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Link¨

oping Studies in Science and Technology

Dissertations, No 1181

Multipole Moments

of

Stationary Spacetimes

Thomas B¨

ackdahl

Division of Applied Mathematics

Department of Mathematics

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Multipole Moments of Stationary Spacetimes

Copyright c 2008 Thomas B¨ackdahl, unless otherwise noted. Matematiska institutionen

Link¨opings universitet SE-581 83 Link¨oping, Sweden thbac@mai.liu.se

Link¨oping Studies in Science and Technology Dissertations, No 1181

ISBN 978-91-7393-902-7 ISSN 0345-7524

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11519

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Abstract

In this thesis we study the relativistic multipole moments for stationary asymptotically flat spacetimes as introduced by Geroch and Hansen. These multipole moments give an asymptotic description of the gravitational field in a coordinate independent way.

Due to this good description of the spacetimes, it is natural to try to construct a spacetime from only the set of multipole moments. Here we present a simple method to do this for the static axisymmetric case. We also give explicit solutions for the cases where the number of non-zero multipole moments are finite. In addition, for the general stationary axisymmetric case, we present methods to generate solutions.

It has been a long standing conjecture that the multipole moments give a complete characterization of the stationary spacetimes. Much progress toward a proof has been made over the years. However, there is one re-maining difficult task: to prove that a spacetime exists with an a-priori given arbitrary set of multipole moments subject to some given condition. Here we present such a condition for the axisymmetric case, and prove that it is both necessary and sufficient. We also extend this condition to the general case without axisymmetry, but in this case we only prove the necessity of our condition.

Acknowledgements

First of all I would like to thank my supervisor docent Magnus Herberthson for his support and good collaboration. It has been a joy to work with him. I would also like to thank my second supervisor professor Brian Edgar for his support in everything from finding references to language corrections and international contacts. Thanks to Dr Bengt Ove Turesson for lengthy discussions and many fruitful ideas. Thanks to docent Anders Bj¨orn for his suggestions regarding typesetting. Thanks to everyone at the department of mathematics who have helped me in one way or another.

Link¨oping, 16 May 2008 Thomas B¨ackdahl

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Popul¨

arvetenskaplig sammanfattning

En viktig fr˚agest¨allning inom den allm¨anna relativitetsteorin ¨ar hur man beskriver isolerade system. Exempel p˚a s˚adana ¨ar stj¨arnor, galaxer och svarta h˚al omgivna av vakuum. Att betrakta system som isolerade ¨ar det naturliga s¨attet att bortse fr˚an p˚averkan av systemet utifr˚an och v¨antas ge goda approximationer ¨aven f¨or fall d˚a omgivningen av systemet inte ¨ar helt tom. Det matematiska s¨attet att beskriva isolerade system ¨ar att kr¨ava att rumtiden ¨ar asymptotiskt plan.

N¨asta naturliga f¨orenklingssteg ¨ar att studera system som inte f¨or¨ an-dras n¨ar tiden g˚ar. S˚adana system kallas station¨ara. Ett typiskt exempel ¨

ar ett roterande svart h˚al eller en idealiserad modell av en galax. I vissa fall g¨or vi en lite h˚ardare restriktion och studerar det statiska fallet. Detta inneb¨ar att inte n˚agon r¨orelse f¨orekommer. Det ¨ar asymptotiskt plana statiska eller station¨ara rumtider som behandlas i den h¨ar avhandlingen.

Den stora skillnaden mellan Newtonsk mekanik och allm¨an relativitets-teori ¨ar att man i den senare saknar en fix bakgrund att relatera till. Matematiskt inneb¨ar det h¨ar att man m˚aste formulera fysiken p˚a ett koor-dinatoberoende s¨att. Genom att anv¨anda rent geometriska beskrivningar f˚ar man automatiskt koordinatoberoende beskrivningar. D¨arf¨or formulerar man sig g¨arna geometriskt n¨ar man jobbar med allm¨an relativitetsteori.

Arbetet med att ge en enkel geometrisk beskrivning av station¨ara asymptotiskt plana rumtider resulterade i att Geroch och Hansen p˚a 1970-talet definierade multipolmoment f¨or s˚adana rumtider. Multipolmomenten ¨

ar en f¨oljd av tensorer som ger en beskrivning av gravitationsf¨altet p˚a stora avst˚and. De ¨ar dessutom konstruerade p˚a ett s˚adant s¨att att de liknar de multipolmoment som anv¨ands i Newtonsk mekanik.

M˚anga viktiga egenskaper har under ˚aren visats f¨or multipolmomenten, men en viktig fr˚aga har l¨ange varit ¨oppen. Om man har en upps¨attning multipolmoment, hur kan man d˚a avg¨ora om det finns en rumtid med dessa multipolmoment eller inte? Det ¨ar den h¨ar fr˚agan som avhandlingen utg˚ar ifr˚an.

I det allm¨anna fallet tas ett n¨odv¨andigt villkor p˚a multipolmomenten fram f¨or existens av en rumtid med de givna multipolmomenten. I m˚anga till¨ampningar studerar man system som dessutom ¨ar symmetriska kring en axel, och under den extra f¨oruts¨attningen l¨oses ovanst˚aende problem fullst¨andigt. Det vill s¨aga, ett b˚ade n¨odv¨andigt och tillr¨ackligt villkor p˚a multipolmomenten ges f¨or existens av en l¨osning.

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Contents

Introduction 1

1 Background 1

2 Preliminaries 2

2.1 Notation and conventions . . . 2

2.2 Symmetries and Killing vectors . . . 2

2.3 Static and stationary spacetimes . . . 3

2.4 Field equations for the static and stationary case . . . . 4

2.5 Conformal compactifications . . . 5

3 Definition of multipole moments 5 3.1 Newtonian case . . . 6

3.2 Different relativistic definitions . . . 7

3.3 The Geroch–Hansen definition . . . 7

4 Earlier results 8 4.1 Centre of mass . . . 8

4.2 Different potentials . . . 8

4.3 Uniqueness of the moments . . . 8

4.4 Analyticity . . . 9

4.5 Injectivity . . . 9

4.6 Construction of solutions . . . 9

4.7 General existence proofs . . . 10

4.8 Simplification of multipole computation . . . 10

5 Summary of papers 11 5.1 Summary of paper 1 . . . 11 5.2 Summary of paper 2 . . . 12 5.3 Summary of paper 3 . . . 13 5.4 Summary of paper 4 . . . 13 6 Conclusion 14 Paper 1: Static axisymmetric spacetimes with prescribed mul-tipole moments 21 Thomas B¨ackdahl and Magnus Herberthson 1 Introduction 21 2 Explicit moments of the axisymmetric Weyl solutions 22 2.1 The Schwarzschild solution . . . 24

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3 General multipole 25

3.1 On a multipole conjecture due to Geroch . . . 27 3.2 Finite number of moments . . . 28

4 Solving algebraic equations 29

5 Powers of the root 30

6 Examples 33

6.1 Pure 2n-pole . . . 34 6.2 Monopole - 2n-pole . . . . 35

6.3 The monopole-quadrupole conjecture of

Hern´andez-Pastora and Mart´ın . . . 36

7 Conclusions and discussion 37

Paper 2: Explicit multipole moments of stationary

axisym-metric spacetimes 41

Thomas B¨ackdahl and Magnus Herberthson

1 Introduction 41

2 Multipole moments of stationary spacetimes 42 2.1 Multipole moments of axisymmetric spacetimes . . . 43

3 Multipole moments through a scalar recursion on R2 44 4 Multipole moments through a scalar recursion on R 46 5 All moments from one scalar function 47

6 The Kerr solution 49

7 Potentials 50

8 Discussion 51

Paper 3: Calculation of, and bounds for, the multipole mo-ments of stationary spacetimes 57

Thomas B¨ackdahl and Magnus Herberthson

1 Introduction 57

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3 Multipole moments through a scalar recursion on R2 59

3.1 Simplified calculation of the moments . . . 62

4 Bounds on the moments 65

5 Discussion 68

Paper 4: Axisymmetric stationary solutions with arbitrary

multipole moments 73

Thomas B¨ackdahl

1 Introduction 73

2 Field equations and potentials 74

3 Definition of multipole moments 75

4 Axisymmetry 75

5 Axisymmetric field equations 76

6 Specified multipole moments 77 6.1 Solving the field equations . . . 81

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Introduction

1 Background

The subject of this thesis has its origin in the problem of how to describe time independent isolated systems in general relativity. For instance these can model stars, galaxies or black holes. To be acceptable, such a descrip-tion can only depend on the manifold structure of the spacetime. If one has to make any arbitrary choices, one has to prove that these do not affect the description in the end. Therefore, intrinsic geometric descriptions are preferred to coordinate descriptions.

The multipole moments as defined by Geroch [14] and Hansen [17] give one such description for stationary asymptotically flat spacetimes. The stationarity is the mathematical formulation of time independence, while the asymptotic flatness is the natural setting to describe isolated systems. Together with a generalized definition of centre of mass, the multipole mo-ments give a unique description of the spacetime in terms of a sequence of tensors. Thus the spacetime can be described by a comparatively small set of data. The major remaining open question concerning this topic is to what extent the multipole moments can be arbitrarily prescribed. This is important because if they cannot be arbitrarily prescribed under a cer-tain convergence condition, there might be a smaller set of data that can describe the spacetime.

This problem of proving existence of a spacetime with a-priori specified multipole moments is the main topic of this thesis. For the general case, we give efficient tools for computing the multipole moments and a necessary convergence condition for the existence of a solution, whereas the problem is completely solved in the axisymmetric case.

In the stationary axisymmetric case we also give a recursive method for computing the metric components from the multipole moments. For the static axisymmetric case this is simpler, and we also give explicit solutions in terms of A-hypergeometric series for the case with finite number of non-zero multipole moments.

The results of the thesis have been published in four papers [6, 7, 8, 9], where the first three papers were jointly written by B¨ackdahl and Herberthson, while the last paper was written by B¨ackdahl alone.

This introduction is devoted to the technical background and a sum-mary of the papers.

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2 Preliminaries

In this section, we specify our conventions and recall the definitions of symmetries and conformal compactifications.

2.1 Notation and conventions

We use the abstract index notation [38]. Symmetrization of indices is de-noted by round brackets (. . . ) and anti-symmetrization by square brackets [. . . ].

A spacetime is given by a C∞ (unless otherwise noted) manifold M of dimension 4 and equipped with a metric gab of Lorentzian signature

(−, +, +, +). The connection, ∇a, is always the torsion-free Levi–Civita

connection. We denote the tangent space of M at the point p by TpM .

The Riemann curvature tensor is defined by

(∇a∇b− ∇b∇a)vc= −Rabdcvd. (1)

Einstein’s equations are given by

Rab−

1

2Rgab= 8πTab, (2) where Rab is the Ricci tensor, R is the Ricci scalar and, Tabis the

energy-momentum tensor. In this thesis Raband R is used for the Ricci tensor and

Ricci scalar for a 3-dimensional manifold with a positive definite metric. We use geometrized units with G = c = 1. In this thesis we will only study vacuum solutions, that is Tab= 0.

2.2 Symmetries and Killing vectors

In general it is difficult to compare tensors at different points on a manifold, but if we have a diffeomorphism that maps one point to another, we can use this to ‘move’ the tensor field and compare it to the original tensor field. The concept of transformation symmetries of a tensor field is defined using these ideas.

Let M and N be two manifolds, (possibly the same manifold). Let φ : M → N be a C∞ map. The differential map φ∗ : TpM → Tφ(p)N is

defined by the relation

(φ∗ν)a∇af = νa∇a(f ◦ φ) for all f ∈ C∞(N, R), (3)

where νa ∈ T

pM is arbitrary.

We also define the pullback φ∗: Tφ(p)∗ N → Tp∗M via

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3 (φ∗ν)a TpM p νa M N φ Tφ(p)N φ(p)

Figure 1: Definition of the differential map.

If, furthermore, φ is a diffeomorphism we can extend φ∗ to tensors of

type (m, n) (φ∗T )b1...bma1...an(µ1)b1. . . (µm)bm(ν1) a1. . . (ν n)an = Tb1...bm a1...an(φ ∗µ 1)b1. . . (φ ∗µ m)bm((φ −1) ∗ν1)a1. . . ((φ−1)∗νn)an for all (µi)b∈ Tφ(p)∗ N, (νi)a∈ Tφ(p)N. (5)

If φ is a diffeomorphism from the manifold to itself, we can compare a tensor field T with the tensor field φ∗T . When φ∗T = T we call φ a

symmetry transformation for T . This means that if we move T with φ, the tensor field T stays the same. In this thesis we will only study the special case when φ∗gab= gab. In this case φ is called an isometry.

If we have a smooth vector field ξa on a manifold M we can define a

one-parameter family of diffeomorphisms φt : M → M , by following the

flow of ξafor a ‘distance’ t. If φ

tis a group of isometries we call ξa a Killing

vector field.

For a more complete treatment of symmetries and Killing vector fields, see chapter 5 in [29].

2.3 Static and stationary spacetimes

The intuitive picture of a stationary spacetime is that ‘nothing happens’ when time evolves. Another way of saying this is that time evolution should be a symmetry transformation. The formal definition of a stationary space-time is that it admits a space-timelike Killing vector field ξa. We see that this

gives a one parameter group of isometries; these isometries correspond to time translation. We can also use the parameter as a time coordinate.

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If ξa is also orthogonal to a spacelike hyper-surface we call the

solu-tion static. A necessary and sufficient condisolu-tion that ξa is hyper-surface

orthogonal is (cf. [38])

ξ[a∇bξc]= 0. (6)

The spacetime describing the exterior of a rigid massive object in vac-uum that does not rotate is static. If the object spins around a symmetry axis the corresponding spacetime is stationary (outside the object). This gives an intuitive picture of the difference between stationary and static. Observe that this difference does not affect the gravitational potential in Newtonian mechanics.

Because of the independence of time in the stationary case we can ‘fac-tor out’ the time. We do this by considering the 3-manifold of trajec‘fac-tories of the timelike Killing vector field ξa. In the general case the correspondence between tensor fields on this 3-manifold and the spacetime was investigated in detail by Geroch [15]. In the static case however, this is simpler, because here the 3-manifold is equivalent to the spacelike hyper-surface that is or-thogonal to ξa.

2.4 Field equations for the static and stationary case

In this thesis we restrict ourselves to stationary or static vacuum space-times. Thus it is sufficient to consider the ‘spacelike part’ of a spacetime. We will therefore study the 3-manifold of trajectories of the timelike Killing vector field ξa. Call this 3-manifold V . We let λ = −ξaξ

a be the norm of

ξa. Intuitively the norm describes in a sense how fast the time elapses at

different points. Furthermore, we define the twist ωaby ωa= εabcdξb∇cξd;

the twist describes how much ξa differs from being hyper-surface

orthogo-nal. Due to the fact that we only consider vacuum solutions, we can define a twist potential ω via ∇aω = ωa, since ∇[bωa]= 0 in vacuum.

The metric gab(with signature (−, +, +, +)) on the spacetime induces

the positive definite metric

hab= λgab+ ξaξb

on V .

In the stationary vacuum case, Einstein’s field equations (2) induce field equations on V (see for instance (7) in [15]),

Rab= 1 2λ2((Daλ)Dbλ + (Daω)Dbω) (7a) DaDaλ = 1 λ((D aλ)D aλ − (Daω)Daω) (7b) DaDaω = 2 λ(D aλ)D aω, (7c)

where Rab and Da are the Ricci tensor and the covariant derivative

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and can use φ = 12ln λ as a potential, and the field equations reduce to Rab= 2(Daφ)Dbφ and DaDaφ = 0, i.e., a covariant version of the Laplace

equation. In the general stationary case we can introduce the potential φ =1−λ−iω1+λ+iω. With this potential, the field equations reduce to

Rab= 2 (φ ¯φ − 1)2(D(aφ)Db)φ¯ (8a) DaDaφ = 2 ¯φ φ ¯φ − 1(D a φ)Daφ. (8b)

In the axisymmetric case, using Weyl-coordinates the equation (8b) becomes the Ernst equation [11]. The non-linearity of this equation makes it much more difficult to solve than the Laplace equation. However, several methods to generate solutions have been developed. See section 4.6, below for more details.

2.5 Conformal compactifications

To be able to treat asymptotic behaviour near infinity, we use the method of conformal compactification. In principle, this means that we make a conformal rescaling to bring infinity to a finite point that is added to the manifold.

Let hab be the positive definite induced metric on V . We then call V

asymptotically flat if there exists a 3-manifold eV and a conformal factor Ω satisfying

(i) eV = V ∪ Λ, where Λ is a single point;

(ii) ˜hab= Ω2hab is a smooth metric on eV ;

(iii) at Λ, Ω = 0, eDaΩ = 0, eDaDebΩ = 2˜hab;

where eDa is the derivative operator associated with ˜hab. The point Λ is

called the infinity point.

As an example, we see that with the flat metric in R3 we can simply invert in the unit sphere. Λ will then correspond to the new origin. We also see that Ω = 1

x2+y2+z2 gives the correct behaviour for asymptotic flatness.

3 Definition of multipole moments

In this section we use the Newtonian definition of multipole moments to motivate the relativistic definition of Geroch and Hansen. We also present a list of different relativistic definitions and conclude with the Geroch–Hansen definition.

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3.1 Newtonian case

In this section we will describe multipole moments in Newtonian gravita-tion. We will also reformulate this description to fit the relativistic setting. This gives a motivation for the relativistic definition.

In the static Newtonian case the gravitational field Ga outside an

iso-lated object can be described by a potential V , such that Ga = −∇aV ,

0 = ∇aG

a = −∇a∇aV , and V vanishes at infinity. If we fix an origin

and use spherical coordinates we can expand the harmonic function V in powers of 1 R: V (R, θ, ϕ) = ∞ X k=0 k X l=−k ck,lYkl(θ, ϕ)R−k−1, (9) where Yl

k are the spherical harmonics. The coefficients ck,l can be used to

describe the multipole moments.

We can get a simpler expansion if we change the radial variable to r =R1 and rescale, e V (r, θ, ϕ) = V ( 1 r, θ, ϕ) r = ∞ X k=0 k X l=−k ck,lYkl(θ, ϕ)rk. (10)

Note that eV is harmonic with respect to the spherical coordinates (r, θ, ϕ). With x1= r sin θ cos ϕ, x2= r sin θ sin ϕ, x3= r cos θ, we see that rkYl

k(θ, ϕ)

is a homogeneous polynomial in x1, x2, x3of order k. Taylor’s formula gives

e V (xa) = ∞ X k=0 xa1. . . xak k! ∇a1. . . ∇akVe 0. (11)

This means that there is a direct relation between {ck,l}lk=−l and the

ten-sors ∇a1. . . ∇akVe

0. Due to flatness and the smoothness of the field, the

derivative operators commute. This means that Pa1...ak= ∇a1. . . ∇akV ise

totally symmetric. We also have that

gak−1akP

a1...ak= ∇a1. . . ∇ak−2∇ b

bV = 0,e (12)

i.e., Pa1...ak is totally symmetric and trace-free. Note that Pa1...ak is

ob-tained recursively via Pa1...ak = ∇a1Pa2...ak.

Now the multipole moments are the totally symmetric and trace-free tensors Pa1...akat r = 0. Given eV , this definition is coordinate independent,

and is therefore easier to generalize than the ck,l-description. We also see

that this picture fits well with the concept of conformal compactifications with Ω = 1/r2.

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3.2 Different relativistic definitions

Often multipole moments are defined through series expansions in special coordinate systems. A drawback with this kind of definition is that extra care has to be taken to obtain coordinate invariance. A paper by van der Burg [5] contains an early version of multipole moments for stationary solutions using series expansions. Thorne [37] also used the same kind of ideas, and defined multipole moments as expansions in asymptotically Cartesian and mass-centred coordinate systems. Another definition of this kind was made by Beig and Simon [35], and in the same paper they proved it to be equivalent to the Geroch–Hansen moments, defined below. All of these definitions use the physical spacetime. The conformally rescaled metric components on the other hand, were used by Kundu [24] when he treated multipole moments in terms of power series expansions in the conformally rescaled normal coordinates.

3.3 The Geroch–Hansen definition

Geroch [14] defined the multipole moments for a static spacetime, and later Hansen [17] generalized the definition to the stationary case. In contrast to the Newtonian case, there is much freedom in the way we can choose the gravitational potential. However, following Hansen, we use the scalar potential φ on V φ = φM + iφJ, φM = λ2+ ω2− 1 4λ , φJ= ω 2λ, (13) where φM is a potential for the mass part of the field, whereas φJ describes

the angular momentum part. The only essential difference compared to Geroch’s definition is this choice of potential, which also incorporates the angular momentum parts for stationary spacetimes. These different poten-tials do not coincide in the static case, although they produce the same multipole moments. The multipole moments are defined on eV as certain derivatives of the scalar potential ˜φ = φ/√Ω at Λ, similar to the Newtonian case. More explicitly let eRabdenote the Ricci tensor of eV , and let P = ˜φ.

Define the sequence P, Pa1, Pa1a2, . . . of tensors recursively:

Pa1...an= C  e Da1Pa2...an− (n − 1)(2n − 3) 2 Rea1a2Pa3...an  , (14)

where C[·] stands for taking the totally symmetric and trace-free part. The multipole moments are then defined as the tensors {Pa1...an(Λ)}

n=0. Here

we see that the curvature comes into play. The coefficient in the second term of (14) is chosen this way to ensure the same behaviour under translations as the Newtonian counterpart and independence of the higher order terms of the conformal factor. We also see that we have to symmetrize ‘by hand’ and subtract the traces. This scheme is computationally very cumbersome, so in paper 2 we simplify this computation for the axisymmetric case.

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4 Earlier results

Over the years a number of important relevant results have been obtained. Here we briefly describe a few of them. For a review of the earlier results see also [32].

4.1 Centre of mass

In Newtonian theory it is easy to see that the choice of origin affects the multipole expansion. The corresponding choice in the geometric relativistic definition is the freedom in the choice of conformal factor. For instance, in the Newtonian case we can use Ω = 1

R2, which is dependent of the choice

of origin point. Also in the Newtonian case we see that if we have a mass, and place the origin in the centre of the mass, we will not get any dipole moment.

In the relativistic case, the dipole moment can also be equated to zero, with a careful choice of conformal factor Ω if the mass is non-zero. This can be generalized even if there is no mass, as was demonstrated by Beig [1]. If Pa1...an is the first non-vanishing moment, this is done by choosing the

conformal factor such that Pa1...anP

a1...an+1 = 0. In the axisymmetric case

this implies that Pa1...an+1 = 0.

The advantage of this approach is that it is a geometric way to define a generalized centre of mass without using coordinates. Thus, this is a good way to define it in general relativity. This choice of conformal factor together with the geometric definition of multipole moments of Geroch [14] and Hansen [17] gives a coordinate independent definition of mass centred multipole moments.

4.2 Different potentials

In the definitions of Geroch and Hansen, specific choices of the potential are used. It is natural to ask if other choices of the potential can be made, and if the resulting multipole moments will be the same. It turns out that many different potentials produce the same multipole moments. This is discussed at the end of the paper [35] by Simon and Beig. We also treat this problem in paper 2 [7].

4.3 Uniqueness of the moments

It is important that the multipole moments are unique, so regardless of how they are computed they are the same. Due to the geometric nature of the Geroch–Hansen definition and the independence of the conformal factor, this is achieved.

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4.4 Analyticity

M¨uller zum Hagen [27, 28] proved that the metric functions of the phys-ical metric are analytic when harmonic coordinates are used. This im-plies that the metric components can be developed in a convergent power series around each point. He also proved that, when using harmonic co-ordinates, the transformation between overlapping coordinate patches is analytic. These facts might be useful when one wants to extend solutions from a neighbourhood of a point to a larger region. However, some care needs to be taken as regards the global structure.

Beig and Simon [2] proved analyticity for the static case when the mass is non-zero and some differentiability conditions are satisfied. They later proved the same thing for stationary solutions [3]. This was also independently proved by Kundu [25]. Kennefick and ´O Murchadha [23] weakened the fall-off conditions, but still got the analyticity results of Beig and Simon.

Reula [34] also gave a proof of analyticity at infinity, but instead of a multipole problem, he studied the problem of prescribing data on a sphere and extending this data analytically to infinity.

4.5 Injectivity

Xanthopoulos [39] proved that a static spacetime is flat if and only if all multipole moments vanish. Beig and Simon [2] proved for the static case that two solutions with the same multipole moments are identical at least in some neighbourhood of infinity. Kundu [25] also proved that the multipole moments uniquely determine the local structure of a stationary, asymptot-ically flat, vacuum metric.

In this context it is also worth noting that Xanthopoulos [39] proved that a stationary spacetime is static if and only if all its angular moments vanish. G¨ursel [16] also proved that a stationary spacetime is axisymmetric if and only if all its multipole moments are axisymmetric.

4.6 Construction of solutions

Beig and Simon [2] describe in principle a method for generating static solutions with prescribed multipole moments. However, this method is computationally complicated and does not give any general recursion for-mulae. They also leave open the question as to whether the multipole moments can be arbitrarily specified and the solution still satisfies the field equations. The suitable fall-off conditions on the multipole moments for convergence of the method was also left as an unsolved issue.

In [33] Quevedo describes a method of transforming one solution of the Einstein-Maxwell equation to another, with a-priori specified change in the gravitational as well as in the electromagnetic multipole moments. This is done explicitly. However, all solutions cannot be obtained from

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static solutions. An algorithm, for computing the multipole moments of a stationary asymptotically flat axisymmetric vacuum space-time, can be found in the paper [12] by Fodor, Hoenselaers and Perj´es. This method can also be used to reconstruct a spacetime knowing only the multipole moments. However, the convergence of this method is not clear.

For the stationary axisymmetric case, several techniques are available to generate explicit solutions. Unfortunately, the set of multipole moments are seldom known beforehand. These methods are based on solution gener-ating techniques for the Ernst equation. These techniques use known solu-tions as seeds, and then apply certain transformasolu-tions to generate new so-lutions. Harrison [18] and Neugebauer [30] found B¨acklund transformations that map an asymptotically flat solutions to another. HKX-transformations were introduced by Hoenselaers, Kinnersley and Xanthopoulos [21, 22]. It has also been shown that the Ernst equation is completely integrable [26], and that the inverse scattering method can be used to produce solutions to the Ernst equation [4]. For an overview of the technique, see [31]. All these methods can be used repeatedly to generate large classes of solutions.

Many interesting solutions can be generated by any of these methods. Therefore it is natural to investigate how these methods are related. This was done by Cosgrove [10]. One can also find a review of these solution generating techniques in [32].

In this thesis we do not use any of these explicit solution generating techniques because it is difficult to generate solutions with a-priori specified multipole moments using these techniques.

4.7 General existence proofs

In a recent paper [13] Helmut Friedrich proves the existence of static space-times with a-priori specified sets of data. He gives a condition for conver-gence of the formal series solutions. However, although the data he uses are related to the multipole moments, yet they are different enough to make the convergence proof difficult to translate into the multipole moments setting.

4.8 Simplification of multipole computation

In the paper [19] Herberthson describes a way of simplifying the computa-tions of the multipole moments for static axisymmetric spacetimes. This is done in several steps. First the metric is cast into Weyl coordinates in order to have a fixed starting point. The 3-manifold orthogonal to the timelike Killing vector field is then studied. A specific conformal transformation, such that the rescaled metric component ˜α satisfies Laplace’s equation in Euclidean 3-space, is then used. The conformally rescaled metric for the 3-manifold then becomes

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The new idea introduced in this paper, is to contract Pa1a2...an with

the special complex vector (∂r)a+ri(∂θ)a to get a scalar. It then turns out

that the recursion of tensor fields due to Geroch reduces to a recursion of scalar fields. The next step is to get rid of the angular dependence. Due to the fact that ˜α is analytic and satisfies the axisymmetric Laplace equation in Euclidean 3-space one has

˜ α(r, θ) = ∞ X n=0 anrnPn(cos θ), (16)

where Pn(z) are the Legendre polynomials. Furthermore, the polynomial

Pn(z) is completely determined by its leading order term (2n)! 2nn!2z

n. Thus it

seems reasonable to believe that the computation can be done with just the leading order terms. This turns out to be true, and after some computations this reduces the recursion of scalar fields to a recursion of scalar functions of one variable. Thus instead of ˜α(r, θ) one uses

Y (r) = ∞ X n=0 anrn (2n)! 2nn!2. (17)

The final step is then to use the remaining freedom in the conformal factor to simplify the recursion in such a way that it can be completely solved, and the multipole moments can be read off as derivatives at 0 of a scalar function y(ρ) in one variable.

5 Summary of papers

In this section we summarize the papers presented in this thesis.

5.1 Summary of paper 1

In paper 1 [6], we study the problem of finding the static axisymmetric spacetime corresponding to an a-priori specified set of multipole moments. To achieve this goal we need to simplify the computation of the multipole moments. This can be done by Herberthson’s method as described above. Now, one can turn this argument around. Due to the axisymmetry condition, for every order, the multipole moment has only one degree of freedom. Therefore, it is completely specified by a scalar mn. From the set

of scalars {mn}∞n=0one forms the function

y(ρ) = ∞ X n=0 mn n! ρ n. (18)

The relations in [19] then give the relation Y (r) = q

ρ(r)

r y(ρ(r)), where

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would be able to get Y (r), and then reconstruct ˜α. One such relation is ρ = reκ(r). Unfortunately, one does not at this stage know κ(r). However,

lemma 2 gives us an implicit relation between ρ and r.

0 = −ρ r + ρ Z ρ 0 1 σ2 Z σ 0 2ρ (y + 2ρyρ)2dρ dσ + 1 + κ0(0)ρ. (19)

One can then use this to get Y (r) and then ˜α(r, θ).

Unfortunately, from the relation (19) it can be difficult to find explicit formulae for ρ(r). Yet, it turns out that if the number of moments are finite, the relation (19) becomes an algebraic polynomial equation. It is well known that such equations in general cannot be solved in terms of radicals. Nevertheless, Sturmfels [36] expresses all solutions of such equations in terms of A-hypergeometric series. This give us explicit formulae for ρ(r), but to get Y (r) we need to calculate the square root and other half-integer powers of these series. This is in general awkward, but these can also be described in terms of A-hypergeometric series. Hence, explicit formulae in terms of power series can be found for all solutions with finite number of multipole moments.

To illustrate the ideas the paper concludes with some examples. The metric component ˜α for the pure 2n-pole as well as the monopole-2n-pole is explicitly calculated. The latter is then specialized to the monopole-quadrupole and rewritten to the form conjectured by Hern´andez-Pastora and Mart´ın [20].

5.2 Summary of paper 2

In paper 2 [7], we consider the problem of computing the multipole moments of a stationary axisymmetric spacetime. The method is a generalization of the method in [19], but developed in a more elegant way. The main difference is that the potential and the multipole moments are now complex. We do not demand that the metric is given in Weyl–Papapetru coordinates, only that the conformally compactified 3-manifold of Killing trajectories has the special form (15). Thus, one does not always have to rewrite the metric in Weyl–Papapetru coordinates first. Instead of contracting Pa1...an

with the special vector (∂r)a+ri(∂θ)a, we can contract with any complex

vector ηa with the following properties:

(a) for all tensors Ta1...an, η

a1. . . ηanT a1...an= η a1. . . ηanC[T a1...an], (b) at Λ, Pa1...an is determined by η a1. . . ηanP a1...an, (c) ηaDeaηb is parallel to ηb.

Note that (a) and (c) are satisfied if ηa is the tangent vector of a complex

null-geodesic. This property is exploited more in paper 3 [8]. After this contraction the recursion collapses to a recursion of scalar functions of

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two variables. Due to analyticity, one can express the functions as power series in the two variables. Following the method above, one would like to extract something resembling the leading order terms of the Legendre polynomials above. It turns out that one can define the leading order part by complexifying the variables in the power series, and then study the function along a special complex curve. This allows us to reduce the recursion to a recursion of scalar functions of one variable. The resulting recursion can then be solved in a similar way to the static case, using the remaining conformal freedom.

To exemplify the method, the multipole moments for the Kerr metric are computed. The paper is concluded by a study of different potentials that will give the same multipole moments.

5.3 Summary of paper 3

In paper 3 [8], we study general stationary spacetimes, without extra sym-metries. The main result is a necessary fall-off condition on the multipole moments for existence of a solution. To achieve this we use the ideas from the previous papers to simplify the calculation of the multipole moments. In this general case, one cannot choose coordinates such that the rescaled 3-metric satisfies (15). Therefore, we choose normal coordinates with re-spect to the rescaled 3-metric. In the axisymmetric case, we were able to reconstruct Pa1...an at Λ from the contraction η

a1. . . ηanP

a1...an where η a

was a single vector. In the general case, this cannot be done because the degrees of freedom is 2n + 1 for Pa1...an at Λ compared to 1 in the

axi-symmetric case. However, if one lets ηa be a vector field on a 2-surface S

spanned by complex null geodesics, this can in fact be done in the general case. It also turns out that corresponding to taking the leading order part in the axisymmetric case is to restrict the fields to S. Therefore one can obtain a recursion on S instead of a recursion on the rescaled 3-manifold. As before, this recursion can be solved by using the remaining freedom in the conformal factor. This is more technical than the axisymmetric case but the same general ideas apply.

The paper concludes with a proof that the multipole moments can be collected in a function g in a neighbourhood of Λ, where g is flat harmonic. Hence, the multipole moments must satisfy the same fall-off conditions as an harmonic function, generating the multipole moments. This condition is proved to be necessary and we conjecture it to be sufficient.

5.4 Summary of paper 4

In paper 4 [9], we study the existence problem of stationary axisymmetric spacetimes with arbitrarily specified multipole moments. This is a gener-alization of what was done in paper 1 [6]. However, this problem is much harder due to the fact that the relevant field equation is the non-linear Ernst-equation instead of the linear Laplace equation. The leading order

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terms of the potential can be constructed in a way very similar to the one in paper 1 [6]. A recursive method to find the values of the remaining coefficients is then presented. To then prove that this give a convergent series, the coefficients are majorized by coefficients defined though another recursion. This new recursion corresponds to a similar but somewhat nicer differential equation. This differential equation is then transformed to a lin-ear one by a non-linlin-ear transformation. A solution to this linlin-ear equation then produces a solution that majorizes the formal series solution, thus, proving that the formal series solution converges in a neighbourhood of Λ. The majorizing solution also gives estimates on the terms in the formal power series solution as well as estimates of the radius of convergence.

6 Conclusion

In this thesis, we have studied multipole moments for stationary spacetimes. The definition of Geroch and Hansen was used as a starting point. A scheme was developed for simplification of the computation procedure for the multipole moments. This scheme is based on the methods for the static axisymmetric case presented in paper [19]. This method was generalized to the stationary axisymmetric case in paper [7] and the general stationary case in paper [8]. A clearer, more geometrical picture of the method has now been achieved.

This computational scheme was used when we considered the inverse problem, i.e., a-priori specifying the multipole moments, and using only these to compute the metric components. In the axisymmetric case, this problem was proved to have a solution if the multipole moments satisfy a simple convergence condition [6, 9]. We also presented methods to com-pute these solutions. In the static axisymmetric case with a finite number of non-zero multipole moments, the solutions were expressed explicitly in terms of power series. For the general stationary case, we proved that the appropriate convergence condition on the multipole moments is necessary. The task to prove that the convergence condition is sufficient is still to be completed. Herberthson is currently working on this for the static case. The convergence proof for the asymptotic expansions of Friedrich [13] also strengthens the belief that the convergence conditions are sufficient.

There are many reasons why these results are interesting. From a more theoretical point of view one obtains a complete characterization of the stationary spacetimes in terms of the multipole moments, hence a simple description for this class of solutions. There are also many possible applica-tions in physics. When constructing soluapplica-tions it is important to know the physical interpretation beforehand. Unfortunately, with most solution gen-erating techniques this is not the case. The multipole moments, however, can give such an interpretation. Hence, our solution generating techniques might be important for making relativistic models for astrophysical objects like galaxies, stars or planets.

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References

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[18] Harrison, B. K., B¨acklund Transformation for the Ernst Equation of General Relativity, Phys. Rev. Lett. 41 (1978), 1197–1200.

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[22] Hoenselaers, C., Kinnersley, W. and Xanthopoulos, B. C., Generation of Asymptotically Flat, Stationary Space-Times with Any Number of Parameters, Phys. Rev. Lett. 42 (1979), 481–482.

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