B.35) tt ,z − B − ω ,z z B,z ω,z 2 2 2λ Γ = ν + + ρ B e− ω, (B.36) tφ − ,z B 2ω z B,z 2 2 2λ Γ = ν ρ B e− , (B.37) φφ ,z − B Γz = λ ν , (B.38) ρρ ,z − ,z Γz = λ ν , (B.39) ρz ,ρ − ,ρ Γz = λ ν . (B.40) zz ,z − ,z

82 C. Properties of Legendre / Gegenbauer functions

1 (3/2) (3/2) Legendre functions Pn(x), Qn(x) and Gegenbauer functions Cn (x), Dn (x) (λ) (λ) are special cases of general Gegenbauer functions Cn (x) and Dn (x). Due to this generality of their deﬁnition (using associated Legendre functions) and their properties are focus of this appendix. Unless cited the results are based on so far unpublished paper about series of Gegenbauer functions. This appendix summarizes relevant content of this unﬁnished article and sketches corresponding proofs2.

C.1 Deﬁnition of Gegenbauer functions

(λ) Gegenbauer polynomials Cn (x) are polynomial solutions of ultraspheric diﬀer- ential equation [DLMF, section 18.8]

d 2 λ+ 1 dy(x) 2 λ 1 (1 x ) 2 + n(n + 2λ)(1 x ) − 2 y(x) = 0, (C.1) dx − dx − + 1 where λ > 0 and n Z0 . Special case when λ = /2 reduces it to Legendre diﬀerential equation (see∈ [DLMF, (14.2.1)]). The second solution (diverging at (λ) x = 1) is going to be denoted Gegenbauer function of the second kind Dj (x) in analogy with Legendre function of second kind Qj(x). Formally they can be deﬁned using associated Legendre functions

1 2 λ 1 2λ P −1 (x) when x 1 (λ) 2 − √2πΓ(n + 2λ) +λ+n x x 4 − 2 | | ≤ , Cn ( ) 1 λ 1 λ (C.2) ≡ − 2 Γ(λ)Γ(n + 1)P 2 − x x 1 +λ+n( ) when 1  − 2 ≥ 1 2 λ 1 2λ Q −1 (x) when x <1 − √2πΓ(n + 2λ)  (λ) 2 4 2 +λ+n D (x) 1 x − 1 | | ,(C.3) n λ iπ λ 1 λ ≡ − 2 Γ(λ)Γ(n + 1)e ( − 2 )Q 2 − x x> 1 +λ+n( ) when 1  − 2

µ µ µ µ where Pν (x), Pν (x), Qν (x) and Qν (x) denotes associated Legendre functions of first (second) kind. The pairs differs in position of its cut in complex plane. First and third one jumps when x +1; ) and the remaining two are discontinuous at x 1; +1 . The notation∈ h ∞ of associated Legendre polynomials is in accordance∈ with h− [DLMF,i chapter 14]. The split of definitions on real axis is in order to have real valued function there (with possible discontinuity at x = 1). Generally assuming x C and removing absolute value (in accordance with value of x) they can be regarded∈ as definition of two functions of complex variable differing by complex phase (and

1In this appendix x denotes arbitrary variable. It is not linked to coordinate x deﬁned by (2.27). Analogously all other variables (notable example λ) does not have its physical meaning. 2Some technical aspects can be omitted here. However main points are given and reader can ﬁnish formal proofs.

83 3 (λ) position of cut). It should be remarked that in case of polynomials Cn (x) there is no singularity at x = 1 and both deﬁnitions coincide. 1 3 Main interest of this thesis are special cases λ = /2 and λ = /2. The former one leads to well known Legendre functions and in the latter case (and assuming + n Z0 ) deﬁnitions (C.2) and (C.3) reads ∈ 1 (3/2) (n + 2)(n + 1) Pn−+1(x) when x 1 Cn (x) = 1 | | ≤ , (C.4) 1 x2 Pn−+1(x) when x 1 | − | ≥ n n 1 (3/2) ( p+ 2)( + 1) Qn−+1(x) when x < 1 Dn (x) = 1 | | . (C.5) 1 x2 Qn−+1(x) when x > 1 | − | − p C.2 Basic properties

C.2.1 Values at special points Values of Gegenbauer polynomials take simple form at several special points. + Based on [DLMF, Table 18.6.1] and assuming n Z0 ∈ Γ(n + 2λ) C(λ)(1) = , (C.6) n n!Γ(2λ) Γ(n + λ) C(λ) (0) = 0, C(λ)(0) = ( 1)n . (C.7) 2n+1 2n − n!Γ(λ) In case of Legendre polynomials (C.6) and (C.7) reads

Pn(1) = 1, (C.8) (2n)! P (0) = 0, P (0) = ( 1)n (C.9) 2n+1 2n − 4n(n!)2

3 and if λ = /2 (n + 2)(n + 1) C(3/2)(1) = , (C.10) n 2 (2n + 1)! C(3/2) (0) = 0, C(3/2)(0) = ( 1)n . (C.11) 2n+1 2n − 4n(n!)2

(λ) To get values at x = 1 one can use parity of Cn (x). It is determined by parity of n, i.e. − C(λ)( x) = ( 1)nC(λ)(x). (C.12) n − − n See [DLMF, Table 18.6.1].

C.2.2 Orthogonality (λ) Gegenbauer polynomials Cn (x) forms basis of space of analytic functions of variable x. Also they follows relation of orthogonality [DLMF, section 18.2]

1 1 2λ 1 2 − πΓ(n + 2λ) (λ) (λ) 2 λ 2 Cm (x)Cn (x)(1 x ) − dx = δmn 2 . (C.13) 1 − (n + λ)n! [Γ(λ)] Z− 3This fact is exploited in chapter 2.

84 Two special cases we are interested in read 1 2 Pm(x)Pn(x)dx = δmn , (C.14) 1 2n + 1 − 1 Z (3/2) (3/2) 2 2(n + 1)(n + 2) Cm (x)Cn (x)(1 x )dx = δmn . (C.15) 1 − 2n + 3 Z− C.2.3 Asymptotic behaviour Inﬁnite order As n + the functions behaves like4 → ∞ λ 1 (λ) λ 2n − 1 C (cos θ) = (sin θ)− cos ((λ + n)θ) + O , (C.16) n 2λΓ(λ) n λ 1 (λ) λ πn − π 1 D (cos θ) = (sin θ)− sin (λ + n)θ λ + O ,(C.17) n 2λΓ(λ) − − 2 n where θ (0; π) and ∈ λ 1 λ n − λ+n 1 C(λ)(x) = x2 1 − 2 x + √x2 1 1 + O , (C.18) n − 2λΓ(λ) − n λ 1 λ πn − λ n 1 D(λ)(x) = x2 1 − 2 x + √x2 1 − − 1 + O , (C.19) n − 2λΓ(λ) − n assuming x > 1. (λ) Also when Cn (x) does not diverge at x = 1 the asymptotic behaviour there is given by (C.6) n2λ 1 1 C(λ)(1) = − 1 + O . (C.20) n Γ(2λ) n Inﬁnite argument As x the functions behaves like5 → ∞ 2nΓ(n + λ) C(λ)(x) xn, (C.21) n ≈ Γ(λ)Γ(n + 1)

(λ) πΓ(n + 2λ) n 2λ D (x) x− − , (C.22) n ≈ 22λ+nΓ(λ)Γ(n + λ + 1) assuming n 0 and λ > 0. Special cases we are primary interested in read ≥ Γ(2n + 1) n Pn(x) x , (C.23) ≈ [Γ(n + 1)]2 2n n 2 2 [Γ(n + 1)] n 1 Q (x) x− − 0, (C.24) n ≈ Γ(2n + 2) → Γ(2n + 2) C(3/2)(x) xn, (C.25) n ≈ [Γ(n + 1)]2 2n n+1 (3/2) 2 Γ(n + 2)Γ(n + 3) n 3 D (x) x− − 0. (C.26) n ≈ Γ(2n + 4) → 4Also see [DLMF, (18.15.10)] for Gegenbauer polynomials. 5See [DLMF, sec. 14.8(iii)] and deﬁnitions of Gegenbauer functions.

85 Behaviour at 1 In the case when x 1+ it can be shown 6 → (λ) Γ(n + 2λ) lim Cn (x) = , (C.27) x 1+ n λ → Γ( + 1)Γ(2 ) 1 1 √πΓ λ (λ) 2 λ 2 lim Dn (x) (2x 2) − − , (C.28) x 1+ ≈ − 2Γ(λ) → (λ) where it is assumed n 0 (in both cases) and λ > 0 (in the case of Cn (x)) or 1 ≥(λ) 1 7 λ > /2 (in the case of Dn (x)). Legendre functions (λ = /2) goes to

lim Pn(x) = 1, (C.29) x 1+ → 1 lim Qn(x) ln(x 1) + O (1) (C.30) x 1+ → ≈ 2 − 3 and Gegenbauer polynomials with λ = /2

(3/2) (n + 1)(n + 2) lim Cn (x) = , (C.31) x 1+ → 2 (3/2) 1 lim Dn (x) . (C.32) x 1+ ≈ 2(x 1) → − C.2.4 Recurrence relation Recurrence relation of Gegenbauer polynomials is well known. Analogous relation for the Gegenbauer functions of the second kind hold as well. Generally (λ) (λ) (λ) (n + 1) n+1(x) + (n + 2λ 1) n 1(x) = 2(n + λ)x n (x), (C.33) A − A − A (λ) where n (x) stands for an arbitrary linear combination of Gegenbauer functions of theA ﬁrst and second kind8.

C.2.5 Sum of double product One consequence of recurrence relation given earlier is n Γ(l + 1) (x t) 2(l + λ) (λ)(x) (λ)(t) = − Γ(l + 2λ)Al Bl Xl=0 Γ(n + 2) = (λ) (x) (λ)(t) (λ)(x) (λ) (t) + (λ)(x, t), (C.34) Γ(n + 2λ) An+1 Bn − An Bn+1 C h i + (λ) (λ) where n Z0 , x, t R (or C). Series of functions n (x) and n (x) are assumed to obey∈ relation (C.33)∈ and (λ)(x, t) denotes suitableA functionB 9 independent on n. C 6Combine [DLMF, sec. 14.8(ii)] with deﬁnitions (C.2) and (C.3). 7 1 In the latter case it is based on[DLMF, (14.8.9)] because (C.28) fails when λ = /2. 8It can be proven using recurrence relations of (associated) Legendre functions and linearity of (C.33). 9 This function can be obtained, for example, putting n = 0 1 (λ)(x, t) = 2λ(x t) (λ)(x) (λ)(t) (λ)(x) (λ)(t) + (λ)(x) (λ)(t) . (C.35) C Γ(2λ) − A0 B0 − A1 B0 A0 B1 h i

86 Proof. Due to the existence of (λ)(x, t) in formula (C.34) it is sufficient to prove C that the the difference of the r.h.s of (C.34) (let us denote it n) with n and n 1 equals the difference of the l.h.s. (i.e. the additional termR in the sum). (C.33)− ensures that

Γ(n + 2) (λ) (λ) Γ(n + 1) (λ) (λ) n+1(x) n (t) n (x) n 1(t) = (C.36) Γ(n + 2λ)A B − Γ(n + 2λ 1)A B − − Γ(n + 1) (λ) (λ) (λ) (λ) = (n + 1) n+1(x) n (t) (n + 2λ 1) n (x) n 1(t) Γ(n + 2λ) A B − − A B − Γ(n + 1) h i = 2x(n + λ) (λ)(x) (λ)(t) Γ(n + 2λ) An Bn − (λ) (λ) (λ) (λ) (n + 2λ 1) n 1(x) n (t) + n (x) n 1(t) . − − A − B A B − i And the diﬀerence of the r.h.s of (C.34) with n and n 1 takes form −

Γ(n + 2) (λ) (λ) Γ(n + 1) (λ) (λ) n n 1 = n+1(x) n (t) n (x) n 1(t) R − R − Γ(n + 2λ)A B − Γ(n + 2λ 1)A B − − − Γ(n + 2) (λ) (λ) Γ(n + 1) (λ) (λ) n (x) n+1(t) n 1(x) n (t) = − Γ(n + 2λ)A B − Γ(n + 2λ 1)A − B − Γ(n + 1) = 2x(n + λ) (λ)(x) (λ)(t) Γ(n + 2λ) An Bn − (λ) (λ) (λ) (λ) (n + 2λ 1) n 1(x) n (t) + n (x) n 1(t) − − A − B A B − − Γ(n + 1) i 2t(n + λ) (λ)(x) (λ)(t) −Γ(n + 2λ) An Bn − (λ) (λ) (λ) (λ) (n + 2λ 1) n (x) n 1(t) + n 1(x) n (t) = − − A B − A − B Γ(n + 1) i = 2(x t)(n + λ) (λ)(x) (λ)(t). (C.37) Γ(n + 2λ) − An Bn

Comparing this with the diﬀerence of the l.h.s. of (C.34) completes the proof. There are two notable special (extreme) cases of relation (C.34). Sum involving only Gegenbauer polynomials

n Γ(l + 1) (x t) 2(l + λ) C(λ)(x)C(λ)(t) = − Γ(l + 2λ) l l Xl=0 Γ(n + 2) = C(λ) (x)C(λ)(t) C(λ)(x)C(λ) (t) (C.38) Γ(n + 2λ) n+1 n − n n+1 h i and “mixed” product of Gegenbauer functions of the ﬁrst and second kind

n Γ(l + 1) (x t) 2(l + λ) C(λ)(x)D(λ)(t) = (C.39) − Γ(l + 2λ) l l Xl=0 1 λ Γ(n + 2) 2π t2 1 2 − = C(λ) (x)D(λ)(t) C(λ)(x)D(λ) (t) | − | . Γ(n + 2λ) n+1 n − n n+1 − 4λΓ2(λ) h i

87 Both series are generalization of well known equalities of Legendre polynomials [DLMF, (14.18.6-7)] n (x t) (2l + 1)P (x)P (t) = (n + 1) [P (x)P (t) P (x)P (t)] , (C.40) − l l n+1 n − n n+1 l=0 Xn (x t) (2l + 1)P (x)Q (t) = (n + 1) [P (x)Q (t) P (x)Q (t)] 1.(C.41) − l l n+1 n − n n+1 − Xl=0 3 And when λ = /2 (C.38) and (C.39) becomes

n (3/2) (3/2) (3/2) (3/2) (3/2) (3/2) (2l+3)C (x)C (t) C (x)Cn (t) Cn (x)C (t) (x t) l l = n+1 − n+1 (C.42) − (l + 1)(l + 2) n + 2 l=0 Xn (3/2) (3/2) (3/2) (3/2) (3/2) (3/2) (2l+3)C (x)D (t) C (x)Dn (t) Cn (x)D (t) (x t) l l = n+1 − n+1 − (l + 1)(l + 2) n + 2 − l=0 X 1 . (C.43) − t2 1 | − | Proof of (C.38). Exact knowledge of low-order Gegenbauer polynomials (see for example [Erdelyi, 1981, page 175, (12)])

(λ) C0 (x) = 1, (C.44) (λ) C1 (x) = 2λx, (C.45) can be used to prove (x) = 0 using (C.38) with n = 0, QED. C Proof of (C.39). (C.44) and (C.45) lead to

C(λ)(x)D(λ)(x) C(λ)(x)D(λ)(x) = 2λxD(λ)(x) D(λ)(x). (C.46) 1 0 − 0 1 0 − 1 On the other hand the cross-products of associated Legendre functions are given by literature, for example [DLMF, (14.2.5),(14.2.11)], Γ(ν + µ 1) Pµ (x)Qµ(x) Pµ(x)Qµ (x) = − , (C.47) ν+1 ν − ν ν+1 Γ(ν µ + 2) − Γ(ν + µ 1) P µ (x)Qµ(x) P µ(x)Qµ (x) = eµπi − . (C.48) ν+1 ν − ν ν+1 Γ(ν µ + 2) − So

2 1 λ 2πΓ(2λ) (1 x ) 2 when x < 1 (λ) x (λ) x (λ) x (λ) x − C1 ( )D0 ( ) C0 ( )D1 ( ) = 2λ 2 2− 1 λ | | − 2 Γ (λ) (x 1) 2 − when x > 1 ( − 2πΓ(2λ) 1 λ = 1 x2 2 − , (C.49) 22λΓ2(λ) −

where the absolute value makes sense on the real axis10 . Combining (C.46) and (C.49) leads to

(λ) (λ) 2πΓ(2λ) 1 λ D (x) = 2λxD (x) x2 1 2 − . (C.50) 1 0 − 4λΓ2(λ) −

10If complex x is considered the absolute value is to be understood in the sense of corresponding branch.

88 and, surprisingly, function can be calculated as given by (C.39). C 2π 1 λ (λ)(x, t) = t2 1 2 − (C.51) C −4λΓ2(λ) −

C.3 Miscilaneous relations

C.3.1 Relation between integration and differentiation One of the consequences of ultraspherical differential equation (C.1) can be used to exchange differentiation and integration in special case. Assuming λ > 0

x 2 λ+ 1 (λ) 2 λ 1 (λ) (x 1) 2 dCn (x) (z 1) − 2 C (z)dz = − (C.52) − n n(n + 2λ) dx Z1

1 When λ = /2 it takes form

x (x2 1) dP (x) P (z)dz = − n (C.53) n n(n + 1) dx Z1

3 and if λ = /2

x 2 2 (3/2) (x 1) dCn (x) (z2 1)C(3/2)(z)dz = − . (C.54) − n n(n + 3) dx Z1

C.3.2 Product simpliﬁcation Product of two Gegenbauer polynomials can be expressed using integral of single polynomial [DLMF, (18.17.5)].

(λ) (λ) Cn (cos θ1)Cn (cos θ2) = (C.55) π 2 Γ(n + 2λ) (λ) 2λ 1 = C (cos θ cos θ + sin θ sin θ cos φ)(sin φ) − dφ, 22λΓ2(λ) Γ(n + 1) n 1 2 1 2 Z0 where θ1, θ2 0; π . This formula is crucial expressing sums of triple/quadruple products ∈ h i

C.3.3 Generalized Neumann integral The Gegenbauer function of the second kind can be obtained using integral relation 1 (λ) 2 λ 1 (λ) 1 2 1 λ Cn (x)(1 x ) − 2 D (t) = (t 1) 2 − − dx. (C.56) n 2 − t x Z1 − −

89 1 where t > 1 and λ > 0. When λ = /2 equation (C.56) reduces to the Neumann’s integral [DLMF, (14.12.13)]

1 1 P (x) Q (t) = n dx. (C.57) n 2 t x Z1 − − 3 and assuming λ = /2

1 (3/2) 2 1 Cn (x)(1 x ) D(3/2)(t) = − dx. (C.58) n 2(t2 1) t x − Z1 − − Proof. Using orthogonality of Gegenbauer polynomials (C.13) l.h.s. of (C.39) (divided by (x t)) leads to − 1 2 m (n+λ)Γ (λ)Γ(n+1) (λ) 2 λ 1 Γ(l+1) (λ) (λ) x x − 2 l λ x t x 1 2λ Cn ( )(1 ) 2( + ) Cl ( )Dl ( )d = 2 − πΓ(n+2λ) − Γ(l+2λ) Z1 l=0 − X 0 when m < n = 2(n+λ)Γ(n+1) (λ) . (C.59) Dn (t) when m n ( Γ(n+2λ) ≥ As m + it goes (up to some normalization) to Gegenbauer function of the second→ kind.∞ So 1 2 m Γ (λ) 1 Γ(l + 1) (λ) (λ) (λ) 2 λ 2 (λ) lim Cn (x)(1 x ) − 2(l+λ) C (x)D (t)dx = Dn (t). m + 2 2λπ l λ l l → ∞ 2 − − Γ( + 2 ) Z1 Xl=0 − (C.60) The r.h.s. of (C.39) (divided by (x t)) will be split into two parts. The one, containing Gegenbauer functions, vanishes− as m . The other forms r.h.s. of (C.56). Let us begin with the vanishing part. Using→ ∞ integral analogous to (C.60) gives11

1 2 (λ) (λ) Γ (λ) Γ(m + 2) Cm+1(x)Dm (t) 2 λ 1 x − 2 x 2 2λ (1 ) d 2 − π Γ(m + 2λ) x t − ≈ Z1 − − 1 2 2 2λ λ 1 Γ (λ)m − 2m − 2 λ λ m x t − 2 2 2λ λ cos [( + ) arccos ]( 1) ≈ 2 − π 2 Γ(λ) − × Z1 − λ 1 λ 1 2 − πm − λ m (1 x ) 2 t + √t2 1 − − − dx × 2λΓ(λ) − x t ≤ − λ λ m 1 2 2 2 λ 1 (t 1)− 2 t + √t 1 − − (1 x ) −2 − − cos [(λ + m ) arccos x] − dx ≤ 2 x t ≤ Z1 − − 11f(m) g(m) is to be understood as f(m) behaves like g(m) as m + , i.e. ≈ → ∞ f(m) lim = 1. (C.61) m + → ∞ g(m)

90 λ λ m 1 2 2 2 3 (t 1)− 2 t + √t 1 − − (1 x )− 4 − − − dx = ≤ 2 t 1 Z1 − − 3 1 λ λ m Γ Γ 2 2 − − 2 4 = (t 1)− 2 t + √t 1 0. (C.62) − − 2Γ 3 (t 1) → 4 − All limits are considered m (with the assumption t > 1 and usage of (C.17) and (C.16)). Analogous relation→ ∞ can be found for the part of (C.39) involving (λ) (λ) Cm (x)Dm+1(t). The rest

1 1 2 2 λ (λ) 2 λ 1 Γ (λ) 2π (t 1) 2 − C (x)(1 x ) 2 n − x 2 2λ − λ 2− − d = 2 − π 4 Γ (λ) x t Z1 − − 1 (λ) 1 x 1 1 2 2 λ Cn ( ) 2 λ = t 1 − (1 x ) − 2 dx, (C.63) 2 − t x − Z1 − − forms the r.h.s. of (C.56).

C.4 Triple product

C.4.1 CCC Let us begin with (inﬁnite) generalization of (C.38). Assuming x, y, z ( 1; +1), (1 x2 y2 z2)2 4x2y2z2 = 0 and λ > 0 ∈ − − − − − 6 2 ∞ Γ(l + 1) 2(l + λ) C(λ)(x)C(λ)(y)C(λ)(z) = (C.64) Γ(l + 2λ) l l l l=0 X 2 2 2 λ 1 π[1 x y z + 2xyz] − = − − − H 1 x2 y2 z2 + 2xyz , λ 1 4 λ 1 2 2 2 2 − − − Γ (λ)4 − [(1 x )(1 y )(1 z )] − − − − holds12. When 1 x2 y2 z2 + 2xyz < 0 the sum will vanish (as indicated by Heaviside function)− regardless− − on possible ill–deﬁniteness of power term. However the convergence is only conditional when λ 1. The proof is carried out later (see page 94) because there are several lemmas≤ needed for that. 1 In the case of Legendre polynomials (λ = /2) (C.64) reduces to result [Bara- nov, 2006, (2.21)]

∞ 1 H (1 x2 y2 z2 + 2xyz) n + Pn(x)Pn(y)Pn(z) = − − − (C.65) 2 2 2 2 n=0 π 1 x y z + 2xyz X − − − 3 p and when λ = /2 it simpliﬁes to

∞ 2l + 3 C(3/2)(x)C(3/2)(y)C(3/2)(z) = (C.66) (l + 1)2(l + 2)2 l l l Xl=0 12H(x) stands for Heaviside step function as stated earlier.

91 8 1 x2 y2 z2 + 2xyz = − − − H 1 x2 y2 z2 + 2xyz . π(1 x2)(1 y2)(1 z2) − − − p − − − Starting with asymptotic behaviour of l.h.s. of (C.64). By direct usage of (C.16) it can be shown

Γ(l + 1) 2 2(l + λ) C(λ)(cos θ )C(λ)(cos θ )C(λ)(cos θ ) = Γ(l + 2λ) l 1 l 2 l 3 16 3 sin((λ + n)θ ) 1 = i + O , (C.67) 8λΓ3(λ)nλ sin θ n "i=1 i # Y where λ > 0 and θ1, θ2, θ3 (0; π). Moreover asymptotic behaviour of convolution of general function with (rapidly∈ oscilating) sine is required. Let q(θ) denote a function with bounded ﬁrst (and second when = 0) derivatives in the interval 6 x; y , 0; 1), θ0 x; y and ψ R. Then h i ∈ h ∈ h i ∈ y q(θ) 1 θ ψ θ O sin (Λ( )) d = 1 (C.68) (θ θ0) − Λ − Zx − as Λ . → ∞ Proof. Let us begin with = 0 (i.e. without the singular part). Deﬁning

q(θ) when θ x; y Q(θ) ∈ h i (C.69) ≡ (0 otherwise

(C.68) takes form

y + ∞ q(θ) sin (Λ(θ ψ)) dθ = Q(θ) sin (Λ(θ ψ)) dθ = − − Zx Z −∞ + 1 ∞ π = Q(θ) Q θ sin (Λ(θ ψ)) dθ, (C.70) 2 − − Λ − Z −∞ h i where the fact that sin(x π) = sin x has been used. This integral can be split into three parts. In one of− them− both Q(θ) and Q(θ π/Λ) vanishes. Such part clearly does not contribute. − In the second one (spanning over two intervals) only one of them is non-zero. However length of the integration intervals is 2π/Λ. Because the function q(ϑ) is bounded (consequence of its continuity over the interval x; y ) this part of the 1 h i integral vanishes like O (Λ− ). The last part have both Q(θ) and Q(θ π/Λ) are non-zero. Using mean value theorem it can be shown −

y 1 π Q(θ) Q θ sin (Λ(θ ψ)) dθ = 2 − − Λ − Zπ x+ Λ h i

92 y π dQ = (ξ(θ)) sin (Λ(θ ψ)) dθ, (C.71) 2Λ dθ − Zπ x+ Λ where ξ(θ) is suitable function. Because the ﬁrst derivative of q(θ) is bounded and the integration interval is ﬁnite the integral itself is bounded as well. On the 1 other hand factor in front of it vanishes like O (Λ− ). 1 Each part disappears at least as fast as O (Λ− ) so the sum vanishes similarly, QED. When (0; 1) the situation is a bit more tricky. However we can split the ratio ∈

q(θ) q(θ0) q(θ) q(θ0) q(θ0) 1 dq = + − = + (θ θ ) − (ξ(θ)), (C.72) (θ θ ) (θ θ ) (θ θ ) (θ θ ) − 0 dθ − 0 − 0 − 0 − 0 where ξ(θ) is suitable function given by mean value theorem. The term containing derivative of q is regular in the integration interval and has bounded derivative13. Using result for = 0 proven earlier

y 1 dq 1 (θ θ ) − (ξ(θ)) sin (Λ(θ ψ)) dθ = O (C.73) − 0 dθ − Λ Zx as Λ takes care of second part of the split (C.72). The rest is described by → ∞ y y θ0 q θ − ( 0) iΛ(θ0 ψ) iΛθ θ ψ θ q θ e − θ− e θ sin (Λ( )) d = ( 0) d = (θ θ0) − =   Zx − x Zθ0  −  1 i − iΛ(θ0 ψ)   = q(θ ) e − [γ(1 ; iΛy) γ(1 ; iΛx)] = 0 = Λ − − − − − ( ) 1 = O , (C.74) Λ1 − where γ(a, z) = Γ(a) Γ(a, z) stands for the (in)complete gamma functions. Asymptotic behaviour− of the last one, according to [Olver, 1974, p. 109-110] 1 Γ(a, iz) = (iz)ae(iz) 1 + O = O (za) , (C.75) z has been used. This justiﬁes the last equality in C.74 and so it completes the proof. A bit more complicated relation holds for convolusion of function with pole 1 behaving like x− . Once again let q(θ) denote a function with bounded ﬁrst and second derivative in the interval x; y . Then h i y q(θ) q(0)π + O 1 when 0 (x; y) sin(Λθ)dθ = Λ ∈ (C.76) θ 1 (O Λ when 0 / x; y Zx ∈ h i as Λ . → ∞ 13This is why bounded second derivative is required.

93 Proof. If 0 / (x; y) relation (C.68) can be applied (q(θ)/θ has bounded derivatives ∈ 1 up to second in the integration interval, . . . ). So it has to vanish like O (Λ− ). In the other case, i.e. x < 0 < y, it is useful to separate “singular part” of the integrand

y y y q(θ) q(θ) q(0) q(0) sin(Λθ)dθ = − sin(Λθ)dθ + sin(Λθ)dθ. (C.77) θ θ θ Zx Zx Zx As we can see (q(θ) q(0))/θ is regular in the vicinity of θ = 0 and it has bounded ﬁrst derivative (assuming− the value at zero is deﬁned suitably). Therefore we can 1 again use (C.68) to prove that it vanishes like O (Λ− ) as Λ . The rest is given by → ∞

y Λy q(0) sin(ϑ) sin(Λθ)dθ = q(0) dϑ = q(0) [Si (Λy) Si (Λx)] = θ ϑ − Zx ΛZx 1 = q(0) π + O , (C.78) Λ where Si (a) denotes sine integral and its asymptotic behaviour has been used — π 1 it tends to /2 as a (up to the part vanishing like O (Λ− ), see [DLMF, (6.2.14)]). ± → ±∞ Proof of (C.64). Let us begin with the question of convergence. As it is seen in λ (C.67) terms of the series decreases (grows) like n− . Therefore when λ 0 the series diverges14. On the other hand if λ > 1 the series converges absolutely.≤ The situation is less clear when λ (0; 1 . The proof of conditional convergence will be carried out using Dirichlet’s∈ criterion.i Let 16 a = , (C.79) n 8λΓ3(λ)nλ 3 sin((λ + n)θ ) b = i . n sin θ i=1 i Y λ N Then an vanishes monotonously as n− when n . Moreover n=1 bn is 15 → ∞ bounded . So ∞ anbn converges. However an does not converge absolutely n=0 P and bn does not vanish as n . Therefore only conditional convergence takes place. P → ∞

14This forms the requirement of positive λ. 15Because

3 sin((λ + n)θ ) sin ((λ + n)( θ + θ + θ )) + sin ((λ + n)(θ θ + θ )) i = − 1 2 3 1 − 2 3 sin θ 4 sin θ sin θ sin θ i=1 i 1 2 3 Y sin ((λ + n)(θ + θ θ )) + sin ((λ + n)(θ + θ + θ )) + 1 2 − 3 1 2 3 (C.80) 4 sin θ1 sin θ2 sin θ3 and

N N sin((λ + n)Θ) = ei(λ+n)Θ = = " # n=1 n=1 X X

94 The r.h.s. of (C.67) contains also remainder disappearing at least as fast as 1 λ n− − . This remainder is clearly absolutely convergent and so it does not change behaviour of the series as n . → ∞ Denoting y = cos θ1 and z = cos θ2 and using (C.55) it can be shown that

2 ∞ Γ(l + 1) 2(l + λ) C(λ)(x)C(λ)(cos θ )C(λ)(cos θ ) = (C.82) Γ(l + 2λ) l l 1 l 2 Xl=0 N 2 Γ(l + 1) (λ) (λ) (λ) = lim 2(l + λ) Cl (x)Cl (cos θ1)Cl (cos θ2) = N l λ →∞ Γ( + 2 ) Xl=0 π N 2 2λ 1 Γ(l + 1) (λ) (λ) = lim (sin φ) − 2(l + λ) Cl (x)Cl (ξ) dφ = N 2λ 2 λ l λ →∞ 2 Γ ( ) ( Γ( + 2 ) ) Z0 Xl=0 π (λ) (λ) (λ) (λ) 2 Γ(N + 2) 2λ 1 Cn+1(x)CN (ξ) CN (ξ)CN+1(ξ) = lim (sin φ) − − dφ = N 2λ 2 λ N λ x ξ →∞ 2 Γ ( ) Γ( + 2 ) Z0 −

cos(θ1 θ2) − λ 1 2 Γ(N + 2) [(cos(θ θ ) ξ)(ξ cos(θ + θ ))] − = lim 1 − 2 − − 1 2 N 2λ 2 λ N λ 2λ 1 →∞ 2 Γ ( ) Γ( + 2 ) [sin θ1 sin θ2] − cos(θZ1+θ2) C(λ) (x)C(λ)(ξ) C(λ)(ξ)C(λ) (ξ) n+1 N − N N+1 dξ, × x ξ − where ξ = cos θ1 cos θ2 + sin θ1 sin θ2 cos φ. Let us make a few notes concerning regularity of the integrated term. The denominator of the last fraction vanishes when ξ = x. However there is no (λ) (λ) (λ) (λ) singularity. The nominator Cn+1(x)CN (ξ) CN (ξ)CN+1(ξ) vanishes there as + − well. Moreover because N Z0 the nominator is polynomial it can be written in the form (x ξ)P oly(x, ξ∈), where P oly(x, ξ) stands for some other (suitable) polynomial of x−and ξ. λ 1 Another potential threat arises from [(cos(θ1 θ2) ξ)(ξ cos(θ1 + θ2))] − . With λ < 1 the term is singular at both ends of− the integration− − interval16. How- x ever such singularities are of the type 0 x− dx with (0; 1) and so their integral is ﬁnite. ∈ R (λ) (λ) To proceed further examination of asymptotic behaviour of C (x)C (ξ) n+1 N − 1 0 when Θ = 2kπ λ = k0 + 2 ∧ 1 =  when Θ = 2kπ λ = k0 + . ±∞ ∧ 6 2  iλΘ 1 eiΘ(N+1) 2  e −1 eiΘ 1 cos Θ otherwise = − ≤ − h i  The condition of convergence read θ1 θ2 θ3 = kπ for any choice of signs. Equivalently it can be given by ± ± 6

0 = sin (θ + θ + θ ) sin ( θ + θ + θ ) sin (θ θ + θ ) sin (θ + θ θ ) = 6 1 2 3 − 1 2 3 1 − 2 3 1 2 − 3 = (1 cos2 θ cos2 θ cos2 θ )2 4 cos2 θ cos2 θ cos2 θ . (C.81) − 1 − 2 − 3 − 1 2 3

16It should be noted the power is well deﬁned because the bracket is not negative inside the integration interval.

95 C(λ)(ξ)C(λ) (ξ) as N is necessary. Using (C.16) yields N N+1 → ∞ C(λ) (x)C(λ)(ξ) C(λ)(ξ)C(λ) (ξ) = R (ψ, ϑ) + (C.83) n+1 N − N N+1 N 2λ 2 N − 1 ϑ ψ + sin λ + N + (ψ + ϑ) sin − + 22λ 2Γ2(λ) sinλ ψ sinλ ϑ 2 2 − 1 ψ + ϑ + sin λ + N + (ϑ ψ) sin , 2 − 2 2λ 3 where RN (ψ, ϑ) = O N − and ψ, ϑ (0; π) are deﬁned by x = cos ψ, ξ = cos ϑ. The l.h.s. of equation (C.83) vanishes∈ at least as fast as x ξ when x ξ. The r.h.s., without the remainder R (ψ, ϑ), behaves similarly.− Therefore → N remainder RN (ψ, ϑ) must vanish there as well. Inserting this into (C.82) leads to (omitting the limit and WLOG assuming θ θ ) 1 ≥ 2 N Γ(l + 1) 2 2(l + λ) C(λ)(x)C(λ)(cos θ )C(λ)(cos θ ) = (C.84) Γ(l + 2λ) l l 2 l 2 Xl=0 cos(θ1 θ2) − λ 1 2 Γ(N + 2) [(cos(θ θ ) ξ)(ξ cos(θ + θ ))] − R (ψ, ϑ) 1 2 1 2 N ξ = 2λ 2 − − − 2λ 1 d + 2 Γ (λ) Γ(N + 2λ) [sin θ1 sin θ2] − x ξ cos(θZ1+θ2) −

θ1+θ2 2λ 2 λ 1 8N Γ(N + 2) [(cos(θ θ ) cos ϑ)(cos ϑ cos(θ + θ ))] − + − 1 2 1 2 4λ 4 − − 2λ 1 λ− λ 1 2 Γ (λ) Γ(N + 2λ) [sin θ1 sin θ2] − sin ψ sin − ϑ × θ1Zθ2 − 1 ϑ ψ 1 ψ+ϑ sin λ+N + (ψ+ϑ) sin − + sin λ+N + (ϑ ψ) sin 2 2 2 − 2 dϑ = × cos ψ cos ϑ − 41 2λ = − 4 2λ 1 λ Γ (λ) [sin θ1 sin θ2] − sin ψ ×

θ1+θ2 sin [Λ(ψ + ϑ)] sin [Λ(ϑ ψ)] 1 T (ϑ) + − dϑ + O , × ϑ+ψ ϑ ψ N ( sin 2 sin −2 ) θ1Zθ2 − 1 where Λ = n + λ + /2,

λ 1 (cos(θ θ ) cos ϑ)(cos ϑ cos(θ + θ )) − T (ϑ) = 1 − 2 − − 1 2 (C.85) sin ϑ and the upper limit of the integral assumes17 θ + θ π. Moreover we have 1 2 ≤ used the fact, that the integral containing remainder RN (ψ, ϑ) is regular (see 1 previous paragraph) and it vanishes like O (N − ) when considered with rest of the integrand. The expression T (ϑ) is potentially singular. When λ > 1 the problem causes sin ϑ in the denominator. This term vanishes only when ϑ = 0 or ϑ = π. Based on bounds of the integration ϑ can have such value only if θ1 = θ2 or θ1 + θ2 = π

17 If θ1 + θ2 > π then it should be replaced by 2π θ1 θ2 in the upper integration limit and the rest of the proof has to be modiﬁed accordingly.− Because− nothing new is necessary for this modiﬁcation explicit discussion is not given here.

96 ensuring regularity of T (ϑ). In the other case, λ (0; 1), the divergence is of the ∈ order x− and its discussion was performed above — its integral is ﬁnite. 1 λ ϑ+ψ 18 Because T (ϑ) diverges at most like x − and sin 2 is regular (C.68) can be used to prove θ1+θ2 sin [Λ(ψ + ϑ)] min(1,λ) T (ϑ) ϑ+ψ dϑ = O Λ− (C.86) sin 2 θ1Zθ2 − The second part of integral (containing ϑ ψ) is a bit more complicated due to the (possible) existence of singularity when− ψ = ϑ. First let us isolate that singularity 1 2 ϑ ψ = + f(ϑ, ψ), (C.87) sin − ϑ ψ 2 − where f(ϑ, ψ) is suitable function that does not tend to inﬁnity when ϑ ψ. Therefore →

θ1+θ2 sin [Λ(ϑ ψ)] T (ϑ) ϑ −ψ dϑ = sin −2 θ1Zθ2 − θ1+θ2 θ1+θ2 sin [Λ(ϑ ψ)] = 2T (ϑ) − dϑ + T (ϑ)f(ϑ, ψ) sin [Λ(ϑ ψ)] dϑ = ϑ ψ − θ1Zθ2 − θ1Zθ2 − − 2πT (ψ) + O Λ min(1,λ) when ψ (θ θ ; θ + θ ) − 1 2 1 2 , = min(1,λ) ∈ − (C.88) O Λ− when ψ / θ θ ; θ + θ ( ∈ h 1 − 2 1 2i where we have used (C.68) to prove that the second integral vanishes and (C.76) to find result of the first integral (in the limit N, Λ + ). All together forms relation → ∞ n Γ(l + 1) 2 2(l + λ) C(λ)cos ψC(λ)cos θ C(λ)cos θ = (C.89) Γ(l + 2λ) n n 1 n 2 l=0 X λ 1 θ1+θ2+ψ θ1+θ2+ψ θ1 θ2+ψ θ1+θ2 ψ π sin sin − sin − sin − − = 2 2 2 2 4 2λ 1 Γ (λ) [2 sin θ1sin θ2 sin ψ] − × 1 H θ2 (ψ θ )2 + O . × 2 − − 1 nmin(1,λ) 1 In case of Legendre functions (λ = /2) the result correspond exactly to [Baranov, 2006, (2.21)]. However at the first glance the result is not symmetric with respect to the x, cos θ1, cos θ2 as it should be. Also we have assumed θ1 + θ2 π. Careful analysis reveals the argument of the Heaviside function can be replaced≤ with the content of the square bracket with sines and it whichs symmetry is evident. Moreover this symmetric form need not to assume θ1 + θ2 π (the proof is analogous) and therefore the final result, after using limit n ≤ , states → ∞ 2 ∞ Γ(l + 1) 2(l + λ) C(λ)(x)C(λ)(y)C(λ)(z) = (C.90) Γ(l + 2λ) n n n Xl=0 18ψ 0; π and ϑ (0; π). ∈ h i ∈ 97 2 2 2 λ 1 π[1 x y z + 2xyz] − = − − − H 1 x2 y2 z2 + 2xyz , λ 1 4 λ 1 2 2 2 2 − − − Γ (λ)4 − [(1 x )(1 y )(1 z )] − − − − where, as mentioned above, x, y, z ( 1; 1). The convergence is as fast as min(1,λ) ∈ − O N − . C.4.2 DCC Triple product of Gegenbauer functions (DCC) reads

2 ∞ Γ(l + 1) 2(l + λ) D(λ)(x)C(λ)(y)C(λ)(z) = Γ(l + 2λ) l l l l=0 X 1 2λ π (x2 1) −2 1 , 1 1 y2 z2 + y2z2 = − F 2 ; − − , (C.91) 22λ 1Γ2(λ)Γ(2λ)(x yz) 2 1 λ + 1 (x yz)2 − − 2 − a, b where F ; x stands for the Gauss hypergeometric function, x > 1 + , 2 1 c y, z 1; 1 andλ (0; + ) and the convergence is absolute and uniform. ∈ h− i ∈ ∞ 1 One can also check the formula with λ = /2 corresponds to the result obtained by Baranov Baranov [2006]. Speciﬁc cases of (C.91) that are main focus are given by ∞ 1 (2l + 1)Ql(x)Pl(y)Pl(z) = , (C.92) x2 + y2 + z2 2xyz 1 Xl=0 − − abd p

∞ 2l + 3 D(3/2)(x)C(3/2)(y)C(3/2)(z) = (l + 1)2(l + 2)2 l l l Xl=0 x yz x2 + y2 + z2 2xyz 1 = − − − − . (C.93) (x2 1)(1 y2)(1 z2) p− − − During proof of (C.91) integral

π 2λ 1 1 1 2 (sin φ) − B λ, , 1 b φ 2 2 , d = 2F1 1 ; 2 (C.94) a b cos φ a λ + 2 a Z0 − where B(x, y) denotes beta function is encountered. Such relation holds when b > a > 0. | | | | Proof.

π π l 2λ 1 ∞ (sin φ) − 1 b 2λ 1 l dφ = (sin φ) − (cos φ) dφ = (C.95) a b cos φ a a Z0 − Z0 Xl=0 π l ∞ 1 b 2λ 1 l = (sin φ) − (cos φ) dφ = a a Xl=0 Z0

98 π 2l ∞ 1 b 2λ 1 2l = (sin φ) − (cos φ) dφ = a a Xl=0 Z0 2l 1 1 ∞ b 1 B λ, 1 , 1 b2 = B λ, l + = 2 F 2 ; , a a 2 a 2 1 λ + 1 a2 l=0 2 X where the fact that series decreases exponentially fast allows to use Fubini’s theorem. Proof of (C.91). Starting with convergence [DLMF, (18.10.4)] can be used to (λ) (λ) prove Cn (ξ) < Xλ Cn (1) for suitable constants Xλ. Assuming l large enough 19, | | | |

2 Γ(l + 1) (λ) (λ) (λ) 2(l + λ) Dl (x)Cl (y)Cl (z) < Γ(l + 2λ) λ πX2 λ l < x2 − 2 λ lλ x √x2 − − . 1 λ 3 2 + 1 (C.96) − 2 − Γ(λ)Γ (2λ) − The last (exponential) part dominate behaviour of the series and so it the original sum (C.91) is absolutely convergent. Moreover x2 1 > 2 and x+√x2 1 > 1+. Therefore (assuming 1 < p q and large enough−20) − ≤ q 2 Γ(l + 1) (λ) (λ) (λ) 2(l + λ) Dl (x)Cl (y)Cl (z) < Γ(l + 2λ) Xl=p 2 p 16πXλ λ < 3λ λ+2 p 1 , (C.97) 2 2 2 λ 1 − 1(1 + ) 2 Γ(λ)Γ (2λ) (1 + ) − which clearly vanishes as p . Also it does not depend on x, y and z. Cauchy criterion proves uniform convergence→ ∞ of the series. Again relation (C.55) was employed to convert single Gegenbauer polynomial in (C.39) into product of two of them. n Γ(l + 1) 2 2(l + λ) D(λ)(x)C(λ)(cos θ )C(λ)(cos θ ) = (C.98) Γ(l + 2λ) l l 1 l 2 Xl=0 π n 2 2λ 1 Γ(l + 1) (λ) (λ) = (sin φ) − 2(l + λ) D (x)C (ξ)dφ = 22λΓ2(λ) Γ(l + 2λ) l l Z0 Xl=0 π 2 (sin φ)2λ 1 Γ(n + 2) = − C(λ) (ξ)D(λ)(x) 22λΓ2(λ) ξ x Γ(n + 2λ) n+1 n − Z0 − h 1 2λ 2π x2 1 −2 (λ) ξ (λ) x φ Cn ( )Dn+1( ) | λ −2 | d = − − 4 Γ (λ) ) i cos(θ1+θ2) λ 1 2 [(cos(θ1 θ2) ξ)(ξ cos(θ1 + θ2)] − Γ(n + 2) = 2λ 2 − − 2λ−1 2 Γ (λ) [sin θ1 sin θ2] − (ξ x) Γ(n + 2λ) × cos(θZ1+θ2) −

19 1 l > λ and the remainder in (C.17) O l < 1. | | λ 20That is that assumptions of (C.96) holds and p > 1 , i.e. l+1 < 1 + . (1+/2)1/λ 1 l 2 − 99 1 2λ 2π x2 1 −2 (λ) ξ (λ) x (λ) ξ (λ) x ξ, Cn+1( )Dn ( ) Cn ( )Dn+1( ) | λ −2 | d × − − 4 Γ (λ) ) h i where ξ = cos θ1 cos θ2 + sin θ1 sin θ2 cos φ (as in the previous proof). The integral can be separated into two parts — one containing cross product of Gegenbauer functions of the ﬁrst and second kind21 and the one independent on n. Now for the evaluation of the limit itself. Because we are interested in the (λ) limit n + the ﬁrst part of the integral vanish (exponential decrease Dn (x) dominate→ its∞ behaviour). Therefore

2 ∞ Γ(l + 1) 2(l + λ) D(λ)(x)C(λ)(cos θ )C(λ)(cos θ ) = Γ(l + 2λ) l l 1 l 2 l=0 X 1 2λ π π x2 1 −2 (sin φ)2λ 1 − φ. = |4λ −2 4| d = 2 − Γ (λ) x cos θ1 cos θ2 sin θ1 sin θ2 cos φ Z0 − − 1 2λ π (x2 1) −2 1 , 1 1 y2 z2 + y2z2 = − F 2 ; − − , 22λ 1Γ2(λ)Γ(2λ)(x yz) 2 1 λ + 1 (x yz)2 − − 2 − where (C.94) has been used. It could be useful to note that the restriction posed on arguments of (C.91) can be loosened — it holds as long as y, z ( 1; + ) and a > max(y, 1) max(z, 1) + (max(y, 1)2 1)(max(z, 1)2 1). As∈ n− ∞the function summed in (C.91) vanishes exponentially.− Due− to that is can→ ∞ be easily seen that partial sums p l ( n=1 ... with arbitrary l) are uniformly convergent (as a function of y and z). This holds even when y and z are complex numbers where above criterion holdsP for their real part and their imaginary part is suﬃciently small (i.e. in the vicinity of real axis). Also partial sums are analytic functions of y and z (partial sums are just polynomials). Therefore the series is convergent and the result is analytic function of (complex) variables y and z. Uniqueness theorem for functions of complex variables allows us to loosen the requirement y, z ( 1; 1). ∈ − C.4.3 DDC, DDD The other two kinds of series containing triple product of Gegenbauer functions, i.e. 2 ∞ Γ(l + 1) 2(l + λ) D(λ)(x)D(λ)(y)C(λ)(z) (C.99) Γ(l + 2λ) l l l Xl=0 and 2 ∞ Γ(l + 1) 2(l + λ) D(λ)(x)D(λ)(y)D(λ)(z) (C.100) Γ(l + 2λ) l l l Xl=0 can be obtained using (C.56) on (C.91) once (twice). No additional problem arises from the exchange of order of sum and integral, because the series convergence is uniform22. Because of straightforwardness of the procedure and not using them

21(ξ x) in the denominator does not introduce singularity. It can be shown analogously to the proof− of regularity of the integrand in proof of (C.64). 22This is in contrast with (C.64).

100 in the thesis explicit relations is not given. Reader can ﬁnish them (including proof) if necessary.

C.5 Quadruple product

C.5.1 CCCC Assume x, y, z, t ( 1; +1). Then ∈ − 3 ∞ Γ(l + 1) 2(l + λ) C(λ)(x)C(λ)(y)C(λ)(z)C(λ)(t) = (C.101) Γ(l + 2λ) l l l l l=0 X 2λ 1 λ 1 < > − > < > > − 16π M M+ M+ M+ M M+ = − − − − − 6λ 4 λ 1 2 Γ (λ)Γ(2λ) 2 × (1 x2)(1 y2)(1 z2)(1 t2) 1 M >2 − − − − − − + h 1 1 i F(4) λ; 1 λ, 1 λ, λ , λ ; 2λ; × D − − − 2 − 2 > < < > < > < > M+ M M M+ M M+ M M+ ; > − −< , −> − > , − − > , − − > , M+ M+ M M+ − 1 + M+ 1 M+ − − − − (4) where FD (a; b1, b2, b3, b4; c; x1, x2, x3, x4) denotes Lauricella function of four vari- ables23, M >, M < stands for ± ± M > = max xy (1 x2)(1 y2), t (1 z2)(1 t2) , (C.102) ± ± − − ± − − h i M < = min xy p(1 x2)(1 y2), t p(1 z2)(1 t2) (C.103) ± ± − − ± − − h i p 1 p 1 and its convergence is absolute when λ > /2. If λ (0; /2 the convergence is conditional (Dirichlet’s criterion is employed) and additional∈ i condition(s) arccos x arccos y arccos z arccos t = kπ (for all combinations of signs) must hold. ± Proof± of this theorem± is not6 given here. It follows closely the proof of (C.64), only using (C.55) twice.

C.5.2 DCCC Let a > 1, x, y, z ( 1; 1) and ∈ − 3 2 1 λ ∞ Γ(l + 1) π(a 1) 2 l λ (λ) a (λ) x (λ) y (λ) z − 2( + ) Dl ( )Cl ( )Cl ( )Cl ( ) = 4λ 2 4− Γ(l + 2λ) 2 − Γ (λ)Γ(2λ) · Xl=0 23

∞ (a) (b ) (b ) (b ) (b ) F(4) (a; b , b , b , b ; c; x , x , x , x ) i1+i2+i3+i4 1 i1 2 i2 3 i3 4 i4 xi1 xi2 xi3 xi4, D 1 2 3 4 1 2 3 4 ≡ (c ) (c ) (c ) (c ) i !i !i !i ! 1 2 3 4 i ,i ,i ,i =0 1 i1 2 i2 3 i3 4 i4 1 2 3 4 1 2X3 4 where (a)n = Γ(a + n)/Γ(a) denotes Pochhammer Symbol. In this appendix integral relation

1 a 1 c a 1 (4) Γ(c) t − (1 t) − − dt FD (a; b1, b2, b3, b4; c; x1, x2, x3, x4) = − Γ(a)Γ(c a) (1 x t)b1 (1 x t)b2 (1 x t)b3 (1 x t)b4 − Z0 − 1 − 2 − 3 − 4 have signiﬁcant role.

101 yz+√(1 y2)(1 z2) − − 1 λ 1 · 2 2 2 × [(1 y )(1 z )] − Z − − yz √(1 y2)(1 z2) − − − λ 1 1 , 1 (1 x2)(1 ξ2) [1 y2 z2 ξ2 + 2yzξ] − F 2 ; − − − − − dξ, (C.104) × 2 1 λ + 1 (a xξ)2 a xξ 2 − − (λ) and the convergence is absolute (once again exponential decrease of Dl (a) dominates behaviour). To prove this relation it is suﬃcient to apply relation (C.55) on (C.91) and uniform convergence of triple product24. 1 Special case of Legendre functions (λ = /2) was already considered in Baranov [2006] (including result in closed from instead of integral (C.104)). It reads

∞ (2l + 1)Ql(a)Pl(x)Pl(y)Pl(z) = (C.105) Xl=0 yz+√(1 y2)(1 z2) − − dξ = π a2 + x2 + ξ2 1 2axξ 1 y2 z2 ξ2 + 2yzξ Z yz √(1 y2)(1 z2) − − − − − − − − p p 3 and for λ = /2 (C.104) reduces to

∞ 2l + 3 D(3/2)(a)C(3/2)(x)C(3/2)(y)C(3/2)(z) = (l + 1)3(l + 2)3 l l l l Xl=0 yz+√(1 y2)(1 z2) − − = 1 y2 z2 ξ2 + 2yzξ − − − × Z yz √(1 y2)(1 z2) p − − − a xξ + a2 + x2 + ξ2 1 2axξ − − − dξ. (C.106) ×π(ha2 1)(1 px2)(1 y2)(1 z2)(1 ξi2) − − − − − The requirements for (C.104) can be loosened. Analogously to the discussion performed for (C.91) the series converges as long as a + √a2 1 > max(x + √x2 1, 1) max(y + y2 1, 1) max(z + √z2 1, 1). − − − −(C.107) p Similarly (C.104) holds as long as the criterion above holds25.

C.5.3 DDCC, DDDC, DDDD Other series including four Gegenbauer functions can be solved using the same approach given by section C.4.3. Applying (multiple times) (C.56) on (C.104) and considering exponential convergence (to use Foubini’s theorem) lead to the desired result. Once again due to the straightforwardness of the procedure and that these sums are not used in this thesis these integrals are nor given here explicitly. 24In the case of quadruple product uniform convergence is present as well. 25The argumentation to prove that is analogous as well.

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103 DLMF. NIST Handbook of Mathematical Functions, 2010. URL http://dlmf. nist.gov/. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds. P C´ıˇzek.Linearˇ perturbations of a Schwarzschild black hole by thin disc. Journal of Physics: Conference Series, 314(1):012071, 2011. P. C´ıˇzekandˇ O. Semerák. Thin-disc perturbation of Schwarzschild black hole. In J. Safránkováandˇ J. Pavl˚u,editors, WDS’09 Proceedings of Contributed Papers: Part III - Physics, pages 38–45, Prague, 2009. MatfyzPress. ISBN 978-80-7378-103-3. P. C´ıˇzekandˇ O. Semerák.Linear perturbations of a Schwarzschild blackhole by thin disc - convergence. AIP Conference Proceedings, 1458(1):351–354, 2012. P. C´ıˇzekandˇ O. Semerák. Perturbation of a Schwarzschild black hole due to a rotating thin disc. Astrophysical Journal, 2017. Submitted. Arthur Erdelyi. Higher Transcendental Functions, volume 2. Krieger Publishing Company, Malabar, Florida, 1981. ISBN 0-89874-069-X. F. J. Ernst. New Formulation of the Axially Symmetric Gravitational Field Problem. Physical Review, 167:1175–1178, 1968. P. Grandclément and J. Novak. Spectral Methods for Numerical Relativity. Liv- ing Reviews in Relativity, 12(1), 2009. URL http://dx.doi.org/10.12942/ lrr-2009-1. J. T. Jebsen. Uber¨ die allgemeinen kugelsymmetrischen Lösungen der Einstein- schen Gravitationsgleichungen im Vakuum. Arkiv förMatematik, Astronomi och Fysik, 15(18):1–9, 1921. R. P. Kerr. Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Physical Review Letters, 11:237–238, 1963. P. W. Kuchel, Bulliman B. T., and Fackerell E. D. Bi-Cyclide and Flat-Ring Cyclide Coordinate Surfaces:Correction of Two Expressions. Mathematics of Computation, 49(160):607–613, 1987. W. Kundt and M. Trümper. Orthogonal decomposition of axi-symmetric stationary spacetimes. Zeitschrift fürPhysik, 192:419–422, 1966. H. Lass and L. Blitzer. The gravitational potential due to uniform disks and rings. Celestial Mechanics, 30(3):225–228, 1983. T. Morgan and L. Morgan. The Gravitational Field of a Disk. Physical Review, 183:1097–1101, 1969. G. Neugebauer and D. Kramer. Einstein-Maxwell solitons. Journal of Physics A: Mathematical and General, 16(9):1927, 1983. G. Neugebauer and R. Meinel. General Relativistic Gravitational Field of a Rigidly Rotating Disk of Dust: Axis Potential, Disk Metric, and Surface Mass Density. Physical Review Letters, 73:2166–2168, 1994.

104 G. Neugebauer and R. Meinel. General Relativistic Gravitational Field of a Rigidly Rotating Disk of Dust: Solution in Terms of Ultraelliptic Functions. Physical Review Letters, 75:3046–3047, 1995.

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S. A. Teukolsky. Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino ﬁeld perturbations. Astrophys- ical Journal, 185:635–647, 1973.

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C. M. Will. Perturbation of a slowly rotating black hole by a stationary axisymmetric ring of matter. I. Equilibrium conﬁgurations. The Astrophysical Journal, 191:521–531, 1974.

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105 Attachments

Papers created during my doctoral studies relevant to this thesis are attached below. C´ıˇzekandˇ Semer´ak[2012] is given as it was submitted (and accepted) only due to technical problems with obtaining published version.

Thin-disc Perturbation of a Schwarzschild Black Hole P. Cˇ´ıˇzek and O. Semer´ak Charles University in Prague, Faculty of Mathematics and Physics, Czech Republic

Abstract. The perturbation scheme used by Will (1974) is revisited in order to ﬁnd the metric for an (originally) Schwarzschild black hole surrounded by a light and slowly rotating ring. Focusing on linear order, we show that the method might also be adapted to the case of a thin disc. However, the expansions included are not much suitable for numerical processing due to a rather bad convergence.

Introduction Disc-like structures around very compact bodies play a key role in such highly active astrophysical sources like active galactic nuclei, X-ray binaries, supernovas and gamma-ray bursts. Very compact objects generate very strong and inhomogeneous gravitational fields, so general relativity has to be used in their description. The matter (ring or disc) around should also be included in this description, because its gravitational effect may not always be negligible, and because the properties of the accretion system can be very sensitive to the precise shape of the field. However, general relativity is non-linear and the fields of multiple sources cannot be obtained by a simple superposition. Recently, such more complicated fields are being successfully treated numerically (this even applies to strongly time-dependent cases including collisions and gravitational collapse), but, for the present, the compass of explicit “analytical” solution terminates at systems with very high degree of symmetry, factually at static and axially symmetric cases. It would be most desirable to extend this compass to stationary cases, namely those admitting rotation. The book by Klein & Richter (2005) summarises the exact-solution prospects in this direction. If the gravity of external matter is weak, the problem may be tackled as a small (stationary and axisymmetric) perturbation of the central-source metric, determined by linearised Einstein equations. The method can be iterated; in a limit case (many iterations), it goes over to a solution in terms of series. The result then need not any longer represent a tiny variation of some “almost right” metric — it may even be put together on Minkowski, with the “strong” part (e.g. a black hole) “dissolved” within the fundamental systems. The main problem here is convergence and meaning of the series. It is for 35 years now that the paper by Will (1974) remains the major reference in the subject. It provided the field of a slowly rotating and light thin equatorial ring around a black hole by mass-and- rotational perturbation of the Schwarzschild metric. Unfortunately, the perturbation strategy is strongly dependent on how “nice” the background metric is — and the Schwarzschild metric is exceptional in this respect: the Will’s procedure can neither be simply followed for the Kerr background, nor for a more complicated static one. Thus only partial questions have been answered explicitly in this direction, in particular the one of deformation of the black-hole horizon (Demianski 1976; Chrzanowski 1976). (Interestingly, these results do not agree on certain points, mainly in the limit of an extreme black hole.) In the present note, we check whether the Will’s scheme can be adapted to the case of a (finite, annular) thin disc. Although suggesting a positive answer, we also indicate that the expansions that come out in this approach converge rather badly and their numerical processing is problematic.

Einstein equations and boundary conditions for the hole–disc system We consider asymptotically ﬂat space-times (without the cosmological term) that are stationary, axially symmetric and circular (orthogonally transitive), i.e. where a time-like and a space-like Killing µ µ vector ﬁelds ηµ ∂x , ξµ ∂x exist and commute, and where the tangent planes to meridional ≡ ∂t ≡ ∂φ directions (orthogonal to both Killing vectors) are integrable. In the isotropic-type spheroidal coordinates the metric with these properties can e.g. be written in the “Bardeen-Carter” form1 (e.g. Bardeen 1973)

2 2ν 2 2 2 2ν 2 2 2ζ 2ν 2 2 2 ds = e dt + r B e− sin θ(dφ ωdt) + e − (dr + r dθ ), (1) − − where the unknown functions ν, B, ω, ζ only depend on meridional coordinates r and θ.

1Metric signature is ( +++) and geometrised units are used in which c = G = 1, Greek indices run 0-3 and partial derivative is denoted by a− comma.

38 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

Apart from the asymptotic flatness, the boundary conditions include those on the symmetry axis, on the black-hole horizon and on the external-source surface. Regularity of the axis requires that eζ B ζ ν→ there. Should the circumferential radius √gφφ grow linearly with proper cylindrical radius ρ[e − ]ρ=0 2 (thus with ρ r sin θ), there must be gφφ O(ρ ), and from finiteness of ω one also has gtφ = gφφω = 2 ≡ ∝ 2ν − O(ρ ). The stationary horizon is characterised by e = 0 and ω = const ωH (thus ω,θ = 0). Regularity 2ζ ≡ of gφφ and grr, gθθ then requires rB = 0 and e = 0 on the horizon. More specifically, in order that the azimuthal and latitudinal circumferences of the horizon be positive and finite, the latter has to be valid 2ν 2ζ 2ν for Be− and e − . (Let us add that one of the field equations implies that ω,r/B and ν,rω,r have to be finite on a regular horizon, so ω,r has to vanish there, too.) Let the external matter have the form of an infinitesimally thin disc in the equatorial plane, stretching over some interval of radii. We will assume that it bears neither charge nor current (no EM fields) and that the space-time is reflection symmetric with respect to its plane (z r cos θ = 0). The metric is ≡ then continuous everywhere, but has finite jumps in first normal derivatives gµν,z across the disc. The functions B, ν, ω, ζ are even in z, their z-derivatives are odd in z and their even powers and multiples (for example, B,zν,z) are even in z (therefore, the latter have no jumps on z = 0). In order that the space-time be stationary, axially symmetric and orthogonally transitive, the disc elements must only move along surfaces spanned by Killing fields, namely they must follow spatially dφ µ t circular orbits with steady angular velocity Ω = dt . This corresponds to four-velocity u = u (1, 0, 0, Ω), t t 2 2ν 2 2ν uα = u (gtt + gtφΩ, 0, 0, gtφ + gφφΩ), where (u )− = e (1 v ) with v = ρBe− (Ω ω) the linear velocity with respect to the local zero-angular-momentum observer− (ZAMO). For the thin− discs without ρ radial pressure (Tρ = 0), the surface energy-momentum tensor

z=0+ ∞ µ µ 2ζ 2ν µ µ 2ζ 2ν µ Tν gzz dz = Tν e − dz Sν (ρ) Tν e − Sν (ρ)δ(z) (2) z=0 ≡ ⇔ ≡ Z−∞ Z − t t φ φ t 2 t φ has only three non-zero components (St ,Sφ,Sφ ). If (Sφ St ) + 4SφSt 0, it can be diagonalised to µν µ ν µ ν −ν ζ ν ζ≥ S = σu u + P w w , where σ and P (more precisely, σe − and P e − ) stand for the surface density and azimuthal pressure in a co-moving frame and wµ is the unit “azimuthal” vector perpendicular to uµ, in components wµ = 1 (u , 0, 0, u ), w = ρB( uφ, 0, 0, ut). Hence the surface-tensor components ρB φ − t α − t φ t t φ φ read St = σ (σ + P )u uφ, Sφ = (σ + P )u uφ, Sφ = P + (σ + P )u uφ. Circular− space-times− are described by 5 independent Einstein equations. For our thin disc they read (ρ B) = 0 , (3) ∇ · ∇ 2 3 2 ρ B 2 2ζ 2ν φ t t (σ + P )(1 + v ) (B ν) ( ω) = 4πBe − (T 2ωT T ) = 4πB δ(z) , (4) ∇ · ∇ − 2e4ν ∇ φ − φ − t 1 v2 −

2 3 4ν 2ζ 2ν t 2 (σ + P ) v (ρ B e− ω) = 16πBe − T = 16πρB δ(z) , (5) ∇ · ∇ − φ − 1e2ν ( v2) − plus two equations for ζ,ρ and ζ,z (or, ζ,r and ζ,θ) which are integrable provided that the vacuum equations for ν and ω are satisfied; and denote gradient and divergence in fictive Euclidean three-space. The axis boundary condition ∇eζ = B∇·implies that the ζ function can elsewhere be obtained according to θ ζ(r, θ) = 0 ζ,θdθ + ln B, where ζ,θ follows from the last two field equations. µ Relation between the gµν,z jumps across the disc and Sν can now be obtained by integrating the R + field equations over the infinitesimal interval [z = 0−, z = 0 ]. Only the terms proportional to δ(z) (i.e. the source terms on the r.h. side and the terms linear in B,zz , ν,zz , ω,zz and ζ,zz on the l.h. side) z=0+ + + contribute according to ν,zzdz = 2ν,z(z = 0 ) (etc.), so we have (at z = 0 ) z=0− R (σ + P )(1 + v2) (σ + P )(Ω ω) σv2 + P ν (0+) = 2π , ω (0+) = 8π − , ζ (0+) = 4π . (6) ,z 1 v2 ,z − 1 v2 ,z 1 v2 − − − Thin discs in a counter-rotating interpretation In interpreting the counterweight of its (and the black hole’s) gravity, the disc may either be considered as a solid structure (a set of circular hoops), or, to the contrary, as a non-coherent mix of azimuthal streams. In the astrophysical context, one usually adheres to the latter extreme possibility: that the disc is composed of two non-interacting streams of particles following circular orbits in opposite azimuthal

39 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

ρ directions (e.g. Lynden-Bell & Pineault 1978; Ledvinka 1998). In the case without radial stress (Tρ = 0), these orbits are geodesic. The surface energy-momentum tensor is thus decomposed as µν µ ν µ ν S = σ+u+u+ + σ u u , (7) − − − where the +/ signs indicate the stream orbiting in a positive/negative sense of φ (with respect to uµ). The four-velocities− are of the uµ = ut (1, 0, 0, Ω ) form again, now with Ω corresponding to free ± ± 2 ± 2 ± motion, i.e. to the roots of gtt,µ + 2gtφ,µΩ + gφφ,µΩ = 0. Explicitly, ρ2ω + 2ρω(1 ρν ) ρ4(ω )2 + 4ρν e4ν (1 ρν ) Ω = ,ρ − ,ρ ± ,ρ ,ρ − ,ρ . (8) ± 12ρ( ρν ) p− ,ρ Comparing the two forms of the energy-momentum tensor, the parameters of the counter-rotating picture (σ+, σ ,Ω ) are found to be related to the “total”, one-stream parameters (σ, P , Ω) by − ± µ κ 2 P u+wµ u wκ 2 σ + P 1 + s − σ+ + σ = σ P, = ν λ ( s ) = = 2 , (9) − − σ − u+uν u uλ ≡ ⇒ σ+ + σ 1 s − − − tφ tt t 2 S Ω S σ (u ) = − ∓ . (10) ± ± ± Ω+ Ω − − Perturbation scheme k2 3 It is convenient to choose B = 1 4r2 as a solution of equation (3). With such a choice, the horizon lies where B = 0, hence on r = k/2.− The main task remains: to solve the coupled equations (4) and (5). We will expand the metric functions ν, ω and ζ as

∞ j ∞ j ∞ j ν = νjλ , ω = ωjλ , ζ = ζjλ , (11) j=0 j=0 j=0 X X X where the coeﬃcients νj, ωj and ζj depend on r and θ and the parameter λ is proportional to the disc mass. More precisely, let it be related by 1 1 π (σ + P )δ(z) λΣ(ρ)δ(z) = λΣ(r) δ(cos θ) = λΣ(r) δ θ , (12) ≡ r − r − 2 where δ denotes δ-distribution. The functions ν0, ω0 and ζ0 represent the black-hole background, i.e. they are given by the Schwarzschild metric (in isotropic coordinates) 2r M 2 M 4 ds2 = − dt2 + 1 + dr2 + r2dθ2 + r2 sin2 θdφ2 , (13) − 2r + M 2r thus 2r M ν = ln − , ω = 0, ζ = ln B. (14) 0 2r + M 0 0 Substituting (11), (12) into equations (4), (5), one obtains

k 2 k l 1 ∞ ∞ − ∞ − − k k 1 2 2 3 4ν0 j λ (B νk) = λ r sin θB e− exp 4 νjλ ωm ωk l m + ∇ · ∇ 2  −  ∇ · ∇ − −  k=0 k=2 l=0 j=1 m=1 X X  X X l X   2   1 ∞ k+1 1 + v  + 4πBΣ(r) δ (cos θ) λ 2 , (15) r 1 v k Xk=0 −

k 1 ∞ k 2 2 3 4ν ∞ k − 2 2 3 4ν ∞ j λ r sin θB e− 0 ω = λ r sin θB e− 0 exp 4 ν λ ω ∇ · ∇ k − ∇ ·   − j  ∇ l − k=0 k=2 l=1 j=1 X X X X k l     −  2 ∞ k+1 2ν v 16π sin θB Σ(r)δ (cos θ) λ e− 2 , (16) − 1 v k Xk=0 −

2The interpretation is only possible where the expression under the square root is non-negative; physically, this is not satisfied for discs with “too much matter on larger radii”. 3Different solutions are possible, but changing B generally does not make a real physical difference anyway — it in fact 1 corresponds to a certain re-definition of coordinates. Only the B = r sin θ choice leads to different, plane-wave solutions.

40 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

k where [f]k means the coefficient standing at λ in f. If the background is static, the first-order equations only contain mass-energy terms multiplied by the background metric on the r.h. sides, because the first-line sums do not contribute. In higher orders, the r.h. sides only contain lower-order terms then. (For non-static backgrounds the equations do not decouple so easily.) Now, the eigenfunctions with respect to θ of the operator on the l.h. side of 3/2 (15) and (16) are Legendre polynomials Pl(cos θ) and Gegenbauer polynomials Tl (cos θ), respectively. r M 2 Introducing a dimensionless radius x = M 1 + 4r2 , we may thus write ∞ ∞ 3/2 ν1 = ν1l(x)Pl(cos θ), ω1 = ω1l(x)Tl (cos θ) . (17) Xl=0 Xl=0 In the first λ-order, equations (15) and (16) then appear as

∞ d d 1 + v2 (x2 1) ν l(l + 1)ν P (cos θ) = 4πrΣ(r) δ(cos θ) , (18) dx − dx 1l − 1l l 1 v2 Xl=0 − ∞ d d (x+1)3 π(2r + M)3Σ(r) v δ (cos θ) (x+1)4 ω l(l+3)ω T 3/2(cos θ)= . (19) dx dx 1l − x 1 1l l − 24M 2( r M) 1 v2 sin θ Xl=0 − − − The source terms on the r.h. sides of (15), (16) (let us denote them R(r, θ), S(r, θ)) can also be decomposed thanks to the orthogonality relations valid for the polynomials; the l-th order terms read

1 2l + 1 1 + v2 R = R(r, θ)P (cos θ) d(cos θ) = 2(2l + 1)πP (0)rΣ(r) , (20) l 2 l l 1 v2 Z1 − − 1 2l + 3 πT 3/2(0)(2l + 3) (2r + M)3 Σ(r)v S = S(r, θ)T 3/2(cos θ) sin2 θ d(cos θ)= l .(21) l 2(l + 1)(l + 2) l − 8M 2(l + 1)(l + 2) (2r M) 1 v2 Z1 − − − Demanding the equality for each and every order, one thus arrives at equations (for every integer l 0) ≥ d d (x2 1) ν l(l + 1)ν = R , (22) dx − dx 1l − 1l l d d (x + 1)3 (x + 1)4 ω l(l + 3)ω = S . (23) dx dx 1l − x 1 1l l − Now, a general solution of a linear diﬀerential equation can be written as a sum of a particular solution and some function from its fundamental system. In our case the fundamental systems are vector spaces with dimension 2. Let us ﬁnd their bases. It is easy to verify that

∞ dξ Pl(x) and Ql(x) = Pl(x) 2 2 (24) ][Pl(ξ) (ξ 1) Zx − are two independent solutions of (22) without the r.h. side. Similarly (though not so obviously),

x + 1 ∞ dξ Fl(x) = 2F1 l, l + 3; 4; and Gl(x) = Fl(x) 2 4 (25) − 2 ][Fl(ξ) (ξ + 1) Zx represent the desired couple for (23); here 2F1(α, β; γ; z) stands for the hypergeometric function. Asymp- l l 1 l l 3 totically, for r , the basis polynomials behave as P r , Q r− − , F r and G r− − . → ∞ l → l → l → l → Near the horizon (r = M/2, i.e. x = 1), Pl and Fl are regular, but Ql and Gl diverge (except for G0). With the above fundamental system and the requirement of asymptotic ﬂatness and regularity on the horizon, the Green functions ( ) of the equations (22) and (23) have to look G ν Ql(x)Pl(x′) for x > x′ l (x, x′) = − , (26) G Pl(x)Ql(x′) for x < x′ −

41 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

ω Gl(x)Fl(x′) for x > x′ l (x, x′) = − . (27) G Fl(x)Gl(x′) for x < x′ − 4 For a thin ring, the surface density is Σ(r) = τ(1 + 2M/r) δ(r rring), where τ is the linear density (constant due to the axial symmetry). Substituting this into (20)− and (21) and subsequently into (22) and (23), the ﬁrst-order perturbation due to the ring ﬁnally comes out as

∞ 1 + v2 (2r + M)4 ν = (2l + 1)πτ P (0) ring ν (x, x )P (cos θ) , 1 1 v2 l 8r3 Gl ring l j=0 ring X − 3/2 ∞ πτT (0)(2l + 3) v (2r + M)7 ω = l ring ω(x, x )T 3/2(cos θ) . 1 − 8M 2(l + 1)(l + 2) 1 v2 r4 (2r M) Gl ring l j=0 ring ring X − − Two important properties can immediately be noticed. First, the perturbation of dragging is constant ω over the horizon (x = 1). Actually, Fl(1) = 0 for l > 0, so l (1, x′) = 0 for any x′ > 1; and, for l = 0, 3/2 ν G ω Tl (cos θ) = 1 independently of θ. Second, unlike l , the l is only uniquely defined (by boundary G G 3/2 conditions) for l > 0. Namely, it is possible to modify the perturbation to ω1′ = ω1 + CG0(x)T0 (cos θ) 1 (with C constant), because G0(x) = 3(x+1)3 which is regular on the horizon and vanishes at infinity. This modification satisfies the vacuum Einstein equation everywhere outside the black hole and is independent of θ as well. The constant C can be chosen in such a way, for instance, that ω1′ = 0 at the horizon (so the hole is kept non-rotating with respect to infinity). The mass and angular momentum of the system can be determined from Komar integrals which also allow to separate the contributions of the hole and of the disc. For the disc part, their original expression ρ in terms of Killing vector fields may be rewritten in various ways using the field equations. If Tρ = 0 z and Tz = 0, one has 1 + v2 M 2 = 2π (Sφ St) ρB dρ = 2π Σ(r) 1 r dr , (28) M φ − t 1 v2 − 4r2 discZ discZ − v (2r + M)4 = 2π St ρB dρ = 2π Σ(r) dr . (29) J φ 1 v2 16r2 discZ discZ − Perturbation by an annular thin disc Knowing the Green functions, one might attempt to find the thin-disc perturbation by “integrating over the rings”. However, this leads to complicated integrals, so we will rather try a different way. part part If we know some particular solution νl , ωl which satisfies equations (22), (23) between certain radii rin < r < rout above the horizon, we may try to modify it in such a way that it could be extended to the whole space-time, that it will satisfy the boundary conditions on the horizon and at infinity and that the respective energy-momentum tensor be interpretable as a thin finite (annular) equatorial disc with in out surface density Σ(r) between rin and rout. A natural guess is νl = Al Pl(x) at r < rin, νl = Bl Ql(x) = = part in out = = at r > rout and νl = Al Pl(x) + Bl Ql(x) + νl in between. The constants Al , Bl and Al , Bl should be fixed so that the function νl and its first derivative be continuous on the shells rin and rout. A suitable solution for ωl could be searched for in a similar manner. With the l-dependences adopted from (20), (21), we start from the r.h. sides 2l + 1 2l + 3 (x + 1)3 R = C P (0) ,S = C T 3/2(0) , (30) l R 2 l l S 2(l + 1)(l + 2) l )(x 1 − 4 where CR and CS are some constants. The respective particular solutions of (22), (23) are 0 for l odd part 2l+1 ν = CR Pl(0) for l > 0 even , l  − 2l(l+1) 1 C ln(x2 1) for l = 0  4 R − 0 for l odd  3/2 part 2l+3 CS 2l(l+1)(l+2)(l+3) Tl (0) for l > 0 even ωl =  − . (31)  C 1 1 + 1 + 1 + 2 ln(x 1) for l = 0  − S 4 1x+ (x+1)2 − 4 (x+1)3 −  h i 4  CR is dimensionless, while CS has the dimension of 1/length. They should be chosen so that the physical requirements on the corresponding energy-momentum tensor (energy conditions, in particular) be satisfied in the disc region.

42 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

Now we must ensure, in the above manner, that the solutions composed of these particular solutions and the respective fundamental systems match smoothly enough (C1) at the disc rims. Also, we demand the r.h. sides (30) to be interpretable as generated by a thin disc in the chosen radial range, i.e. we put them equal to (20), (21) and check how the radial functions Σ(r), v(r) thus determined look. Alternatively, in a counter-rotating picture of the disc, the functions to determine are surface densities of the geodesic streams σ+, σ , their velocities being ﬁxed to the Schwarzschild circular-geodesic value. − The explicit relations between (σ+, σ , v ) and (σ, P , v) are rather complicated in general, but in the Schwarzschild ﬁeld, where the two geodesic− orbital± velocities Ω are just opposite and the corresponding ± √Mr ZAMO-measured linear velocities are v = r M , they reduce to ± ± − 2 2 2 Σ + σ+ + σ Σ σ+ σ K √K 4L σ = − , P = − − − , v = − − , (32) 2 2 2L where

2 2 √K 4L 2 Σ = − 2 , K (σ+ + σ )(1 + v ) ,L (σ+ σ ) v+ . 1 v ≡ − ± ≡ − − − ± Due to them, the r.h. sides of (20), (21) can be rewritten, respectively,

2 3/2 3 1 + v πTl (0)(2l + 3) (2r + M) (σ+ σ )v+ 2(2l + 1)πPl(0)r(σ+ + σ ) ± , − − , − − 1 v2 − 8M 2(l + 1)(l + 2) (2r M) 1 v2 − ± − − ± which is then to be equated with the r.h. sides of (30) in order to find σ+ and σ corresponding to some − specific choice of CR and CS. Substituting into equations (28) and (29), we find the mass and angular momentum of the disc,

rout 2 2 CR M CR M = 1 2 dr = (rout rin) 1 , (33) M 2 − 4r 2 − − 4rinrout rZin rout C (2r + M)7 = S dr . (34) J − 32 2r4( r M) rZin −

Numerical example Now we give a specific example of the first-order annular-disc perturbation. We place the disc 1 between rin = 5M and rout = 6M and choose the constants in (30) as CR = 0.03, CS = 0.001M − . The corresponding disc mass and angular momentum are = 0.015M and = 0.12M 2.− Should the disc be interpreted as made of two counter-rotating geodesicM streams, the necessaryJ surface densities come out as shown in figure 1. The corresponding parameters of the “total” one-stream interpretation are drawn in figure 2. The energy conditions are clearly satisfied. Finally we will compute the linear metric perturbations ν1 and ω1 as described above. Restricting the infinite sums in orthogonal polynomials to their 14-th order, the total gravitational potential and the dragging angular velocity are obtained as shown in figure 3. Unfortunately, the terms of these sums (the magnitude of which is mainly given by the functions Ql and Gl, see (24) and (25)) decrease rather slowly and oscillate while almost compensating each other. Thus even with high precision numerics (the figures spent about a day on a handsome PC5) the resulting curves do not look satisfactory in detail (notice the anomaly near the z-axis in the ω-plot, in particular). This mainly applies to the radii where the disc exists. In addition, the latter’s profile in the θ-direction being δ-function, it would require very high frequency terms (i.e. high orders of the Legendre and Gegenbauer polynomials) in order to be well rendered by the decompositions. The problems are not so serious in the case of gravitational potential, because this is anyway dominated by the black hole. But exactly the opposite is true with dragging which is purely generated by the perturbing agent. We learned, by increasing the number of series terms, that the contours smooth out only very reluctantly.

5In order to make this statement observer independent, let us specify the PC has a 3GHz processor.

43 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

0.5 0.00014

0.00012 0.4

0.0001

0.3 8e±05

0.2 6e±05

4e±05

0.1

2e±05

0 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 0 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 r r

Figure 1. Left: Velocity magnitude v+ = v of the counter-rotating components, as measured by ZAMO. (The gray line indicates Keplerian− orbital− velocity in a pure Schwarzschild ﬁeld.) Right: ν ζ ν ζ Respective “real” surface densities σ+e − (upper curve) and σ e − (lower one). The radius is in the 1 − units of M, the surface densities in M − .

0.07 0.00032 0.0003 7e±05 0.06 0.00028 0.00026 6e±05 0.00024 0.05 0.00022 5e±05 0.0002 0.04 0.00018 4e±05 0.00016 0.03 0.00014 0.00012 3e±05 0.0001 0.02 8e±05 2e±05 6e±05 0.01 4e±05 1e±05 2e±05

0 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 0 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 0 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 r r r

ν ζ ν ζ Figure 2. Left: “Total” speed v. Middle: “Real” surface density σe − . Right: “Real” pressure P e − . 1 The radius is in the units of M and the surface density and pressure in M − .

6 6 z z

4 4

2 2

2 4 6 8 10 2 4 6 8 10 rho rho

Figure 3. Left: Contours of the gravitational potential ν = νSchwarzschild + ν1 in the meridional plane (ρ = r sin θ, z = r cos θ). The lines correspond to the potential levels 1.0, 0.9,..., 0.1 (as going − − − outwards from the centre). Right: Contours of the dragging angular velocity ω = ω1 (remember that 5 1 5 1 ω0 ωSchwarzschild = 0) in the same plane. The lines correspond to the values 9 10− M − ... 1 10− M − (as≡ going away from the disc at z = 0, 5M < ρ < 6M). The axes are in the units· of M. ·

Concluding remarks We have shown that the method used by Will (1974) in order to perturb the Schwarzschild metric by a light and slowly rotating ring can be adapted to the case of a thin annular disc. Focusing on the

44 Cˇ´IZˇEK AND SEMERAK:´ THIN DISC AROUND A SCHWARZSCHILD BLACK HOLE

first-order perturbation, we found that the formulas useful for analytic/symbolic calculations (angular decompositions, in particular) are less suitable for numerical evaluation. Actually, a very slow convergence of the series and instabilities encountered in counting up “the second-kind functions” may explain why no explicit results of this type have yet appeared in the literature (as far as we are aware). There remains a number of points that should be clarified in future, for instance the dependence of solutions on the parameters CR, CS, rin and rout (as illustrated by more numerical examples) or the effect of the perturbation on various properties of the field (like deformation of the horizon). Also, stability of the above obtained disc sources should be analysed in order to judge whether the solutions might have some (astro)physical relevance. A special question is the validity of the linear approximation. It cannot be answered in few general inequalities, it will be necessary to check, for each particular solution, whether the ν1(r, θ)- and ω1(r, θ)- terms are really negligible as compared with the ν0(r, θ)-terms. For the moment we can only add to our example that both the values encountered in Fig. 3 and the total mass of the disc seem to indicate that these conditions are satisfied in this case.

Acknowledgments. We thank for support from the grants GAUK-86508 and GACR-205/09/H033 (P.C.)ˇ and GACR-202/09/0772 (O.S.).

References Bardeen J. M., 1973, Rapidly Rotating Stars, Disks, and Black Holes, in Black Holes, Les Houches 1972, eds. C. DeWitt and B. S. DeWitt (New York: Gordon and Breach), p. 241 Bardeen J. M. and Wagoner R. V., 1971, Relativistic disks I. Uniform rotation, Astrophys. J. 167, 359 Chrzanowski P. L., 1976, Applications of metric perturbations of a rotating black hole: Distortion of the event horizon, Phys. Rev. D 13, 806 Demianski M., 1976, Stationary axially symmetric perturbations of a rotating black hole, Gen. Rel. Grav. 7, 551 Klein C., Richter O., 2005, Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods, Lect. Notes Phys. 685 (Springer, Berlin) Ledvinka T., 1998, Thin disks as sources of stationary axisymmetric electrovacuum spacetimes, Ph.D. Thesis (Charles Univ., Prague) Lynden-Bell D., Pineault S., 1978, Relativistic disks — I. Counter-rotating disks, Mon. Not. R. Astron. Soc. 185, 679 Will C. M., 1974, Perturbation of a slowly rotating black hole by a stationary ring of matter. I. Equilibrium conﬁgurations Astrophys. J. 191, 521

45 Spanish Relativity Meeting (ERE 2010): Gravity as a Crossroad in Physics IOP Publishing Journal of Physics: Conference Series 314 (2011) 012071 doi:10.1088/1742-6596/314/1/012071

Linear perturbations of a Schwarzschild black hole by thin disc

P. C´ıˇzekˇ Institute of theoretical physics, Faculty of Mathematics and Physics, Charles University E-mail: [email protected]

Abstract. The article written by C.M.Will (1974) describes a method how to obtain a perturbation of an (originally) Schwarzschild metric by a slowly rotating light ring. This scheme is revisited in order to adapt it to the case of a thin disc.It turns out for some class of mass distribution on the disc bounded between some radii the solution (or at least its decomposition into spherical harmonics) can be found explicitly.

1. Introduction Many of the very compact active astrophysical objects (for example active galactic nuclei and X-ray binaries) are believed to contain disc-like structures. However these very compact objects generates strong and inhomogenous gravitational field so Newton’s theory is not sufficient to describe them. On the other hand general relativity is non-linear and the simple superposition cannot be used. Recently these and even more complicated (for example time dependant) fields are successfully treated numerically. However so far the only known “analytical” solutions of this problem contains a very high degree of symmetry, usually they are axisymmetric and stationary. Even in this “symmetric” model the equations are quite complicated so it is a bit surprising in the static case the superposition can be used and the equations solved explicitly (Weyl class of solutions). The fundamental treatment of stationary axisymmetric thin discs can be found in the paper by Bardeen [1]. It was succeeded by the article by Will [2], which remains a major reference in this subject even now.

2. Basic concepts Considering axisymmetric stationary case we can write metric in form

2 2ν 2 2 2 2ν 2 2 2ζ 2ν 2 2 2 ds = e dt + r B e− sin θ(dϕ ωdt) + e − dr + r dθ , (1) − − where ν, ω, B and ζ stands for an unknown metric functions of coordinates r and θ. The energy-momentum tensor of thin dust disc can expressed

α 2ν 2ζ α 1 Tβ = σe − u uβr− δ(cos θ), (2)

Published under licence by IOP Publishing Ltd 1 Spanish Relativity Meeting (ERE 2010): Gravity as a Crossroad in Physics IOP Publishing Journal of Physics: Conference Series 314 (2011) 012071 doi:10.1088/1742-6596/314/1/012071

where σ describes surface mass density and uα is four-velocity of the fluid. It is convenient to ν α e− dφ write four-velocity in the form u = 1 v2 (1, 0, 0, Ω), where Ω = dt is coordinate angular velocity 2ν − 1 and v = r sin θBe− (Ω ω) linear velocity of the fluid. It can be shown function− B does not contain physically relevant information. As long as it satisfies the equation (r sin θ B) = 0(where the divergence ( ) and gradient ( ) operators are the same as in the∇ virtual · Euclidean∇ space-time with spherical∇· coordinates r, θ and∇ φ) it can be chosen arbitrary2 and the solutions for different B will only differ in the choice of coordinates. The important pair Einstein equations is

2 1 2 2 4ν 1 + v 1 (B ν) r sin θe− ω ω = 4πBσ δ(cos θ), (3) ∇ · ∇ − 2 ∇ · ∇ 1 v2 r 2 2 3 4ν 2− 2ν v r sin θB e− ω = 16πB σe− δ(cos θ) (4) ∇ · ∇ − 1 v2 − and the last two independent equations can be used to express ζ using known ν and ω. Now let us assume the background metric is perturbed by a thin dust disc. The metric function can be written in form ν = ν0 + ν1 + where the lower index corresponds with the power of the disc mass it contains (and analogously··· for the functions ω and ζ). In the subsequent text we will work only with the linear perturbations. The Schwarzschild black hole in the isotropic coordinates takes form

2r M 2 M 4 ds2 = − dt2 + 1 + dr2 + r2dθ2 + r2 sin2 θdφ2 , (5) − 2r + M 2r so the background metric functions are

M 2 2r M M 2 B = 1 , ν = ln − , ω = 0, ζ = ln 1 . (6) 4r2 0 2r + M 0 0 4r2 − − ! The linear perturbation of this black hole by disc will lead us to the equations

∂ ∂ν 1 ∂ ∂ν 1 + v2 (x2 1) 1 + sin θ 1 = 4πrσ δ(cos θ), (7) ∂x − ∂x sin θ ∂θ ∂θ 1 v2 − ∂ ∂ω (x + 1)3 1 ∂ ∂ω π (2r + M)3 v (x + 1)4 1 + sin3 θ 1 = σ δ(cos θ),(8) ∂x ∂x x 1 sin3 θ ∂θ ∂θ −4M 2 2r M 1 v2 − − − r 2 2 where x = M 1 + M /(4r ) is new radial coordinate. Considering the particles are moving on geodesics their velocity can be written3 as v = (x 1) 1/2. − − To solve these equations we will use expansion into the Legendre (for the function ν1) and 4 Gegenbauer (ω1) polynomials . So let us deﬁne

∞ ∞ 3/2 ν1 = ν1 n(x)Pn(cos θ), ω1 = ω1 n(x)Tn (cos θ). (9) nX=0 nX=0 1 With respect to the zero angular momentum observer. 2 1 With the exception B = r sin θ , which represents plane waves solutions instead of stationary axisymmetric ones. 3 In the zeroth order of expansion. We need not to consider ﬁrst order since the velocity is used in the source term of the equations, which is clearly proportional to the disc mass solely because of σ. 4 It can be shown the series converges absolutely anywhere besides the axis. On the axis ν1 converges absolutely, however in the case of ω1 it is still not clear. The conjecture is ω1 converges, but not absolutely.

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Moreover we will write the r.h.s. of Einstein equations using these polynomials 2 1 + v ∞ 4πrσ δ(cos θ) = R P (cos θ), where R =(4n + 2)πP (0)xχ(x) and (10) 1 v2 n n n n − nX=0 3 3/2 3/2 π (2r+M) σv ∞ πT (0)(2n+3)(x+1) χ(x) − δ(cos θ) = S T 3/2(cos θ), where S = .(11) 4M 2 2r M 1 v2 n n n 2M(n+1)(n+2) − − nX=0 − In both expressions χ(x) = rσ/(x 2) stands for the new function used to describe mass density. − Also we can see Rn = Sn = 0 when n is odd. After substitution into linearized Einstein equations and their separation we will obtain d dν (x2 1) 1 n n(n + 1)ν = R , (12) dx dx 1 n n − − d dω (x + 1)3 (x + 1)4 1 n n(n + 3) ω = S . (13) dx dx − x 1 1 n n − The base of fundamental system of the ﬁrst equation is the n-th Legendre polynomial Pn(x) and the Legendre function of the second kind Qn(x). Analogously we can obtain fundamental system of the second equation. It consists of the polynomial Fn(x) and the “function of the second kind” Gn(x) deﬁned as ∞ dξ Fn(x) = 2F1 ( n, n + 3, 4; (x + 1)/2),Gn(x) = Fn(x) 2 4 , (14) − [Fn(ξ)] (ξ + 1) Zx

where 2F1 (a, b, c; z) denotes hypergeometric function. Assuming asymptotic flatness and regularity of the perturbation one can construct corresponding Green functions ν ω (x, x′)= P [min(x, x′)]Q [max(x, x′)] and (x, x′)= F [min(x, x′)]G [max(x, x′)]. (15) Gn − n n Gn − n n These can be used to calculate first perturbation, however the integrals are quite complicated. In special case, where χ(x) is an even polynomial, particular solution can be found explicitly. Part ν1 n is exactly m m c xχ(x) = c P (x) νPart = 2l+1 P (x), (16) 2l+1 2l+1 ⇒ 1 n 2(l + 1)(2l + 1) n(n + 1) 2l+1 Xl=0 Xl=0 − Part where m and cj are some constants. The case of ω1 n is slightly more difficult, but it takes form

Part An(x) √x + 1 + √2 ω1 n = + CnFn(x) ln , (17) √x + 1 √x + 1 √2! − where An(x) stands for a suitable polynomial and Cn is some constant. However we would like to obtain solution with disc spanned between some radii rmin and rmax and we want the solution to be regular on the horizon and asymptotically ﬂat. Its uniqueness is clear from the uniqueness of the Green functions. It can be also described by

A n ≥ max where A<, A=, B= and B> are constants obtained by the the requirement of continuity of metric functions and their ﬁrst derivatives at the radii of the rims of the disc (and analogously we can express ω1 n).

3. Numerical examples Let us present some numerical examples of the procedure described above. In both cases we take M = 1, rmin = 5 and rmax = 6 and we consider only ﬁrst 30 orders in the “harmonic” 3/2 expansion (i.e. Pn(cos θ) and Tn (cos θ)).

3 Spanish Relativity Meeting (ERE 2010): Gravity as a Crossroad in Physics IOP Publishing Journal of Physics: Conference Series 314 (2011) 012071 doi:10.1088/1742-6596/314/1/012071

The ﬁrst one (Fig. 1 and 2) deals with χ = (40 x2) 105 and the second one (Fig. 3 and 4) 7 − · with χ = 10− .

5 8

3 z z 4 2

1 2

0 1 2 3 4 5 rho 2 4 6 8 10 rho Figure 1. Gravitational potential Figure 2. Dragging ω = ω0 + ω1. ν = ν0 + ν1. The lines corresponds with The lines corresponds with the values the potential values 1.5, 1.4, , 0.1 8 10 6, 7 10 6, , 1 10 6 . . − − ··· − · − · − ··· · −

8 10

8 6

z 6 z 4

2 2

0 2 4 6 8 10 12 rho 0 2 4 6 8 10 rho Figure 3. Gravitational poten- Figure 4. Dragging ω = ω + ω . tial ν = ν + ν . The lines cor- 0 1 0 1 The lines corresponds with the values responds with the potential values 10 10 9, 9 10 9, , 1 10 9 . 0.20, 0.18, , 0.04 . · − · − ··· · − − − ··· − 4. Conclusion Analogously to the paper by Will [2] we were able to obtain procedure calculating the ﬁrst order perturbations of Schwarzchild black hole by a thin dust disc for some class of matter distributions. However mentioned procedure is not suited well for second or higher order expansion in the mass because of the problems associated with multiplication of spherical harmonics. Moreover it is not well suited for the numerical calculations (as we can see in the numerical examples).

5. Acknowledgements I thank Oldˇrich Semer´akfor ideas and many enlightening conversations concerning this topic and Marcus Ansorg for the remarks concerning convergence of the series. Beside that I would like to acknowledge ﬁnancial support from the grants GAUK-86508 and GACR-205/09/H033.

References [1] Bardeen J.M., Wagoner R.V. 1971 Relativistic disks. I. Uniform rotation Ap.J. 167 359-423 [2] Will C.M. 1974 Perturbation of a slowly rotating black hole by a stationary axisymmetric ring of matter. I. equilibrium conﬁgurations Ap.J. 191 521-531

4 Linear perturbations of a Schwarzschild black hole by thin disc - convergence

P. Cížek,ˇ O. Semerák

Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague; E-mail: [email protected]

Abstract. In order to ﬁnd the perturbation of a Schwarzschild space-time due to a rotating thin disc, we try to adjust the method used by [4] in the case of perturbation by a one-dimensional ring. This involves solution of stationary axisymmetric Einstein’s equations in terms of spherical- harmonic expansions whose convergence however turned out questionable in numerical examples. Here we show, analytically, that the series are almost everywhere convergent, but in some regions the convergence is not absolute. Keywords: general relativity, black holes, perturbation techniques, accretion discs PACS: 04.70.Bw, 04.25.-g, 02.30.Lt , 04.20.-q >

LINEAR PERTURBATION OF BLACK HOLE

The orthogonally transitive, stationary and axisymmetric space-time can be described by the metric [1]

2 2ν 2 2 2 2ν 2 2 2ζ 2ν 2 2 2 ds = e dt + r B e− sin θ(dϕ ωdt) + e − dr + r dθ , (1) − − wwhere t, φ are Killing coordinates and r, θ are isotropic coordinates covering the meridional planes; B, ν, ω and ζ denote functions of r and θ which are determined by Einstein’s equations. In the thin-disc case, the energy-momentum tensor reads To calculate them we must specify the energy-momentum tensor. In the case of thin dust disc it can be expressed as

α 2ζ 2ν α 1 T = σe u uβ δ(cosθ), (2) β − r where σ is a surface density and uα is the disc-matter four-velocity which can be expressed as ν α e u = − (1,0,0,Ω) (3) 1 v2 − in terms of linear velocity with respect to the zero-angular-momentum observer v = 2ν r sinθBe− (Ω ω) (Ω = dφ/dt is the corresponding angular velocity at inﬁnity). The ﬁeld equation− for B reads

∇ (r sinθ∇B) = 0, (4) where ∇ and ∇ denote gradient and divergence in a Euclidean 3D space (represented in spherical coordinates r, θ, φ). Otherwise B can be chosen arbitrarily1. The most important Einstein equations are those for the dragging and gravitational potentials, ν and ω,

2 1 2 2 4ν 1 + v 1 ∇ (B∇ν) r sin θe− ∇ω ∇ω = 4πBσ δ(cosθ), (5) − 2 1 v2 r 2 2 3 4ν 2− 2ν v ∇ r sin θB e− ∇ω = 16πB σe− δ(cosθ). (6) − 1 v2 − Knowing B, ν and ω, the last function ζ can be obtained by line integration of the remaining two relevant ﬁeld equations. We are interested in a perturbation of the Schwarzschild metric which in isotropic coordinates reads

2r M 2 M 4 ds2 = − dt2 + 1 + dr2 + r2dθ 2 + r2 sin2 θdφ 2 (7) − 2r + M 2r ν 2r M ω 2 2 also it corresponds to = ln 2r+−M and = 0. We start from choosing B = 1 M /(4r ). Then one can rewrite the equations (5), (6) using the perturbation expansions−

2r M ∞ ν = ln − + ∑ δν (x)P (cosθ) + O(ε2), (8) 2r + M n n n=0 ∞ 3/2 2 ω = ∑ δωn(x)Cn (cosθ) + O(ε ), (9) n=0 (with the O-remainders omitted), where ε = disc mass/M is mass expansion parameter, r M2 3/2 x 1 + new "radial" coordinate and Pn and Cn are Legendre and Gegenbauer ≡ M 4r2 polynomials, respectively, while decomposing the source terms on the r.h. sides in the same manner. Demanding equality in each n-order, the equations (5), (6) leads to

2 d 2 d 1 + v (x 1) δνn n(n + 1)δνn = 2(2n + 1)πPn(0)rσ(r) (10) dx − dx − 1 v2 − 3 π 3/2 3 σ d 4 d (x + 1) Cn (0)(2n + 3)(2r + M) (r)v (x + 1) δωn n(n + 3)δωn = . dx dx − x 1 −8M2(n + 1)(n + 2)(2r M) 1 v2 − − − (11) This set can be solved to obtain expansions of the linear perturbation of functions ν and ω which are induced by the chosen source (thin disc) and which can be made regular both at the horizon and at radial inﬁnity2; see e.g. [2].

1 With the exception of the B = 1/(r sinθ) case which leads to plane-wave space-times. 2 One can choose freely one constant which represents angular velocity of the horizon. We will omit it since it is not important for convergence. Besides that the system is determined. TABLE 1. Coefﬁcients of Green functions of the problem

αβ γ δ Nn(x,x′) Ln(x,x′) Σ(r) γ 2 ν 0 0 1 20 1 2π 2n 1 C 0 rσ r 1+v Gn / ( + ) n( ) ( ) 1 v2 − γ 3/2 − ω n+3 π(2n+3)(n+3)Cn(0)(x 1)√x 1(x +1) v 1 3 3 2 1 x 1 x 1 − ′− ′ rσ r Gn / ( )( ′ ) n 2Mn(n+1)(n+2) ( ) 1 v2 − − −

However, numerical illustrations of the results revealed unsatisfactory behaviour in the region close to the axis, mainly of the ω function. Suspecting bad convergence of the employed perturbation series, we have tried to check this issue analytically. As brieﬂy summarized below, we found that the ω-expansion does not converge absolutely in certain regions and that its convergence may become problematic at certain parts of the axis.

CONVERGENCE OF THE SERIES

The general disc solution can be obtained convolving rings of matter. This ring solution (see [4]) corresponds (up to the numerical factor) to the Green function of equation,

(α,β) (α,β) Pn δ (x)Qn δ (x′) x < x′ Gn(x,x′) = Nn(x,x′) (−α,β) (−α,β) , (12) Pn δ (x′)Qn δ (x) x′ < x − − (α,β) (α,β) where Pn , Qn are Jacobi functions of the ﬁrst and second kind and the coefﬁcients α, β, δ and the function N(x,x′) are written explicitly in the table 1. The whole disc solution takes form ∞ θ γ θ Σ (α,β) (α,β) f (r, ) = ∑ Cn(cos )Ln(x,x′) (x′)Pn δ (min(x,x′))Qn δ (max(x,x′))dx′, (13) n=0 disc − − where f r θ is ν or ω and functions and constants are chosen in accordance to the table ( , ) γ 1. It should be noted Cn(0) = 0 when n is odd so in the rest of this paper we will consider only "even" contributions to the Green functions. To analyze asymptotic behaviour (with respect to n ∞) it is convenient to express Jacobi function in terms of Legendre functions: →

4 4 1 P(1,3)(x)Q(1,3)(x ) = H(x,x ) ∑ ∑ X 1 + O P (x)Q (x ), (14) n n ′ ′ kl n n+k n+l ′ k=0 l=0 where H(x,x′) is rational function of x and x′ and Xkl are constants. Using relations for modiﬁed Bessel functions I0 and K0 (see [3]) we can write

ζ ξ ξζ 2n+2k+1 ζ 2n+1 ξ 1 Pn+k(cosh )Qn(cosh ) = sinhξ sinhζ I0 2 K0 2 1 + O n =(15) 1 ζk (ζ ξ)(n+ 1 ) 1 = e e − 2 1 + O . 2n√sinhξ sinhζ n We will also need to know asymptotic behaviour of the "spherical harmonics"

γ 1 γ θ π γ 1 2n − [cos((n+ ) )+O( )] − 2 n when 0 < θ < π γ θ Γ(γ) Cn(cos ) = 2γ 1 . (16)  n − 1 + O 1 when cosθ = 1  Γ(2γ) n This expression can be also used to ﬁnd properties of Ln(x,x′) when n ∞. Taking all together we can conclude that → • At radii, where σ(r) = 0 (i.e. without matter) the exponential in (15) will dominate remaining terms and so there is exponential convergence. • At the radii with σ(r) = 0 the behaviour depends on the actual position. – Between axis and equatorial plane there will be conditional convergence (as 1 2 n− ) in the case of ring and absolute convergence (as n− ) in the case of disc. – At the equatorial plane there will be logarithmic divergence in the case of ring 2 and absolute convergence (as n− ) in the case of disc. – On the axis the situation is much more complicated. When the source is ring, 1/2 there is conditional convergence (as n− ) when considering gravitational potential and divergence (as n1/2) for dragging. In the disc case convergence is 3/2 1/2 one order faster, i.e. absolute for ν (as n− ) and conditional for ω (as n− ).

CONCLUSIONS

The method used by Will [4] when considering perturbation of a Schwarzschild black hole by a light slowly rotating ring can be extended to also enable disc perturbation. On the other hand, it involves expansions which do not behave well numerically. We have shown here that the series used in the ﬁrst perturbation order are convergent almost everywhere, but the convergence is indeed slow at radii where the source is present (in the equatorial plane).

ACKNOWLEDGMENTS

P.C.ˇ thanks Marcus Ansorg for stimulating remarks. We are grateful for support from the grants GAUK-428011, GACR-205/09/H033 and SVV-265301 (P.C.),ˇ GACR-202/09/0772 and MSM0021610860 (O.S.).

REFERENCES

1. Bardeen J.M., Wagoner R.V. Relativistic disks. I. Uniform rotation Ap.J. 167 359-423 (1971) 2. P. Cížek,Linearˇ perturbations of a Schwarzschild black hole by thin disc Journal of Physics: Confer- ence Series 314 (2011) 3. Olver F.J.W. et al. (eds.), NIST handbook of Mathematical Functions Cambridge University Press (2010) 4. Will C.M. Perturbation of a slowly rotating black hole by a stationary axisymmetric ring of matter. I. equilibrium conﬁgurations Ap.J. 191 521-531 (1974) Draft version July 28, 2017 Preprint typeset using LATEX style emulateapj v. 01/23/15

PERTURBATION OF A SCHWARZSCHILD BLACK HOLE DUE TO A ROTATING THIN DISC P. Cˇ´ıˇzek and O. Semerak´ Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic Draft version July 28, 2017

ABSTRACT Will (1974) treated the perturbation of a Schwarzschild black hole due to a slowly rotating light concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to the perturbation by a thin disc (which is more relevant astrophysically), but, due to a rather bad convergence properties, the resulting expansions are not suitable for specific (numerical) computations. However, we show that Green’s functions represented by the Will’s result can be expressed in a closed form (without multipole expansion) which is more useful. In particular, they can be integrated out over the source (thin disc in our case), to yield well converging series both for the gravitational potential and for the dragging angular velocity. The procedure is demonstrated, in the first perturbation order, on the simplest case of a constant-density disc, including physical interpretation of the results in terms of a one-component perfect fluid or a two-component dust on circular orbits about the central black hole. Free parameters are chosen in such a way that the resulting black hole has zero angular momentum but non-zero angular velocity, being just carried along by the dragging effect of the disc. Keywords: gravitation, black hole physics, accretion discs

1. INTRODUCTION Disc-like structures around very compact bodies are likely to play a key role in the most energetic astrophysical sources like active galactic nuclei, X-ray binaries, supernovas and gamma-ray bursts. Analytical modeling of such structures relies on various simplifying assumptions, the basic ones being their stationarity, axial symmetry and test (non-gravitating) nature (see e.g. Kato et. al 2008). The last assumption is justified by two arguments: (i) in the above astrophysical systems, the disc mass is typically much smaller than that of the black hole (or neutron star) in their center, so the latter surely dominate the gravitational potential as well as the “radial” field; (ii) black holes are the strongest possible (extended) gravitational sources (and neutron stars are just slightly less compact), so they would - at least in a certain region - even dominate the field if their mass was less than that of the matter around. However, such arguments need not hold for a latitudinal component of the field (namely perpendicular to the disc),1 and, most importantly, the additional matter may in fact dominate the second and higher derivatives of the metric (curvature). These higher derivatives are in turn crucial for stability of the matter’s motion, and thus the tricky issue of self-gravity enters the problem. Actually, a real, massive matter may thus assume quite a different configuration than a test matter (Abramowicz et. al 1984). One should also add that even if the accreting matter really had only a tiny effect on the geometry, it could still change the observational record of the source significantly, in particular, it may change the long-term dynamics of bodies orbiting in the system (e.g. Suková& Semerák 2013 and references therein). Hence, the properties of accretion systems may be sensitive to the precise shape of the field (Semerák 2003, 2004). Unfortunately, general relativity is non-linear and the fields of multiple sources mostly cannot be obtained by a simple superposition. Such more complicated fields are being successfully treated numerically (this even applies to strongly time-dependent cases including gravitational collapse, collisions and waves), but, for the present, the compass of explicit analytical solution terminates at systems with very high degree of symmetry, practically at static and axially symmetric cases. It would be most desirable to extend this to stationary cases, namely those admitting rotation. Stationary axisymmetric problem is usually represented in the form of the Ernst equation, but actually being tackled is the corresponding linear problem (Lax pair of equations whose integrability condition is just the Ernst equation). Exact solutions of this problem have been searched for in several ways. Klein & Richter (2005) and Meinel et al. (2008) summarized “straightforward” but rather involved treatment of the respective boundary-value problem, providing both the black-hole and the thin-disc solutions, plus prospects of how to also obtain their non-linear superpositions. Other attempts employed the “solution-generating” techniques - mathematical procedures which transform one stationary axisymmetric metric into another and can in principle provide any solution of this type. The practical power of these methods strongly depends on how simple is the “seed” metric, so usually a static one is started from. Using the soliton (inverse-scattering) method of Belinsky & Zakharov, Tomimatsu (1984), Krori & Bhattacharjee (1990), Chaudhuri & Das (1997) and Zellerin & Semerák (2000) generated black holes immersed in external fields, but at least the case corresponding to a hole surrounded by a thin disc (Zellerin & Semerák 2000) turned out to be unphysical (Semerák 2002). More successful seem to have been Bretónet. al (1997) who started from a different representation oldrich.semerak@mff.cuni.cz 1 For a Schwarzschild black hole which is spherically symmetric, there is no latitudinal field of course, so any additional source would automatically dominate this component. 2 P. Cˇ´ıˇzek and O. Semerak´ of the static axisymmetric seed and managed to “install” a rotating black hole in it (see also Bretónet. al (1998) for a charged generalization). If the external-matter gravity is weak, the problem may be treated as a small perturbation of the central-source metric, determined by linearized Einstein equations. In doing so, one can restrict to a special type of perturbations, for example, to stationary and axisymmetric ones. The method can be iterated; in a limit case (many iterations), it goes over to a solution in terms of series. The result then need not any longer represent a tiny variation of any “almost right” metric: it may even be put together on Minkowski background, with the “strong” part (e.g. a black hole) “dissolved” within the fundamental systems of the equations. The main problem of this scheme is convergence and meaning of the series. No less than 43 years ago, the paper by Will (1974), published in this journal, became a seminal reference in the subject. It provided the gravitational field of a light and slowly rotating thin equatorial ring around an (originally Schwarzschild) black hole by mass-and-rotational perturbation of the Schwarzschild metric. (See also the following paper Will (1975) where basic properties of the obtained solution were discussed.) Unfortunately, the perturbation- scheme success depends strongly on how simple the background metric is – and the Schwarzschild metric is exceptionally simple: the Will’s procedure cannot be simply extended to a Kerr background. Consequently, only partial questions have been answered explicitly in these directions, in particular the one of deformation of the Kerr black-hole horizon (Demianski 1976; Chrzanowski 1976) (interestingly, these two results do not agree on certain points, mainly in the limit of an extreme black hole).2 Recently the Will’s black-hole–ring problem has been revisited by Sano & Tagoshi (2014), but using the perturbation approach of Chrzanowski, Cohen and Kegeles in which the metric is found on the basis of solution of the Teukolsky equation for the Weyl scalars. The Will’s results have also been followed by Hod who analyzed the behaviour of the innermost stable circular orbit in the black-hole–ring field (Hod 2014) and the relation between the angular velocity of the horizon and the black-hole and ring angular momenta (Hod 2015). In the present note, we check whether the Will’s scheme can be adapted to the case of a thin equatorial stationary and axisymmetric disc. [Preliminary results were presented in C´ıˇzek&ˇ Semerák (2009) and C´ıˇzekˇ (2011).] In converging to a positive answer, we first observed that the expansions in spherical harmonics that typically come out in this approach (even in computing just the linear terms) converge rather badly and their numerical processing is problematic. Much more effective is the usage of Green functions of the problem, namely the perturbations generated by an infinitesimal ring. We have been able to express the Green functions in such a manner that the linear perturbation due to a thin disc can be obtained in a closed form. The paper is organized as follows. In section 2 we introduce equations describing the gravitational field of a thin disc. Section 3 summarizes Will’s approach and section 4 discusses its (un)suitability for a numerical treatment. In section 5 we compute Green functions of the problem in a closed form and in section 6 we show, on linear perturbation by a thin annular concentric disc, that they can be integrated in order to obtain a perturbation generated by a given (stationary and axisymmetric) distribution of mass. The resulting series converge much better and allow to compute specific configurations explicitely. Notation and conventions: our metric signature is ( +++) and geometrized units are used in which c = G = 1; Greek indices run 0-3 and partial derivative is denoted− by a comma. Complete elliptic integrals are given in terms of the modulus k, so π π π 2 dα 2 2 dα K(k) := ,E(k) := 1 k2 sin2 α dα , Π(n, k) := . 2 2 2 0 1 k2 sin α 0 − 0 (1 n sin α) 1 k2 sin α ∫ − ∫ √ ∫ − − 2. BLACK-HOLE√ & THIN-DISC SYSTEM: EINSTEIN EQUATIONS AND BOUNDARY CONDITIONS√ We will search for the black-hole–disc field by perturbation of the Schwarzschild metric, while restricting to the simplest space-times which can host rotating sources, namely to those which are stationary and axially symmetric. In addition, we will consider asymptotically flat space-times, without cosmological term, and will require their orthogonal transitivity (i.e., the motion of sources will be limited to stationary circular orbits). In such space-times, the time and µ ∂xµ µ ∂xµ axial Killing vector fields η = ∂t and ξ = ∂ϕ exist and commute, and the tangent planes to meridional directions (locally orthogonal to both Killing vectors) are integrable. Needless to say, it is assumed that there exists an axis of the space-like symmetry, namely a connected 2D (time-like) set of fixed points of the space-like isometry. In isotropic-type spheroidal coordinates (t,r,θ,ϕ) (of which t and ϕ have been chosen as parameters of the Killing symmetries), the metric with these properties can – for instance – be written in the “Carter-Thorne-Bardeen” form (e.g. Bardeen 1973) 2 2ν 2 2 2 2ν 2 2 2ζ 2ν 2 2 2 ds = e dt + B r e− sin θ (dϕ ωdt) + e − (dr + r dθ ) , (1) − − where the unknown functions ν, B, ω and ζ depend only on r and θ covering the meridional surfaces. Besides the above coordinates, we will also occasionally use the Weyl-type cylindrical coordinates ρ=r sin θ and z =r cos θ. Apart from the asymptotic flatness, the boundary conditions have to be fixed on the symmetry axis, on the black-hole horizon and on the external-source surface. Regularity of the axis (local flatness of the orthogonal surfaces z = const at ρ = 0) requires that eζ B at ρ 0+. The invariants g = g ηαηβ, g = g ηαξβ and g = g ξαξβ have to be → → tt αβ tϕ αβ ϕϕ αβ 2 Note that another approximation possibility is the post-Newtonian expansion. The composition of a rotating gravitational center with a massive ring in Keplerian rotation was tackled, using the gravito-electromagnetic analogy, by Ruggiero (2016). Disc perturbation of a Schwarzschild black hole 3 even functions of ρ (in order not to induce a conical singularity on the axis). Should the circumferential radius √gϕϕ ζ ν 2 grow linearly with proper cylindrical radius ρ [e − ]ρ=0, thus with ρ, there must be gϕϕ O(ρ ), and, demanding the 2 ≈ finiteness of ω, also gtϕ =gϕϕω O(ρ ). − ≈ 2ν The stationary horizon is characterized by e =0 and ω =const=:ωH (in our coordinates it specifically means that ω,θ = 0 there). In order that the azimuthal and latitudinal circumferences of the horizon be positive and finite, the 2ν 2ζ 2ν 2ζ functions Bre− and e − have to be such; the latter ensures regularity of grr as well. Hence, Br = 0 and e = 0 on the horizon. (Let us add in advance that the field equations also imply that ω,r/B and ν,rω,r have to be finite on the horizon, so ω,r has to vanish there as well.) Now for boundary conditions on the external source. We assume that the latter has the form of an infinitesimally thin disc in the equatorial plane z =0, stretching over some interval of radii lying above the central black-hole horizon. We assume that the disc bears neither charge nor current (there are no EM fields) and that the space-time is reflection symmetric with respect to its plane. The metric is then continuous everywhere, but has finite jumps in the first normal derivatives gαβ,z across the disc. The functions ν, B, ω and ζ must be even in z, their z-derivatives are odd in z, and even powers and multiples of derivatives (for example, B,zν,z) are even in z (therefore they do not jump across z =0). In order that the space-time be stationary, axially symmetric and orthogonally transitive, the disc elements must only move along surfaces spanned by the Killing fields, namely they must follow spatially circular orbits with steady dϕ angular velocity Ω= dt . This corresponds to four-velocity ηα + Ωξα uα = = ut(1, 0, 0, Ω) , u = ute2ν δt + utBρv ( ω, 0, 0, 1) (2) ηα + Ωξα α − α − | | with 2ν 2ν t 2 e− e− 2ν ν (u ) = = , where v := Bρe− (Ω ω) = √gϕϕ e− (Ω ω) 1 B2ρ2e 4ν (Ω ω)2 1 v2 − − − − − − represents linear velocity with respect to the local zero-angular-momentum observer (ZAMO). For the thin discs z ρ ρ (Tz =0, Tz =0) without radial pressure (Tρ =0), the surface energy-momentum tensor

z=0+ ∞ α α 2ζ 2ν α α 2ζ 2ν α T g dz = T e − dz =: S (ρ) T e − =: S (ρ)δ(z) (3) β zz β β ⇐⇒ β β ∫ z=0∫ −∞ − t t ϕ has only three non-zero components (St ,Sϕ,Sϕ ), representing energy density, orbital-momentum density and azimuthal ϕ t 2 t ϕ αβ α β α β pressure, respectively. If (Sϕ St ) + 4SϕSt 0, it can be diagonalized to S = σu u + P w w , where σ and ν ζ − ν ζ ≥ P (more precisely, σe − and P e − ) stand for the surface density and azimuthal pressure in a co-moving frame and wα is the “azimuthal” vector perpendicular to uα, with components wα = 1 (u , 0, 0, u ), w = ρB ( uϕ, 0, 0, ut). ρB ϕ − t α − Hence, the surface-tensor components read St = σ (σ + P ) uϕu ,St = (σ + P ) utu ,Sϕ = P + (σ + P ) uϕu . (4) t − − ϕ ϕ ϕ ϕ ϕ Orthogonally transitive stationary and axisymmetric space-times are described by 5 independent Einstein equations. t t ϕ 3 In our case of a thin disc, the energy-momentum tensor has only Tt , Tϕ and Tϕ components and the equations read (ρ B) = 0 , (5) ∇· ∇ 3 2 2 B ρ 2 2ζ 2ν ϕ t t 1 + v (B ν) ( ω) = 4πBe − T 2ωT T = 4πB (σ + P ) δ(z) , (6) ∇· ∇ − 2e4ν ∇ ϕ − ϕ − t 1 v2 − 3 2 4ν 2ζ 2ν t ( 2 2ν ) v (B ρ e− ω) = 16πBe − T = 16πB ρe− (σ + P ) δ(z) , (7) ∇· ∇ − ϕ − 1 v2 2 2 − 2 2 2 3B ρ 2 2 2ζ 2ν ϕ t σv + P ζ + ζ + (ν ) + (ν ) (ω ) + (ω ) = 8πe − T ωT = 8π δ(z) , (8) ,ρρ ,zz ,ρ ,z − 4e4ν ,ρ ,z ϕ − ϕ 1 v2 ( ) − 2 [ 2 1 ] 1 3 3 4ν 2 2 ζ (Bρ) ζ (Bρ) = Bρ (ν ) (ν ) [(Bρ) (Bρ) ] + B ρ e− (ω ) (ω ) , (9) ,ρ ,ρ − ,z ,z − ,ρ − ,z − 2 ,ρρ − ,zz 4 ,ρ − ,z [ ] 1 3 3 4ν [ ] ζ (Bρ) + ζ (Bρ) = 2Bρν ν (Bρ) + B ρ e− ω ω , (10) ,ρ ,z ,z ,ρ − ,ρ ,z − ,ρz 2 ,ρ ,z where and denote gradient and divergence in an (auxiliary) Euclidean three-space. The last two equations (for ζ) are∇ integrable,∇· provided the ﬁrst three vacuum equations hold. The axis boundary condition eζ = B implies that θ the ζ function can elsewhere be obtained according to ζ(r, θ) = 0 ζ,θdζ + ln B, where ζ,θ follows from the last two ﬁeld equations. ∫ 3 Equation (8) is not independent, but we include it here since it provides the jump of ζ,z across the equatorial plane given later in section 2.2. 4 P. Cˇ´ıˇzek and O. Semerak´

The treatment of the stationary axisymmetric problem (5)–(10) usually starts from a suitable solution of the first equation (5). In the parametrization (coordinates) we use here, it is convenient to choose4 k2 B = 1 . (11) − 4r2 With such a choice, the horizon lies where B =0, hence on r =k/2. This reveals the meaning of the constant k (which is supposed to be positive), in particular, for Schwarzschild one has k = M; for Kerr it would be k = M +√M 2 a2 , with a the centre’s specific angular momentum (k = 0 would correspond to an extreme black hole, or to a Minkowski− space-time). The main task is to solve the coupled equations (6) and (7), and then to integrate equations (9) and (10) for ζ. With 2 the choice B =1 k (thus with B =0) and written out explicitly in the (t,ρ,z,ϕ) coordinates, these equations read − 4r2 ,θ B2r2 1 + v2 (r2ν ) + r2ν (ln B) + ν + ν cot θ = sin2 θ r2(ω )2 + (ω )2 + 4πr2(σ + P ) δ(z) , (12) ,r ,r ,r ,r ,θθ ,θ 2e4ν ,r ,θ 1 v2 − [ ]16πre2ν v r2ω + 4rω (1 rν ) + 3r2ω (ln B) + ω + 3ω cot θ 4ω ν = (σ + P ) δ(z) , (13) ,rr ,r − ,r ,r ,r ,θθ ,θ − ,θ ,θ − B sin θ 1 v2 − 2 2 2 1 3 2 4ν 2 2 2 2 (2 B)rζ Bζ cot θ = B r (ν ) (ν ) + 2B 2 B r e− sin θ r (ω ) (ω ) , (14) − ,r − ,θ ,r − ,θ − − 4 ,r − ,θ [ ] 1 3 3 4ν 2[ ] Brζ cot θ + (2 B)ζ = 2Brν ν + 2(1 B) cot θ B r e− ω ω sin θ . (15) ,r − ,θ ,r ,θ − − 2 ,r ,θ 2.1. Counter-rotating interpretation of thin discs When asking about counterbalance to its (and black holes) gravity, the disc may either be considered as a solid structure (a set of circular hoops), or, to the contrary, as a non-coherent mix of azimuthal streams. In the astrophysical context, one usually adheres to the latter extreme possibility: that the disc is composed of two non-interacting streams of particles which follow stationary circular orbits in opposite azimuthal directions (Morgan & Morgan 1969; Lynden-Bell & Pineault 1978; Lamberti & Hamity 1989; Biˇcáket al. 1993; Biˇcák& Ledvinka 1993; Klein & Richter 1999; González& Espitia 2003; Garc´ıa-Reyes & González 2004). These orbits are geodesic if there is no radial stress ρ acting within the disc (Tρ =0). The surface energy-momentum tensor is thus decomposed as αβ α β α β S = σ+u+u+ + σ u u , (16) − − − where the +/ signs indicate the stream orbiting in a positive/negative sense of ϕ, as taken with respect to the “average” fluid− four-velocity uα. The four-velocities are of the uα =uα (1, 0, 0, Ω ) form again, so ± ± ± 2 2 t 2ν σ+v+ σ v ϕ t σ+v+ σ v t t σ+ σ − − − − Sϕ = Bρe− 2 + 2 ,Sϕ ωSϕ = 2 + −2 , St ωSϕ = 2 + 2 , (17) 1 v+ 1 v − 1 v+ 1 v − − 1 v+ 1 v ( − − − ) − − − − − − now with Ω (and the corresponding v ) given by free circular motion, i.e. by the roots of equation gtt,α + 2gtϕ,αΩ + 2 ± ± gϕϕ,αΩ = 0. Such a motion is only possible in the equatorial plane (z =0) in general, where just the radial component of acceleration remains relevant, vanishing if

g ω g ω 2 g Ω = Ω = ω + ϕϕ ,ρ ω + ϕϕ ,ρ tt,ρ ; (18) ± g ± g − g ϕϕ,ρ √( ϕϕ,ρ ) ϕϕ,ρ in particular, with the B =1 choice, this expression reduces to ρ2ω (ρ2ω )2 + 4e4ν ρν (1 ρν ) Ω = ω + ,ρ ± ,ρ ,ρ − ,ρ . (19) ± 2ρ(1 ρν ) √ − ,ρ The interpretation is only possible where the expression under the square root is non-negative; physically, this is not satisﬁed for discs with “too much matter on larger radii”: in such a case, the total gravitational pull at a given location points outwards, so “no angular velocity is low enough” to admit Keplerian orbiting there. Parameters of the counter-rotating picture (σ ,Ω ) and the “total”, one-stream parameters (σ,P ,Ω) are related by comparing the ± ± α β α β α β α β respective two forms of the energy-momentum tensor, σu u + P w w = σ+u+u+ + σ u u . From the trace of this − − − equation and from its projections onto uαuβ and wαwβ, while using a suitable expression of the scalar products

α vv 1 α v v α β v+v 1 u uα = ± − , u wα = ± − , gαβu+u = − − , (20) ± 2 2 ± 2 2 − 2 2 √1 v 1 v √1 v 1 v 1 v+ 1 v 4 − − ± − − ± − − − When working in the Weyl-type√ coordinates (t,ρ,z,ϕ) and with the√ corresponding, Weyl-Lewis-Papapetrou√ √ form of the metric, the B-equation is usually being satisfied by B = 1. This choice is advantageous in most respects, but it makes the horizon “degenerate” to a central segment of the symmetry axis ρ=0, which is not suitable for discussion of its properties. Different solutions for B are also possible for the “Carter-Thorne-Bardeen” form of the metric we use here; however, changing B generally does not imply a real physical difference, it effectively corresponds to a certain re-definition of coordinates, cf. section 6.1 (only the B = 1/ρ choice leads to different, plane-wave solutions). Disc perturbation of a Schwarzschild black hole 5 one finds 2 2 2 2 σ+ (vv+ 1) σ (vv 1) σ+ (v+ v) σ (v v) − − − − σ P = σ+ + σ , σ = 2 − 2 + 2 − 2 ,P = 2 − 2 + 2 − 2 , (21) − − (1 v )(1 v+) (1 v )(1 v ) (1 v )(1 v+) (1 v )(1 v ) − − − − − − − − − − α λ P P u+wα u wλ (v+ v)(v v ) σ + P 1 + σ − − = β κ = − − = = P = ..., (22) σ u uβ u uκ (1 vv+)(1 vv ) ⇒ σ+ + σ 1 + − − − − − − σ tϕ tt t 2 S Ω S σ σ(v v) P v(1 vv ) ± ± ± ± σ (u ) = − = 2 = − 2− − . (23) ± ± ± Ω+ Ω ⇒ 1 v ± (1 v )(v+ v ) − − − ± − − − αβ We may also, for example, compare expressions for SαβS , finding the relation 2 2 2 2 2 α β 2 2 2 (1 v+v ) − σ + P = σ+ + σ + 2σ+σ (gαβu+u ) = σ+ + σ + 2σ+σ −2 2 , (24) − − − − − (1 v+)(1 v ) − − − which in combination with σ P = σ+ + σ also leads to − − 2 2 (v+ v ) 2 2 (1 v+v ) − − σP = σ+σ 2− 2 , (σ+ σ ) = (σ + P ) 4σP − 2 . (25) − (1 v+)(1 v ) − − − (v+ v ) − − − − − 2.2. One-stream and two-stream interpretations: integrating jumps in the field equations α Relation between the jumps of gαβ,µ across the disc and Sβ are obtained by integrating the field equations (5)– + (8) over the infinitesimal interval z = 0−, z = 0 . Only the terms proportional to δ(z) (i.e. the source terms on ⟨ ⟩ the right-hand sides and the terms linear in B,zz, ν,zz, ω,zz and ζ,zz on the left-hand side) contribute according to + z=0 + + ν,zzdz = 2ν,z(z =0 ) (etc.), so we have B,z(z =0 ) = 0 and z=0− 2 2 2 ∫ + ϕ t t 1 + v 1 + v+ 1 + v − ν,z(z =0 ) = 2π Sϕ 2ωSϕ St = 2π(σ + P ) 2 = 2π σ+ 2 + σ 2 , (26) − − 1 v 1 v+ − 1 v − ( − − − ) ( t ) + 8π Sϕ Ω ω Ω+ ω Ω ω − ω,z(z =0 ) = 2 2 4ν = 8π(σ + P ) − 2 = 8π σ+ −2 + σ −2 , (27) −B ρ e− − 1 v − 1 v+ − 1 v − ( − − − ) 2 2 2 + ϕ t σv + P σ+v+ σ v − − ζ,z(z =0 ) = 4π Sϕ ωSϕ = 4π 2 = 4π 2 + 2 , (28) − 1 v 1 v+ 1 v ( ) − ( − − − ) α where we have expressed the results in terms of the one-stream as well as two-stream form of Sβ .

3. PERTURBATION SCHEME We will look for a solution of equations (6) and (7) in the form of series, expanding

∞ j ∞ j ∞ j ν = νjλ , ω = ωjλ , ζ = ζjλ , (29) j=0 j=0 j=0 ∑ ∑ ∑ where the coefficients νj, ωj and ζj depend on r and θ (or ρ and z) and the dimensionless parameter λ is proportional to the ratio of the disc mass to the black-hole mass M. More specifically, let it be related by 1 1 (σ + P )δ(z) λΣ(ρ)δ(z) = λΣ(r) δ(cos θ) = λΣ(r) δ(θ π/2), (30) ≡ r − r − where δ denotes the δ-distribution and λΣ σ +P is an “effective” surface density. The functions ν0, ω0 and ζ0 represent the black-hole background, i.e. the≡ Schwarzschild metric which in isotropic coordinates (recall that ρ=r sin θ and z =r cos θ) reads 2r M 2 M 4 ds2 = − dt2 + 1 + dr2 + r2dθ2 + r2 sin2 θ dϕ2 , (31) − 2r + M 2r ( ) ( ) hence ( ) 2r M M 2 ν = ln − , ω = 0,B = eζ0 = 1 , (32) 0 2r + M 0 − 4r2 and the corresponding orbital velocity is 8 √Mr3 2 √Mr Ω = , v = . (33) 0 (2r + M)3 0 2r M − 6 P. Cˇ´ıˇzek and O. Semerak´

Substituting (29) and (30) into (6) and (7) and subtracting the pure-Schwarzschild terms, one obtains

7 2 k 2 k l 1 ∞ k ∞ k (2r + M) sin θ − ∞ j − − λ (B νk) = λ exp 4 λ νj ωm ωk l m + ∇· ∇  27r4(2r M)  −  ∇ · ∇ − −  k=1 k=2 l=0 j=1 m=1 ∑ ∑  − ∑ ( ∑ ) l ∑   2  B ∞ k+1 1 + v  + 4π Σ δ(cos θ) λ 2 , (34) r 1 v k k∑=0 [ − ]

k 1 ∞ (2r + M)7 sin2 θ ∞ − (2r + M)7 sin2 θ ∞ λk ω = λk exp 4 λjν ω ∇· 26r4(2r M) ∇ k − ∇· 26r4(2r M)  − j  ∇ l − k=1 k=2 l=1 j=1 { − }  − ( ) k l  ∑ ∑ ∑ ∑ − 4   M  ∞ k+1 v  16π 1 + Σ δ(cos θ) λ 2 , (35) − 2r 1 v k ( ) k∑=0 [ − ] k where [f]k means the coefficient standing at λ in Taylor expansion of f. Since the background is static, the first-order equations only contain mass-energy terms multiplied by the background metric on the right-hand sides, because the first-line sums do not contribute. In higher orders, the right-hand sides contain only lower-order terms. (For non-static backgrounds the equations do not decouple so easily.) Now, the eigen-functions with respect to θ of the operator on the left-hand sides of (34) and (35) are the Legendre (3/2) 5 polynomials Pl(cos θ) and the Gegenbauer polynomials Cl (cos θ), respectively. Introducing a dimensionless radius r M 2 x:= M 1+ 4r2 , we may thus write ( ) ∞ ∞ (3/2) νl = νlj (x) Pj(cos θ), ωl = ωlj (x) Cj (cos θ) . (36) j=0 j=0 ∑ ∑ Substituting this into (34) and (35) leads to

∞ d dν (x2 1) lj j(j + 1)ν P (cos θ) = R (x, θ) , (37) dx − dx − lj j l j=0 ∑ { [ ] } ∞ d dω (x2 1) (x + 1)4 lj (x + 1)4j(j + 3)ω C(3/2)(cos θ) = S (x, θ) , (38) − dx dx − lj j l j=0 ∑ { [ ] } where Rl(x, θ) and Sl(x, θ) stand, up to an l-independent multiplication factor, for the coeﬃcients of expansion (with respect to λ) of the right-hand sides of equations (34) and (35); speciﬁcally,

2 4 ∞ l r ∞ l Br Rl(x, θ)λ = [r.h.s. of (34)] , Sl(x, θ)λ = [r.h.s. of (35)] . (39) B M 4 sin2 θ ∑l=0 ∑l=0 Provided that Rl and Sl do not diverge on the axis θ =0, π, these coeﬃcients can be also decomposed as

∞ ∞ (3/2) Rl(x, θ) = Rlj(x) Pj(cos θ) ,Sl(x, θ) = Slj (x) Cj (cos θ) , (40) j=0 j=0 ∑ ∑ where 1 2j + 1 R (x) = R (x, θ) P (cos θ) d(cos θ) , (41) lj 2 l j ∫1 − 1 2j + 3 S (x) = S (x, θ) C(3/2)(cos θ) sin2 θ d(cos θ) . (42) lj 2(j + 1)(j + 2) l j ∫1 − Demanding that the equations (37) and (38) hold for each order (multipole moment) separately, one obtains a system

5 3/2 In Will’s article the latter are denoted by Tl (cos θ). Disc perturbation of a Schwarzschild black hole 7 of independent ordinary diﬀerential equations

d dν (x2 1) lj j(j + 1) ν = R , (43) dx − dx − lj lj [ ] d dω (x2 1) (x + 1)4 lj (x + 1)4j(j + 3) ω = S , (44) − dx dx − lj lj [ ] + where l N, j Z . They only contain lower-order source terms (assumed to be given) for every l and are to be ∈ ∈ 0 supplemented by boundary conditions on the horizon x=1 (νlj and ωlj are supposed to be regular there) and at spatial infinity (νlj and ωlj should vanish there). Of many techniques available for such equations, we shall focus on finding their Green functions. Let us start from fundamental systems of equations (43)(44). The first one is the Legendre differential equation whose fundamental system can be expressed as linear combination of Legendre functions of the first and second kinds

∞ dξ Pj(x) and Qj(x) = Pj(x) . (45) 2 2 (ξ 1) [Pj(ξ)] ∫x − For the second equation, Will (1974) used the substitution t = (x + 1)/2 which transforms it to the hypergeometric diﬀerential equation, having two generators of the fundamental system,

x + 1 ∞ dξ Fj(x) = 2F1 j, j + 3; 4; and Gj(x) = Fj(x) , (46) 4 2 − 2 (ξ + 1) [Fj(ξ)] ( ) ∫x

+ where 2F1(a, b; c; ξ) denotes the Gauss hypergeometric function. Note that since j Z0 , F (x) is in fact a polynomial j l 1 j ∈ j 3 of degree j. Asymptotically (as x ), P (x) x , Q (x) x− − , F (x) x and G(x) x− − . At the horizon → ∞ j ∼ j ∼ ∼ ∼ (x=1), Qj(x) and Gj(x) diverge (except G0(x) which will be discussed later). Given the above boundary conditions, the Green functions of equations (37), (38) can be found in the form

ν Qj(x) Pj(x′) for x x′ j (x, x′) = − ≥ , (47) G Pj(x) Qj(x′) for x x′ { − ≤ ω Gj(x) Fj(x′) for x x′ j (x, x′) = − ≥ , (48) G Fj(x) Gj(x′) for x x′ { − ≤ and their inhomogeneous solutions as

∞ ν ν (x) = R (x′) (x, x′) dx′ , (49) lj lj Gj ∫1 ∞ 0 ω Jl δj ω (x) = S (x′) (x, x′) dx′ + , (50) lj lj Gj (x + 1)3 ∫1 where Jl are arbitrary constants, representing a choice of the black-hole spin (see section 3.1). Such a solution is unique up to a coordinate transformation. Note that one could add to the Green functions terms proportional to P0(x) = F0(x) = 1 which is everywhere ﬁnite, but such an addition only corresponds to a rescaling of time and (thus) of the coordinate angular velocity, so it has no invariant physical eﬀect. Using (36), we may write

1 ∞ ∞ ∞ ∞ ν ν ν (x, θ) = ν (x) P (cos θ) = R (x′) (x, x′) P (cos θ) dx′ = R (x′, θ′) (x, θ, x′, θ′) dx′ d(cos θ′) , l lj j lj Gj j l G j=0 j=0 ∑ ∫1 ∑ ∫1 ∫1 − (51) 1 ∞ ∞ ∞ (3/2) ∞ ω (3/2) ω ω (x, θ) = ω (x) C (cos θ) = S (x′) (x, x′) C (cos θ) dx′ = S (x′, θ′) (x, θ, x′, θ′) dx′ d(cos θ′) , l lj j lj Gj j l G j=0 j=0 ∑ ∫1 ∑ ∫1 ∫1 − (52) 8 P. Cˇ´ıˇzek and O. Semerak´ where

ν ∞ 2j + 1 (x, θ, x′, θ′) := P (min(x, x′)) Q (max(x, x′)) P (cos θ) P (cos θ′)) , (53) G − 2 j j j j j=0 ∑ ω ∞ 2j + 3 (3/2) (3/2) Jl (x, θ, x′, θ′) := F (min(x, x′)) G (max(x, x′)) C (cos θ) C (cos θ′) + (54) G − 2(j + 1)(j + 2) j j j j (x + 1)3 j=0 ∑ [ ] are Green’s functions of homogeneous parts of equations (34) and (35). These represent a perturbation by an in- ﬁnitesimal (2D) circular ring placed at x′, θ′ in the ﬁrst order in λ. (We omit the Jl term in the following, see below.)

3.1. Differences from the Will’s article Will (1974) found the Green functions of equations (47) and (48) or, more precisely, he proposed the inhomogeneous solutions (49) and (50). He employed three expansions of the solutions: with respect to the linear mass density of the ring, with respect to the angular velocity of the horizon, and the multipole expansion performed to convert partial differential equations into the ordinary ones. In contrast, we use only two expansions: with respect to the disc density and the multipole expansion. Below (see section 5), we will even be able to drop the multipole expansion and work solely with expansion with respect to the disc (ring) density. Namely, the rotational expansion can be “reconstructed” using the constants Jl (not employed in the Will’s paper). This is possible due to Fj(1) = 0 for j > 0 (see, for example, (67)), which implies that higher multipole moments do not contribute to the black-hole angular velocity at l all, the only important entering terms (in a given order λ of the mass perturbation) being proportional to F0(x)=1, 3 G0(x) = (x + 1)− or ωl0(x), where ωl0 is the inhomogeneous solution of (44) given by (50) (with Jl = 0). The first term F0(x) = 1 only adds constant coordinate angular velocity, so it is not a physical degree of freedom (it can be cancelled out by a coordinate transformation). The meaning of the remaining two terms is revealed by their behaviour in a vicinity of the disc (ring): G0 is smooth everywhere, but ωl0 jumps in its first derivative (in the disc case it thus contributes to the density of the disc energy-momentum). Hence, the term proportional to G0 can be interpreted as the black hole’s “own” angular velocity (perturbation towards the Kerr solution), while the terms proportional to ωl0 represent rotational perturbation due to the presence of the rotating disc (ring). Needless to say, such an interpretation is not unique, because ωl0 + ClG0 (Cl are arbitrary constants) is also a solution of (44) contributing to the source. Since there is no clear way how to say which choice of Cl is the “correct” one, we will adhere to Cl =0 for simplicity. Having prescribed the black-hole angular velocity

∞ l ωH = βlλ , (55) ∑l=1 one can ﬁnd the Jl constants according to

∞ 8Sl0(x′) Jl = 8 βl ωl0(1) = 8βl + dx′ , (56) − |Jl=0 (x + 1)3 ∫ ′ [ ] 1 where ω stands for the right-hand side of (50) with J =0, and S contains lower-order terms of the expansion lj|Jl=0 l l0 in λ (i.e., it also contains Jk with k

4. CONVERGENCE AND RELATED ISSUES The main attribute of the above procedure is its speed of convergence to the desired solution. The problem is familiar from spectral methods (multipole expansion can actually be viewed as a spectral method): the numerics works well when the desired result is smooth, otherwise convergence problems arise. More specifically, one can expect exponential convergence for analytical function, whereas at most a power-law one for functions having some derivatives discontinuous (see, for example, Grandclément & Novak 2008). Let us begin with the potential for the ring case. Focusing on the region outside the black hole, 1 x< x>, and regarding the asymptotic behaviour of special functions (see, for example, Olver et. al. 2010), one can,≤ after≤ lengthy calculations, find that

1 +j 2 2 2j + 1 1 x< + x< 1 P (cos θ) P (x ) Q (x ) P (0) − [f(x, x′) + O(1/j)] , (57) j j < j > j 2 2 ≈ j (x> + x> 1) √ − where f(x, x′) is a suitable function independent of j, x< := min(√x, x′) and x> := max(x, x′). This implies that the sum (51) exponentially converges outside the ring radius and conditionally converges (like 1/j) just on the radius of the ring. Practically, this means that near the ring radius one has to include quite many terms (in j) in order to get reasonably small oscillations. (Convergence could be expected to be slow near a singular source.) Disc perturbation of a Schwarzschild black hole 9

In the disc case, the situation is better and worse at the same time. After a lengthy calculation again, one ﬁnds that

xR+ 1 +j 2 2 2j + 1 1 x<′ + x<′ 1 R (x ) P (cos θ) P (x ) Q (x ) P (0) dx < − [f(x, x′) + O(1/j)] , (58) 2 l R j j < j > j R j2 2 (x>′ + √x>′ 1) xR∫ − −

√ where x< and x> have the same meaning as above, i.e. x< := min(x, xR) and x> := max(x, xR), and x<′ , x>′ stand for analogous quantities taken with respect to the nearest radius lying inside the disc (xR′ ), i.e.,

x when x xout xin when x xin x<′ = ≤ , x>′ = ≤ , (59) xout when x xout x when x xin { ≥ { ≥ where xin and xout represent inner and outer rim of the disc. Equation (58) factually yields a lower bound of the convergence speed. (One can already obtain a speed estimate by checking the simplest example, i.e. constant Slj .) We can thus conclude that in the disc case the convergence is 2 exponentially fast outside the radii of the disc and polynomial (like j− ) at the radii “within” the disc. Therefore, it is generally impossible to improve the polynomial convergence using interpolation in some infinitesimally small area. Calculation of dragging (ωjl) shows similar behaviour, i.e. exponential convergence outside the radii of the ring (disc) 1 2 and polynomial one like j− (j− ) at the radii of the source. It is somewhat longer to prove this, so let us only say that (γ) (γ+1) (γ) one can proceed as follows: (i) prove that dDn (x)/dx = 2γ Dn 1 (x), where Dn (x) is the second independent solution of the ultraspherical differential equation (the “second− Gegenbauer− function”, see section 5 below), (ii) express Fj(x) and Gj(x) using (67), and (iii) consider the asymptotic behaviour of the Gegenbauer functions. There is one more challenge for numerical precision: integrals (45) and (46), when expressed using elementary functions, have a form of two terms almost cancelling each other. This problem can be managed by recalling that the Legendre function of the second kind Qj(x) can be expressed in terms of elementary functions and then evaluated efficiently. Analogously, Gj(x) can be expressed as

( 1)j(j + 2)! (j + 3)! 2 j+3 2 G (x) = − F j, j + 3; 2j + 4; (60) j 48 (2j + 3)! x + 1 2 1 x + 1 ( ) ( ) which converges very well after the Gauss hypergeometric function is expanded suitably. Let us add that we do not know how to prove the equivalence of (60) and (46) directly. However, it is possible to check that both expressions of G (x) solve the homogeneous part of (44) and that both vanish at x . Equation (44) is a second-order ordinary j → ∞ linear differential equation, so its fundamental space is two-dimensional. And since the other solution Fj(x) does not vanish at infinity, both expressions of Gj(x) have to be proportional to each other. Comparing their leading terms (of expansion in 1/x), one finally concludes that the functions exactly coincide.

5. GREEN FUNCTIONS To overcome the convergence problems, one can try to avoid angular expansion. A straightforward way to do so is ν ω to express the Green functions (x, θ, x′, θ′) and (x, θ, x′, θ′) in a closed form. This section is devoted to this task G G and uses relations which we will justify in detail elsewhere (C´ıˇzekˇ in preparation). As an advertisement for the usage of the closed-form Green functions, see figure 1 where, on an example of the ring perturbation, results are compared computed i) by the usual expansions and ii) using the closed-form Green functions. The first 30 terms of multipole expansion are summed there, but the closed-form Green functions provide better results. See also Tables 1 and 2 where the convergence of the multipole expansion towards exact result (given by the closed-form Green functions) is illustrated. One should admit that the multipole series of this type can usually be re-summed in such a way that their convergence is much better even close to the source radius. Anyway, the main advantage of the closed-form Green functions is that one can better integrate them over the source, as exemplified in section 6 below.

5.1. Green function for the gravitational potential In order to ﬁnd the closed form of the Green function (53), one can start from a more general relation which will be proven in an accompanying paper by C´ıˇzekˇ (in preparation):6

∞ Γ3(j + 1) 2(j + γ) D(γ)(a) C(γ)(u) C(γ)(v) C(γ)(w) = Γ3(j + 2γ) j j j j j=0 ∑ vw+√(1 v2)(1 w2) 2 2 2 2 2 γ 1 − − 1 1 (1 u )(1 ξ ) (1 v w ξ +2vwξ) − π 2F1 , 1; γ + ; − −2 − − − dξ vw √(1 v2)(1 w2) 2 2 (a uξ) a uξ = − − − − − , (61) ( ) γ 1 ∫ 24γ 2Γ4(γ)Γ(2γ) [(a2 1)(1 v2)(1 w2)] − 2 − − − − 6 It is also shown there that the presented sum converges absolutely when γ =3/2. 10 P. Cˇ´ıˇzek and O. Semerak´ z z

ρ ρ

z z

ρ ρ

Figure 1. Meridional-plane contours of the gravitational potential (ν1; left column) and of the linear dragging term (ω1; right column), generated by a thin ring of matter located at r = 3 and having rest mass 0.01 (where is some mass scale; note that here the disc is the only source, there is no black hole). In theM ﬁrst row, the metric functionsM are calculatedM summing the ﬁrst 30 orders of multipole expansion, yet still there are obvious problems near the ring radius. In the second row, the same are computed (with a clearly better accuracy) using the closed-form Green functions derived in section 5. The coordinate axes are given in the units of M.

(γ) (γ) where u, v, w ( 1; 1), a should be greater then 1, and Cj (x) and Dj (x) denote, respectively, Gegenbauer functions ∈ − (γ) (γ) of the first and of the second kind. Namely, Cj (x) and Dj (x) are two independent solutions of the ultraspherical differential equation d2y(x) dy(x) (1 x2) (2γ + 1) x + j(j + 2γ) y(x) = 0 . (62) − dx2 − dx The first solution can be written as 1 2 γ 1 2γ √ P −1 (x) when x < 1 (γ) 2 −4 2π Γ(n + 2γ) +γ+n − 2 | | Cn (x) = 1 x γ 1 γ ; (63) − 2 Γ(γ)Γ(n + 1) ×  2 − P 1 +γ+n(x) when x 1  − 2 ≥ it reduces to Gegenbauer polynomials when n is a non-negative integer and to Legendre functions of the first kind (actually Legendre polynomials) for γ = 1/2. The second solution can be written analogously,

1 2 γ 1 2γ √ Q −1 (x) when x < 1 (γ) 2 −4 2π Γ(n + 2γ) +γ+n − 2 | | Dn (x) = 1 x γ 1 1 γ ; (64) − 2 Γ(γ)Γ(n + 1) ×  iπ(γ 2 ) 2 − e − Q 1 +γ+n(x) when x 1  − 2 ≥ it reduces to Legendre functions of the second kind for γ = 1/2 (hence how we call it). Above, we have employed the µ µ µ µ notation Pν (x), Pν (x), Qν (x) and Qν (x), used in Olver et. al. (2010), chapter 14, in order to distinguish the Ferrers functions of the first and of the second kinds (i.e., Legendre functions defined on the cut x < 1; written in roman) from the associated Legendre functions of the first and of the second kinds (defined on x>1;| | written in italic). One more remark to the formula (61) is at place, namely that it requires u to lie within ( 1; 1), while for a ring outside of the horizon actually u>1. However, a function of complex variable has a unique extension− (up to a Riemann folding), so the solution valid for u ( 1; 1), when extended into the complex plane, should also yield a (unique) solution of the respective differential∈ equation− with different boundary conditions. Disc perturbation of a Schwarzschild black hole 11

number of terms added in (53) θ x exact value 10 20 30 40 50 2.00 0.19807 0.19813 0.19813 0.19813 0.19813 0.19813 − − − − − − 2.50 0.22468 0.22664 0.22680 0.22682 0.22682 0.22682 − − − − − − 2.90 0.26750 0.28772 0.29629 0.30058 0.30293 0.30636 π − − − − − − 2 3.00 0.28370 0.31884 0.34032 0.35583 0.36797 − − − − − −∞ 3.10 0.25857 0.27846 0.28700 0.29134 0.29374 0.29738 − − − − − − 3.50 0.19552 0.19818 0.19849 0.19853 0.19854 0.19854 − − − − − − 4.00 0.15445 0.15478 0.15479 0.15479 0.15479 0.15479 − − − − − − 2.00 0.17340 0.17344 0.17344 0.17344 0.17344 0.17344 − − − − − − 2.50 0.16688 0.16747 0.16754 0.16755 0.16755 0.16755 − − − − − − 2.90 0.15563 0.15601 0.15780 0.15881 0.15894 0.15858 π − − − − − − 3 3.00 0.15196 0.15038 0.15329 0.15671 0.15813 0.15585 − − − − − − 3.10 0.15012 0.15044 0.15221 0.15323 0.15337 0.15299 − − − − − − 3.50 0.14007 0.14071 0.14085 0.14086 0.14086 0.14086 − − − − − − 4.00 0.12584 0.12599 0.12600 0.12600 0.12600 0.12600 − − − − − − 2.00 0.16020 0.16016 0.16016 0.16016 0.16016 0.16016 − − − − − − 2.50 0.15061 0.15003 0.15008 0.15008 0.15008 0.15008 − − − − − − 2.90 0.14321 0.13879 0.14019 0.14108 0.14074 0.14062 π − − − − − − 4 3.00 0.14178 0.13419 0.13694 0.14025 0.13889 0.13816 − − − − − − 3.10 0.13822 0.13388 0.13526 0.13617 0.13582 0.13570 − − − − − − 3.50 0.12655 0.12582 0.12591 0.12592 0.12592 0.12592 − − − − − − 4.00 0.11458 0.11444 0.11445 0.11445 0.11445 0.11445 − − − − − − 2.00 0.15110 0.15113 0.15113 0.15113 0.15113 0.15113 − − − − − − 2.50 0.13920 0.13971 0.13975 0.13974 0.13974 0.13974 − − − − − − 2.90 0.12758 0.12952 0.13141 0.13019 0.13032 0.13046 π − − − − − − 6 3.00 0.12402 0.12610 0.13108 0.12705 0.12729 0.12817 − − − − − − 3.10 0.12310 0.12499 0.12687 0.12564 0.12577 0.12591 − − − − − − 3.50 0.11656 0.11716 0.11722 0.11721 0.11721 0.11721 − − − − − − 4.00 0.10712 0.10725 0.10725 0.10725 0.10725 0.10725 − − − − − − 2.00 0.14420 0.14434 0.14434 0.14434 0.14434 0.14434 − − − − − − 2.50 0.13010 0.13270 0.13242 0.13246 0.13245 0.13245 − − − − − − 2.90 0.10950 0.13056 0.11932 0.12593 0.12186 0.12343 0 − − − − − − 3.00 0.10054 0.13646 0.10871 0.13221 0.11144 0.12127 − − − − − − 3.10 0.10552 0.12621 0.11502 0.12169 0.11754 0.11915 − − − − − − 3.50 0.10822 0.11153 0.11104 0.11112 0.11111 0.11111 − − − − − − 4.00 0.10153 0.10208 0.10206 0.10206 0.10206 0.10206 − − − − − − Table 1 Convergence of the multipole expansion of the Green function for gravitational potential ν computed for a ring with radius x =3 located G ′ in the equatorial plane (θ′ =π/2). Values at several latitudes θ are computed in the radial region close to the ring (from x=2 to x=4).

ν The relation (61) provides (x, θ, x′, θ′) (up to a multiplication factor) if putting γ = 1/2, a = max(x, x′), u = G min(x, x′), v = cos θ and w = cos θ′. Namely, as will be shown in C´ıˇzekˇ (in preparation) (the result follows from Baranov 2006),

4 √(a2 1)(u2 1)(1 v2)(1 w2) 2 K − − − − 2 (au vw)2 √(a2 1)(u2 1) √(1 v2)(1 w2) ∞ [√ − − − − − − − ] (2j + 1) Qj(a) Pj(u) Pj(v) Pj(w) = ( ) , (65) 2 1/2 j=0 ∑ π (au vw)2 (a2 1)(u2 1) (1 v2)(1 w2) − − − − − − − [ ] (√ √ ) hence, comparing this with (53), we get

2 2 4 √(x 1)(x′ 1) sin θ sin θ′ K − − 2 2 2 2 (xx′ cos θ cos θ′) √(x 1)(x′ 1) sin θ sin θ′ ν [√ − − − − − ] (x, θ, x′, θ′) = ( ) . (66) G − 2 1/2 π (xx cos θ cos θ )2 (x2 1)(x 2 1) sin θ sin θ ′ − ′ − − ′ − − ′ [ ] (√ ) This is actually an expected result: the Green function corresponds to the potential due to a thin-ring source (situated at x′, θ′), which is familiar to be given by the complete elliptic integral K(k) (within general relativity, such a source is known as the Bach-Weyl ring, see e.g. Semer´ak 2016). Note in passing that equation (6) is quadratic in dragging (ω) and the background metric is static (Schwarzschild solution), so dragging has to be proportional to the perturbation (λ), i.e. the correction from the ( ω)2 term is at least of order λ2 and does not contribute in the linear order (in higher orders, it behaves like a source∇ term). Without dragging, equation (6) is the same as in the static case (or even a Newtonian one) and one reaches the above result (up to a coordinate transformation, see (80)). 12 P. Cˇ´ıˇzek and O. Semerak´

number of terms added in (54) θ x exact value 10 20 30 40 50 2.00 0.0043591 0.0043605 0.0043605 0.0043605 0.0043605 0.0043605 − − − − − − 2.50 0.0050673 0.0051175 0.0051218 0.0051222 0.0051223 0.0051223 − − − − − − 2.90 0.0063084 0.0068904 0.0071424 0.0072701 0.0073402 0.0074433 π − − − − − − 2 3.00 0.0067968 0.0078323 0.0084794 0.0089513 0.0093229 − − − − − −∞ 3.10 0.0059006 0.0064596 0.0067049 0.0068309 0.0069010 0.0070079 − − − − − − 3.50 0.0037462 0.0038087 0.0038162 0.0038173 0.0038174 0.0038175 − − − − − − 4.00 0.0024485 0.0024550 0.0024551 0.0024551 0.0024551 0.0024551 − − − − − − 2.00 0.0037082 0.0037087 0.0037087 0.0037087 0.0037087 0.0037087 − − − − − − 2.50 0.0033726 0.0033758 0.0033779 0.0033781 0.0033782 0.0033781 − − − − − − 2.90 0.0029896 0.0029065 0.0029373 0.0029732 0.0029866 0.0029767 π − − − − − − 3 3.00 0.0029039 0.0027174 0.0027375 0.0028423 0.0029251 0.0028643 − − − − − − 3.10 0.0027625 0.0026820 0.0027111 0.0027462 0.0027598 0.0027498 − − − − − − 3.50 0.0022901 0.0022911 0.0022943 0.0022948 0.0022948 0.0022948 − − − − − − 4.00 0.0017923 0.0017936 0.0017938 0.0017938 0.0017938 0.0017938 − − − − − − 2.00 0.0033785 0.0033783 0.0033783 0.0033783 0.0033783 0.0033783 − − − − − − 2.50 0.0029105 0.0029114 0.0029139 0.0029139 0.0029139 0.0029139 − − − − − − 2.90 0.0024557 0.0024287 0.0025224 0.0025304 0.0025057 0.0025097 π − − − − − − 4 3.00 0.0023163 0.0022347 0.0024540 0.0025018 0.0023813 0.0024098 − − − − − − 3.10 0.0022598 0.0022333 0.0023241 0.0023322 0.0023075 0.0023116 − − − − − − 3.50 0.0019376 0.0019385 0.0019426 0.0019426 0.0019425 0.0019425 − − − − − − 4.00 0.0015499 0.0015498 0.0015500 0.0015500 0.0015500 0.0015500 − − − − − − 2.00 0.0031638 0.0031633 0.0031633 0.0031633 0.0031633 0.0031633 − − − − − − 2.50 0.0026684 0.0026508 0.0026545 0.0026543 0.0026542 0.0026543 − − − − − − 2.90 0.0023803 0.0021449 0.0022923 0.0022750 0.0022750 0.0022613 π − − − − − − 6 3.00 0.0023586 0.0019140 0.0022626 0.0022328 0.0020593 0.0021686 − − − − − − 3.10 0.0021926 0.0019660 0.0021091 0.0020924 0.0020616 0.0020787 − − − − − − 3.50 0.0017658 0.0017434 0.0017494 0.0017490 0.0017489 0.0017490 − − − − − − 4.00 0.0014080 0.0014058 0.0014060 0.0014060 0.0014060 0.0014060 − − − − − − 2.00 0.0029876 0.0030073 0.0030070 0.0030070 0.0030070 0.0030070 − − − − − − 2.50 0.0020835 0.0025531 0.0024665 0.0024810 0.0024787 0.0024790 − − − − − − 2.90 +0.0004811 0.0045000 0.0000854 0.0037008 0.0008545 0.0020972 0 − − − − − 3.00 +0.0019023 0.0072430 +0.0042764 0.0091957 +0.0059767 0.0020094 − − − 3.10 +0.0005373 0.0042505 +0.0000482 0.0035188 0.0006733 0.0019249 − − − − 3.50 0.0011763 0.0017360 0.0015918 0.0016257 0.0016180 0.0016194 − − − − − − 4.00 0.0012397 0.0013108 0.0013062 0.0013065 0.0013064 0.0013064 − − − − − − Table 2 Convergence of the multipole expansion of the Green function for dragging ω computed for a ring with radius x =3 located in the G ′ equatorial plane (θ′ =π/2). Values at several latitudes θ are computed in the radial region close to the ring (from x=2 to x=4).

5.2. Green function for dragging Now we proceed to the second Green function (54). Considering relations

12 ( 1)j ( 1)j F (x) = − Oˆ C(3/2)(x) ,G (x) = − Oˆ D(3/2)(x) , (67) j j(j + 1)2(j + 2)2(j + 3) x j j 12 x j d2 d d where Oˆ := (x 1) (x 1) = (x 1)2 , (68) x − dx2 − dx − dx Disc perturbation of a Schwarzschild black hole 13 we can rewrite (54) as

3 ∞ ω j + 2 (3/2) (3/2) (3/2) (3/2) = ∆ Oˆ C (x ) Oˆ D (x ) C (cos θ) C (cos θ′) = G − j(j + 1)3(j + 2)3(j + 3) x< j < x> j > j j j=1 ∑ 3 ∞ d 2 j + 2 (3/2) d (3/2) (3/2) (3/2) = ∆ Oˆ (x 1) D (x ) C (x ) C (cos θ) C (cos θ′) = − x> dx < − j(j + 1)3(j + 2)3(j + 3) j > dx j < j j < j=1 < ∑ x 3 < ∞ d 2 j + 2 2 (3/2) (3/2) (3/2) (3/2) = ∆ Oˆ (x + 1)− (ζ 1) D (x ) C (ζ) C (cos θ) C (cos θ′) dζ = − x> dx < (j + 1)3(j + 2)3 − j > j j j < j=0 ∑ ∫1 e x< 2 d ζ 1 ∞ 2j + 3 (3/2) (3/2) (3/2) (3/2) = ∆ Oˆ − D (x ) C (ζ) C (cos θ) C (cos θ′) dζ = (69) − x> dx 2(x + 1)2 (j + 1)3(j + 2)3 j > j j j < < j=0 ∫1 ∑ e x< cos(θ θ′) 2 − d ζ 1 [cos(θ θ′) ξ][ξ cos(θ + θ′)] = ∆ Oˆ x − − − − dξdζ , > 2 2 2 2 2 2 2 − dx< 2π(x< +1) (x> 1) sin θ sin θ′ x> √ζξ + (x> ζξ) (1 ζ )(1 ξ ) ∫1 − cos(θ∫+θ′) − − − − − e √ where

3 (3/2) (3/2) 1 ∆ := F0(x<) G0(x>) C0 (cos θ) C0 (cos θ′) = 3 , (70) −4 −4 (x> + 1) x< ˆ d 3 2 (3/2) (3/2) (3/2) (3/2) ∆ := ∆ + Ox> 2 (ζ 1) D0 (x>) C0 (ζ) C0 (cos θ) C0 (cos θ′) dζ = dx< 16 (x< + 1) − ∫1 e 2 = 3 3 . (71) −(x + 1) (x′ + 1)

We have deliberately switched the order of summation, integration and diﬀerentiation, which might in fact be an issue, but we will later check that the result is really a Green function of the original equation (35) with given boundary conditions. Note that the above result is simple thanks to its symmetry with respect to x< x>, and also due to the relation ↔

x 2 2 (3/2) (x 1) dCn (x) (z2 1) C(3/2)(z) dz = − (72) − n n(n + 3) dx ∫1 which is a direct consequence of the ultraspherical differential equation. Calculation of the integral (69) leads to logarithms or elliptic functions, but, “miraculously”, when one applies the ˆ operator Ox> first and only then integrates by ζ, such difficulties do not occur, because

x< 2 5 d ˆ (ζ 1) dζ ˜ Ox> − = k(x′, θ′) Ik(x′, ξ) , dx< 2π(x +1)2(x2 1) sin2 θ sin2 θ x ζξ + (x ζξ)2 (1 ζ2)(1 ξ2) P ∫1 < > − ′ > − > − − − − k∑=0 [ √ ] (73) 14 P. Cˇ´ıˇzek and O. Semerak´ where we have denoted 2 (x′, θ′) := − , I˜ (x′, ξ) := 1 , 0 3 3 2 2 0 P π(x + 1) (x′ + 1) sin θ sin θ′ (x 1)(x′ 1) 1 (x′, θ′) := − − − , I˜ (x′, ξ) := , 1 3 3 2 2 1 P π(x + 1) (x′ + 1) sin θ sin θ′ ξ + 1 2xx′ + (x 1)(x′ 1) 1 2(x′, θ′) := − − , I˜2(x′, ξ) := , 3 3 2 2 2 2 2 P π(x + 1) (x′ + 1) sin θ sin θ′ x + x + ξ 1 2xx ξ ′ − − ′ 2 ξ 3(x′, θ′) := − , I˜3(x′, ξ) := √ , 3 3 2 2 2 2 2 P π(x + 1) (x′ + 1) sin θ sin θ′ x + x + ξ 1 2xx ξ ′ − − ′ (x′ 1)(x 1)(x′ + x) √ 1 4(x′, θ′) := − − , I˜4(x′, ξ) := , 3 3 2 2 2 2 2 P π(x + 1) (x′ + 1) sin θ sin θ′ (ξ + 1) x + x + ξ 1 2xx ξ ′ − − ′ (x 1)(x′ 1) ˜ √ 1 5(x′, θ′) := − − 2 2 , I5(x′, ξ) := 3/2 . (74) P 2π(x + 1)(x′ + 1) sin θ sin θ′ (x2 + x 2 + ξ2 1 2xx ξ) ′ − − ′ Therefore, we can ﬁnally conclude that

cos(θ θ′) − 5 ω (x, θ, x′, θ′) = ∆(x, x′) [cos(θ θ ) ξ][ξ cos(θ+θ )] (x′, θ′) I˜ (x′, ξ) dξ = G − − ′ − − ′ Pk k ∫ k=0 cos(θ+θ′)√ ∑ e 5 = ∆(x, x′) (x′, θ′) I (x′, θ′) , (75) − Pk k k∑=0 where e π 2 2 I (x′, θ′) := sin θ sin θ′ , 0 2 I (x′, θ′) := π (1 cos θ )(1 cos θ′ ) , 1 − | | − | | a31 a42 a43 I (x′, θ′) := a K(k) a E(k) + a 1 + Π , k , 2 a − 41 − 42 41 a −a √ 42 [ ( 31 ) ( 31 )] 2 2 2 a41 a31 a42 a21 a43 +2s21 2s21s43 a43 I (x′, θ′) := (s +s +2a )K(k) + (s 3s )E(k) + − − Π , k , 3 4 a − 21 43 21 a 43 − 21 a −a √ 42 [ 41 31 ( 31 )] 2a41 K(k) a43 a3 + 1 (a1 + 1)a43 I4(x′, θ′) := + Π , k Π , k , √a a −a + 1 −a − a + 1 −(a + 1)a 31 42 [ 1 ( 31 ) 1 ( 4 31 )] 2 √a31a42 2 I5(x′, θ′) := 2 (2 k )K(k) 2E(k) , (76) a21 − − with [ ] 2 2 2 2 a := xx′ + (x 1)(x 1) , a := xx′ (x 1)(x 1) , a := cos(θ θ′), a := cos(θ + θ′) (77) 1 − ′ − 2 − − ′ − 3 − 4 and √ √ a a a := a a , s := a + a , k := 21 43 . (78) rs s − r rs s r a a √ 31 42 On the horizon the above Green function simpliﬁes considerably,

ω 1 (x=1, θ, x′, θ′) = 3 . (79) G −4(x′ + 1) Note that this is independent of the position on the horizon (i.e. of the angle θ), consistently with the well known “rigid rotation” of stationary horizons.

6. DISC SOLUTION Perturbation equations can be solved by integrating the Green functions over the source mass distribution. Due to the presence of (complete) elliptic integrals in the above concise expression of the Green functions, a numerical integration has to be employed in general. Integration is of course simpler for thin sources (surface distribution) than for extended ones. Below we consider the case of a stationary and axially symmetric thin disc, encircling the central black hole between some two ﬁnite radii x x r + M x in a concentric manner (thus lying in the equatorial in ≤ ≡ M 4r ≤ out Disc perturbation of a Schwarzschild black hole 15 plane). We will show how to obtain explicit results for the linear perturbation of the metric functions and for the corresponding basic characteristics (density, pressure and velocity distribution) of the source.

6.1. Gravitational potential

First, let us ﬁnd the linear perturbation of potential, ν1. It is advantageous to choose B = 1 here, since this makes equation (34) the Laplace/Poisson equation, so one can use its solutions known from the classical ﬁeld theory. Let us denote by t, ϕ,ρ ˜ andz ˜ the coordinates corresponding to this choice (time t and azimuth ϕ do not depend on the 2 choice of B). Their counterparts (ρ, z) corresponding to B =1 M are related by − 4r2

ρ (M 2 4ρ2 4z2) (4r2 M 2) sin θ ρ˜ = − − = − = M x2 1 sin θ , 4(ρ2 + z2) 4r − z (M 2 + 4ρ2 + 4z2) (4r2 + M 2) cos θ √ z˜ = = = Mx cos θ , (80) 4(ρ2 + z2) 4r 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ˜ 1 (4ρ + 4z +M ) 16M z 1 (4r +M ) 16r M cos θ 1 4 (x cos θ) ζ = ζ ln 2 2−2 = ζ ln − 4 = ζ ln − 2 − 2 16(ρ + z ) − 2 16r − 2 x + √x2 1 [ ] [ ] − ( ) 2ζ 2ν (to recall the function ζ, let us remind that gρρ = grr = e − ). Of the known disc solutions of the Laplace equation, we will choose the one describing a disc which extends from the origin (˜ρ=z ˜=0) to some ﬁnite equatorial radius x′ (orρ ˜′ when expressed in the Weyl radius corresponding to the B = 1 choice) and whose Newtonian surface density S is constant in the unperturbed B = 1 coordinate system. Such a solution was obtained by Lass & Blitzer (1983),

V (x′; x, θ) =

2S 2 2 2 2 2 ρ˜′ ρ˜ 4˜ρ′ρ˜ = 2πS z˜ H(˜ρ′ ρ˜) z˜ + (˜ρ′ +ρ ˜) E(k) + (˜ρ′ ρ˜ ) K(k) +z ˜ − Π , k = | | − − 2 2 − ρ˜ +ρ ˜ (˜ρ +ρ ˜)2 z˜ + (˜ρ′ +ρ ˜) { ′ [ ′ ]} 2 2 2 2[ ] = 2πMSx cos θ H(x′√ cos θ x sin θ) | | − − − 2 2 2 2 2 2 2 2 (a41a32 x cos θ) x cos θ 2a21 sin θ a a E(k) + (x′ cos θ x sin θ) K(k) + 2 − 2 2 2 Π 2 2 , k 31 42 x cos θ x sin θ a31a42 x cos θ 2MS − − ′ − − − , (81) − √a31a42 ( )

where H(x) stands for the Heaviside function, and the ars(x′) and k(x′) symbols are given by (77) and (78) evaluated at θ′ =π/2,

2 2 2 2 a (x′) = xx′ + (x 1)(x 1) , a (x′) = xx′ (x 1)(x 1) , a (x′) = sin θ, a (x′) = sin θ, 1 − ′ − 2 − − ′ − 3 4 − 2 2 √ 2 2a21 sin θ √ 4 (x 1)(x′ 1) sin θ 4˜ρ′ρ˜ a (x′) = a (x′) a (x′), k (x′) = = − − = . (82) rs s − r a a 2 (˜ρ +ρ ˜)2 +z ˜2 31 42 x2x 2 √ (x2 1)(x 2 1) sin θ ′ ′ − − ′ − − [√ ] The potential of a disc extending between two ﬁnite radii x x x (which we are interested in) is obtained by in ≤ ≤ out subtraction of two potentials (81) (with the same density S) with outer radii at xout and xin,

ν (x, θ) = V (x′ =x ; x, θ) V (x′ =x ; x, θ) . (83) 1 out − in

6.2. Dragging In order to ﬁnd the perturbation of dragging angular velocity, one has to integrate (69) over the “density” of the disc W (x) (here assumed to be constant, W ). Since, as already noticed, (69) is symmetric with respect to the exchange 16 P. Cˇ´ıˇzek and O. Semerak´ x x , we can substitute x by x′ and x by x without changing the result. < ↔ > < > xout ω ω (x, θ) = W (x, ω, x′, π/2) dx′ = 1 − G x∫in

xout xout x′ sin θ 2 2 2 d ζ 1 sin θ ξ dξ dζ dx′ = W ∆(x, x′) dx′ + W Oˆ x − − = 2 2 2 2 2 2 − dx′ π(x′ + 1) sin θ (x 1) x ζξ +√ (x ζξ) (1 ζ )(1 ξ ) x∫in x∫in ∫1 − sin∫ θ − − − − − − e xout sin θ x′ √ (x x )(x + x + 2) W Oˆ sin2 θ ξ2 (ζ2 1) dζ = out − in out in + W x − − dξ = 3 2 2 2 2 2 2 2 2 (x + 1) (xin + 1) (xout + 1) 2π sin θ x ζξ+ (x√ ζξ) (1 ζ )(1 ξ ) (x′ +1) (x 1) sin∫ θ ∫1 − − − − − − x =x − ′ in sin θ  √  (x x )(x + x + 2) W 7 out in out in 2 2 ˜ ˜ = − 3 2 2 + W sin θ ξ Qj(xout) Ij(xout, ξ) Qj(xin) Ij(xin, ξ) dξ = (x + 1) (xin + 1) (xout + 1) − − sin∫ θ √ j=0 − ∑ [ ] (x x )(x + x + 2) W 7 = out − in out in + W [Q (x , θ) I (x , θ) Q (x , θ) I (x , θ)] , (84) (x + 1)3(x + 1)2(x + 1)2 j out j out − j in j in in out j=0 ∑ where xin and xout mark, again, the inner and outer rims of the disc (in the coordinate x), I˜0,..., I˜5 are deﬁned by (74), 1 1 I˜6(x′, ξ) = , I˜7(x′, ξ) = , ξ 1 (ξ 1) x2 + x 2 + ξ2 1 2xx ξ − ′ ′ − 2 − − 1 (1√ x)(x′ 1) Q (x′) = , Q (x′) = − − , 0 3 2 2 1 3 2 2 π (x + 1) (x′ + 1) sin θ 4π (x + 1) (x′ + 1) sin θ x′(1 2x) 1 Q (x′) = − , Q (x′) = , 2 3 2 2 3 3 2 2 π (x + 1) (x′ + 1) sin θ π (x + 1) (x′ + 1) sin θ 2 (x + x′)(x 1)(x′ 1) Q (x′) = − − , Q (x′) = 0 , 4 3 2 2 5 4π (x + 1) (x′ + 1) sin θ (1 x) (1 x)(x′ x) Q6(x′) = − , Q7(x′) = − − , (85) 4π (x + 1)3 sin2 θ 4π (x + 1)3 sin2 θ and Ik(x′) correspond to (76) taken at θ′ =π/2,

π 2 I (x′) = sin θ , 0 2 I (x′) = π (1 cos θ ) , 1 − | | a31 a41 2 sin θ I (x′) = a K(k) a E(k) + 2xx′ Π , k , 2 a − 41 − 42 a − a √ 42 [ 31 ( 31 )] 2 2 2 2 a31 a2 3a1 3a42 x x′ + cos θ (x x′) 2 sin θ I (x′) = a − K(k) xx′E(k) + − − Π , k , 3 41 a 4 − 2a a − a √ 42 [ 41 31 ( 31 )] 2a41 a31 2 sin θ 1 + sin θ 2 sin θ (a1 + 1) I4(x′) = K(k) + Π , k Π , k , √a a −a + 1 − a − a + 1 − a (1 sin θ) 31 42 { 1 ( 31 ) 1 [ 31 − ]} √a31a42 2 I (x′) = (2 k ) K(k) 2E(k) , 5 2(x2 1)(x 2 1) − − − ′ − I (x′) = π ( cos θ 1) , [ ] 6 | | − 2a41 a31 2 sin θ 1 sin θ 2 sin θ (a1 1) I7(x′) = K(k) + Π , k + − Π − , k , (86) √a a −a 1 − a a 1 a (1 + sin θ) 31 42 { 1 − ( 31 ) 1 − [ 31 ]} where the argument (x′) of ai, ars and k (given by (82)) has been omitted.

7. BEHAVIOUR OF THE SOLUTION AT SIGNIFICANT LOCATIONS Regarding the axial and reflectional symmetry, one naturally asks whether and how the solution simplifies on the 2 axis and in the equatorial plane. On the axis, sin θ =0 and cos θ =1, so, in the potential (81), a3 =0, a4 =0, k =0, so Disc perturbation of a Schwarzschild black hole 17 all the elliptic integrals there reduce to π/2 and 2 2 a a = a a = a a = x′ + x 1 . 31 42 41 32 1 2 − 2 Finally, considering that the disc has to lie above the horizon, i.e. at x′ >1, we have H(x′ 1) = 1, so the first term − of (81) is the same for both x′ =xin and x′ =xout (and thus cancels out), and hence

ν (x, θ =0) = 2πMS x2 + x2 1 x2 + x2 1 . (87) 1 − out − − in − (√ √ ) Also the equatorial form of the potential (81) is somewhat simpler, namely the ﬁrst term and the one with the Π integral vanish, so one is left with

x′=x 2 2 out (a1 1)(a2 + 1) E(k) + (x′ x ) K(k) ν1(x, θ =π/2) = 2MS − − . (88) − [ (a1 1)(a2 + 1) ] − x′=xin At radial inﬁnity (r ) the disc potential falls oﬀ as √ →∞ πS M 4 1 1 ν (r ) r2 r2 (89) 1 →∞ ∝ − r out − in − 16 r2 − r2 [ ( in out )] and at the horizon (x=1) it assumes the value

ν (x=1, θ) = 2πMS x2 sin2 θ x2 sin2 θ . (90) 1 − out − − in − (√ √ ) The formula for the dragging perturbation ω1(x, θ) is more tricky and does not reduce so much on the axis and in the equatorial plane (however, figure 3 shows that it behaves reasonably). Very simple limits can still be obtained at radial infinity where it falls off as W x x WM 2 M 2 1 1 ω (x ) out − in = ω (r ) r r + , (91) 1 →∞ ∝ 4 x3 ⇒ 1 →∞ ∝ 4r3 out − in 4 r − r [ ( out in )] and on the horizon where it is everywhere the same (independent of θ) and having its sign given by W ,

W (xout xin)(xout + xin + 2) ω1(x=1) ωH = − 2 2 . (92) ≡ 8 (xout + 1) (xin + 1) 8. PARAMETERS OF THE DISC SOURCE In order to calculate the physical characteristics of the disc, let us realize that all the densities and one-stream pressure (σ, P, σ+, σ ) are themselves small, namely of linear perturbation order (see equation (30)) and that the geodesic orbital velocities− (18) are given by their unperturbed Schwarzschild values ( Ω , to which corresponds v = ± 0 + v = v0 = 2√Mr/(2r M)) plus terms O(ω), where ω is (of course) linearly small. Consequently, up to the linear −order,− relations (26)–(28−) between normal jumps of the metric gradients across the disc and its physical parameters reduce to 2 2 + ϕ t 1 + v 1 + v0 ν1,z(z =0 ) = 2π Sϕ St = 2π(σ + P ) 2 = 2π(σ+ + σ ) 2 , (93) − 1 v − 1 v0 ( t ) − − + 8π Sϕ Ω Ω0 ω1,z(z =0 ) = = 8π(σ + P ) = 8π(σ+ σ ) , (94) 2 2 4ν0 2 2 −B ρ e− − 1 v − − − 1 v0 2 − 2 − + ϕ σv + P v0 ζ1,z(z =0 ) = 4πS = 4π = 4π(σ+ + σ ) . (95) ϕ 1 v2 − 1 v2 − − 0 + Diﬀerentiating the potential (83), (81) across the disc plane and regarding that ν,θ(θ = π/2−) = rν,z(z = 0 ), we ﬁnd − 2 2 ∂V + M 2πSM 1 + v0 = 2πMSx H(x′ x) = ν1,z(z =0 ) = 2πS 1 + 2 = 2 , (96) ∂θ θ= π − − ⇒ 4r r v0 2 − ( )

r M (the result only applies where the disc actually lies, i.e. at xin M + 4r xout, elsewhere it is zero of course), hence, according to (93), ≤ ≤ + 2 ν1,z(z =0 ) 1 v0 S 2 2 σ+ + σ = − 2 = 2 (4r 8Mr+M ) . (97) − 2π 1 + v0 4r −

More involved is to find the equatorial limit of the normal gradient of ω1 (84). One finds that solely the term Q7I7 contributes eventually, because Q5I5 is zero from the beginning, the first separate term of (84) as well as Q0I0 are 18 P. Cˇ´ıˇzek and O. Semerak´ z z

ρ ρ

z z

ρ ρ Figure 2. Meridional-plane contours of the gravitational potential ν, given by sum of the Schwarzschild expression and the contribution from the disc. The four examples shown represent an equatorial disc stretching from ρ = 5M to ρ = 7M (it is indicated by a thick black line), with potential (81) scaled by S =0.01 (top left), S =0.026 (top right), S =0.1 (bottom left) and S =1.0 (bottom right) (such a series corresponds to a more and more massive disc). The potential is everywhere negative, with light/dark shading indicating shallow/deeper values (the potential diverges to at the horizon, while the “weakest” levels reached at top right corners of the plots amount to 0.20 −∞ − at top left, to 0.34 at top right, to 1.00 at bottom left and to 8.96 at bottom right); the black-hole horizon (at ρ2 + z2 = M 2/4) is represented by− the white quarter-circle.− Both axes are given in the− units of M.

ρ Figure 3. Meridional-plane contours of the dragging angular velocity ω, as entirely given by the ﬁrst-order perturbation due to the disc (ω1). The disc again stretches from ρ = 5M to ρ = 7M (as indicated by the thick black line). The angular velocity is everywhere positive, with light/dark shading indicating smaller/larger values (they reach about 0.0268/M at the disc and fall oﬀ to some 0.0047/M at top right 2 2 2 of the plot); the black quarter-disc at ρ + z M /4 represents the black hole. Both axes are given in the units of M. Since ω = ω1 is proportional to W , the isolines have the same shape≤ for any W , only their values scale with this rotational parameter.

independent of θ, Q6I6 is independent of x′ (and thus its “out” and “in” values subtract to zero), the Q2I2 and Q3I3 terms vanish in the θ (π/2)− limit, and contributions of the terms Q I and Q I exactly cancel out each other in → 1 1 4 4 Disc perturbation of a Schwarzschild black hole 19 this limit. From (Q7I7),θ one speciﬁcally obtains, in the above limit, non-zero terms

2Q7 (a1 + 1) ∂Π(n, k) 2 sin θ (a1 1) Π(n, k) cos θ + (1 sin θ) , where n = n(x, x′, θ) := − (a 1)3(a + 1) − − ∂θ (1 + sin θ)(a1 sin θ) 1 − 2 [ ] − √ Π(n,k) ∂n(x,x′,θ) and where the derivative of Π(n, k) contributes solely through the term 2(1 n) ∂θ . Using the asymptotics − π √n Π(n, k) + O(1) valid for n 1− ∝ 2 √n k2 √1 n → − − in both terms and making the equatorial limit, one ﬁnally ﬁnds a simple result ∂ω ∂ω W x 1 (2r M)2 lim 1 = r lim 1 = − = 8WM 2r2 − , (98) θ π ∂θ − z 0+ ∂z 2 (x + 1)3 (2r + M)6 → 2 − ( → ) which, when compared against

2 ∂ω1 Ω0 8r √Mr (2r M) lim = 8π(σ+ σ ) 2 = 8π(σ+ σ ) − z 0+ ∂z − − − 1 v − − − (2r + M)3 4r2 8Mr + M 2 → − 0 − given by (94), implies that WM 2 4r2 8Mr + M 2 σ+ σ = − . (99) − − 8π √Mr (2r + M)3 By adding (97) and (99), we have then 4r2 8Mr + M 2 W (Mr)3/2 σ = − S . (100) ± 8r2 ± 2π (2r + M)3 [ ] To find the one-stream disc characteristics, one first combines (93) and (95), which yields σv2 P 4Mrσ (2r M)2P v2 = 0 − = − − . (101) σ P v2 (2r M)2σ 4MrP − 0 − − Substituting this for v2 in (94) leads to 2 2 2 2 (σv0 P )(σ P v0) = (σ+ σ ) v0 − − − − and from there, using σ P = σ+ +σ , we find − − 2 2 2 2 σ+ +σ σ+ +σ 4 σ+σ v0 σ+ +σ σ+ +σ 16Mr (2r M) σ+σ σ = + − + − + − = + − + − + − − , (102) 2 √ 2 (1 v2)2 2 √ 2 (4r2 8Mr+M 2)2 ( ) − 0 ( ) − 2 2 2 2 σ+ +σ σ+ +σ 4 σ+σ v0 σ+ +σ σ+ +σ 16Mr (2r M) σ+σ P = − + − + − = − + − + − − . (103) − 2 √ 2 (1 v2)2 − 2 √ 2 (4r2 8Mr+M 2)2 ( ) − 0 ( ) − Note that when substituting these σ and P back to (101), the resulting v given by square root of the latter is to be taken with +/ sign in case that σ+ >σ / σ+ <σ . In order to demonstrate− that the procedure− really− works and, in particular, to illustrate the role of the parameters S and W , let us consider a disc spanning between the radii ρ =5M and ρ+ =7M, surrounding a black hole of mass M, and let us plot the results for several different values of S and/or− W . Note that both S and W have the dimension of 1/length, practically 1/M. Note also that in order to emphasize their effect, we ignore here the assumption that the perturbation should be very small. More precisely, the first-order perturbation of the gravitational potential is in fact not restricted by this assumption, because rotation/dragging only enters in the second order, so the change of ν can be understood as an exact superposition within the non-rotating, static class. The potential ν (the sum of the Schwarzschild background ν0 and the perturbation ν1) is shown in figure 2; it behaves in an expected manner, namely the disc effect grows with increasing S (and it is independent of W ). The dragging angular velocity amounts to ω =ω1 (since ω0 = 0), so the level-contour shape is in fact fixed and does not change with parameters (only the level values do change, in particular they scale with W ); this is shown in figure 3. Now for parameters of the double-stream and single-stream interpretations, i.e., σ and v , and σ, P and v, respectively. In fact the orbital velocities v of the double-stream picture need not be computed,± ± namely, in the first perturbation order they are the same as in± a pure Schwarzschild field, because the field equations (6) and (7) have right-hand sides proportional to the surface densities which are themselves of linear order. The other quantities are plotted in figure 4 for several choices of the S and W parameters. One sees there that an increasing value of W makes σ+ and σ more and more different, with σ finally becoming negative, which marks limits of the (counter-)rotating − − 20 P. Cˇ´ıˇzek and O. Semerak´ interpretation. Physically, such a situation means that, for a given mass, the disc has too much angular momentum. Naturally, this is also accompanied by a need for too high orbital velocity; such a circumstance can be “remedied” by increasing the mass, i.e. S.

1st column: W = 25/M 2nd column: W = 50/M 3rd column: W = 90/M 4th column: W = 150/M

0.12 σ+ solid, σ− dotted 0.08 0.04 0 -0.04 ρ ρ ρ ρ 0.16 σ solid, P dotted 0.12 0.08 0.04 0 -0.04 ρ ρ ρ ρ 1 v 0.75

0.50

0.25

0 5 5.5 6 6.5 75 5.5 6 6.5 75 5.5 6 6.5 75 5.5 6 6.5 7 Figure 4. Parameters of the disc’s double-stream and single-stream interpretations, plotted as functions of the Weyl radius for a disc stretching between ρ = 5M and ρ = 7M and for several combinations of the parameters S (Newtonian density) and W (scaling the disc rotation). The first/second/third/fourth columns represent the cases given by W =25/50/90/150, with the first row showing densities σ+ (solid line), σ (dotted line) of the double-stream, counter-rotating dust interpretation, the second row showing density σ (solid line) and azimuthal pressure− P (dotted line) of the single-stream fluid interpretation, and the third row showing the corresponding “bulk” velocity of the fluid v (solid). For each of the values of W , three different values of S have been chosen, corresponding to discs of different mass – S =0.04, 0.12 and 0.2. The density and pressure curves obtained for higher S are higher (given by larger values), whereas the corresponding bulk velocities decrease with growing S. The ranges of the chosen W and S are largely out of the scope of the linear perturbation, but this is in order to illustrate clearly what they represent. Since all the parameters (densities, pressure, velocity) should be real and positive, and the velocity v has to be < 1 in addition, one sees immediately that i) the discs with a given mass cannot bear however large angular momentum (with growing W , the density σ of the double-stream interpretation – and consequently also the single-stream parameters – tend to be negative); and, ii) since large W−implies the need for high orbital velocities, for too large W the orbital interpretation would have to involve superluminal motion (in the bottom row, from left to right, one sees that all three / two / two / one of the shown cases are “physical” in this respect). The densities S, W , σ+, σ , σ as well as the pressure P have the dimension of 1/length and their values are in the units of 1/M, the speed v is dimensionless. −

8.1. Mass and angular momentum of the disc The total mass and angular momentum of a stationary and axially symmetric space-time can be found from Komar µ ∂xµ µ ∂xµ integrals, given by the Killing vector ﬁelds η = ∂t and ξ = ∂ϕ . In our case (involving two sources) the integrals over spatial inﬁnity can be split into contributions from the black hole M and J (given by integration over the horizon H) and from the disc (integrated over some space-like hypersurface Σ covering the black-hole exterior),

1 µ;ν 1 µ;ν µ ν µ i t 3 mass = lim η dSµν = η dSµν (2Tν η T η ) dΣµ = M + (Ti Tt )√ g d x , (104) 8π S 8π − − − − →∞ IS HI Σ>∫H Σ>∫H

1 µ;ν 1 µ;ν 1 µ ν µ t 3 ang.m. = lim ξ dSµν = ξ dSµν + (2Tν ξ T ξ ) dΣµ = J + Tϕ √ g d x , (105) −16π S −16π 2 − − →∞ IS HI Σ>∫H Σ>∫H

µ µ µ µ where we have ﬁnally employed the “Killing” coordinates t and ϕ in which η = δt , ξ = δϕ , and a natural space-like t 3 i t 2λ 2ν i t section Σ = t = konst (which corresponds to dΣµ = δµ√ g d x). Clearly (Ti Tt ) – or actually e − (Ti Tt ) – plays the same{ role as} mass density in the Newtonian theory.− Note also that the− second part of the mass integral− represents what is sometimes called Tolman’s formula. 2 4λ 4ν Choosing B =1, the metric determinant reads g = ρ e − , so, if considering the thin z =0 -disc as source and thus having T µ√ g = Sµρ δ(z), the disc mass and− angular momentum (denoted by and{ ) come} out as ν − ν M J = 2π (Sϕ St) ρ dρ , = 2π St ρ dρ . M ϕ − t J ϕ disc∫ disc∫ Disc perturbation of a Schwarzschild black hole 21

Substituting here for (4) with uϕ = utΩ and u = g ut(Ω ω), we have ϕ ϕϕ − St = (σ + P )(ut)2g (Ω ω),Sϕ St = (σ + P )(ut)2 e2ν + g (Ω2 ω2) = σ + P + 2ΩSt , (106) ϕ ϕϕ − ϕ − t ϕϕ − ϕ hence [ ]

= 2π (σ + P + 2ΩSt ) ρ dρ , (107) M ϕ disc∫ 2ν ρe− (Ω ω) = 2π (σ + P ) 2 4ν − 2 gϕϕ dρ ; (108) J 1 ρ e− (Ω ω) disc∫ − − in the special case of Ω=const the ﬁrst integral amounts to

= 2π (σ + P ) ρ dρ + 2Ω . M J disc∫ µ One can alternatively express, in the integrals, Sν in terms of jumps of the normal ﬁelds, according to (26)–(27). Setting B =1 again, we thus have

t 1 2 4ν ϕ t ν,z t S = ρ e− ω ,S S = + 2ωS , ϕ −8π ,z ϕ − t 2π ϕ hence 1 2 4ν 1 3 4ν = ν ρ e− ωω ρ dρ , = ρ e− ω dρ (109) M ,z − 2 ,z J −4 ,z disc∫ ( ) disc∫ + 2 (where ω,z and ν,z are evaluated at z 0 ). Since the second term in is O(λ ), in the linear order one is left with just → M + 1 3 4ν + = ν (z 0 ) ρ dρ , = ρ e− 0 ω (z 0 ) dρ . (110) M1 1,z → J1 −4 1,z → disc∫ disc∫

Finally, one has to realize that the above formulas hold for B =1, whereas our results for ν1 and ω1 have been derived M 2 with B = 1 4r2 . However, adapting the integrals to the latter choice practically means just to write Bρ instead of ρ in both integrands,− reaching

+ 1 3 3 4ν + = Bρ ν (z 0 ) dρ , = B ρ e− 0 ω (z 0 ) dρ . (111) M1 1,z → J1 −4 1,z → disc∫ disc∫

Considering speciﬁcally the above constant-density disc, it is clear from (81) and (84) that 1 is scaled by the free M + parameter S while is scaled by the second parameter W . We have from (96) and (98), at z 0 (i.e., θ π/2−), J1 → → M 2 (2r M)2 ν ρ = ν = πS 2r + , ω ρ = ω = 8WM 2r2 − , (112) 1,z − 1,θ 2r 1,z − 1,θ − (2r + M)6 ( ) 4 4ν0 (2r+M) which yields, after using ρ(θ =π/2) = r and e− = (2r M)4 , − rout 2 2 4 M M 2 2 M 1 1 1 = πS 2r + 1 2 dr = πS rout rin 2 2 , (113) M 2r − 4r − − 16 rin − rout r∫in ( )( ) [ ( )]

rout WM 2 M 2 WM 2 M 2 1 1 1 = 1 2 dr = rout rin . (114) J 8 − 4r 8 − − 4 rin − rout r∫in ( ) [ ( )] The same results follow from the asymptotics (89) and (91), if regarding the general behaviour (in an asymptotically flat space-time) M + J + ν(r ) 1 M1 , ω(r ) 2 1 J1 . → ∞ ∝ − r → ∞ ∝ r3 The last statement implies that i) the black-hole mass remains M (which should be so in the linear perturbation order), and that ii) the whole angular momentum of the system (inferred from the asymptotic behaviour of ω) is being carried by the disc, hence that the angular momentum of the hole remains zero, J = 0. Actually, we can verify this 22 P. Cˇ´ıˇzek and O. Semerak´ directly by computing the respective Komar integral over the horizon. It is known that the latter can be rewritten (see e.g. Will 1974, equation (21)) as π 4 k 3 4ν 3 J = B ω e− sin θ dθ , − 27 ,r r=k/2 ∫ 0 [ ] where in our first-order case one takes ν =ν0, ω =ω1 and k =M. Inspecting the radial gradient ω1,r of (84), one finds that all the terms of this expression individually vanish at the horizon (irrespectively of θ), so the above formula really yields J =0. Hence, the perturbed black hole is rotating with respect to the asymtotic inertial frame with the non-zero (positive) angular velocity (92), but has a zero angular momentum, which means that it is just being “carried along” by dragging primarily induced by the disc. We stress that this feature is not a necessary outcome of the perturbation procedure, namely it is a consequence of our choice of the constants Jl (namely Jl = 0) which can be employed to fix the black-hole spin in the solution (50) of the inhomogeneous perturbation equation (44); see the discussion in section 3.1.

9. CONCLUDING REMARKS We have shown that the procedure suggested by Will (1974), originally employed to determine the gravitational perturbation of a Schwarzschild black hole by a slowly rotating and light thin ring, can be also applied to the perturbation due to a thin disc. However, concerning the bad numerical properties of the series involved, we have dropped the angular expansion and rather expressed the Green functions (perturbations due to a thin ring) in a closed form. Such expressions bring more complex special functions, but these can be evaluated effectively with rapidly converging algorithms, so the resulting numerical convergence is much better. Using the proposed closed-form Green functions, one can in principle obtain an arbitrary order of the perturbation. However, a numerical treatment is necessary in order to analyse specific results, as illustrated on a simple example of the “uniform-density” disc in section 6. An important point has been to show that the series involved in computation of the Green functions converge (section 4 and the accompanying paper by C´ıˇzekˇ (in preparation)). However, what still remains to be answered is whether the perturbation scheme is effective to any order, namely whether the perturbation expansion (29) with parameter proportional to the external-source mass converges. Regarding the structure of the Green functions and the speed of their convergence to zero at infinity, as well as the structure of source terms of higher perturbation orders, one conjectures that the expansion converges at least for some positive disc masses. Various properties of the obtained solution could be studied now, for instance, deformation of the geometry (as represented by curvature invariants, in particular), deformation of the (originally spherical) horizon, perturbation of the properties of stationary circular motion or influence on geodesic structure. In particular, it will be interesting to see how the perturbation influences the location of important circular equatorial geodesics (mainly of the innermost stable one, usually abbreviated as ISCO), because this should indicate how the actual quasi-stationary accretion disc may differ from its test-matter model. Also, àpropos, one could consider a different type of disc (a different density distribution) – preferably close to what follows from models of accretion onto astrophysical black holes – and compare the results with what has been found here for the simplest case of constant density. 9.1. Comparison with black-hole–disc configurations found numerically Another obvious option is to compare the perturbative solution with the results of numerical treatment of similar source configurations. The most similar of these – a hole with a thin finite annular equatorial disc – was studied by Lanza (1992), while Nishida & Eriguchi (1994) considered a hole with a thick toroid. More recently, Ansorg & Petroff (2005) used a different numerical method to compute stationary and axisymmetric configurations of uniformly rotating constant-density toroids around black holes. They specifically used these solutions to demonstrate that both the central hole and the surrounding toroid may in some cases have negative Komar masses (Ansorg & Petroff 2006). Yet another codes for studying self-gravitating matter around black holes have been developed, and specifically used to find stationary thick-toroid configurations, by Shibata (2007), Montero et al. (2008) and Stergioulas (2011). Finally, Karkowski et al. (2016) analyzed such rotating black-hole–toroid systems in the first post-Newtonian approximation. The possibility to compare our results with the above “exact” numerical configurations (in particular the thin discs by Lanza) is very limited, mainly because our first-order perturbation only represents gravitation of the disc, not its self-gravitation (no back effect of the source on itself through its field); more generally speaking, the solution does not incorporate any non-linearity of the Einstein equations. There are also other, more definite differences. The numerical results of Lanza (1992) were obtained by numerical “relaxation” of initial configurations provided by “squeezing” the well known (analytical) test thick-disc models in a given Kerr background. More specifically, assuming constant ratio of angular momentum to energy (with respect to infinity) throughout the disc, in our notation 1 u g + g Ω e2ν − const = ϕ = tϕ ϕϕ = ω + , u − g + g Ω Bρ v − t tt tϕ ( ) and choosing the inner disc radius and the proportionality constant appearing in the polytropic equation of the disc-gas state, the initial surface density and outer radius of the disc are obtained. Generally, with the increase of that constant, Disc perturbation of a Schwarzschild black hole 23 the surface density (as well as pressure) and mass of the resulting disc rise quickly, while the outer disc radius grows with the specific angular momentum of the disc matter. In contrast to Lanza’s constraint of constant specific angular momentum, and the disc’s surface density and outer radius derived accordingly, we have exemplified our perturbation procedure on a disc with prescribed and constant Newtonian surface density (while non-constant angular momentum) and with both inner and outer radii prescribed as well. Lanza illustrated his numerical scheme on two groups of configurations sequences, one containing a rapidly rotating hole (specified by its horizon area and angular momentum) and one with a slowly rotating hole (in this case specified by its horizon angular velocity ωH and isotropic radius k/2). Focusing naturally only to the latter, one finds solution sequences for three different rotation choices in the cited paper: ωH =0.00025/M, ωH =0.0025/M and ωH =0.025/M, in all cases with k =M and rin =8M; each sequence is characterized by a certain fixed value of the angular momentum to energy ratio (constant throughout the disc) and was generated by gradual increase of the disc mass. With the increasing disc mass, also the total mass of the system were found to increase (almost linearly), while the total angular momentum and the horizon area AH were increasing slightly faster, with the horizon surface gravity decreasing consequently according to the generic relation κH = 4πk/AH. One special point Lanza examined on slowly rotating configurations was that the black-hole rotational angular momentum may decrease to negative values when the disc angular momentum is being increased (while ωH is kept fixed). This happens when the disc is “overtaking” the black hole in the sense that the combined rotational-dragging effect is stronger than the effect due to the hole alone (such a circumstance was already pointed out by Will 1974). Let us remind that in our case the black hole was effectively set to keep zero angular momentum (while acquiring non-zero angular velocity) in the perturbation, as confirmed at the end of section 8.1. Regarding the differences between assumptions of the above numerical treatment and our perturbative one, the comparison of results obtained by these two methods is going to be rather problematic. Anyway, we now plan to compute various more specific parameters of the disc considered as an example in section 6 (and the following ones), in order to possibly return to this point. Let us thus conclude with one particular surprising observation made by Lanza (1992) which should be simple to compare: for slowly rotating black holes, Lanza found that the presence of the disc can make the horizon’s polar proper circumference larger than the equatorial one, which would suggest that the horizon becomes prolate (along the symmetry axis). This goes against common experience that the black holes stretch towards external sources of gravity (cf. Nishida & Eriguchi 1994 and Ansorg & Petroff 2005 who always got oblate horizons).

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