Master’s Thesis
The rotating mass shell in the general theory of relativity
Florian Atteneder Graz, October 2019
supervised by Univ.-Prof. Dr.rer.nat Reinhard Alkofer Abstract
The model of a rotating mass shell (RMS) was initially introduced to judge if rotation has only relative meaning. It comprises a general relativistic description of a spacetime with an energy-matter content that is assembled in a rotating quasi-spherical shell with zero radial extension. The concept of relativity of rotation goes back to Mach’s principle and latest results obtained by perturbation theory (PT) calculations have shown that it is indeed realized in such a spacetime. However, because this conclusion was based on PT, its validity is limited to slowly RMSs. This thesis pursues a numerical treatment of the problem which can provide insight into the validity of Mach’s idea of relativity of rotation also for rapidly RMSs. The mathematical formulation of the RMS problem involves a splitting of spacetime into a region that is flat and another region that is asymptotically flat, where the latter is used as a reference to define relative rotation. Under these assumptions, a RMS forms at the common boundary of these two regions. On the basis of previous work, we formulate Einstein’s equations as a free-boundary value problem and solve them numerically using a pseudo-spectral method. As a result we obtain a three-parameter solution that is characterized by the shell’s polar radius R, its total gravitational mass M and angular momentum J. The existence of the solution is enough to positively answer the question if Mach’s idea of relativity of rotation can be extended for rapidly RMSs.
2 Kurzfassung
Das Modell einer rotierenden Massenschale (RMS) wurde entwickelt um zu testen, ob Rotation nur eine relative Bedeutung besitzt. Dies geschieht im Sinne einer allgemein relativistischen Beschreibung einer Raumzeit in welcher der Energie-Masse Gehalt durch eine rotierende und quasi-sphärischen Schale mit verschwindender radialer Ausdehnung betrachtet wird. Das Konzept von relativer Rotation geht auf das Mach’sche Prinzip zurück und jüngste Ergebnisse einer störungstheoretischen Behandlung des RMS Prob- lems zeigten, dass dies tatsächlich realisiert werden kann in einer solchen Raumzeit. Da diese Schlussfolgerung jedoch auf einer stöhrungstheoretischen Rechnung gründete, ist diese auf langsam rotierende Massenschale beschränkt. In dieser Arbeit wird ein numerischer Zugang des Problems entwickelt, welcher klären soll, ob Machs Idee der rel- ativen Rotation auch auf schnell rotierende Massenschalen ausgedehnt werden kann. Die mathematische Formulierung des RMS Problems erfolgt durch eine Teilung der Raumzeit in einen flachen sowie asymptotisch-flachen Anteil, wobei letzterer als Referenze zurBes- timmung der relativen Rotation herangezogen wird. Aus diesen Annahmen folgt, dass sich die RMS an der gemeinsamen Grenze der beiden Teilbereiche bildet. Aufbauend auf einer vorrangegangenen Arbeit formulieren wir die Einstein Gleichungen als ein freies Randwertproblem und lösen dieses numerisch mittels einer pseudo-spektralen Methode. Als Ergebnis erhalten wir eine numerische Lösung, welche durch die drei Parameter polarer Schalenradius R, gravitative Gesamtmasse M und den Drehimpuls J charak- terisiert ist. Die Existenz der numerischen Lösung erlaubt eine positive Beanwortung der Frage nach der Realisierung der Mach’schen Idee von relativer Rotation für schnell rotierende Massenschalen.
3 Acknowledgement
I would like to thank my supervisor Univ.-Prof. Dr.rer.nat Reinhard Alkofer for introducing me to the topic of rotating mass shells. I am also thankful to Dr.rer.nat. Helios Sanchis-Alepuz for co-supervising this project. Many thanks must go to Tobias Benjamin Russ. I very much appreciate the time he took to explain clarifying details to me about his previous work on rotating mass shells. I would also like to thank Fabian and Martin for proofreading the first draft of this work. Lastly, I am very thankful for the financial support by the Paul-Urban-Stipendienstiftung of the University of Graz.
4 To my family. Contents
Notation8
1. Introduction9
2. The system under study 11 2.1. Previous work...... 11 2.2. Assumptions...... 12 2.3. Coordinates and metric tensor...... 13 2.4. Einstein’s field equations...... 16 2.4.1. Solution strategy...... 16 2.4.2. Vacuum field equations...... 17 2.5. Boundary and regularity conditions...... 18 2.5.1. Behavior at spatial infinity...... 18 2.5.2. Behavior on the axis of symmetry and across the equatorial plane 21 2.5.3. Conditions on the shell...... 23
3. Numerical treatment 24 3.1. Introduction to pseudo-spectral methods...... 24 3.2. Series expansions...... 25 3.2.1. Ansätze for U, Ω,V ...... 25 3.2.2. Analytic solution for W ...... 27 3.2.3. Ansatz for f ...... 28 3.3. Putting everything on the computer...... 28 3.3.1. Guess and extrapolation...... 31
4. Results 33 4.1. Centrifugal deformation and differential rotation...... 33 4.2. Surface-stress-energy tensor...... 36 4.3. Error estimation...... 37
5. Conclusion 44 5.1. Discussion...... 44 5.2. Summary and Outlook...... 45
A. Thin shell description 47 A.1. Definition and notation...... 47 A.2. Description of the shell...... 47
6 A.3. Induced metric and first junction conditions...... 48 A.4. Extrinsic curvature and surface-stress-energy tensor...... 49 A.4.1. Eigenvalues and eigenvectors...... 50 A.4.2. Energy conditions...... 51
B. Komar integrals 53
C. Details on numerical implementation 55 C.1. Rational Chebyshev functions...... 55 C.2. Collocation points...... 56 C.3. Orthogonality relations...... 57 C.4. Code and external libraries...... 58
D. Perturbation theory 59
Bibliography 63
7 Notation
Throughout this work we apply the metric sign convection ( + ++). Greek indices α, β, γ, ... run over 4 dimensional spacetime coordinates.− Latin indices a, b, c, ... run over 3 dimensional hypersurface coordinates.
∂µ represents partial derivative operation and when acting on a tensor it is abbreviated α... α... as ∂µ X β... = X β... ,µ. µ represents covariant derivative operation and when acting on a tensor it is abbrevi- ∇ α... α... ated as µX β... = X β... ;µ. Geometric∇ units are employed, that is, G = c = 1 is used. For the definition of the Riemann and Ricci tensor we follow closely the convention that is employed in [1,2].
8 1. Introduction
General relativity (GR) was published by Einstein in 1915 [3]. It was developed to circumvent the shortcomings of a Newtonian description of gravity, which suffered from incompatibility with the previously established theory of special relativity [4]. GR is a geometric theory in which the gravitational interaction is mediated via the curvature of space and time, or spacetime. The content of this theory is embodied in Einstein’s field equations. They relate the spacetime’s curvature to the energy and momentumof the matter and radiation content. For the most general case, these equations are ten coupled, nonlinear, second-order partial differential equations (PDEs). Solving Einstein’s equations for a rotating mass shell (RMS), utilizing numerical meth- ods, is the topic of this work. A RMS may be imagined as a rotating and mas- sive quasi-spherical shell of zero radial extension. The model was introduced first by Einstein [2, section 4.1], and the reason why he was interested in studying it, was that it shall allow calculating what is nowadays known as (frame) dragging effects, see [5, chapter XII.5] for an example. These effects refer to the gravitational influence of the RMS on local inertial frames. Ultimately, according to [1,2,6], based on such a calculation it can be made plausible that rotation has only a relative meaning in such a spacetime and that [1] "(...) it should be impossible to decide, in principle, whether an observer rotates relative to the fixed stars or all the stars and galaxies in the universe rotate relative to him." This idea often goes under the name Mach’s principle of relativity of rotation. In par- ticular, it is this principle one was interested initially to test with the RMS model and to check whether it is included in GR or not. This is a controversy topic, because no precise definition of Mach’s principle has been given so far and for further details onthis topic we refer to [2] and the references listed there. On the trail of Einstein’s RMS model, over the years many physicists have worked on refining his initial perturbation theory (PT) calculations. The most recent results found in the literature were conducted by Pfister et al. [1], with a revision of this work published in [2, appendix B]. Their conclusion states that Mach’s idea of relativity of rotation is completely realized in general relativity within the RMS model. The initial question that has led to the introduction of the RMS has thus been answered. So why would one bother and reexamine their work using numerical methods? Firstly, the conclusion of Pfister et al. about Mach’s idea is based on a PT calculation and
9 therefore of limited validity. Strictly speaking, the conclusion is only proven in the perturbation regime. A successful extension of these calculations via numerical methods beyond the PT regime can guarantee that the conclusion of Pfister et al. is applicable to all RMS configurations, and not only those which are rotating slowly. Another reason that may justify the expense has been provided in [7]. Pfister et al. have proven that for given parameters total gravitational mass M and initial sphere radius R, there exists exactly one solution for a RMS (with a flat interior spacetime, discussed below) to any order of their PT expansion. If this result holds true beyond PT, then one can construct numerically a family of solutions to Einstein’s field equations that is characterized exactly by three parameters. The actual mathematical problem associated with the RMS model and treated by Pfister et al. can be stated as follows [1,2,8]: Assume that spacetime is partitioned by a hypersurface Σ into two distinct regions + and −, corresponding to an exterior and + − V V + − interior region, respectively. Each , is equipped with a metric g µν, g µν expressed in a system of coordinates x+µ, x−Vµ. TheV task is to join + and − smoothly at Σ, so + − V V that the union of the metrics g αβ and g αβ forms a valid solution of Einstein’s field equations. It is demanded that the interior metric is flat and the exterior metric is asymptotically flat. This setup together with certain assumptions discussed in themain text allows to turn the Einstein’s field equations into a free boundary value problem. In this work we tackle this problem with numerical methods. The present thesis is structured as follows: In chapter 2 we elaborate on how the RMS problem is approached. This includes a discussion about the choice of coordinates as well as an ansatz for the metric tensor. Based on this, the Einstein field equations for the situation at hand are presented and a set of PDEs, complemented with boundary conditions, is established. In chapter 3 we introduce the method of choice - the pseudo- spectral method - for the numerical treatment of the PDEs. Based on this, ansätze for the pseudo-spectral treatment are constructed and, subsequently, we discuss how the actual implementation on the computer is carried out. In chapter 4, numerical results are presented where emphasis is made on a comparison with the PT results of Pfister et al.. Additionally, we present characteristic properties of the associated energy-matter content of the numerical results and check if they obey the dominant energy condition. To conclude the analysis, an error estimation of the results is presented. In chapter 5 we conclude the thesis with a discussion and an outlook. To spare technical details about certain computations, we have outsourced some of the content to appendices: Appendix A gives details about the utilized description of the RMS and the compu- tation of its stress-energy tensor; Appendix B lists the detailed calculation of Komar quantities, which are used for the error estimation; Appendix C provides a discussion about subtleties of the numerical treatment, such as properties of utilized functions; Appendix D gives a brief summary of the most important findings and formulae of the most recent work by Pfister et al. [2].
10 2. The system under study
2.1. Previous work
The RMS problem was extensively studied with PT over the years. Among these studies were the two famous papers [9, 10] that predicted the Lense-Thirring effect which was experimentally verified around 100 years after its discovery [11]. Since this work aims at extending the most recent PT calculations of Pfister et al. [1,2], below we provide a brief and qualitative overview of their approach and results. In Appendix D we list details about the result that are too involved for this section. The work of Pfister et al. was based on a perturbation expansion around the Schwarzschild solution. It was carried out in the parameter ωPT R, which can be interpreted as the circumferential velocity of the RMS’s constituents moving in the equatorial plane mea- sured by a distant observer. The expansion is valid unless ωPT R 1, e.g. much smaller than the speed of light. Their results are valid for arbitrary, physical≪ ratios of mass and radius M/R of the shell. In their work two observations were made that have been key to deduce their results. In [2, section 4.2] they state: "Any physically realistic, rotating body will suffer a centrifugal deformation to • order ω2 and higher, and cannot be expected to keep its spherical shape." "If we aim and expect to realize quasi-Newtonian conditions with the ’correct’ • Coriolis and centrifugal forces (and no other forces!) in the interior of the rotating mass shell, this interior obviously has to be a flat piece of spacetime. (...)"
The above mentioned parameter ω is called ωPT in the present thesis in order to distinct it from the true angular velocity which is labeled as ω. Based on these observations, Pfister et al. then solved the field equations in PT and presented the following results:
1. Allowing a latitude dependent deformation of the RMS, Pfister et al. found that for all physical reasonable ratios 0 < M/R < 2 an initially spherical RMS deforms to a prolate ellipsoid if in rotation. Furthermore, it was found that the strength of this deformation reaches its maximum for a ratio of M/R = 8/5. It was also claimed that the deformation vanishes in the collapse limit of M/R 2. These 2 → conclusions are consistent with PT up to order ωPT .
11 2. The analysis also unveiled that the RMS is differentially rotating, which means that the angular velocity is not constant for all elements constituting the RMS. In particular, they state that the angular velocity of the shell’s constituents is higher for those moving at the poles compared to the ones moving in the equatorial plane only for configurations where M/R < 4/7. However, for cases where 2 > M/R > 4/7 the opposite is true; they state that the angular velocity of the poles are relatively smaller compared to the equatorial plane. Furthermore, in the limit of M/R 2 they conclude that differential rotation transitions to rigid rotation, meaning→ that a constant angular velocity along the shell is restored. These findings 3 are based on the PT results from order ωPT . In total, Pfister et al. could solve the RMS problem with a flat interior consistently up 2 to order ωPT . Based on this, they claim that Mach’s idea of relativity of rotation is 2 3 completely realized in the RMS model [1,2]. As their analysis in order ωPT and ωPT has shown, this can only be possible if the shell deforms to a ellipsoid and is differential rotating. Most recently, Russ attempted to solve the problem studied by Pfister et al. by utilizing numerical methods [8]. Although this work provides a formidable preparation of the analytic description of the RMS problem, the numerical results presented in this thesis suffered from a systematic disagreement with the PT results ofPfister et al. in the slow rotation limit. In this thesis we build upon this preliminary work, extend some of his arguments, provide clarifying details where necessary and present numerical results that are in accordance with the solutions of the PT calculations of Pfister et al. up to order 2 ωPT within the slow rotation limit.
2.2. Assumptions
The spacetime under study shall obey the properties of [8] (a) Stationarity, (b) Axisymmetry, (c) Asymptotic flatness, (d) Reflection symmetry with respect to the equator, (e) Flatness in the interior region of the RMS. Properties (a), (b) together with (c) guarantee the existence of two conserved currents with charges corresponding to the total gravitational mass M and angular momentum J of the RMS, respectively [12]. Assumption (d) is carried over from the PT calculations of Pfister et al., where it was assumed that the perturbation corrections to the initially non-rotating and spherically symmetric configuration are symmetric with respect to the
12 equatorial plane. Property (e) is also carried over from the PT calculations and is required to test Mach’s idea of relativity of rotation [1,2]. In the subsequent section we make a choice of coordinate system, an ansatz for the metric tensor and discuss the relevant field equations following [8]. All these decisions are tied to the assumptions provided in the above list. Furthermore, it shall be mentioned that all assumptions, except property (e), are com- monly used in studies of rotating stars in GR [13–16].
2.3. Coordinates and metric tensor
An ansatz for the metric tensor that incorporates properties (a) and (b) was provided by Pfister et al. [1,2] and it reads
ds2 = e2U dt2 + e−2U (︁e2V (︁dr2 + r2 dθ2)︁ + W 2 (dφ Ω dt)2)︁ . (2.1) − − Ansatz (2.1) is given in quasi-isotropic coordinates which are denoted by (xµ) = (t, r, θ, φ) throughout this work. The unknown functions U, V, W, Ω are functions of r, θ only. For a derivation and a detailed discussion of this ansatz as well as quasi-isotropic coordinates one is referred to [12]. In the following only the most important characteristics of these coordinates and the ansatz are listed. Quasi-isotropic coordinates are spherical-like coordinates. This is to be understood such that the parameters t, r, θ, φ describe points on the manifold similar to how Schwarzschild coordinates1 do this: for an asymptotically flat spacetime the parameter t ( , ) agrees with proper time as measured by an asymptotically inertial observer∈ who−∞ is∞ at rest with respect to the central region of spacetime, r [0, ) measures the radial distance from the coordinate system’s origin and θ [0, π∈), φ ∞[0, 2π) label the polar and azimuthal angle, respectively. ∈ ∈ A transformation that relates quasi-isotropic coordinates with Schwarzschild coordinates can only be given for static spacetimes. In this case ansatz (2.1) has to reduce to Schwarzschild’s solution (D.1), due to the Jebsen-Birkhoff theorem17 [ , 18]. The metric functions constituting (2.1) then reduce to (D.3)-(D.5), Ω = 0 and the radial coordinates are related by
2 1 (︂ √︁ )︂ (︃ M )︃ r = ρ M + ρ2 2Mρ , ρ = r 1 + . 2 − − ⇔ 2r
1By Schwarzschild coordinates we refer to the spherical coordinate system (xµ) = (t, ρ, θ, φ) in which the Schwarzschild solution is of the form
(︃ 2M )︃ (︃ 2M )︃−1 ds2 = 1 dt2 + 1 (︁dρ2 + ρ2 dθ2 + ρ2 sin2(θ) dφ2)︁ . − − ρ − ρ
13 As mentioned in [12], there is another natural choice of coordinate system called the Lewis-Papapetrou coordinates (xµ) = (t, ρ, z, φ). They are related to quasi-isotropic coordinates via the transformation
ρ = r sin(θ) z = r cos(θ). (2.2)
In fact, this is simply a transformation from a spherical-like to cylindrical-like coordi- nate system in the same sense as described above. Since we do not make use of these coordinates a further discussion about these coordinates is not provided. However, note that they are also frequently used in studies of rotating stars, cf. [12–15, 19]. Because ansatz (2.1) is stationary and axisymmetric, it admits two Killing vector fields2. These are the asymptotically timelike3 Killing field ξµ and the spacelike Killing field χµ. In quasi-isotropic coordinates they are given by
µ µ ξ = δ t, (2.4) µ µ χ = δ φ. (2.5)
Note that these vectors are determined up to a normalization. If spacetime is asymp- totically flat and if the angular coordinate φ takes values within [0, 2π), then the Killing fields are determined uniquely and(2.4), (2.5) are already properly normalized [12, chapter 2]. Last but not least, the metric potentials U, W, Ω can be given a physical meaning. Hereto, the term zero angular momentum observer (ZAMO), or sometimes called locally non- rotating observer [15], must be introduced. They are defined as those observers whose worldlines are orthogonal to the t = const. hyperplanes. In an axisymmetric and station- ary spacetime that is generate by a rotating object a ZAMO maintains its r, θ coordinates while moving in a circular motion around the axis of symmetry, due to dragging effects. Its 4-velocity is then given by
uµ = e−U (ξµ + Ωχµ) . (2.6)
From this expression one deduces that e−U is the time dilation factor and Ω is the angular velocity of the ZAMO that relate to an asymptotic observer. The name ZAMO stems µ from the fact that the angular momentum (per unit mass) l := uµχ vanishes locally at some fixed r, θ [5, chapter XII.5].
2A Killing vector field ξµ is a vector field which obeys Killing’s equations:
ξ + ξ = 0. (2.3) ∇µ ν ∇ν µ
3 µ By asymptotically timelike it is meant that ξ ξµ approaches 1 at least in the asymptotically flat re- gion of spacetime, which corresponds to r in quasi-isotropic− coordinates. Otherwise, according to the definition of a stationary spacetime→ ∞ given in[12, chapter 2], Schwarzschild’s solution would not be stationary, cf. [17, section 2.9].
14 To deduce the meaning of W another quantity must be introduced. For a stationary and axisymmetric spacetime, define the proper circumference of a circle centered around the axis of symmetry as its positive (or possibly zero) metricC arc length. Additionally, define the proper radius := /2π. Next, let us calculate and using the metric R C C R (2.1) in quasi-isotropic coordinates. For an instant in time t0, a circle centered around µ the axis of symmetry may be parameterized by (x ) = (t0, r0, θ0, φ), where t0, r0, θ0 are fixed and φ [0, 2π). It then follows that ∈ ∫︂ √︁ ∫︂ 2π 0 = ds2 = dφ e−U W = 2πe−U W, (2.7) ≤ C 0 = e−U W. (2.8) R Because e−U measures time dilation between a ZAMO and an observer at infinity, it follows that W is the length contracted proper radius of the circle which is measured in the ZAMO’s rest frame. To verify that is the proper radius measured by an observer at infinity, one may evaluate it in the Rstatic case where the metric is that of Schwarzschild. It follows that = ρ sin(θ), where ρ is the radius in Schwarzschild coordinates. R To conclude this section we mention that the functions U, W, Ω can be invariantly char- acterized by the spacetime’s Killing vector fields. According to [13, 15, 19, 20] such a characterization is given by
2 µ 2U W 2 µ µ µ 2 ξ χµ e = µ ,W = ξ ξµ χ χµ + (ξ χµ) , Ω = µ . (2.9) χ χµ − −χ χµ The metric function V is a conformal factor, meaning that it depends on the chosen coordinate system. Because of this, there is no such simple characterization for V .
Flat spacetime
Assumption (e) states that the region of spacetime interior to the RMS has to be flat. A flat spacetime is characterized by a metric tensor whose components are diffeomorphic to the ones of a Minkowski metric. For this purpose we cast ansatz (2.1) into the form (︃ )︃ ̂ ̂ ̂ (︂ )︂2 ds2 = e2U dt2 + e−2U e2V (︁dr2 + r2 dθ2)︁ + Ŵ 2 dφ Ω̂ dt − − (︃ )︃ ̂ ̂ ̂ (︂ )︂2 = e2U dt2 + e−2(U−V ) dr2 + r2 dθ2 + r2 sin2(θ) dφ Ω̂ dt , (2.10) − − where U,̂ V,̂ Ω̂ are constants and Ŵ = eV̂ r sin(θ). A change of coordinates according to
̂ ̂ ̂ t′ = eU t, r′ = eV −U r, θ′ = θ, φ′ = φ Ω̂ t, (2.11) − then shows that (2.10) is indeed the metric of flat spacetime.
15 2.4. Einstein’s field equations
Having established the coordinates and ansatz for the metric tensor it is straightforward to compute the components of the Einstein tensor according to 1 Gµ := Rµ δµ R, (2.12) ν ν − 2 ν µ λ where R ν are the components of the Ricci tensor and R := R λ is the Ricci scalar. Usually, evaluation of the Einstein tensor is a lengthy computation. Luckily, for the case at hand these expressions have already been presented in [1,2]. We utilize the following useful linear combinations of (2.12) for the study of the RMS:
2(V −U) r 1 1 − W e G = W,θθ + W,r W U, U r r2 r − { } 1 + V,W − + W 3e−4U Ω, Ω − , (2.13) { } 4 { } 2(V −U) r 1 W e G = W,rθ W,θ + 2WU,rU,θ − θ − r 1 3 −4U W,rV,θ W,θV,r W e Ω,rΩ,θ, (2.14) − − − 2 ∆W e2(V −U) (︁Gr + Gθ )︁ = , (2.15) r θ W 2 e2(V +U)Gt = W ∆Ω + 3 Ω,W + 4W Ω,U + , (2.16) −W φ { } − { } (︃1 )︃ W e2(V −U) (︁Gt Gr Gθ Gφ )︁ + ΩGt − 2 t − r − θ − φ φ 1 = W ∆U + U, W + e−4U W 3 Ω, Ω + . (2.17) { } − 2 { }
± 1 Furthermore, the abbreviations X,Y = X,rY,r + r2 X,θY,θ as well as the flat spacetime { }1 1 two-dimensional Laplacian ∆ = ∂rr + r ∂r + r2 ∂θθ have been introduced [8].
2.4.1. Solution strategy
To recap, the RMS serves as a substitute for all the masses of the distant stars as seen from earth to approximate their gravitational influence onto observers located inside the shell. This shell is assumed to be of zero radial extension. In the language of GR such a configuration is characterized by a delta-type stress-energy tensor. Thatis,
µ λ µ λ T ν = δ(Φ(x ))t ν(y ), (2.18) where Φ(xµ) = 0 is a suitable scalar function [21, section 3.7]. It serves to describe the λ µ location of the RMS in terms of a hypersurface using coordinates (tensorx ). The t ν are the components of a tensor that describes energy and momentum of the matter and
16 radiation content, and they solely depend on coordinates (yλ) that are intrinsic to the hypersurface. The goal is to solve Einstein’s field equations
µ µ G ν = 8πT ν (2.19) with a stress-energy tensor of the form (2.18). To achieve this the following strategy is employed: Spacetime is split along the common interface, defined by Φ = 0, into an interior and exterior region where no energy-matter content is present. Because the stress-energy tensor is of the form (2.18), within these regions (2.19) reduces to the vacuum field equations
µ G ν = 0. (2.20)
To incorporate the presence of energy-matter at Φ = 0 the solutions to the respective vacuum field equations are then coupled to one another by imposing appropriate con- ditions at their common boundary. These boundary conditions (BCs) are chosen such that the unification of the interior and exterior solution solves(2.19) for a configuration characterized by (2.18). The formalism proposed in [22, 23] is utilized to compute the relevant conditions. De- tails about the application of the formalism to the situation at hand can be found in Appendix A.
2.4.2. Vacuum field equations
µ The vacuum field equations are simply obtained by putting G ν = 0 in (2.13)-(2.17) which then yields:
1 1 − − 1 3 −4U − 0 = W,θθ + W,r W U, U + V,W + W e Ω, Ω , (2.21) r2 r − { } { } 4 { } 1 1 3 −4U 0 = W,rθ W,θ + 2WU,rU,θ W,rV,θ W,θV,r W e Ω,rΩ,θ, (2.22) − r − − − 2 ∆W 0 = , (2.23) W 0 = W ∆Ω + 3 Ω,W + 4W Ω,U + , (2.24) { } −1 { } 0 = W ∆U + U, W + e−4U W 3 Ω, Ω + . (2.25) { } − 2 { }
At first glance, it appears that these equations have to be solved for U, V, W, Ω simulta- neously. As noted in [8], the functions V,r,V,θ only contribute linearly in (2.21), (2.22). Thus, they can be combined with (2.23) to give
17 (︃ )︃ 1 (︁ +)︁ W W,θ V,r = log W, W + 2 A + W,rB , (2.26) 2 { } ,r W, W + r2 { } 1 (︁ +)︁ W V,θ = log W, W + (2W,rA W,θB) , (2.27) 2 { } ,θ W, W + − { } 1 2 −4U A = U,rU,θ W e Ω,rΩ,θ, (2.28) − 4 1 B = U, U − W 2e−4U Ω, Ω − . (2.29) { } − 4 { }
Observe that V,r,V,θ are completely determined by U, W, Ω and their derivatives. Also note that (2.23)-(2.25) are independent of V,r,V,θ. This means that it is possible to only solve (2.23)-(2.25) for U, W, Ω in a first calculation. After having found solutions for U, W, Ω, the remaining unknown V can be constructed such that the equations (2.21), (2.22) are fulfilled. This may be done using a line integral, provided that V,rθ = V,θr holds. Because V is a conformal factor and, thus, of no physical importance, we spare out an explicit calculation of it and only solve the vacuum field equations for U, W, Ω.
2.5. Boundary and regularity conditions
Equations (2.23)-(2.25) constitute a system of PDEs for U, W, Ω that is nonlinear, cou- pled and of second order in the variables r, θ. Being of second order implies that solutions to these equations are determined up to two integration constants4 per variable and per unknown. In total this amounts to 12 integration constants, whereby 6 of them are associated with the r-dependence and the other 6 with the θ-dependence. Below follows a discussion about the BCs that are used to determine the integration constants.
2.5.1. Behavior at spatial infinity
Assumption (c) regarding asymptotic flatness of spacetime is commonly applied in stud- ies of massive rotating objects [13–16, 19]. A rigorous treatment of asymptotic flatness is a very involved topic, see [18, chapter 11]. Here follows a coordinate dependent discus- sion of asymptotic flatness and the conditions for the metric tensor that are later implied onto the metric potentials in order to guarantee asymptotic flatness of spacetime. A spacetime is said to be asymptotically flat if the corresponding metric tensor ap- proaches the Minkowski metric tensor, the metric tensor of flat spacetime, for large
4In the context of PDEs integration constants are functions rather than constant numbers. They may freely depend on variables other than the one they are associated with. E.g. a general solution to u,x(x, y) = 0 is u(x, y) = f(y) with f(y) being the integration constant. A special solution is obtained by explicitly specifying f(y).
18 distances from the central region. A metric represented in polar-like coordinates, e.g. quasi-isotropic coordinates, shall then behave as
r→∞ ds2 = g dxµ dxν η dxµ dxν = dt2 + dr2 + r2 dθ2 + r2 sin2(θ) dφ2 . (2.30) µν −−−→ µν − When comparing this with ansatz (2.1) one concludes that the functions U, V, Ω,W shall approach
U 0,V 0, Ω 0,W r sin(θ), (2.31) → → → → for r to guarantee asymptotic flatness. These are four conditions, and they may be used→ as ∞ BCs on the outer boundary of the domain when solving the system of PDEs previously discussed. One can also work out the leading order behavior of U, V, W, Ω that leads to conditions (2.31) in an expansion in powers of r as r . This is based on a study of the gravitational field far from a stationary→ source ∞ by solving Einstein’s field equations in the weak field regime of an asymptotically flat spacetime. The resulting metric tensor is then a combination of the flat spacetime metric and a small perturbation. The latter tells how the stationary and possibly strong source manifests itself in the weak field region. The actual calculation of the gravitational field far from the source is quite involved. Good presentations of this topic can be found in [17, pp. 272-275,5, pp. 574-576] 5. For this work we adopted the results of [17] for the metric tensor’s components. They are expressed in a rectangular coordinate system (xµ) = (t, x, y, z) and have been expanded √︁ in powers of r = x2 + y2 + z2 to give
(︃ 2 )︃ 2M M (︁ −3)︁ gtt = 1 + 2 + r , (2.32) − − r r2 O j k J x (︁ −3)︁ git = 2εijk + r , (2.33) − r3 O (︃ )︃ 2M (︁ −2)︁ gij = 1 + δij + r , (2.34) r O where i, j, k x, y, z . The parameter M is the total gravitational mass as measured by an observer∈ in { the weak} field region. The components J i constitute a vector in a three- dimensional Euclidean space, which behaves as the angular momentum of an object as measured by an observer at infinity. εijk is the total anti-symmetric Levi-Cevita symbol. In order to compare ansatz (2.1) with the far field results to deduce the leading order behavior of U, V, W, Ω, a change of coordinate system has to be carried out. Because the
5Care must be taken when contacting this reference, because the final result is missing a factor of2. Nevertheless, this resource provides a comprehensive discussion.
19 components (2.32)-(2.34) are already expanded in powers of r, a transformation from a rectangular to spherical polar coordinate system according to
x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), z = r cos(θ), (2.35)
j z j seems to fulfill its purpose. If it is assumed that J = J δz, e.g. the object’s axis of rotation aligns with the z axis of an observer at spatial infinity measuring the fields, one then arrives at the line element (︃ (︃ 2M )︃ )︃ ds2 = 1 + (︁r−2)︁ dt2 − − r O (︃ 2J )︃ + r2 sin2(θ) + (︁r−3)︁ dφ dt − r3 O (︃(︃ 2M )︃ )︃ + 1 + + (︁r−2)︁ (︁dr2 + r2 dθ2 + r2 sin2(θ) dφ2)︁ . (2.36) r O
Note that the r−2-term in the first bracket in (2.32) has been absorbed into (r−2) for easier comparison. Finally, comparing this with (2.1) allows to deduce the leadingO order behavior of the metric functions in a consistent manner as M U = + (r−2), (2.37) − r O V = (r−2), (2.38) O W = r sin(θ) + (r−1), (2.39) 2J O Ω = + (r−4). (2.40) r3 O According to [5, 17, 19] the numbers M,J characterize the source of the gravitational fields. M,J as presented here are known as the ADM mass and ADM angular mo- mentum, respectively, of a stationary, axisymmetric spacetime that is asymptotically flat. They arise in a systematic study where general relativity is formulated viaHamil- ton’s principle, which is known as the Arnowitt-Deser-Misner (ADM) formalism [24]. A comprehensive discussion about this technically involved topic can be found in [21]. Similar to (2.31), expressions (2.37)-(2.40) are four conditions that may be used as BCs on the outer boundary of the domain when solving the system of PDEs. In this work we made the decision to use the latter conditions over the former ones. The reason for this is that it allows to identify two of the three necessary parameters for the characterization of the spacetime to be M,J. If one would stick with conditions (2.31) instead, then the characterization has to be made in a different way, e.g. via the junction conditions (2.51) discussed below. Also note that it is structurally not more complicated to impose (2.37)-(2.40) instead of (2.31), especially when pursuing a numerical approach. The reason is that the actual implementation of these conditions requires analytic treatment of the functions U, V, W, Ω, because of the domain being at r . In chapter 3 it will become clear that a (pseudo-)spectral method is very well suited→ ∞ for this approach.
20 2.5.2. Behavior on the axis of symmetry and across the equatorial plane
The quasi-isotropic coordinates utilized are polar-like coordinates. They allow to exploit the axisymmetry of the model and therefore reduce the complexity of the problem. However, this comes at the cost that points on the rotation axis cannot be uniquely described6 via these coordinates. To be specific, a single point on the rotation axis with coordinates (t, r > 0, θ = 0, φ) may be assigned any value of φ [0, 2π). The same holds true for θ = π. ∈ To account for this ambiguity, regularity conditions for the components of the metric tensor are imposed. Since the ansatz is determined by the four unknown functions U, V, W, Ω, it is sufficient to impose regularity conditions onto these functions. The actual regularity conditions are deduced from symmetry transformations. Recap that ansatz (2.1) is axisymmetric which means that it is invariant under the trans- formation φ φ + const.. It is easy to check that in spherical-like coordinates the transformation→θ θ has the same effect as a rotation φ φ + π. Therefore, in order that the metric remains→ − invariant we impose the conditions→
2 X,θ = 0,X U, Ω,W ,V , (2.41) ∈ { } which shall hold for all points (r > 0, θ = 0) and (r > 0, θ = π) Assumption (d) demands that the metric is also symmetric with respect to reflections across the equatorial plane. This requirement is satisfied if the functions U, Ω,W 2 fulfill conditions (2.41) evaluated at θ = π/2. Furthermore, this reduces the domain of the polar angle θ from [0, π) to [0, π/2). In the case of X = W 2, condition (2.41) can be reduced to
WW,θ = 0. (2.42)
To fulfill this either W , or W,θ, or both may be zero independently on the pole and the equator. To judge which of these possibilities applies we recall the calculation of the proper radius (2.8). It is clear that shall be only zero if a circle has shrunken to a point on the axis of rotation. In this caseR θ = 0 and one deduces that W = 0. This may also have been deduced from the ansatz (2.1) itself, because W = 0 guarantees that no ambiguity along the axis of symmetry occurs due to changes in φ. Furthermore, local flatness of spacetime demands that W,θ = 0 at the pole, which is demonstrated below. On the other hand, if θ = 0, then ≯ 0 shall hold, which yields that W > 0 and ̸ R consequently W,θ = 0 must hold if θ = π/2.
6In fact, also the coordinate system’s origin where r = 0 would need careful treatment. Because the flat metric (2.10) used in the interior of the RMS is already regular at this point, a further discussion of this is spared out.
21 To summarize, the unknowns U, Ω,V shall obey conditions (2.41) on the pole and equator and
W (r, θ = 0) = 0,W,θ(r, θ = π/2) = 0 (2.43) must hold. To conclude this section we examine the criterion of local flatness of spacetime along the axis of symmetry. In quasi-isotropic coordinates this property translates to the requirement that the proper radius shall agree with the metric length of the circle’s radius R [15]. Hereto, consider aR fixed point on the axis described by coordinates (t0, r0, θ = 0, φ0). With this, R can be computed as
∫︂ θ ∫︂ θ (︃ )︃ ′ ′ ′ ′ dr R √ 2 −U(r(θ ),θ )+V (r(θ ),θ ) ′ = ds = e ′ + 1 dθ . (2.44) 0 0 dθ In the limit of θ 1 the integral kernel may be Taylor expanded around θ = 0. The corresponding expansions≪ yield
dr tan(θ) (︁ 3)︁ = r0 = r0(θ + θ ), (2.45) dθ − cos(θ) − O (︁ 2)︁ X = X(r0, 0) + (X,θ(r0, 0) + X,r(r0, 0)r0θ) θ + θ O (︁ 2)︁ = X(r0, 0) + θ , (2.46) O where (2.41) has been utilized in the last line. The integral then simplifies to
∫︂ θ −U(r0,0)+V (r0,0) −U(r0,0)+V (r0,0) (︁ 2)︁ R = e r0(1 + (θ)) = e r0θ + θ . (2.47) 0 O O On the other hand, the circle’s proper radius (2.8) in the case of θ 1 is expanded as ≪
−U(r(θ),θ) −U(r0,0) (︁ 2)︁ = e W (r(θ), θ) = e W,θ(r0, 0)θ + θ . (2.48) R O Equality of both radii then reveals the relation
V (r0,0) e r0 = W,θ(r0, 0), (2.49) which shows a non zero angular derivative of W at θ = 0. It shall be mentioned that reference [12, section 2] provides a similar explanation which finally yields the identity (︃ )︃ ⃓ V (r,0) W ⃓ e = ⃓ . (2.50) r sin(θ) ⃓θ→0 Also references [15, 16] come to the same conclusion.
22 2.5.3. Conditions on the shell
As mentioned in chapter 1, the BCs imposed at the common interface between the exte- rior and interior region of the RMS encode the energy-matter content of the spacetime. In Appendix A these conditions are deduced using the geometric formalism that was proposed in [22] and it summarizes results presented in [8]. The relevant conditions for this study are the first junction conditions (A.7) applied to the situation at hand. For a flat interior metric (2.10) the conditions reduce to ⃓ ⃓ X⃓ := X(f(θ), θ) = X̂ = const., X U, V, Ω , (2.51) ⃓Σ ∈ { } and (︃ )︃ ⃓ W ⃓ V̂ ⃓ = e = const. (2.52) r sin(θ) ⃓Σ ⃓ where X⃓Σ = X(f(θ), θ) and f is an additional unknown function that has to be deter- mined when solving the set of PDEs. The function f(θ) serves to describe the latitude dependent radius of the RMS. At this point one may ask how to determine f when one also has to solve for U, W, Ω,V at the same time? In fact, one only has to impose the conditions (2.51) and (2.52) as they are. However, when pursuing the approach where one first solves only the PDEs that involve U, W, Ω, then the condition (2.51) for X = V has to be rewritten. This is because condition (2.51) for X = V cannot be evaluated in this approach, because V has been eliminated. Nevertheless, to guarantee that V̂ is a constant number, we demand that V̂,θ = 0, which yields the condition ⃓ ⃓ ′ ⃓ ⃓ f V,r⃓ + V,θ⃓ = 0. (2.53) ⃓Σ ⃓Σ
Because V,r,V,θ can be solely expressed by U, W, Ω and their derivatives due to (2.26), (2.27), this is merely a constraint for U, W, Ω and their derivatives as well as f. If for some reason, one attempts to solve the full set of PDEs (2.21)-(2.25) for U, W, V, Ω, one does not need condition (2.53). This can be understood by recognizing that the differential equations that involve V are only of first order. This means that only one integration constant is required to determine V , compared to two for U, W, Ω. As a consequence, because V can be also directly evaluated in this approach, one can stick to conditions (2.51) and (2.52).
23 3. Numerical treatment
Einstein’s field equations for a RMS problem have been reduced to a system ofPDEsthat are complemented with certain BCs. As it is the nature of Einstein’s equations, these PDEs are nonlinear and coupled. For the case at hand they are two-dimensional and homogeneous equations. Furthermore, the domain where the BCs have to be imposed is unknown and has to be determined consistently when solving the equations. Such a problem is often referred to as a free-boundary-value problem. Due to the complexity and nonlinearity of the system of equations, it is not expected to find an analytic solution for the RMS problem (beyond PT), at least none has been reported in the literature so far. Because of this, a numerical treatment of the system of equations is pursued. The decision was made to utilize pseudo-spectral methods. Reasons for this included that these methods have already been applied successfully to a diversity of complex (and nonlinear) problems in physics. If utilized correctly, pseudo- spectral methods can outperformed finite-difference schemes in terms of computational efficiency and numerical accuracy, cf.[25] for an overview. Furthermore, the state-of- the-art codes used for studies of rotating relativistic objects also rely on such methods, cf. [13, 14] and [19] for an overview. This chapter provides a very brief introductory to the pseudo-spectral method. Based on this, suitable ansätze for the unknown functions, necessary for the numerical treatment of the equations, are then proposed. The chapter ends with a brief summary about the evaluation strategy of the pseudo-spectral method on the computer.
3.1. Introduction to pseudo-spectral methods
Consider a differential equation for the unknown function u(x) given in the form
L(x)u(x) = f(x), (3.1) where L(x) is a (possibly nonlinear) differential operator and f(x) shall label the equa- tion’s inhomogeneity. The idea of a spectral method is to approximate the true solution u(x) to this equation by a linear combination of the form
N−1 ∑︂ u(x) uN (x) := un ϕn(x), (3.2) ≈ n=0
24 where un is a set of coefficients that corresponds to the setof N basis functions { } ϕn(x) . Define the residual associated with (3.1) as { }
R(x; un ) := L(x)uN (x) f(x). (3.3) { } −
It is clear that if the approximation uN (x) agrees with the true solution u(x) then (3.3) is zero for all points of x. Thus, the goal is to choose the unknown coefficients un for the approximation uN (x) such that R(x; un ) approaches zero (in a numerical implementation,{ } at best up to machine precision).{ } Different strategies for minimizing the residual are available. Among these strategies is the pseudo-spectral method which is characterized by a simple evaluation procedure. This method minimizes (3.3) by imposing N conditions onto the un by demanding that [25] { }
R(xi; un ) = 0, i = 0, ..., N 1. (3.4) { } −
The set of points xi is called collocation grid. Which grid to use depends on the utilized basis functions.{ } For a discussion of this topic see [25, chapter 4]. Advantages of pseudo-spectral methods over finite-difference methods are that the ap- proximation (3.2) and derivatives of it can be evaluated with moderate effort to high precision, usually with computational complexity of (N). Another characteristic is that this method shows a rapid convergence behavior,O if carried out correctly. How- ever, all of this may come at the cost that optimized computational implementations of pseudo-spectral methods are often more involved than finite-difference schemes. Since the pseudo-spectral method is one of many numerical algorithms available, there is no unique strategy to follow in order to solve a certain problem. Most of the choices made in this work have been inspired by discussions that are provided in reference [25].
3.2. Series expansions
One key factor for an efficient numerical treatment of differential equations witha pseudo-spectral method is the choice of series expansion used for the approximations of the unknowns. This is the topic of this section. Firstly, we discuss the expansions of the functions U, Ω,V . The ansatz for W is treated separately, because its corresponding PDE can be solved analytically. The series expansion of f is treated thereafter.
3.2.1. Ansätze for U, Ω,V
The difference between the introductory example and the RMS problem is that the latter system constitutes a set of two-dimensional PDEs. To account for this, the ansätze for
25 U, Ω,V are modeled as two-dimensional functions of the form
N −1 N −1 ∑︂r ∑︂θ X(r, θ) = XnmRn(r)Φm(θ). (3.5) n=0 m=0
The functions Rn(r) and Φm(θ) represent the basis functions that are associated with the r-, respectively, θ-dependence and Xnm label the unknown coefficients. In this text the basis functions Rn are chosen to be variations of the Chebyshev polynomials that are augmented by a suitable change of coordinate. For the Φm the Fourier basis functions are utilized. Modeling the functions according to (3.5) may suffice to solve the system of equations with pseudo-spectral methods. Doing so then may require that the BCs are carefully monitored in each calculation step. However, one can also fine tune the ansatz such that some of the relevant boundary conditions are satisfied already analytically. This may improve the convergence behavior as well as the accuracy of the results themselves. In fact, the behavior of the metric functions in the limit of r requires analytical treatment. To account for the conditions (2.31), an ansatz for→ the ∞ unknowns may take the form m R2 U(r, θ) = + δU(r, θ), (3.6) − r r2 2J R4 Ω(r, θ) = + δΩ(r, θ), (3.7) r3 r4 R2 V (r, θ) = δV (r, θ). (3.8) r2 The functions δX, X U, Ω,V shall describe corrections to the desired far field be- havior of U, V, Ω. By adding∈ { a damping} factor (R/r)i, i = 2, 4 in front of the corrections the asymptotic behavior of these functions is incorporated analytically, provided that the δX reach a constant value in the limit of r . The latter condition can be → ∞ ensured if the δX are taken to be of the form (3.5) and the Rn(r) are chosen to be the rational Chebyshev functions TLn(r). The definition of the TLns as well as some of their properties are briefly discussed in Appendix C. The BCs associated with the angular degree of freedom, (2.41) evaluated at θ = 0, π/2, can also be integrated analytically. Hereto, we assume that each of the basis functions Φm(θ) obey (2.41) by themselves. If the Φm are taken to be Fourier series basis functions, e.g. Φm 1, sin(mθ), cos(mθ) m N , then one can easily deduce that only cosines with an even∈ { integer m are compatible| ∈ } with condition (2.41). In fact, also the BCs at the shell r = f(θ) could have been integrated into the ansätze. This variation has not been tested yet, but it is planned to be added in a future update of the program.
26 To summarize, the improved ansätze for U, V, Ω are given by
N −1 N −1 m R2 ∑︂r ∑︂θ U(r, θ) = + U TL (r) cos(2mθ), (3.9) r r2 nm n − n=0 m=0 N −1 N −1 2J R4 ∑︂r ∑︂θ Ω(r, θ) = + Ω TL (r) cos(2mθ), (3.10) r3 r4 nm n n=0 m=0 N −1 N −1 R2 ∑︂r ∑︂θ V (r, θ) = V TL (r) cos(2mθ). (3.11) r2 nm n n=0 m=0
To conclude this section we shall add a few comments. Firstly, the Ansätze used in this text have been fine-tuned to suit the needs in this study. The ones used in the literature are all of the form of (3.5) and they do not provide special ansätze like they were presented here. One reason for this is that the corresponding numerical implementations make heavy use of fast Fourier transformation algorithms to reduce the computational effort, cf. [13, 14]
3.2.2. Analytic solution for W
The function W shall obey the two-dimensional Laplace equation (2.23), which is ex- pressed in polar-like coordinates. The general solution to this equation can be obtained by making a product ansatz for W (r, θ) = f(r)g(θ). One then arrives at the general solution
∞ ∑︂ k −k W (r, θ) = (a0 + b0 ln(r))c0 + (akr + bkr )(ck cos(kθ) + dk sin(kθ)), (3.12) k=1 where ak, bk, ck, dk are unknown constants. In order to utilize (3.12) in a numerical study the sum needs to be truncated to a finite number of coefficients. Since this ansatz automatically satisfies the Laplace equation it remains to determine the coefficients such that W satisfies the relevant BCs. Two of these conditions are (2.43). Assuming that each of the terms satisfies the BCs by themselves, one then concludes that only sines with an odd integer k in the argument can survive in the ansatz. Hence, (3.12) simplifies to
Nθ−1 ∑︂ 2m+1 −(2m+1) W (r, θ) = (a2m+1r + b2m+1r )d2m+1 sin((2m + 1)θ), (3.13) m=0 where the substitution k = 2m + 1 has been made and the sum was truncated to Nθ contributions.
27 Incorporating the asymptotic behavior (2.39) then dictates that all an must vanish. Because of this, W takes the form
N−1 ∑︂ −(2m+1) W (r, θ) = r sin(θ) + wmr sin((2m + 1)θ), (3.14) m=0 where the coefficients wm := b2m+1d2m+1 have been introduced. This simplification of W was initially presented in [8]. The possibility of solving one of the field equations analytically reduces the computational effort drastically. Itshould be noted that this reduction is only possible in vacuum regions of spacetime where the field equations take the form of(2.23).
3.2.3. Ansatz for f
The one dimensional function f takes the meaning of a latitude dependent radius of the RMS. Due to the assumptions of axisymmetry and reflection symmetry of spacetime it is expected that the RMS also deforms according to these symmetries. Hence, the latitude dependent radius shall also be symmetric with respect to a transformation θ π θ. A suitable ansatz for f that obeys this symmetry is given by → −
N −1 ∑︂θ f(θ) = R + fm(1 cos(2mθ)). (3.15) m=1 − This choice was motivated by the PT calculations of Pfister et al. [2]. Note that in the case where all fm vanish it reduces to f = R, meaning the RMS is a perfect sphere. The solution for{ this} case is known, it is Schwarzschild’s spacetime and it serves as a starting point for the numerical calculations. The second term in (3.15) accounts for a possible latitude dependent correction to an initial spherical shape. The choice that f(0) = R is completely arbitrary and was taken over from the work of Pfister et al. in order to facilitate a comparison of results. Note that this ansatz, which involves Nθ cosine Fourier modes, only uses Nθ 1 coefficients. This is, because we have made the choice that f(0) = R in advance. −
3.3. Putting everything on the computer
Having established the ansätze for the unknown functions it is time to elaborate on how the coefficients are found. As mentioned in the introduction, when utilizing the pseudo-spectral method the unknown coefficients are determined such that they satisfy condition (3.4). Below we discuss how the residuals for this work have been chosen, and we outline the numerical implementation of the procedure for obtaining the unknowns.
28 Firstly, we generalize the introductory example to n differential equations for n unknown functions. To solve these equations one may proceed as follows: From the n equations define n residuals similarly to (3.3). The next step is to set up the ansätze for the unknown functions by specifying the number coefficients m for each ansatz. This number characterizes to which order the series expansion is carried out. A natural choice is to expand all ansätze to the same order. By doing so, the n ansätze are determined by nm coefficients. Then choose m collocation points and evaluate the n residuals at these points according to (3.4). This gives a set of nm nonlinear algebraic equations for the coefficients, however, there will in general be no solution to it. This isbecause special solutions to differential equations need to satisfy certain boundary and/or initial conditions and this information is lacking here. To compensate for this, k of the nm equations need to be substituted by proper BCs, where k depends on the type of problem considered. If the initial system of differential equation is consistent with the provided conditions, then the so constructed set of nonlinear equations for the coefficients shall posses a solution. Finally, solve the set of nonlinear algebraic equations using your favorite nonlinear equation solver. Next, we would like to apply the previously described scheme to the problem at hand. We are dealing with three PDEs (2.21)-(2.23) for the three unknowns U, W, Ω plus the function f to account for centrifugal deformation. Neglecting f for a moment, U, W, Ω are approximated according to (3.9),(3.10),(3.14). The former two ansätze are expanded up to order Nr and Nθ with respect to the r- and θ-dependence, respectively, and the ansatz for W is expanded up to order Nθ. In total, these approximations then contain 2NrNθ + Nθ unknown coefficients.
Ideally, to determine these unknowns a total of 2NrNθ + Nθ residuals shall be used. Hereto, the residuals (2.21), (2.22) may be evaluated at Nr radial times Nθ angular collocation points. Because W has been constructed such that it automatically fulfills (2.23), adding it to the list of residuals would have no effect. This leaves us with a total of 2NrNθ algebraic equations. But BCs have not been taken into account yet. To do so, observe that the utilized ansätze account for all BCs except the ones that have to hold on the shell. Evaluate condition (2.51) for X = U, Ω at Nθ angular collocation points at r = f(θj) and θj. Then substitute them instead of Nθ algebraic equations that were already added to the list of residuals. For W proceed similarly, but instead of substituting them, they are added in addition to the residuals.
The so established 2NrNθ +Nθ algebraic equations would suffice to determine 2NrNθ +Nθ unknowns if a solution for a spherically RMS would exist. Because this is not the case, the latitude dependent shell radius f(θ) was introduced. This adds an additional unknown to the problem. We utilize ansatz (3.15) to approximate f which is expanded up to order Nθ, but only contains Nθ 1 unknowns. To determine these additional Nθ 1 − − unknowns, also Nθ 1 evaluations of condition (2.53) at Nθ 1 angular collocation points are added to− the residuals. Doing so keeps the number of− unknowns and number of algebraic equations in balance.
29 To summarize this lengthy explanation, the unknowns and residual equations are col- lected in table 3.1.
List of unknowns
Function Ansatz Number of coefficients
U(r, θ) (3.9) NrNθ
Ω(r, θ) (3.10) NrNθ
W (r, θ) (3.14) Nθ
f(θ) (3.15) Nθ 1 − List of residuals
Equation type Equation Number of evaluations
PDE (2.21) (Nr 1)Nθ −
PDE (2.22) (Nr 1)Nθ −
BC (2.51) for X = UNθ
BC (2.51) for X = Ω Nθ
BC (2.52) Nθ
BC (2.53) Nθ 1 − Table 3.1.: An overview about the unknown coefficients and residuals. The upper part of the table lists which approximation is used for each unknown as well as the corresponding number of coefficients that has to be determined. The lower part shows which PDEs and BCs are evaluated as a residual and used to arrive at a nonlinear system of algebraic equations for the coefficients. The construction of this table is discussed in thetext.
One thing that has been left unspecified in the preceding discussion was which collocation points are used for the evaluation of the residuals. There is no unique choice and one may utilize different grids for different problems. In this work we used the setofpoints given by (C.9) and (C.10) for the radial and angular grid, respectively. Another detail about the evaluation of the boundary conditions (2.51) for X = U, Ω and (2.52) has to be added: Because the choice was made that parameters M,J are incorporated in the asymptotic behavior of U, V, W, Ω, the constant numbers X,X̂ U, Ω, eV (note that eV = W/f sin(θ)) do not have to be provided for the evaluation∈ { }
30 of these conditions. In fact, these numbers need to be determined consistently, e.g. by substituting
N −1 1 ∑︂θ X̂ := X(f(θ ), θ ), (3.16) N i i θ i=0 or, alternatively,
X̂ := X(f(θref ), θref ) θref / θi . (3.17) ∈ { } in the corresponding conditions for X U, Ω, eV . Note that either choice for X̂ should yield the same final result, because∈ { if X̂ = const.} holds, then they both agree. The condition that θref is not chosen as one of the utilized collocation points is required, ⃓ because otherwise one of the BCs would always yield true no matter X⃓Σ is constant or not.
3.3.1. Guess and extrapolation
The result of the preceding section was a set of nonlinear algebraic equations. Most numerical algorithms, if not all, require an initial guess as input to be able to solve such equations. Below we discuss which guesses we use for the RMS problem and how they were improved to speed up the convergence in our numerical implementation of the problem. Recap that for the setup of the system of nonlinear algebraic equations, inputs for the parameters M, J, R have to be provided by the user. For a set of parameters where J = 0, we already know the solution to the problem analytically, which is the Schwarzschild metric. In this case we can compute the coefficients for U, W, Ω by making use of the orthogonality relations (C.11), (C.13). The so obtained numerical approximation to U, W, Ω for J = 0 can then be used as an educated guess for a calculation where J = 0 has been slightly increased. If a program yields a satisfactory result using the Schwarzschild solution as a guess, then this is a major success. As one may anticipate, the so obtained numerical result for a value of J > 0 can be used as a guess for a calculation where J has been slightly increased again. This procedure can be repeated over and over again till no satisfactory result can be found anymore. In this way, for fixed M,R, one can trace out a family of numerical solutions which are parameterized by values of J. This technique is a variation of the continuation method, discussed in [25, appendix D]. Depending on the type of nonlinear solver utilized, the quality of the guess may influence the rate of convergence of the solver or even decide whether a solution can be found or not. In order to improve the convergence rate, one may either use the above strategy and decrease the increment for J if the convergence rate gets worse. Alternatively, one may utilize more than one previously calculated result to construct an improved guess. The
31 latter approach is pursed in this work. Hereto, we assume that the unknown coefficients depend smoothly on J. We then model the coefficient’s J-dependence by a second order polynomial in the vicinity of some J0, e.g.
x(J) = aJ 2 + bJ + c, (3.18) where a, b, c are constants and x represents the J-dependent coefficient. To use this to construct a guess for some J = J0 + δJ one must proceed as follows: Provide three sets of coefficients xj j = 1, ..., N that correspond to three distinct values of Ji, i = 1, 2, 3 { | } for which the RMS problem could already be solved. For each coefficient xj compute the associated constants ai, bi, ci using the provided reference coefficients from equation (3.18). An extrapolated guess can then be calculated by evaluating formula (3.18) with the previously computed constants ai, bi, ci for the desired value of J0 + δJ. This is a rather computationally inexpensive step compared to the costs of the nonlinear solver. It has also proven to accelerate the performance of our numerical implementation greatly. If needed, it is easy to generalize this procedure to higher order polynomials to take into account more previously calculated results.
32 4. Results
This chapter presents the results obtained by a C++ program that implements the numeri- cal treatment of the RMS problem as described in the previous chapter. As input param- eters for the calculations the numbers R,M,J as well as Nr,Nθ have to be provided. All subsequent discussed results have been computed for R = 1 and Nr = 35,Nθ = 10. This choice then restricts the possible values for the total gravitational mass to 0 < M < 2 for which a total of 719 coefficients have to be determined. According to the utilized solution strategy, the system of equations is solved for different values of the total an- gular momentum, starting with J = 0. This parameter is then gradually increased till no convergent result can be found anymore. To judge whether a calculation yielded a convergent result has to be decided by the user. In this work, the decision was made based on the mean residual error ⃦ ⃦ ⃦ ⃗ ⃦ ⃦R(x, cj )⃦ ϵ := { } . (4.1) N
⃗x is the usual Euclidean norm on an N-dimensional Euclidean space RN . The vector ∥ ∥ R⃗(x, cj ) collects the numeric values of all the residuals listed in Table 3.1 that are evaluated{ } at the discussed collocation points. In each iteration step (4.1) is computed and if it falls below a user-specified threshold ϵmean within a maximal number of Nmax iterations, then the corresponding set of coefficients is said to be a convergent result. −11 The results presented below were computed for ϵmean = 10 and Nmax = 50. Having obtained an approximation to the metric of the RMS that satisfies the field equations up to some user-specified error, a broad range of characteristics can be studied. Below follows a discussion of selected properties as well as an error estimation.
4.1. Centrifugal deformation and differential rotation
By centrifugal deformation we refer to the difference in polar and equatorial radius f(0) = R and f(π/2), respectively. Similarly, differential rotation shall denote the dif- ference between the polar and equatorial angular velocity ω(0) and ω(π/2), respectively. It shall be clear that these numbers are no invariant quantities, but are tied to the chosen coordinate system and metric ansatz. However, they allow for a plausible interpretation of the size and rotation speeds of a RMS as seen from a distant observer. Furthermore,
33 1.00 M/R [ ] − 1.50 0.95 1.40 1.30 1.20 0.90
] 1.10 −
[ 1.00
2) 0.90 (0) π/ ( f 0.85
f 0.80 0.70 0.60 0.80 0.50 0.40 0.75 0.30 0.20 0.10 0.0 0.1 0.2 0.3 0.4 0.5 J M [R]
Figure 4.1.: The ratio of the latitude dependent shell radius f(θ) evaluated at the equator θ = π/2 and pole θ = 0 is plotted as a function of J/M for different configurations characterized by M/R and R = 1. these quantities have also been directly computed by Pfister et al. and, thus, serve as a benchmark for the numerical results. To study centrifugal deformation and differential rotation, the ratios f(π/2)/f(0) and ω(π/2)/ω(0) are plotted as a function of J/M for different values of M in figures 4.1, 4.2. From figure 4.1 two characteristics about the RMS’s shape can be deduced: Firstly, for some fixed M/R, one can read of that the shape of a RMS changes from an initial sphere (at J = 0) towards a prolate ellipsoid when J/M increases, which is indicated by f(π/2)/f(0) < 1. Because f(0) = R has been assumed for all RMS, it is legitimate to directly compare the deformations between different M/R ratios. Secondly, for some fixed J/M, one can read of that the smaller M/R is, the more pronounced the absolute deformation is. From figure 4.2 characteristics about the RMS’s relative angular velocity can be de- duced: For some fixed M/R, the energy-matter content moving in the equatorial plane
34 1.20 M/R [ ] − 1.50 1.16 1.40 1.30 1.20 1.12
] 1.10 −
[ 1.00
2) 0.90 (0) π/ ( ω
ω 1.08 0.80 0.70 0.60 1.04 0.50 0.40 0.30 0.20 1.00 0.10 0.0 0.1 0.2 0.3 0.4 0.5 J M [R]
Figure 4.2.: The ratio of the latitude dependent angular velocity ω(θ) evaluated at the equator θ = π/2 and pole θ = 0 is plotted as a function of J/M for different configura- tions characterized by M/R and R = 1. shows a relative higher angular velocity compared to energy-matter located at the poles, indicated by ω(π/2)/ω(0) > 1. However, note that a certain ratio of ω(π/2)/ω(0) for some fixed J/M will in general translate to distinct absolute values ω(0) ω(π/2) for different M/R configurations. With this in mind, we observe that− forfixed J/M the relative strength of differential rotation is more pronounced, the smaller M/R is. Since this work aims at extending the PT results of Pfister et al., agreement between the numerical results and their work shall be found in the perturbation regime. To this end, we make a comparison of the latitude dependent shell radius f(θ) and fPT (θ) from (D.10). A comparison involving the angular velocity ω(θ) is not given and the reason for this is discussed in Appendix D. In the work of Pfister et al. the strength of rotation was characterized by the expansion parameter ωPT , whereas this work relies on the total angular momentum J. A consistent comparison then requires a mapping between ωPT and J, which is also derived in Appendix D.
The actual comparison of f(θ) and fPT (θ) is presented in figure 4.3. From this plot one
35 deduces that for reasonable small circumferential velocities, e.g. ωPT R below 2% of the speed of light, the difference between the PT and numerical results is well below 0.1%. We also note that the PT result underestimates the deviation from a spherical shell, indicated by a negative relative difference.
0.000 M/R [ ] − 1.50 -0.002 1.40 1.30
] 1.20
− -0.004 [ 1.10 2)
2) 1.00 π/ ( π/
( 0.90 f PT f -0.006 0.80 − 0.70 1 0.60 -0.008 0.50 0.40 -0.010 0.30 0.20 0.10 0.00 0.02 0.04 0.06 0.08 0.10 ω R [ ] PT − Figure 4.3.: The relative difference between the numerical and analytic PT shell radius f(θ) and fPT (θ), respectively, is plotted as a function of the PT parameter ωPT R for different configurations characterized by M/R and R = 1.
4.2. Surface-stress-energy tensor
With the Einstein field equations solved, the surface-stress-energy tensor that charac- terizes the spacetime’s energy-matter content can be computed. How this is done was outlined in Appendix A. Below we present the eigenvalues of the surface-stress-energy tensor for all configurations of M/R that have been presented in the previous section. From figures 4.4-4.7 one can read off if the surface-stress-energy tensor obeys the dom- inant energy conditions (A.32). As it turns out, from the configurations considered in
36 this work, only the ones with M/R = 1.4, 1.5 violate these conditions, because the cor- responding energy-density ρ is smaller than the pressures pv, pw. The transition between satisfaction and violation of this condition can be estimated if one studies the analytic solution for the non-rotating case. The energy density ρ and pressure p := pv = pw for the Schwarzschild solution are given by formulae [8]
2M M ρ = , p = ρ. (4.2) (R + M/2)3 2(R M/2) − An examination of the condition ρ p then reveals that the energy condition is already violated in the case of J = 0 for all≥ values| | of 2 > M/R > 4/3. This is consistent with our finding and has been already reported in[2, section 4.2] as well.
Another characteristic that appears in the plots is that pw = pv for J > 0 and all θ [0, π/2). This indicates that the surface-stress-energy tensor̸ is in general not of ∈ perfect-fluid type, except in the case of J = 0 where pw = pv holds analytically.
4.3. Error estimation
In the case of a stationary, axisymmetric and asymptotically flat spacetime the ADM quantities and Komar quantities have to agree with one another [12, 19, 21]. The former are defined as the first non-vanishing coefficients of a far field series expansion inpowers of 1/r, cf. discussion about behavior at spatial infinity in subsection 2.5.1. The latter numbers are the conserved currents that are associated with the Killing fields (2.4), (2.5). Details about their calculation are provided in Appendix B. Given a result that shall be checked against the Komar quantities, the corresponding integrals (B.14), (B.15) are numerically computed using a Gauss-Chebyshev quadrature rule. In figures 4.8, 4.9 we have plotted the relative difference between the ADM and Komar quantities for all results presented previously. As one can see the relative difference of these results is below 0.1% for all configurations considered. By comparing all plots with one another, one may recognize a rising trend of the error function with increasing J/M. Furthermore, the discrepancy in angular momentum is in general greater than for the gravitational mass.
37 5.289 4.870 ] ] 2 2 − −
R M/R = 1.5 R M/R = 1.4 )[ )[ θ θ
( J/M [R] ( J/M [R] ρ 5.200 ρ 4.697 0.00 0.00
] 7.875 ] 5.553
2 0.04 2 0.05 − 0.08 − 0.10 R R
)[ 0.12 )[ 0.15 θ θ ( 0.16 ( 0.20 w w p 7.265 0.20 p 4.989 0.25 0.24 0.30
] 7.875 ] 5.553
2 0.28 2 0.35 − −
R 0.32 R 0.40
)[ 0.36 )[ 0.45 θ θ ( ( v v p 7.111 p 4.746 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
4.438 4.050 ] ] 2 2 − −
R R M/R = 1.2
)[ M/R = 1.3 )[ θ θ
( ( J/M [R] ρ 4.243 J/M [R] ρ 3.808 0.00 0.00 ] 3.984 ] 2.880
2 2 0.05
− 0.05 − 0.10 R 0.11 R
)[ )[ 0.15 θ 0.16 θ ( ( 0.20 w w
p 0.22 p 3.644 2.662 0.25 0.27 0.30 0.33 ] 3.984 ] 2.880
2 2 0.35
− 0.38 −
R R 0.40 0.44 )[ )[ 0.45 θ θ ( ( v v p 3.409 p 2.409 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
Figure 4.4.: Eigenvalues of the surface-stress-energy tensor ρ, pv, pw plotted as a func- tion of the angular coordinate θ for selected values of J/M and different configurations characterized by M/R and R = 1 (continued on next page). 38 3.666 3.441 ] ] 2 2 − −
R R M/R = 1.0
)[ M/R = 1.1 )[ θ θ
( ( J/M [R] ρ 3.397 J/M [R] ρ 3.000 0.00 0.00 ] 2.084 ] 1.537
2 2 0.06
− 0.05 − 0.11 R 0.11 R
)[ )[ 0.17 θ 0.16 θ ( ( 0.22 w w
p 0.22 p 1.961 1.425 0.28 0.27 0.33 0.33 ] 2.084 ] 1.537
2 2 0.39
− 0.38 −
R R 0.44 0.44 )[ )[ 0.49 θ θ ( ( v v p 1.711 p 1.074 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
3.175 2.907 ] ] 2 2 − −
R M/R = 0.9 R
)[ )[ M/R = 0.8 θ θ
( J/M [R] ( ρ 2.610 ρ 2.240 J/M [R] 0.00 0.00 ] 1.161 ] 0.872
2 0.06 2
− − 0.06 0.11 R R 0.12
)[ 0.16 )[ θ θ 0.18 ( 0.22 ( w w
p p 0.24 1.047 0.27 0.746 0.30 0.33 0.36 ] 1.161 ] 0.872
2 0.38 2
− − 0.42
R 0.44 R 0.48 )[ 0.49 )[ θ θ ( ( v v p 0.663 p 0.368 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
Figure 4.5.: Eigenvalues of the surface-stress-energy tensor ρ, pv, pw plotted as a func- tion of the angular coordinate θ for selected values of J/M and different configurations characterized by M/R and R = 1 (continued on next page). 39 2.377 2.222 ] ] 2 2 − −
R R M/R = 0.6
)[ M/R = 0.7 )[ θ θ
( ( J/M [R] ρ 1.890 J/M [R] ρ 1.560 0.00 0.00 ] 0.597 ] 0.444
2 2 0.04
− 0.05 − 0.09 R 0.10 R
)[ )[ 0.14 θ 0.15 θ ( ( 0.18 w w
p 0.20 p 0.509 0.334 0.23 0.25 0.27 0.30 ] 0.597 ] 0.444
2 2 0.32
− 0.35 −
R R 0.36 0.40 )[ )[ 0.40 θ θ ( ( v v p 0.275 p 0.061 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
1.833 1.434 ] ] 2 2 − −
R M/R = 0.5 R M/R = 0.4 )[ )[ θ θ
( J/M [R] ( J/M [R] ρ 1.250 ρ 0.960 0.00 0.00
] 0.290 ] 0.173
2 0.04 2 0.03 − 0.08 − 0.07 R R
)[ 0.12 )[ 0.10 θ θ ( 0.16 ( 0.14 w w p 0.208 0.20 p 0.120 0.17 0.24 0.21
] 0.290 ] 0.173
2 0.28 2 0.25 − −
R 0.32 R 0.28
)[ 0.36 )[ 0.31 θ θ ( ( v v p -0.006 p -0.038 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
Figure 4.6.: Eigenvalues of the surface-stress-energy tensor ρ, pv, pw plotted as a func- tion of the angular coordinate θ for selected values of J/M and different configurations characterized by M/R and R = 1 (continued on next page). 40 1.030 0.638 ] ] 2 2 − −
R M/R = 0.3 R M/R = 0.2 )[ )[ θ θ
( J/M [R] ( J/M [R] ρ 0.690 ρ 0.440 0.00 0.00
] 0.089 ] 0.035
2 0.03 2 0.02 − 0.06 − 0.05 R R
)[ 0.09 )[ 0.07 θ θ ( 0.12 ( 0.10 w w p 0.061 0.15 p 0.024 0.12 0.18 0.15
] 0.089 ] 0.035
2 0.21 2 0.17 − −
R 0.24 R 0.20
)[ 0.27 )[ 0.22 θ θ ( ( v v p -0.043 p -0.031 π π 3π π π π 3π π 0 8 4 8 2 0 8 4 8 2 θ [ ] θ [ ] − −
0.261 ] 2 − R
)[ M/R = 0.1 θ ( ρ 0.210 J/M [R] 0.00 ] 0.007 2
− 0.02
R 0.04 )[
θ 0.06 ( w p 0.006 0.08 0.10 0.12 ] 0.007 2
− 0.14 R 0.16 )[ θ ( v p -0.008 π π 3π π 0 8 4 8 2 θ [ ] −
Figure 4.7.: Eigenvalues of the surface-stress-energy tensor ρ, pv, pw plotted as a func- tion of the angular coordinate θ for selected values of J/M and different configurations characterized by M/R and R = 1. 41 1 MADM 1 JADM − MKomar − JKomar
3 M/R = 1.5 M/R = 1.1 1 10− ×
5 1 10− ×
7 1 10− × 3 M/R = 1.4 M/R = 1.0 1 10− ×
5 1 10− ×
7 1 10− × 3 M/R = 1.3 M/R = 0.9 1 10− × 5 1 10− × 7 1 10− ×
3 M/R = 1.2 M/R = 0.8 1 10− × 5 1 10− × 7 1 10− × 0.00 0.20 0.40 0.00 0.20 0.40 J J M [R] M [R]
Figure 4.8.: The relative difference between the ADM and Komar quantities M and J are plotted as functions of J/M for different configurations characterized by M/R and R = 1 (continued on next page).
42 1 MADM 1 JADM − MKomar − JKomar
3 M/R = 0.7 M/R = 0.3 1 10− × 5 1 10− × 7 1 10− ×
3 M/R = 0.6 M/R = 0.2 1 10− × 5 1 10− × 7 1 10− ×
3 M/R = 0.5 M/R = 0.1 1 10− × 5 1 10− × 7 1 10− × 0.00 0.20 0.40 3 M/R = 0.4 1 10− J × M [R] 5 1 10− × 7 1 10− × 0.00 0.20 0.40 J M [R]
Figure 4.9.: The relative difference between the ADM and Komar quantities M and J are plotted as functions of J/M for different configurations characterized by M/R and R = 1.
43 5. Conclusion
The obtained numerical results have been presented in the previous chapter. Below we discuss the meaning of our findings and to what extend they differ from previous works. Finally, we conclude this work with an outlook about possible future research based on this thesis.
5.1. Discussion
In section 2.1 we have listed properties of the RMS that have been derived by Pfister et al. on the basis of their (latest) PT calculations [2]. In the following we compare those findings to our results: 1. The numerical results support the claim that a RMS characterized by 0.1 M/R 1.5 does indeed undergo a shape transformation from an initial sphere≤ to a prolate≤ ellipsoid. However, because the current state of the numerical im- plementation is not yet capable of computing reliable results for M/R > 1.5, no statement about the maximum deformation at M/R = 8/5 can be made. The overall trend of the centrifugal deformation lets one suspect that in the limit of M/R 2 the deformation may indeed vanish. The comparison with the PT re- sults has→ shown that in the slow rotation limit the PT results are reproduced by the numerical calculations. 2. The analysis of the angular velocity profiles does not agree with the findings made 3 by Pfister et al. in order ωPT PT. This is because no transition in the angu- lar velocity gradient for the numerical results with M/R = 0.5 and M/R = 0.6 has been observed. According to Pfister et al., the curves plotted in figure 4.2 should have showed ω(π/2)/ω(0) < 1 for configurations with M/R > 4/7. Besides this, based on the numerical results one may suspect that a RMS may be indeed rotating rigidly in the case of M/R 2, assuming that the presented trend of ω(π/2)/ω(0) 1 continuous to hold→ in this limit. This finding may then agree with the conclusion→ of Pfister et al. about the collapse limit and rigid rotation. This comparison leads to the conclusion that the PT calculations of Pfister et al. [2] could not be extended satisfactorily. The reason for the mismatch may be that either the 3 findings of Pfister et al. in order ωPT PT are incorrect, or our numerical results suffer an inconsistency. At the time of writing this thesis the numerical results have been tested
44 for consistency with different methods. Hereto, we performed tests where the residuals of the Einstein equations were evaluated for grid points that differed from the utilized collocation grids to spot any irregularities. Also the Komar integrals can be seen as a kind of consistency check. All tests performed so far do support the numerical results. This raises doubts about the findings of Pfister et al. made beyond second order PT. Concerning the surface-stress-energy tensor and its eigenvalues, one concludes that not any RMS configuration obeys the dominant energy condition. The violation is already evident in the case of no rotation for configurations with M/R > 4/3 and it was already reported in [2]. Additionally, one can conclude that the energy-matter content behaves like a kind of anisotropic-fluid. This differs from a perfect fluid, which is the casefor J = 0 and the Schwarzschild solution, by the fact that pw = pv. ̸ Last but not least, the error estimation showed that the numerical results come with a difference smaller than 0.1% attached to the globally defined quantities M,J. As a benchmark for this error estimation the results presented in reference [14] can be considered. At the time of writing this thesis, their work is the current state-of-the-art for computing massive rotating objects in GR. They provide a list of numerical values for the relative difference between the ADM and Komar quantities, just like we didin this work, for different orders of approximation of their results. The result computed in the lowest order approximation comes with a relative difference error that is of the order of 0.1%. Based on this we conclude that our numerical results have been obtained with a reasonable error, but there is room for further improvement, especially because the RMS problem is believed to be less complex than a calculation of a general relativistic massive rotating object.
5.2. Summary and Outlook
This work in connection with [8] showed that the RMS problem can be treated nu- merically and a three parameter solution to the Einstein field equations was presented. However, a complete and satisfactory agreement between the numerical and PT results was not found so far. In particular, the findings of Pfister et al. [2] made in third order PT could not be confirmed by our results. Nevertheless, the question if Mach’s ideaof relativity of rotation is realized in GR also for rapidly RMS can be answered positively. This is, because it was only necessary to find a solution to Einstein’s equations fora RMS where the interior region is flat and the exterior region is asymptotically flat[1]. Concerning this thesis, it is clear that some work remains to be done to clarify the mismatch with the findings of Pfister et al. [2] and the numerical results. This may include further tests for consistency of the numerical results, e.g. the computation of two general relativistic variants of the viral theorem of Newtonian mechanics [26, 27], also discussed in [12, 19]. On the other hand, one may also repeat the PT calculations of Pfister et al. and give the full solution to third order PT. Doing so would allow to make
45 a comparison for the shell’s angular velocity similar like it was done for the latitude dependent shell radius. At the time of writing this thesis, the numerical implementation was not capable of computing reliable results for RMS configurations near the collapse limit M/R 2. One reason why studying such configurations would be of interest is that PT calculations→ presented in [28] indicated that a RMS may be the source of the Kerr metric. They claim that this can only be the case if the spacetime interior to the RMS is flat and that the RMS must be a rigidly rotating perfect sphere. If such a configuration can be computed numerically, then one could learn about the energy-matter content of a RMS black hole. Besides the study of the collapse limit, one may be also interested in a modification of the assumptions made in this work. What may be of interest would be to relax the condition of a flat vacuum interior spacetime and substitute it by conditions that specify the energy-matter content of the surface-stress-energy tensor instead. It is believed that solving the field equations then in the interior and exterior vacuum region will, ingeneral, lead to a spacetime where the interior region is non-flat. The results presented in this work could then be used as a benchmark and test for such a computation. Last but not least, one may extend the RMS calculations to also include electromag- netism. This problem has been considered in first order PT in [20, 29, 30]. Their main finding is that such a charged RMS configuration posses a gyromagnetic ratio ofaround2 in a large region of its physical parameter space. This ratio makes a connection between the shell’s angular momentum and its magnetic moment. Because this factor is also around 2 for elementary particles, e.g. the free Dirac electron, the authors of this work speculated about a closer connection between general relativity with electromagnetism and quantum theory. Testing this claim with numerical methods, similar to what was done in this work, may provide interesting insights into the structure of Einstein-Maxwell equations.
46 A. Thin shell description
The following situation is presented in the main text: Let spacetime be partitioned via a hypersurface Σ into two distinct regions + and −, corresponding to an exterior and interior region, respectively. Assume thatV each V+, − is equipped with a metric + − +µ V−µ V + − g µν, g µν expressed in a system of coordinates x , x . The task is to join and + − V V smoothly at Σ, so that the union of the metrics g αβ and g αβ forms a valid solution of Einstein’s field equations. In order to achieve this, certain conditions have tobeput onto these metrics. These conditions may be calculated via the geometric formalism proposed in [22, 23]. A modified presentation of this technique may be found in[21]. This appendix reexamines and collects results that have been presented in [8] and cal- culated with the above mentioned formalism.
A.1. Definition and notation
The following calculations rely on the quasi-isotropic coordinate system (xµ) = (t, r, θ, φ) + − introduced in the main text. The metrics g µν, g µν compatible with this coordinates are chosen according to (2.1). In the following all quantities that solely belong to either region + or − are labeled with an additional superscript + or , respectively. E.g. V V + − − the metric functions U, V, W, Ω that are constituting g µν, g µν now carry an additional superscript + or in order to distinguish them from one another. − The following abbreviations are used in the subsequent sections: The jump of a quantity X, defined in + and −, across Σ is abbreviated as V V ⃓ ⃓ +⃓ −⃓ [X] := X ⃓ X ⃓ , (A.1) ⃓Σ − ⃓Σ ⃓ where X⃓Σ := X(f(θ), θ) and with f(θ) being a function that describes the shell’s radial location (defined in the next section).
A.2. Description of the shell
To start with, it is necessary to give a definition for the hypersurface Σ. With this it is possible to set up objects like the unit normal vector to Σ and a coordinate system intrinsic to Σ.
47 According to [1,2] the definition of Σ must involve a dependence on the latitudinal coordinate θ. A suitable definition is
Φ(r, θ) := r f(θ) = 0, (A.2) − where f(θ) is a yet unknown function. f(θ) serves to describe the radial location of the shell and one may think of it as a θ dependent radius. It is assumed that f(θ) is strictly positive and continuously differentiable for all values of θ [0, π]. Since the model under study shall also be symmetric with respect to reflections∈ across the equatorial plane (assumption (d)), at θ = π/2, it must hold f(θ) = f(π θ). − Given (A.2) the unit normal vector nµ to Σ can be calculated. This vector should be µ 1 spacelike, n nµ = 1, as well as pointing in the direction of increasing Φ(r, θ). Taking this into account yields
Φ,µ 1 nµ = αβ = (0, 1, f, 0) , (A.3) − g Φ,αΦ,β α − | | 1/2 (︄ (︃ ′ )︃2)︄ ̂ ̂ f α = eU−V 1 + . (A.4) f
A prime on f denotes differentiation with respect to θ. Finally, let us introduce an intrinsic coordinate system that enables the computation of linear independent and orthogonal tangent vectors to Σ. The intrinsic coordinates are a 2 µ ∂xµ chosen to be (ξ ) = (t, θ, φ). The tangent vectors are computed by ea = ∂ξa and they read
µ µ ′ µ (et ) = (1, 0, 0, 0) , (eθ ) = (0, f , 1, 0) , (eφ ) = (0, 0, 0, 1) . (A.5)
A.3. Induced metric and first junction conditions
The induced metric on Σ is given by ⃓ µ ν ⃓ hab = ea eb gµν ⃓ . (A.6) ⃓Σ
1The latter condition may be phrased as pointing spatially outwards, e.g. from − to +. 2General coordinates intrinsic to Σ shall be given by (ξa) = (τ, ϑ, ϕ). The parametricV equationsV
t = τ, r = f(ϑ), θ = ϑ, φ = ϕ.
then ensure compatibility of the intrinsic coordinates with (A.2) and (xµ) = (t, r, θ, φ), therefore, this choice.
48 Applying this definition to the situation at hand would yield two distinct induced metrics, + + e.g. hab on Σ may be computed using g µν or g µν. Demanding that hab shall be the same when it is computed via either metric yields the first junction conditions as [︁ ]︁ gµν = 0. (A.7)
Explicit evaluation of this using ansatz (2.1) gives
[X] = 0,X U, V, W, Ω , (A.8) ∈ { } and the computation of the induced metric reveals (︃ )︃ ̂ ̂ ̂ (︂ )︂2 ds2 = h ξaξb = e2U dt2 + e−2U e2V (︁f 2 + f ′2)︁ dθ2 + Ŵ 2 dφ Ω̂ dt . (A.9) Σ ab − − ⃓ ⃓ Here X̂ := X+⃓ = X−⃓ ,X U, V, W, Ω is a consequence of (A.7). Note that in Σ Σ ∈ { } general the X̂s depend on θ, but in case that either one of the spacetime regions is flat yields that the X̂s are constant numbers, cf. (2.10).
A.4. Extrinsic curvature and surface-stress-energy tensor
With the objects previously calculated it is possible to compute the extrinsic curvature + − K ab,K ab of Σ. As it turns out [Kab ] = 0, indicating that it is not continuous across Σ. According to [22, 23] such a jump implies̸ that Σ is a singular hypersurface. However, the discontinuities, which due to the junction conditions can only be present in the direction normal to Σ, can be assigned the meaning of a localized non-vanishing stress-energy tensor - the surface-stress-energy tensor. To continue, the jump in the extrinsic curvature is computed via
[Ka ] = [︁Γκ ]︁ n hace µe ν, (A.10) b − µν κ c b κ where Γ µν are the four-dimensional Christoffel symbols. Explicit calculation reveals
[︁ t ]︁ 1 −4Û 2 K = [U,n] Ω̂e Ŵ [Ω,n] , (A.11) t − 2 1 ̂ [︁Kt ]︁ = e−4U Ŵ 2 [Ω ] , (A.12) φ 2 ,n [︁ θ ]︁ K = [V,n] [U,n] , (A.13) θ − [︁ φ ]︁ 1 1 ̂ −4Û ̂ 2 K φ = [U,n] + [W,n] + Ωe W [Ω,n] , (A.14) − Ŵ 2
49 where the abbreviation (︃ ′ )︃ µ 1 2(Û−V̂ ) f X,n := X,µn = e X,r X,θ (A.15) α − f 2 has been introduced. It represents the directional derivative normal to Σ. Since discon- tinuities are only directed normal to Σ it follows that
∂ ∂ (︁ + − )︁ ′ 0 = [X] = X (f(θ), θ) X (f(θ), θ) = f [X,r] + [Xθ] , (A.16) ∂θ ∂θ − and one may conclude
[X,n] = α [X,r] . (A.17)
According to [21–23] the extrinsic curvature is related to the surface-stress-energy tensor via
8πSa = [Ka ] + δa [Kc ] . (A.18) b − b b c Note the similarity of this equation with Einstein’s field equations. A straightforward computation yields
t (︂ θ t )︂ 8πS = 2 [U,n] + [V,n] + 8π S Ω̂S , (A.19) t − θ − φ θ 1 8πS θ = [W,n] , (A.20) Ŵ φ t ̂ 8πS φ = [V,n] + 8πS φ Ω, (A.21)
t 1 −4Û 2 8πS = e Ŵ [Ω,n] , (A.22) φ −2 φ 1 (︁ t )︁ 8πS = [Ω,n] + Ω̂ 8πS [V,n] . (A.23) t 2 t −
A.4.1. Eigenvalues and eigenvectors
The eigenvalues as well as eigenvectors of the surface-stress-energy tensor are used sub- sequently to examine the energy conditions for the gravitational energy-matter source.
a Diagonalization of (S b ) yields the eigenvalues and the components of the corresponding eigenvectors as √︂ 1 t φ 1 φ t 2 t φ a t a a ρ = (S + S ) + (S φ S ) + 4S S , u = u (δ + ωδ ), (A.24) −2 t φ 2 − t φ t t φ √︂ 1 t φ 1 φ t 2 t φ a t a a pv = (S + S ) + (S φ S ) + 4S S , v = v (λδ + δ ), (A.25) 2 t φ 2 − t φ t t φ θ a θ a pw = S θ . w = w δ θ , (A.26)
50 These expressions belong to the eigenvalue equations
a b a a b a a b a S u = ρu ,S v = pvv ,S w = pww . (A.27) b − b b From the eigenvalue equations one obtains the relations
t φ ρ S t pv S φ ω = − −t , λ = −φ , (A.28) S φ S t
t t θ a a a The normalizations u , v , w are chosen such that u ua = 1 and v va = w wa = 1 (for latter convenience) which results in −
−1/2 ut = (︁ h 2ωh ω2h )︁ , (A.29) − tt − tφ − φφ t (︁ 2 )︁−1/2 v = htt + 2λhtφ + λ hφφ , (A.30) θ −1/2 w = (hθθ) . (A.31)
A.4.2. Energy conditions
Reference [21, chapter 2] provides an overview of standard energy conditions used in studies of general relativity. From this reference the dominant energy condition is taken over, which states that ... matter shall flow along timelike or null worldlines. Its precise statement is α α β that if v is an arbitrary, future-directed, timelike vector field, then T β v α −β is a future-directed, timelike or null, vector field. The quantity T β v is the matter’s momentum density as measured by an observer with− four-velocity vα, ... α α T β are the components of the four dimensional stress-energy tensor. If (T β ) has eigenvalues ρ, p1, p2, p3 one can deduce that the dominant energy condition is fulfilled if the conditions{ }
ρ 0, ρ pi , i = 1, 2, 3, (A.32) ≥ ≥ | | are satisfied. In this notation, ρ measures the local mass-energy density as seen by a α comoving observer with 4-velocity v and the pi represent the pressures acting in the spatial directions. To apply the dominant energy condition to the situation at hand, the eigenvalues of the µ stress-energy tensor (T ν ) of four dimensional spacetime that correspond to the surface- a stress-energy tensor (S b ) of Σ must be computed. In reference [21, section 3.4] the a necessary tools to extend (S b ) to four dimensional spacetime are discussed, e.g.
µν ab µ ν T = S ea eb , (A.33)
51 where the intrinsic coordinate basis given by (A.5). Using the diagonal representation of the surface-stress-energy tensor
a a a a S b = ρu ub + pvv vb + pww wb (A.34) then yields
µ µ µ µ T ν = ρu uν + pvv vν + pww wν, (A.35)
µ a µ a µ which is due to x = x ea , xµ = xae µ. Because n is orthogonal to the intrinsic coordinate basis (A.5), it follows that the resulting eigenvalue equations take the form
µ ν µ µ ν T ν u = ρu ,T ν n = 0, (A.36) µ ν − µ µ ν µ T ν v = pvv ,T ν w = pww . (A.37)
Note that no pressure in the direction normal to the RMS, characterized by nµ, can be observed. In general, pw = pv which indicates that the surface-stress-energy tensor is not of perfect fluid type. ̸ Finally, in order to check whether the gravitational energy-matter source of a RMS obeys the dominant energy condition, one can check if the following conditions are satisfied
ρ 0, ρ pv , ρ pw . (A.38) ≥ ≥ | | ≥ | |
52 B. Komar integrals
In order to judge the accuracy of results obtained via numerical computations it is necessary to perform tests with the corresponding results. One of these tests involves a comparison of the ADM quantities with Komar quantities. The former are the numbers M,J used in the main text and the latter are computed via Komar integrals which are the subject of this appendix. Here follows a detailed derivation of the results given in [8]. The claim that is underlying this test can be stated as follows: For axis symmetric and asymptotically flat spacetimes the ADM quantities total gravitational mass M and total angular momentum J agree with the Komar mass MK and Komar angular momentum JK , respectively. A proof of this can be found in [19, chapter 3], [21]. The definitions for Komar mass and Komar angular momentum are taken over from[21, section 4.3]: ∫︂ (︃ )︃ 1 α β√︁ 3 MK = 2 Tαβ T gαβ nt ξ ht d y , (B.1) Σt − 2 | | ∫︂ (︃ )︃ 1 α β√︁ 3 JK = Tαβ T gαβ nt χ ht d y . (B.2) − Σt − 2 | |
Σt represents a hypersurface of constant time with a future directed unit normal vector µ β β nt and ht := det(htab) is the determinant of the induced metric on Σt. ξ , χ are the spacetime’s timelike and rotational Killing field (2.4) and (2.5), respectively, Tαβ are the µ components of the spacetime’s stress-energy tensor and T := T µ denotes its trace.
The surface of constant time t0 is defined by
Σt := t t0. (B.3) − Its future-directed and timelike unit normal vector is given by
1 α 1 tt tφ (nα) = √︁− (1, 0, 0, 0), (n ) = √︁− (g , 0, 0, g ). (B.4) gtt gtt | | | | The integration kernels of (B.1) and (B.2) then simplify to (︃ )︃ (︃ )︃ 1 α β 1 t 1 Tαβ T gαβ n ξ = √︁ T t T , (B.5) − 2 gtt − 2 (︃ )︃ | | 1 α β 1 t Tαβ T gαβ n χ = √︁ T φ . (B.6) − 2 gtt | |
53 a Recap that the surface-stress-energy tensor S b is supported only on Σ (not Σt). We compute its extension to four dimensional spacetime according to (A.33). Explicit eval- t t t t uation shows that T t = S t ,T φ = S φ . Furthermore, from the completeness relation
a b gµν = nµnν + habeµ eν , (B.7) µ µν ab a and (A.33) one concludes that T = T µ = T gµν = S hab = S a = S. a Since S b is localized on Σ in the surrounding spacetime, it comes with an additional delta distribution (︂ )︂ ⃓ µ √︁ µν ⃓ T ν δ(Σ) g Σ,µΣ,ν ⃓ , (B.8) ∼ ⃓Σ where the square root factor ensures invariance under reparametrization of Σ. Next, we compute √︄ √︄⃓ ⃓ ht ⃓grr g gφφ⃓ | | = θθ = e−2U+2V rW, (B.9) gtt gtt | | | | (︄ )︄1/2 (︂ )︂ ⃓ (︂ )︂ ⃓ (︃ ′ )︃2 √︁ µν ⃓ √︁ rr ′ 2 θθ ⃓ Û−V̂ f g Σ,µΣ,ν ⃓ = g + (f ) g ⃓ = e 1 + , (B.10) ⃓Σ ⃓Σ f
t 1 1 −4Û 2 8π(S S) = [U,n] + e Ŵ Ω[Ω̂ ,n]. (B.11) t − 2 − 2 Partial evaluation of the integral yields ∫︂ ∫︂ ∞ ∫︂ π ∫︂ 2π ∫︂ π/2 d3y δ(r f(θ)) ... = dr dθ dφ δ(r f(θ)) ... = 4π dθ ..., (B.12) Σt − 0 0 0 − 0 where it has been used that the integration kernel I(θ) obeys I(θ) = I(π θ). Finally, putting everything together and using (A.17) we arrive at − (︄ )︄ ∫︂ π/2 (︃f ′ )︃2 MK = dθ [U,r]Ŵ f 1 + (B.13) 0 f ∫︂ π/2 (︄ (︃ ′ )︃2)︄ 1 −4Û 3 f dθ Ω[Ω̂ ,r]e Ŵ f 1 + , (B.14) − 2 0 f ∫︂ π/2 (︄ (︃ ′ )︃2)︄ 1 −4Û 3 f JK = dθ [Ω,r]e Ŵ f 1 + . (B.15) −4 0 f Note that in the case of a flat interior metric it follows that (︄ )︄ ∫︂ π/2 (︃f ′ )︃2 MK = dθ U,rŴ f 1 + + 2Ω̂JK . (B.16) 0 f The second term may be interpreted as the ’rotational contribution’ to the total mass [8].
54 C. Details on numerical implementation
In the main text the implementation of the system of equations using pseudo-spectral methods was discussed. A few details about the chosen basis functions, collocation points and actual numerical implementation have been left out. These details are the topic of this appendix.
C.1. Rational Chebyshev functions
The ansätze (3.9)-(3.11) for the metric potentials U, V, Ω have been chosen such that they show agreement with the desired asymptotic behavior (2.37)-(2.40). It is easy to check that this is guaranteed if the corrections δX, X U, V, Ω approach a constant value as r . This can be ensured if one models the radial∈ { dependence} of the unknown δXs via rational→ ∞ Chebyshev functions. These functions are defined via Chebyshev polynomials of the first kind Tn(x) by [25, appendix A] (︃y L)︃ TLn(y; L) := Tn − , y [0, ) (C.1) y + L ∈ ∞ for n = 0, 1, 2, .... The parameter L > 0 allows to adjust the parameterization of the Chebyshev polynomial’s argument to improve convergence if needed [25]. In this work −n we set L = 1 for all calculations. As one can show each TLn(y; L) behaves as y for y , which is the desired property. → ∞ To utilize (C.1) in the ansätze, the domain of y needs to be adjusted. This is done by making the transformation y = r f(θ). The complete parameterization of the Chebyshev polynomial’s argument x together− with its inverse is then given by r f(θ) L x(r; θ, L) = − − , r [f(θ), ), (C.2) r f(θ) + L ∈ ∞ − 1 + x r(x; θ, L) = f(θ) + L , x [ 1, 1]. (C.3) 1 x ∈ − − Because the RMS radius depends on θ, the parameterization also inherits this depen- dency. This shall be of no problem, as long as one always evaluates f(θ) before evaluating the parameterization in the numerical implementation.
55 For the computation of derivatives of TLns we rely on the chain rule: If F (x(r, θ)) is a two times continuous differentiable function, then partial derivatives with respect to α, β r, θ are given by ∈ { }
F,α = x,αF,x (C.4) 2 F,αβ = x,αβF,x + x,αx,βF,xx. (C.5) To this end, the first and second derivatives of the parameterization are listed below: 2L 4L x,r = , x,rr = , (C.6) (r f + L)2 −(r f + L)3 − − 2Lf ′ 2L(f ′′(r f + L) + 2f ′2) x,θ = , x,θθ = − , (C.7) −(r f + L)2 − (r f + L)3 − − 4Lf ′ x = , (C.8) ,rθ (r f + L)3 − where a prime on f denotes differentiation with respect to θ.
The simple definition of the TLns in terms of Chebyshev polynomials of the first kind is practical, because sums of Chebyshev polynomials can be evaluated efficiently using the Clenshaw algorithm [31]. This algorithm is not only computationally inexpensive with costs of (N) function evaluations for a series with N terms, but also guarantees high accuracyO at the boundaries of the Chebyshev polynomial’s argument. For the evaluation of the PDEs, first and second order derivatives of the unknowns are frequently needed. To compute them efficiently, we utilize a generalization of the Clenshaw algorithm, presented in [32]. This scheme, combined with the chain rule for differentiation and the parameterization mentioned above, allows to evaluate the sum of rational Chebyshev series together with its first and second derivatives at costs of order (N). O
C.2. Collocation points
A common choice for collocation points used for Chebyshev pseudo-spectral methods are the roots of the Chebyshev polynomials, or often referred to as Gauss-Chebyshev points used in Gauss-Chebyshev quadratures. They are given by formula [25] (︃(2i + 1)π )︃ xi = cos , i = 0, 1, 2, ..., N 1. (C.9) − 2N − The additional minus sign in this definition is absent in the literature, it has been included to give a ascending order of grid points from -1 to 1 with respect to i. Because rational Chebyshev functions are simply Chebyshev polynomials of the first kind augmented with a specific argument parameterization, transforming the grid points (C.9) by means of (C.3) yields a proper set of collocation points that can be used with the TLns. Note that the grid (C.3) does not contain the end points x = 1, 1. −
56 For the Fourier series which is utilized to model the θ-dependencies of functions, a simple equidistant grid θi can used be used as collocation points [25], e.g. { } π (︃ 1)︃ θi = i + i = 0, 1, 2, ..., N 1. (C.10) N 2 − This is the same grid as one would use for a numerical quadrature with a rectangular rule. Note that the grid has been tailored to the interval θ [0, π/2) and, again, the end points are not included. This is desired, because at θ =∈ 0 the PDEs (2.21)-(2.22) are satisfied identically due to W = 0.1
C.3. Orthogonality relations
The utilized basis functions for this work constitute a set of orthogonal polynomials. Here follow the corresponding orthogonality relations, which may be used for the ap- proximation of the Schwarzschild solution. For the rational Chebyshev functions, the corresponding orthogonality relation is given by [25, appendix A] (︄ √︄ )︄ ∫︂ ∞ 1 L π TL (r; L, θ)TL (r; L, θ) dr = δ (1 + δ ). (C.11) n m r f(θ) + L r f(θ) mn 2 n0 f(θ) − − Note that this relation explicitly depends on θ, because of the utilized parameterization (C.3). However, this shall be of no restriction, because one can always get rid of this dependence by transforming (C.11) back to the initial orthogonality relation for the Chebyshev functions Tn(x), which reads
∫︂ 1 Tn(x)Tm(x) π dx = δmn (1 + δn0). (C.12) 2 −1 √1 x 2 − The orthogonality relations for Fourier basis functions are well known and they read ∫︂ π sin(nx) cos(mx) dx = 0, −π ∫︂ π cos(nx) cos(mx) dx = πδmn(1 + δn0), (C.13) −π ∫︂ π sin(nx) sin(mx) dx = πδmn(1 δn0). −π −
1Of course, one may divide both sides of these equations by W and be able to evaluate them at θ = 0 to arrive at a meaningful result by making use of L’Hôpital’s rule. However, the choice (C.10) allows to easily bypass this additional, analytical expense.
57 C.4. Code and external libraries
The numerical implementation of this work was carried out using the general-purpose programming language C++. I made use of the open source C++ library Eigen [33] for conveniently handling numerical data, matrix and vector manipulations. For solving the nonlinear algebraic equations discussed in the main text, the open source library Ceres Solver [34] was used. This is a nonlinear least-squares solver that comes with various different minimization strategies, e.g. Levenberg-Marquardt, Armijo-Rule, etc. For a very brief overview of the most relevant least-square solver routines the reader is referred to [35]. The reason why I utilized a least-squares solver, instead of a conven- tional nonlinear solver algorithm like the Newton-Raphson algorithm, is that the use of Chebyshev basis functions can often lead to ill-conditioned and unstable Jacobian matrices during the iteration procedure. This may break linear algebra solvers easily. Least-squares solvers allow to bypass this by modifying the Jacobians adaptively during critical iteration steps. Furthermore, like with the linear algebra library, using an ex- ternal library certainly reduces the possibility of introducing errors by writing your own version of an algorithm, which has been implemented many times before. Most of the time they are also optimized very well and usually tested thoroughly.
58 D. Perturbation theory
The RMS problem was extensively studied with perturbation theory over the years and for a historical account we refer to [2]. Since this work aims at extending the latest results beyond the parameter region that is covered by PT, a brief summary of their work and some explicit formulae are provided here. The presented results were taken over from [2]. The work of Pfister et al. was based on a perturbation series of the Schwarzschild solution. The expansion was carried out in the parameter ωPT R 1, where ωPT served to describe the shell’s angular velocity in the slow rotation limit≪1. Their results are valid for arbitrary, physical ratios of mass and radius M/R of the shell. Pfister et al. worked with quasi-isotropic coordinates and utilized the ansatz (2.1). The field equations studied by Pfister et al. were (2.21)-(2.25). The starting point for the work of Pfister et al. was the Schwarzschild solution, expressed in isotropic coordinates (xµ) = (t, r, θ, φ), which reads
(︁ M )︁2 4 1 (︃ M )︃ ds2 = − 2r dt2 + 1 + (︁dr2 + r2 dθ2 + r2 sin2(θ) dφ2)︁ . (D.1) −(︁ M )︁2 2r 1 + 2r The transformation to Schwarzschild coordinates (t, ρ, θ, φ) is given by
2 1 (︂ √︁ )︂ (︃ M )︃ r = ρ M + ρ2 2Mρ , ρ = r 1 + . (D.2) 2 − − ⇔ 2r
Note that the Schwarzschild radius ρS = 2M is located at rS = M/2 in isotropic coordinates. A comparison of (D.1) with ansatz (2.1) allows to read off the metric potentials as
0 (︃x 1)︃ 0 (︃X 1)︃ U(r) = log − for x > X, U 0 = log − for x < X, (D.3) x + 1 X + 1 0 (︃x2 1)︃ 0 (︃X2 1)︃ V (r) = log − for x > X, V 0 = log − for x < X, (D.4) x2 X2 0 0 W (r, θ) = eV (r)r sin(θ), (D.5)
1 We have added the subscript PT to ω, which we use to describe the θ-dependent angular velocity.
59 where x = 2r/M and X = 2R/M. R labels the shell’s radius and M its gravitational mass. Because R r it follows that the shell’s mass is restricted to 0 < M/R < 2. Note that the notation≤ from [2] has been slightly changed for this work: The metric function V in (2.1) was initially labeled by K.
According to [1], in order ωPT only the function A := Ω/ωPT obtains a correction which is independent of θ. It is given by
0 (︂y )︂3 0 (︃Y )︃3 A(r) = λ for y > Y, A(r) = λ for y < Y, (D.6) 4 4 x X y = 4 ,Y = 4 , (D.7) (x + 1)2 (X + 1)2 (X + 1)5(2X 1) λ = 4 − . (D.8) X3(3X 1) −
2 In order ωPT the metric functions U, V, W obtain latitude dependent corrections. The expansions of the metric functions are carried out as
0 2 2 N(r, θ) = N(r, θ) + (ωPT R) N(r, θ) + ... N U, V, W, A . (D.9) ∈ { } The analysis to obtain these corrections is involved, for details see [1,2]. As it turned out, the field equations can only be solved consistently to this order if the shell’sshape deviates from sphericity. To this end, the latitude dependent shell radius was intro- duced
(︁ 2 2 )︁ fPT (θ) = R 1 + (ωPT R) f sin (θ) , (D.10) where f is a constant that depends only on the dimensionless ratio X. It is given by
(X + 1)4 (2X 1)2 [︃(︃ 6 1 )︃ f = − 1 + + (D.11) 2X4 (3X 1)2 X X2 −(︁ 2 (︁ X−1 )︁)︁ ]︄ 32 2X + (X + 1) log X+1 (︁ 2 (︁ 4 2 2 )︁ (︁ X−1 )︁)︁ −3 2X (X + 1) + X + 3 X + 1 log X+1
3 Last but not least, the analysis of Pfister et al. also showed that in order ωPT the RMS has to rotate differently, otherwise the field equations cannot be solved consistently in this order. Differential rotation means that the angular velocity profile obtains a θ-dependence, which was expressed as
(︁ 2 2 )︁ ω̄ = ωPT 1 + (ωPT R) e sin (θ) , (D.12) where e is a very involved constant that depends on X and the other metric potentials evaluated on the shell.
60 In order to judge whether our numerical results extend the PT calculations, agreement between the results shall be found in the slow rotation limit. Hereto, the expansion parameter is restricted by ωPT R 1, which means that the circumferential velocity of the RMS’s equator shall be small≪ compared to the speed of light. Besides the numbers M,R, different RMS are distinguished by their numerical values of ωPT R. In contrast to this, the characterization of different RMS in the main text is done using the parameters M,R as well as J. It is clear that if we would like to make a comparison between the PT and numerical results a relation between ωPT and J must be established. One possibility to make the connection is to compare the asymptotic behavior of the metric function Ω for both results. Hereto, we take (D.6) and expand it in powers of r:
0 (︃ 2 )︃3 λ 2M 2M (︁ −3)︁ Ω(r) = ωPT A(r) = ωPT + r (D.13) 64 r − r2 O 3 λM (︁ −4)︁ = ωPT + r . (D.14) 8r3 O A direct comparison with the coefficient of the r−3 term in (2.40) then yields the rela- tion 16J ω = . (D.15) PT λM 3 In figure D.1 we provide a plot of this relation for selected M/R configurations and R = 1. 2 The mapping (D.15) is consistent up to order ωPT in PT, because Ω obtains its next 3 correction in order ωPT . Because of this, we can make use of this relation to parameterize (D.10) in terms of J for a comparison of the latitude dependent shell radius, as it is done in the main text. Contrary to this, no meaningful comparison for the angular velocities 3 can be made. This is because the first correction to (D.12) is of order ωPT and when parameterized by (D.15) would not be consistent with PT.
61 0.10 M/R [ ] − 1.50 0.08 1.40 1.30 1.20
] 0.06 1.10 − [ 1.00
R 0.90
PT 0.80 ω 0.04 0.70 0.60 0.50 0.02 0.40 0.30 0.20 0.00 0.10 0.0 0.1 0.2 0.3 0.4 0.5 J M [R]
Figure D.1.: Visualization of the mapping (D.15) relating the PT parameter ωPT and the ADM quantity J for selected M/R configurations and R = 1. This mapping is 2 consistent with PT up to order ωPT .
62 Bibliography
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