The Hawking Mass for Ellipsoidal 2-Surfaces in Minkowski and Schwarzschild Spacetimes Daniel Hansevi

Examensarbete

The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild spacetimes Daniel Hansevi

LiTH - MAT - EX - - 08/14 - - SE

The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild spacetimes

Applied Mathematics, Link¨opings Universitet

Daniel Hansevi

LiTH - MAT - EX - - 08/14 - - SE

Examensarbete: 30 hp

Level: D

Supervisor: G¨oranBergqvist, Applied Mathematics, Link¨opings Universitet

Examiner: G¨oranBergqvist, Applied Mathematics, Link¨opings Universitet

Link¨oping: June 2008

Datum Avdelning, Institution Date Division, Department

Matematiska Institutionen June 2008 581 83 LINKOPING¨ SWEDEN

Spr˚ak Rapporttyp ISBN Language Report category ISRN Svenska/Swedish Licentiatavhandling LiTH - MAT - EX - - 08/14 - - SE x Engelska/English x Examensarbete C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering 0348-2960 Ovrig¨ rapport

URL f¨orelektronisk version

Titel The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild space- Title times

F¨orfattare Daniel Hansevi Author

Sammanfattning Abstract In general relativity, the nature of mass is non-local. However, an appropriate definition of mass at a quasi-local level could give a more detailed characterization of the gravitational field around massive bodies. Several attempts have been made to find such a definition. One of the candidates is the Hawking mass. This thesis presents a method for calculating the spin coefficients used in the expression for the Hawking mass, and gives a closed-form expression for the Hawking mass of ellipsoidal 2-surfaces in Minkowski spacetime. Furthermore, the Hawking mass is shown to have the correct limits, both in Minkowski and Schwarzschild, along particular foliations of leaves approaching a metric 2-sphere. Numerical results for Schwarzschild are also presented.

Nyckelord Keyword Hawking mass, Quasi-local mass, General relativity, Ellipsoidal surface. vi Abstract

In general relativity, the nature of mass is non-local. However, an appropriate definition of mass at a quasi-local level could give a more detailed characterization of the gravitational field around massive bodies. Several attempts have been made to find such a definition. One of the candidates is the Hawking mass. This thesis presents a method for calculating the spin coefficients used in the expression for the Hawking mass, and gives a closed-form expression for the Hawking mass of ellipsoidal 2-surfaces in Minkowski spacetime. Further- more, the Hawking mass is shown to have the correct limits, both in Minkowski and Schwarzschild, along particular foliations of leaves approaching a metric 2-sphere. Numerical results for Schwarzschild are also presented. Keywords: Hawking mass, Quasi-local mass, General relativity, Ellipsoidal surface.

Hansevi, 2008. vii viii Contents

1 Introduction 1 1.1 Background ...... 1 1.2 Purpose ...... 1 1.3 Chapter outline ...... 2

2 Mathematical preliminaries 3 2.1 Manifolds ...... 3 2.2 Foliations ...... 4 2.3 Tangent vectors ...... 5 2.4 1-forms ...... 6 2.5 Tensors ...... 7 2.5.1 Abstract notation ...... 7 2.5.2 Component notation ...... 7 2.5.3 Tensor algebra ...... 8 2.5.4 Tensor ﬁelds ...... 8 2.5.5 Abstract index notation ...... 8 2.6 Metric ...... 9 2.7 Curvature ...... 10 2.7.1 Covariant derivative ...... 11 2.7.2 Metric connection ...... 11 2.7.3 Parallel transportation ...... 12 2.7.4 Curvature ...... 12 2.7.5 Geodesics ...... 12 2.8 Tetrad formalism ...... 14 2.9 Newman-Penrose formalism ...... 14 2.10 Spin coeﬃcients ...... 14

3 General relativity 15 3.1 Solutions to the Einstein ﬁeld equation ...... 15 3.1.1 Minkowski spacetime ...... 15 3.1.2 Schwarzschild spacetime ...... 16

4 Mass in general relativity 17 4.1 Gravitational energy/mass ...... 17 4.1.1 Non-locality of mass ...... 17 4.2 Total mass of an isolated system ...... 18 4.2.1 Asymptotically ﬂat spacetimes ...... 18 4.2.2 ADM and Bondi-Sachs mass ...... 18

Hansevi, 2008. ix x Contents

4.3 Quasi-local mass ...... 18

5 The Hawking mass 19 5.1 Deﬁnition ...... 19 5.2 Interpretation ...... 19 5.3 Method for calculating spin coeﬃcients ...... 20 5.4 Hawking mass for a 2-sphere ...... 21 5.4.1 In Minkowski spacetime ...... 21 5.4.2 In Schwarzschild spacetime ...... 21

6 Results 23 6.1 Hawking mass in Minkowski spacetime ...... 24 6.1.1 Closed-form expression of the Hawking mass ...... 25 6.1.2 Limit when approaching a metric sphere ...... 27 6.1.3 Limit along a foliation ...... 28 6.2 Hawking mass in Schwarzschild spacetime ...... 29 6.2.1 Limit along a foliation ...... 30 6.2.2 Numerical evaluations ...... 31

7 Discussion 33 7.1 Conclusions ...... 33 7.2 Future work ...... 33

A Maple Worksheets 37 A.1 Null tetrad ...... 37 A.2 Minkowski ...... 38 A.3 Schwarszchild ...... 39 List of Figures

2.1 A manifold and two overlapping coordinate patches...... 4 2.2 A foliation ofM a manifold ...... 4 2.3 Intuitive picture of a tangentM space...... 5 2.4 Picture of a 1-form...... 6 2.5 Null cone...... 10 2.6 Parallel transportation in a plane and on the surface of a sphere. 10 a 2.7 Deviation vector y between two nearby geodesics λs and λs0 . . . 13 2.8 Spin coeﬃcients...... 14

6.1 Oblate spheroid...... 25 6.2 mH( 1) plotted against parameter ξ...... 27 S p 6.3 A curve given by r = 1 + ε sin2 θ...... 29 6.4 m ( ˜ ) plotted against 4 r 620 for some values of ε...... 32 H Sr ≤ ≤ 6.5 mH( ˜r) plotted against 2.3 r 16 for some values of ε. . . . . 32 6.6 m (S˜ ) plotted against 2.3 ≤ r ≤ 16 for ε = ω/r...... 32 H Sr ≤ ≤

Hansevi, 2008. xi xii List of Figures Chapter 1

Introduction

1.1 Background

In general relativity, the nature of the gravitational field is non-local, and therefore the gravitational field energy/mass cannot be given as a pointwise density. However, there might be possible to find a satisfying definition of mass at a quasi-local level, that is, for the mass within a compact spacelike 2-surface. Several attempts have been made, but the task has proven difficult, and there is still no generally accepted definition.

One of the candidates for a description of quasi-local mass originates from a paper about gravitational radiation written by Stephen Hawking [5] in 1968. The Hawking mass can be viewed as a measure of the bending of outgoing and ingoing light rays orthogonal to the surface of a spacelike 2-sphere, and it has been shown to have various desirable properties [13].

To reach an appropriate deﬁnition for quasi-local mass would certainly be of great value. It could give a more detailed characterization of the gravitational ﬁeld around massive bodies, and it should be helpful for controlling errors in numerical calculations [13].

1.2 Purpose

Even though Hawking’s expression was given for the mass contained in a spacelike 2-sphere, it can be calculated for a general spacelike 2-surface. In this thesis we will calculate the Hawking mass for spacelike ellipsoidal 2-surfaces, both in ﬂat Minkowski spacetime and curved Schwarzschild spacetime.

Hansevi, 2008. 1 2 Chapter 1. Introduction

1.3 Chapter outline

Chapter 2 This chapter provides a short introduction of the mathematics needed. We introduce the notion of a manifold that is used for the model of curved spacetime in general relativity. Structure is imposed on the manifold in the form of a covariant derivative operator and a metric tensor. The concept of geodesics as curves that are ‘as straight as possible’ is introduced along with the definition of the Riemann curvature tensor which is a measure of curvature. We end the chapter with a look at the Newman-Penrose formalism and the spin coefficients which are used in the definition of the Hawking mass. Chapter 3 A very brief presentation of general relativity is given followed by the two solutions (of the field equation) of particular interest in this thesis – Minkowski spacetime and Schwarzschild spacetime. Chapter 4 Discussion of mass in general relativity, and why it cannot be lo- calized. Chapter 5 A closer look at the Hawking mass – definition and interpretation. A method for calculating the spin coefficients used in the expression for the Hawking mass is presented. Chapter 6 In this chapter, we calculate the Hawking mass for ellipsoidal 2- surfaces in both Minkowski spacetime and Schwarzschild spacetime. The main result is the closed-form expression for the Hawk- ing mass of an 2-ellipsoid in Minkowski spacetime. Furthermore, some limits of the Hawking mass are proved. Chapter 7 Conclusions and future work. Appendix Maple worksheets used for some of the calculations performed in chapter 7. Chapter 2

Mathematical preliminaries

This chapter provides a short introduction of the mathematics needed for a mathematical formulation of the general theory of relativity.

2.1 Manifolds

In general relativity, spacetime is curved in the presence of mass, and gravity is a manifestation of curvature. Thus, the model of spacetime must be sufficiently general to allow curvature. An appropriate model is based on the notion of a manifold. A manifold is essentially a space that is locally similar to Euclidean space in that it can be covered by coordinate patches. Globally, however, it may have a different structure, for example, the two-dimensional surface of a sphere is a manifold. Since it is curved, compact and has finite area its global properties are different from those of the Euclidean plane, which is flat, non-compact and has infinite area. Locally, however, they share the property of being able to be covered by coordinate patches. As a mathematical structure a manifold stands on its own, but since it can be covered by coordinate patches, it can be thought of as being constructed by ‘gluing together’ a number of such patches. A Ck n-dimensional manifold is a set together with a maximal Ck M M atlas α φα , that is, the collection of all charts ( α, φα) where φα are bijective {U } U n maps from subsets α to open subsets φα( α) R such that: U ⊂ M U ⊂ (i) cover , that is, each element in lies in at least one ; {Uα} M M Uα (ii) if is non-empty, then the transition map (see figure 2.1, page 4) Uα ∩ Uβ 1 φ φ− : φ ( ) φ ( ), (2.1) β ◦ α α Uα ∩ Uβ → β Uα ∩ Uβ is a Ck map of an open subset of Rn to an open subset of Rn. Each ( , φ ) is a local coordinate patch with coordinates xα (α = 1, . . . , n) Uα α defined by φα. In the overlap α β of two coordinate patches ( α, φα) and α U ∩k U Uβ ( β, φβ), the coordinates x are C functions of the coordinates x , and vice versa.U is said to be Hausdorff 1 if for every distinct p, q (p = q) there exist twoM subsets and such that p , q ∈ M and6 = . Uα ⊂ M Uβ ⊂ M ∈ Uα ∈ Uβ Uα ∩ Uβ ∅ 1Felix Hausdorff (1868-1942), German mathematician

Hansevi, 2008. 3 4 Chapter 2. Mathematical preliminaries

φα xα M α U 1 φβ φ− p ◦ α

β β x U φβ Rn

Figure 2.1: A manifold and two overlapping coordinate patches. M

Furthermore, it is natural to introduce the notion of a function on and the notion of a curve in as follows. M M A (real-valued) function f on a Ck manifold is a map f: R. It is r M 1 nM → said to be of class C (r k) at p if the map f φα− : R R in any coordinate patch ( , φ )≤ holding p is∈ a MCr function of the◦ coordinates→ at p. Uα α A Ck curve in a manifold is a map λ of an open interval I R M n∈ →k M such that for any coordinate patch ( α, φα), the map φα λ: I R is C . U ◦ → Something that is C∞ is usually called smooth. Accordingly, we call a C∞ manifold a smooth manifold, a C∞ function a smooth function and a C∞ curve a smooth curve.

2.2 Foliations

A foliation of a manifold is a decomposition of the manifold into submanifolds. These submanifolds are required to be of the same dimension, and ﬁt together in a ‘nice’ way. More precisely [7], a foliation of codimension m of an n-dimensional manifold is a decomposition of into a union of disjoint connected subsets La a A, Mcalled the leaves of the foliation,M with the property that for every point{ p} ∈ ∈ M there is a coordinate patch ( , φ ) holding p such that for each leaf L , Uα α a 1 m m+1 n n φα:( α La) (x , . . . , x , x , . . . , x ) R , (2.2) U ∩ → ∈ where xm+1, . . . , xn are constants. See ﬁgure 2.2.

n m R − M φα Uα p Rm

Figure 2.2: A foliation of a manifold . M 2.3. Tangent vectors 5

2.3 Tangent vectors

A manifold can be curved and therefore has no global vector space structure. There is no natural way to ‘add’ two points on a sphere and end up with a third point also on the sphere. However, a local vector space structure can be attained. A definition of a vector that only refers to the intrinsic structure of the manifold would be of great value, because such a vector would be independent of an embedding of the manifold in a space of higher dimension. There is a one- to-one correspondence between vectors and directional derivatives in Euclidean space, and since a manifold is locally similar to Euclidean space, a natural definition is provided by the notion of a vector as a differential operator. Let be an n-dimensional manifold and let F be the collection of all smooth,M real-valued functions on .A tangent vector, or vector for short, X at M a point p , is a map X: F R such that for all f, g F and all α, β R: ∈ M → ∈ ∈ (i) X(αf + βg) = αX(f) + βX(g) (linear); (ii) X(fg) = f(p)X(g) + g(p)X(f) (Leibniz’ rule). Let ( , φ) be a coordinate patch, with coordinates xµ, holding p. For µ = U 1, . . . , n define the map Xµ: F R by →

∂ 1 Xµ(f) := µ (f φ− ) . (2.3) ∂x ◦ φ(p)

It is shown in [14], that X1,..., Xn are linearly independent tangent vectors that span an n-dimensional vector space at p. We call this vector space the tangent space at p and denote it by T ( ), or just T if the manifold is given p M p by the context. The basis X1,..., Xn is called a coordinate basis and is usually denoted by ∂/∂x1, .{ . . , ∂/∂xn .} Thus, an arbitrary tangent vector X can be expressed as { }

n n X X ∂ X = XµX =: Xµ , (2.4) µ ∂xµ µ=1 µ=1 where (X1,...,Xn) Rn are the components of X with respect to the coordinate basis. ∈ A tangent space at a point p in a manifold may be intuitively understood as the limiting space when smaller and smaller neighbourhoods of p are viewed at greater and greater magniﬁcation, see ﬁgure 2.3.

X p

Tp( ) M M

Figure 2.3: Intuitive picture of a tangent space. 6 Chapter 2. Mathematical preliminaries

2.4 1-forms

Let be an n-dimensional manifold. Given a point p , let T be the M ∈ M p tangent space at p. Let Tp∗ be the space of all linear maps

ω: Tp R. (2.5) →

Tp∗ is called the dual space of Tp, or the cotangent space at p, and is a vector space of dimension n. Elements of Tp∗ are called dual vectors or covectors. They are also called 1-forms. The number which ω maps a vector X Tp into, is often written as ω, X . ∈ h i If e1,..., en is a basis in Tp, then there exists an associated dual basis 1 { n } 1 n e ,..., e of T ∗ consisting of 1-forms e ,..., e deﬁned by the property { } p 1, µ = ν eµ, e = δµ := . (2.6) h ν i ν 0, µ = ν 6

To see this, for every ω T ∗, we deﬁne ω := ω, e for µ = 1, . . . , n. Let X ∈ p µ h µi be an arbitrary vector in Tp. Then X X X ω, X = ω, Xµe = Xµ ω, e = ω Xµ h i h µi h µi µ µ α µ X (2.6) X = ω Xν δµ = ω Xν eµ, e µ ν µ h ν i µ,ν µ,ν X X X = ω eµ, Xν e = ω eµ, X . (2.7) h µ ν i h µ i µ ν µ Since X was arbitrary, it follows that

X µ ω = ωµe . (2.8) µ

Thus every 1-form can be written as a linear combination of e1,..., en. 1 n Given a coordinate basis ∂/∂x , . . . , ∂/∂x for Tp, the associated dual 1 { n } basis for T ∗ is the basis dx ,..., dx of the so called coordinate differentials. p { } For a given 1-form ω, there is a subspace of Tp defined by all vectors X for which ω, X is constant. Therefore, a 1-form can be pictured as planes, where ω, X his thei number of planes that X is ‘piercing’, see figure 2.4. h i

X ω, X = 3.5 h i

p Y ω, Y = 0 h i

Figure 2.4: Picture of a 1-form. 2.5. Tensors 7

2.5 Tensors

General relativity is formulated in the language of tensors. Tensors summarize sets of equations succinctly and reveal structure. There are two distinct ways of introducing tensors: the abstract approach and the component approach.

2.5.1 Abstract notation Let be an n-dimensional manifold and let p be a point in . The multilinear mapM M S: Tp∗ Tp∗ Tp Tp R (2.9) | × ·{z · · × } × | × ·{z · · × } → r factors s factors is called a tensor, of type or valence (r, s), at p, or just an (r, s)-tensor at p for short. We can see that a 1-form is a tensor of type (0, 1). Since Tp is a ﬁnite dimensional vector space, it is (algebraicly) reﬂexive, and therefore the second (algebraic) dual space Tp∗∗ is isomorphic to Tp. Thus, we can identify every element in Tp∗∗ with a unique element in Tp and we consider a tangent vector as a tensor of type (1, 0).

2.5.2 Component notation

Let ( , φ) and ( 0, φ0) be two overlapping coordinate patches holding a point p U, with coordinatesU related by ∈ M xµ0 = xµ0 (x1, . . . , xn). (2.10)

µ1...µr µ0 ...µ0 An object with components S ν ...ν in ( , φ) and S 1 r in ( 0, φ0) 1 s U ν10 ...νs0 U is called an (r, s)-tensor at p under the transformation xµ xµ0 , if 7→ n X ∂xµ10 ∂xµr0 ∂xν1 ∂xνs Sµ10 ...µr0 = Sµ1...µr ...... (2.11) ν10 ...νs0 ν1...νs µ µ ν ∂x 1 ∂x r ∂xν10 ∂x s0 µ1,...,µr ,ν1,...νs=1 A (0, 1)-tensor (1-form) is often called a covariant vector, and a (1, 0)-tensor (tangent vector) is often called a contravariant vector. Since an (r, s)-tensor S depends linearly on its arguments, it is determined µ1 µr by its components S ··· ν1 νs with respect to a basis. Suppose that S is a (1, 2)-tensor and that eµ , ···e are dual bases. Deﬁne the basis component { } { ν } µ µ S νρ := S(e , eν , eρ) R. (2.12) ∈ P µ P ν P ρ Let ω = µ ωµe Tp∗ and X = ν X eν , Y = ρ Y eρ Tp. Then it follows that ∈ ∈

X µ X ν X ρ S(ω, X, Y) = S( ωµe , X eν , Y eρ) µ ν ρ X µ ν ρ = S(e , eν , eρ) ωµX Y µ,ν,ρ X µ ν ρ = S νρ ωµX Y . (2.13) µ,ν,ρ The components of S satisfy the tensor transformation law (2.11). 8 Chapter 2. Mathematical preliminaries

2.5.3 Tensor algebra Let be an n-dimensional manifold and let p be a point in . Assume that S M M and T are (r, s)-tensors at p and that S0 is an (r0, s0)-tensor at p. Furthermore, i assume that ω T ∗ and X T for i = 1, . . . , r + r0 and j = 1, . . . , s + s0. ∈ p j ∈ p Addition of tensors (of the same type at the same point) and multiplication of a tensor by a scalar α R, are deﬁned in the obvious way: ∈ 1 r (S + T)(ω ,..., ω , X1,..., Xs) = 1 r 1 r S(ω ,..., ω , X1,..., Xs) + T(ω ,..., ω , X1,..., Xs); (2.14)

1 r 1 r (αS)(ω ,..., ω , X1,..., Xs) = α S(ω ,..., ω , X1,..., Xs). (2.15)

With addition and scalar multiplication deﬁned as above, the space of all (r, s)-tensors at p , constitutes a vector space of dimension nr+s. ∈ M The outer product of S and S0, denoted by S S0, is the (r +r0, s+s0)-tensor deﬁned by, ⊗

(S T)(ω1,..., ωr+r0 , X ,..., X ) = ⊗ 1 s+s0 1 r r+1 r+r0 S(ω ,..., ω , X1,..., Xs) S0(ω ,..., ω , Xs+1,..., Xs+s0 ). (2.16)

The contraction with respect to the ith (1-form) and j th (tangent vector) slots is a map from an (r, s)-tensor to an (r 1, s 1)-tensor deﬁned by, − − i (CjS)(ω1,..., ωi 1, ωi+1,..., ωr; X1,..., Xj 1, Xj+1,..., Xs) = − − n X k S(..., e ,... ; ..., ek ,...), (2.17) |{z} |{z} k=1 ith slot jth slot

k where e and e are dual bases of T and T ∗, respectively. { k} { } p p 2.5.4 Tensor fields It is natural to define a Ck tensor field of type (r, s) on a manifold as an assignment of an (r, s)-tensor at each p such that the componentsM with k ∈ M respect to any coordinate basis are C functions. We call a C∞ tensor field a smooth tensor field. A vector field is a tensor field of type (1, 0), and a 1-form field is a tensor field of type (0, 1).

2.5.5 Abstract index notation Equations for tensor components with respect to a particular basis may only be valid in that basis. On the other hand, if we do not specify a basis, the equations we write will be true tensor equations, that is, basis-independent equations that will hold between tensors. It is convenient to introduce a notation called abstract index notation [10]. a1 ar In this notation, an (r, s)-tensor S is written as S ··· b1 bs , where the indices are abstract markers telling us what type of tensor it is.··· Assume that S is a (1, 2)-tensor and that T is a (3, 2)-tensor. In abstract index notation, we write 2.6. Metric 9

a def the outer product of S and T as S bc T gh, and the contraction of S with a respect to the ﬁrst slots as S ab. In order to distinguish between tensors written in the abstract index notation and tensors components, we write the indices of the former with lowercase latin µ letters and indices of the latter with lowercase greek letters, for example, S νρ a denotes a basis component of the (1, 2)-tensor S bc. Given a tensor equation written in the abstract index notation, the corresponding equation (with greek indices) holds for basis components in any basis if a summation over indices that occurs twice in a term, once as a subscript and once as a superscript, is performed.

2.6 Metric

A metric gab on a manifold is a non-singular symmetric tensor ﬁeld. Thus, for every tangent space T ofM : p M (i) g uavb = g uavb for every ua, vb T (symmetric); ab ba ∈ p (ii) g uavb = 0 for every vb T implies that ua = 0 (non-singular). ab ∈ p The metric has the structure of a (not necessarily positive deﬁnite) inner a b product on every tangent space of the manifold. If gabu v = 0, then the vectors ua and vb are said to be orthogonal. a b For any vector v , the metric can be viewed as a linear map gabv : Tp R, → that is, a 1-form. Since gab is non-singular, there is a one-to-one correspondence a between elements of Tp and Tp∗. Given a vector v , we can apply the metric and b get the corresponding 1-form gabv , usually denoted by va in order to make the correspondence with va notationally explicit. Thus, we can ‘raise’ and ‘lower’ indices on tensors by the use of the metric. Particularly, we can write the inner product of two vectors ua and va as

a b a gabu v = uav . (2.18) Assume that we have two coordinate patches overlapping a neighbourhood of a point p and that their coordinates at p are related by xµ0 = xµ0 (xµ). ∈ M Then the basis components of gab are related by (2.19), that is, X ∂xµ ∂xν gµ ν = gµν . (2.19) 0 0 ∂xµ0 ∂xν0 µ,ν

It is always possible to ﬁnd an orthonormal basis v a, . . . , v a such that { 1 n } a i vi vj = δ . (2.20) a ± j The number of basis vectors for which (2.20) equals 1 and the number of basis vectors for which (2.20) equals 1 are basis independent and called the signature of the metric. A metric of signature− (+ ) or ( + +) is called Lorentzian2. We will follow the Landau-Lifshitz− · · ·3 −‘timelike− convention’··· [8] and use a metric of signature (+ ) for spacetime. − − − 2Hendrik Antoon Lorentz (1853-1928), Dutch physicist. 3Lev Davidovich Landau (1908-1968) and Evgeny Mikhailovich Lifshitz (1915-1985), Rus- sian physicists. 10 Chapter 2. Mathematical preliminaries

With a Lorentzian metric on , all non-zero vectors in Tp can be divided into three classes. With our particularM choice of signature, a vector va is said to be a timelike if vav > 0, a null if vav = 0, (2.21) a spacelike if vav < 0.

Thus, a Lorentzian metric deﬁnes a certain structure on each Tp, called a null cone; the set of null vectors form what looks like a double cone if we suppress one spatial dimension, see ﬁgure 2.5.

timelike vector

future cone null vector

spacelike vector

past cone

Figure 2.5: Null cone.

2.7 Curvature

An intrinsic notion of curvature, that can be applied to any manifold without reference to a higher dimensional space in which it might be embedded, can be defined in terms of parallel transport. If one parallel-transports a vector around any closed path in the plane, the final vector always coincides with the initial vector. However, for a sphere, the final vector does not coincide with the initial vector when carried along the curve shown in figure 2.6. Based on this, we characterize the plane as flat and the sphere as curved. Once we know how to parallel transport a vector along a curve, we can use this idea to obtain an intrinsic notion of curvature of any manifold.

p u

q v

Figure 2.6: Parallel transportation in a plane and on the surface of a sphere. 2.7. Curvature 11

Since the tangent spaces at two distinct points are different vector spaces it is not meaningful to say that a vector in the first tangent space equals a vector in the latter. Thus, before we can define parallel transport, we must impose more structure on the manifold. Given a notion of a derivative operator, it is natural to define a vector to be parallel-transported if its derivative along the given curve is zero. The notion of curvature can be defined in terms of the failure of the final vector to coincide with the initial vector when parallel transported around an infinitesimal closed curve, which in turn corresponds to the lack of commutativety of derivatives.

2.7.1 Covariant derivative

A connection, or covariant derivative operator, a on a smooth manifold a ∇ a M assigns to every vector field x on , a differential operator x a that maps a M a b ∇ an arbitrary vector field y on into a vector field x ay such that for all vector fields xa, ya, za and functionsM f on : ∇ M (i) xa (yb + zb) = xa yb + xa zb (linear); ∇a ∇a ∇a (ii) xa (fyb) = (xa f)yb + f(xa yb) (Leibniz’ rule); ∇a ∇a ∇a (iii) xa f = x(f) (consistency with the notion of tangent vectors). ∇a a b a The vector field x ay is called the covariant derivative of y with respect to a ∇ b a a b x , and the (1, 1)-tensor ay mapping x to x ay the covariant derivative of yb. ∇ ∇ The definition of can be extended to apply to any tensor field on by ∇a M the additional requirement that a when acting on contracted products should satisfy Leibniz’ rule. ∇

2.7.2 Metric connection A connection is not uniquely defined by the above conditions. In [14], how- ∇a ever, it is shown that if is endowed with a metric gab, then there exists a unique connection withM the properties that for all smooth functions f on : ∇a M (i) g = 0 (compatible with metric); ∇a bc (ii) f f = 0 (torsion-free). ∇a∇b − ∇b∇a 4 This particular a is called the metric connection, or the Levi-Civita connection, on , and∇ has the properties that for all smooth vector fields ya on : M M yb = ∂ yb + Γb yc and y = ∂ y Γc y , (2.22) ∇a a ac ∇a b a b − ab c 5 c where ∂a is an ordinary derivative, and the Christoffel symbol Γ ab is given by 1 Γc = gcd∂ g + ∂ g ∂ g . (2.23) ab 2 a bd b ad − d ab Thus, in a given coordinate patch ( , φ) with coordinates xµ, the coordinate basis components of yb are given byU ∇a ρ ∂y 1 X ∂gνσ ∂gµσ ∂gµν + gρσ + yν . (2.24) ∂xµ 2 ∂xµ ∂xν − ∂xσ σ,ν

4Tullio Levi-Civita (1873-1941), Italian mathematician. 5Elwin Bruno Christoﬀel (1829-1900), German mathematician. 12 Chapter 2. Mathematical preliminaries

2.7.3 Parallel transportation

Given a derivative operator a we can deﬁne the notion of parallel transport. A smooth vector ﬁeld ya is said∇ to be parallelly transported around a curve with tangent vector xa if the equation

xa yb = 0 (2.25) ∇a is satisﬁed along the curve. a The metric connection has the property that the inner product f = yaz of any two smooth vector ﬁelds ya and za remains unchanged when parallelly transported along any curve with tangent vector xa, since

a a b c x (f) = x a(gbcy z ) a∇ b c a b c a c b = (x agbc)y z + (x ay )gbcz + (x az )gbcy . (2.26) |∇{z } | ∇{z } | ∇{z } =0 =0 =0

2.7.4 Curvature

6 d Let ωa be any smooth 1-form field on . The Riemann curvature tensor Rabc is the tensor field on defined by, M M ( ) ω =: R dω , (2.27) ∇a∇b − ∇b∇a c abc d that is directly related to the failure of a vector to return to its initial value when parallel transported around a small closed curve.7 For any smooth vector field ta on , the corresponding expression is M ( ) tc = R ctd. (2.28) ∇a∇b − ∇b∇a − abd The Ricci 8 tensor is defined by contraction with respect to the second and b fourth slot, Rac := Rabc , and the Ricci scalar curvature is defined by further a contraction, R := Ra . The Ricci tensor and scalar curvature are used to define the Einstein tensor, 1 G := R R g , (2.29) ab ab − 2 ab which is a fundamental tensor in general relativity.

2.7.5 Geodesics Let xa be a smooth vector field on . From the theory of ordinary differential equations, we know that given a pointM p , there exists a unique curve that passes through p and has the property that∈ M for each point on the curve, the tangent vector of the curve coincides with the corresponding vector of xa. Such a curve is called an integral curve. a a b Let x be a smooth vector field such that x ax = 0. Then the integral curves of xa are called geodesics. ∇

6Georg Friedrich Bernhard Riemann (1826-1866), German mathematician. 7Some authors reverse the sign of the left-hand side in the deﬁnition. 8Gregorio Ricci-Curbastro (1853-1925), Italian mathematician. 2.7. Curvature 13

Geodesics are curves that are ‘as straight as possible’, and it can be shown [14] that there is precisely one geodesic through a given point p in a given direction xa T . ∈ M ∈ p Let λs(t) be a one-parameter family of geodesics and consider the two- dimensional surface, with coordinates (t, s), spanned by λs(t). The vector ﬁeld xa = (∂/∂t)a is tangent to the family of geodesics, thus

xa xb = 0, (2.30) ∇a and the vector field ya = (∂/∂s)a represents the deviation vector, which is the displacement from the geodesic λs to an ‘infinitesimally’ nearby geodesic λs0 , see figure 2.7.

λs

λs0 xa xa

a Figure 2.7: Deviation vector y between two nearby geodesics λs and λs0 .

Let f be any smooth function on . Since M (xa yb ya xb) f = xa (yb f) ya (xb f) ∇a − ∇a ∇b ∇a ∇b − ∇a ∇b = x(y(f)) y(x(f)) − ∂2f ∂2f = = 0, (2.31) ∂t∂s − ∂s∂t it follows that xa yb = ya xb. (2.32) ∇a ∇a The relative acceleration za, in the direction of ya, of a nearby geodesic when moving along the direction of xa, is given by

(2.32) za = xc (xb ya) = xc (yb xa) ∇c ∇b ∇c ∇b = (xc yb) xa + yb(xc xa) ∇c ∇b ∇c∇b (2.28) = (yc xb) xa + yb(xc xa) R aybxcxd ∇c ∇b ∇b∇c − cbd = yc (xb xa) R aybxcxd ∇c ∇b − cbd (2.25) = R aybxcxd. (2.33) − cbd Thus, the geodesic deviation, that is, the acceleration of geodesics toward or away from each other, which is a characterization of the curvature of , is determined by the Riemann curvature tensor. is ﬂat if and only if R Md = 0. M abc 14 Chapter 2. Mathematical preliminaries

2.8 Tetrad formalism

The tetrad formalism (or frame formalism) is a useful technique for deriving useful and compact equations in many applications of general relativity. The idea is to use a so called tetrad basis of four linearly independent vector fields, project the relevant quantities onto the basis, and consider the equations satis- fied by them. a Let ei , i = 1, 2, 3, 4, be smooth vector fields that are linearly independent at each point in spacetime ( , gab). Then the Ricci rotation coefficients are defined by M c a γkij := ek ej ei . (2.34) ∇a c It is shown in [2] that

1 a b γijk = (λijk + λkij λjki), where λijk = (∂ ej ∂ ej )ei ek , (2.35) 2 − b a − a b which is an eﬃcient way of calculating the Ricci rotation coeﬃcients, since there is no need to calculate any covariant derivatives.

2.9 Newman-Penrose formalism

The tetrad formalism with the choice of a particular type of null basis, introduced by Ezra Newman and Roger Penrose [9] in 1962, is usually called the Newman-Penrose formalism. The basis la, na, ma, m¯ a consists of null vectors, where la and na are real, ma andm ¯{a are complex conjugates,} satisfying

a a a a la l = na n = ma m =m ¯ a m¯ = 0 (null); a a a a la m = la m¯ = na m = na m¯ = 0 (orthogonal); (2.36) l na = m m¯ a = 1 (normalized). a − a 2.10 Spin coeﬃcients

In the Newman-Penrose formalism, the Ricci rotation coefficients are called spin coefficients, and are given in figure 2.8.

κ = γ κ0 = ν = γ 2α = γ + γ 311 − 422 214 344 ρ = γ ρ0 = µ = γ 2β = γ + γ 314 − 423 213 343 σ = γ σ0 = λ = γ 2γ = γ + γ 313 − 424 212 342 τ = γ τ 0 = π = γ 2ε = γ + γ 312 − 421 211 341

Figure 2.8: Spin coeﬃcients.

The spin coeﬃcients ρ and ρ0 are of particular interest to us, since they are used in the deﬁnition of the Hawking mass. Chapter 3

General relativity

General relativity is the modern geometric theory of space, time and gravitation published by Albert Einstein [4] in 1916. In the theory, space and time are uni- fied into spacetime ( , gab), which is represented by a smooth four-dimensional Hausdorff manifold M endowed with a Lorentzian metric g and a metric con- M ab nection a. The presence of matter ‘warps’ spacetime according to the Einstein field equation,∇ 1 G := R R g = 8π T , (3.1) ab ab − 2 ab ab where Gab is the Einstein tensor that describes the curvature of and Tab is the stress-energy-momentum tensor describing the distributionM of matter.1 Free particles travel along timelike geodesics and light rays travel along null geodesics. Thus gravity is a manifestation of the curvature of spacetime. Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.2

3.1 Solutions to the Einstein ﬁeld equation

The Einstein ﬁeld equation might look simple when written with tensors. How- ever, it constitutes a system of coupled, non-linear partial diﬀerential equations.

3.1.1 Minkowski spacetime The spacetime of special relativity – Minkowski3 spacetime – is a solution to 4 the vacuum field equation Gab = 0. It is the flat spacetime ( , ηab) given by the constant metric M  1 0 0 0   0 1 0 0  (ηab) =  −  . (3.2)  0 0 1 0  0 0− 0 1 − 1Some authors use a different sign in the definition of the Ricci tensor resulting in a minus sign in front of the right-hand side. Furthermore, we use ‘geometrized units’ where the gravitational constant G, and the speed of light c, are set equal to one. 2Brief explanation of general relativity by the ‘student’ portrayed in chapter one of [8]. 3Hermann Minkowski (1864-1909), Russian-born German mathematician. 4 d Rabc = 0.

Hansevi, 2008. 15 16 Chapter 3. General relativity

In the coordinate system implied by ηab, all geodesics appear straight, that is, they take the form, xa(t) = ya + zat.

3.1.2 Schwarzschild spacetime One of the solutions of the vacuum ﬁeld equation was discovered by Karl Schwarzschild5 in 1916, just a couple of months after Einstein published his ﬁeld equation. The Schwarzschild solution is the unique solution that describes the curved spacetime exterior to a static spherically symmetric mass, such as a (non-rotating) star, planet, or black hole, and it remains one of the most impor- tant exact solutions. Schwarzschild spacetime is the curved spacetime ( , gab) given, in Schwarzschild coordinates, by the metric M

 2M  1 r 0 0 0 − 1  2M −   0 1 r 0 0  (gab) =  − −  . (3.3)  0 0 r2 0  − 0 0 0 r2 sin2 θ − In these coordinates, the metric becomes singular at the surface r = 2M, which is called the event horizon. Events inside (or on) this surface cannot aﬀect an outside observer; nothing can escape to the outside, not even light.

5Karl Schwarzschild (1873-1916), German physicist. Chapter 4

Mass in general relativity

The stress-energy-momentum tensor, Tab, is the tensor that describes the density and flux of energy and momentum in spacetime. It represents the energy due to matter and electromagnetic fields. Mass is the source of gravity, and 1 energy is associated with mass, therefore Tab is in the right-hand side of the Einstein field equation ‘telling spacetime how to curve.’

4.1 Gravitational energy/mass

Imagine a system of two massive bodies at rest relative to each other. If they are far apart, then there will be a gravitational potential energy contribution that makes the total energy of the system greater than if they are close to each other. There is a difference in total energy, despite that integrating the energy densities, Tab, yields the same result in both scenarios. That energy difference is the energy attributed to the gravitational field. Since the gravitational field has energy, and therefore mass, it is a source of gravity, hence it is coupled to itself. Mathematically, this is possible because the field equation is non-linear.

4.1.1 Non-locality of mass

The contribution of gravitational mass should be included in a description for total mass in a spacelike volume of spacetime. The stress-energy-momentum tensor Tab is given as a pointwise density and can be integrated over the volume. Can the gravitational mass be given as a point density that we can integrate? The answer is no. At any point in spacetime, one can always find a local coordinate system (Riemann-normal coordinates) in which all Christoffel symbol components vanish, which mean that there is no local gravitational field, hence no local gravitational mass [8, 14]. In such a coordinate system, an observer in ‘free fall’ moves along a straight line, and does not ‘feel’ any gravitational forces.

1According to E = mc2; Einstein’s famous equation [3].

Hansevi, 2008. 17 18 Chapter 4. Mass in general relativity

4.2 Total mass of an isolated system

Despite the fact that the gravitational mass cannot be given as a pointwise density, there exist meaningful notions of the total mass of an ‘isolated system’.

4.2.1 Asymptotically flat spacetimes Say that we want to study the system of the two massive bodies (of section 4.1). Even though no physical system can be truly isolated from the rest of the universe, we can simplify our model by pretending that the system is isolated, ignoring the influence of distant matter. The spacetime of our simplified model will have vanishing curvature at large distances from the two bodies, and we say that it is asymptotically flat. A precise and useful, but rather technical, definition of an asymptotically flat spacetime is given in [14].

4.2.2 ADM and Bondi-Sachs mass In asymptotically flat spacetimes, the total mass can be determined by the asymptotic form of the metric. The ADM2 mass is the total mass measured at spacelike infinity, whereas the Bondi-Sachs3 mass is measured at future null infinity [14].

4.3 Quasi-local mass

A meaningful definition of mass at a quasi-local level, that is, for the mass within a compact spacelike 2-surface, should have certain properties. For example, the quasi-local mass should be uniquely defined for all domains. Furthermore, it should be strictly positive (except in the flat case, where it should be equal to zero). Its limits at spacelike infinity and future null infinity should be the ADM mass and Bondi-Sachs mass, respectively. It should be monotone, that is, the mass for a domain should be greater or equal to the mass for a domain that is contained in the first. One can ask if it is possible to find a satisfying definition of mass at a quasi- local level. Several attempts have been made, for example, the Dougan-Mason mass, the Komar mass, the Penrose mass, and the Hawking mass. However, they fail to agree on the mass for a spacelike cross section of the event horizon of a Kerr4 black hole [1]. There is still no generally accepted definition. To reach an appropriate definition for quasi-local mass would certainly be of great value. It could give a more detailed characterization of the gravitational field around massive bodies, and it should be helpful for controlling errors in numerical calculations [13].

2R. Arnowitt, S. Deser and C. Misner. 3H. Bondi and R. K. Sachs. 4A solution to the Einstein ﬁeld equation found in 1963 by Roy Kerr. It describes spacetime outside a rotating black hole. Chapter 5

The Hawking mass

One of many suggestions that have been made for a definition of quasi-local mass originates from a paper about gravitational radiation written by Stephen Hawking [5] in 1968 . The Hawking mass has been shown to have various desirable properties, for example, the limits at spacelike infinity and future null infinity are the ADM mass and the Bondi-Sachs mass, respectively [13]. Its advantage is its simplicity, calculability and monotonicity for special families of 2-surfaces. In Minkowski spacetime, the Hawking mass vanish for 2-spheres. However, it can give negative results for general 2-surfaces, for example for non-convex 2-surfaces.

5.1 Deﬁnition

Let la and na be, respectively, the outgoing and the ingoing null vectors orthogonal to a spacelike 2-sphere and let ma andm ¯ a be tangent vectors to . Then the Hawking mass is deﬁnedS by S r Area( ) 1 I m ( ) := S 1 + ρρ0d . (5.1) H S 16π 2π S S

5.2 Interpretation

Consider a one-parameter family of null geodesics, that is light rays, that inter- sects a circle in a 2-plane. Following the geodesics in the future direction, the optical scalars, θ and σ, introduced by Rainer Sachs [12] in 1961, can be deﬁned as θ = Re ρ and σ = Im ρ, (5.2) and interpreted as the expansion and rotation, respectively, of the circle [2]. a a Both l and n are orthogonal to the surface, thus both ρ and ρ0 are real [10]. Hence ρ and ρ0 measure the expansion of outgoing and ingoing null geodesics, respectively, and the Hawking mass can be viewed as a measure of the bending of outgoing and ingoing light rays orthogonal to the surface of a spacelike 2-sphere.

Hansevi, 2008. 19 20 Chapter 5. The Hawking mass

5.3 Method for calculating spin coeﬃcients

In this section, a method for calculating the spin coefficients used in the expression for the Hawking mass is provided. In order to calculate the required spin coefficients (section 2.10), introduce • a coordinate system such that for a given r, and t held fixed, 0 θ < π and 0 ϕ < 2π trace out the surface . ≤ ≤ S In the new coordinates, every vector with the first two components van- • ished lies in the tangent plane of , as ma is required to do. The other two vectors, la and na, are parallel toS the outgoing and ingoing null directions, respectively, thus it is convenient to set up a null tetrad (la, na, ma, m¯ a), by making the ansatz  a  l = ( ABC 0 ), na = ( A B C 0 ), (5.3)  ma = ( 0− 0 −X iY ).

By imposing the conditions (2.36) to (5.3), A, B, C, X and Y can be determined by solving the obtained equations in three steps. 1. Solve the system m ma = 0 (null vector) a , (5.4) m m¯ a = 1 (normalization) a − and pick one solution X,Y . { } 2. Solve the system a la l = 0 (null vector) a , (5.5) la n = 1 (normalization) and pick the solution A, B , where A > 0 (and B > 0 if possible). The solution will possibly{ be} dependent of C. 3. Determine C (and A and B if they depend on C) by solving

a la m = 0 (orthogonality) . (5.6) The result of the procedure above is a null tetrad (la, na, ma, m¯ a) satisfying the conditions (2.36). Calculate the spin coeﬃcients with the use of the null tetrad by ﬁrst • calculating the required λijk given by (2.35),

a b λ314 = (∂b la ∂a lb) m m¯ − a b λ431 = (∂b ma ∂a mb)m ¯ l − a b λ143 = (∂b m¯ a ∂a m¯ b) l m − a b , (5.7) λ423 = (∂b na ∂a nb)m ¯ m − a b λ342 = (∂b m¯ a ∂a m¯ b) m n λ = (∂ m − ∂ m ) na m¯ b 234 b a − a b then the spin coeﬃcients follow easily as, ρ = γ = (λ + λ λ )/2 314 314 431 − 143 . (5.8) ρ0 = γ = (λ + λ λ )/2 423 423 342 − 234 5.4. Hawking mass for a 2-sphere 21

5.4 Hawking mass for a 2-sphere

In the following subsections, we demonstrate the method of section 5.3 by calculating the Hawking mass for a 2-sphere.

5.4.1 In Minkowski spacetime Using coordinates (t, r, θ, ϕ), where the spatial part (r, θ, ϕ) is written in spherical polar coordinates, the Minkowski metric (3.2) can be written as,

 1 0 0 0   0 1 0 0  (gab) =  −  . (5.9)  0 0 r2 0  − 0 0 0 r2 sin2 θ − By solving the equations (5.4), (5.5) and (5.6) in three steps, we obtain the null tetrad (la, na, ma, m¯ a), given by

√2 la = 1 1 0 0 , 2 √2 na = 1 1 0 0 , (5.10) 2 − √2 1 i ma = 0 0 − , 2 r r sin θ and then it follows easily from (5.7) and (5.8) that 1 ρρ0 = . (5.11) −2r2 The area of the 2-sphere is the familiar

Z2πZπ Area( ) = r2 sin θ dθ dϕ = 4πr2, (5.12) Sr 0 0 and the integral of the spin coeﬃcients

2π π I Z Z 1 ρρ0d = sin θ dθ dϕ = 2π. (5.13) r S −2 − S 0 0 Finally, we can conﬁrm the well-known result that the Hawking mass vanish for all 2-spheres in Minkowski spacetime, since r 4πr2 1 r m ( ) = 1 + ( 2π) = 1 1 = 0. (5.14) H Sr 16π 2π − 2 −

5.4.2 In Schwarzschild spacetime We repeat the calculations of the previous section, but this time for a centered 2-sphere in Schwarzschild spacetime. The coordinates (t, r, θ, ϕ) of the 22 Chapter 5. The Hawking mass

Schwarzschild metric (3.3) have the property that 0 θ < π and 0 ϕ < 2π trace out the surface of a 2-sphere, thus we do not have≤ to change to≤ another coordinate system. WeS obtain a null tetrad (la, na, ma, m¯ a) given by

√2 √r √r 2M la = − 0 0 , 2 √r 2M √r − √2 √r √r 2M na = − 0 0 , (5.15) 2 √r 2M − √r − √2 1 i ma = 0 0 − , 2 r r sin θ from which it follows that r 2M ρρ0 = − . (5.16) − 2r3 The area of the 2-sphere is 4πr2, and the integral of the spin coeﬃcients

2π π I Z Z r 2M 2π(r 2M) ρρ0d = − sin θ dθ dϕ = − . (5.17) r S − 2r − r S 0 0 Finally, we get the expected result for the Hawking mass of a centered 2-sphere in Schwarzschild spacetime, r 4πr2 1 2π(r 2M) r 2M m ( ) = 1 + − − = 1 1 + = M. (5.18) H Sr 16π 2π r 2 − r Chapter 6

Results

In this chapter, we calculate the Hawking mass, with the aid of Maple1, for ellipsoidal 2-surfaces in both Minkowski and Schwarzschild spacetimes. The calculations for Minkowski are performed symbolically, and the results are presented as a theorem and two corollaries. In Schwarzschild spacetime, the Hawking mass are calculated numerically for approximately ellipsoidal 2-surfaces, and the results are therefore presented with diagrams. First, we prove a lemma that will be useful in section 6.1. Lemma 6.1. Assume that ξ > 1. Then

2 arccosh ξ ξ 1 2(ξ 1) 5/2 p = 1 − + − + (ξ 1) . ξ2 1 − 3 15 O − −

Proof. Consider the equivalence, p p ξ = x2 + 1, x > 0 x = ξ2 1, ξ > 1, (6.1) ⇐⇒ − and the following Maclaurin expansions [11], p x2 + 1 = 1 + x2/2 x4/8 + (x6), (6.2) − O ln(y + 1) = y y2/2 + y3/3 y4/4 + y5/5 + (y6). (6.3) − − O Perform a change of variable,

p 2 arccosh ξ (def.) ln(ξ + ξ2 1) (6.1) ln √x + 1 + x p = p − = , (6.4) ξ2 1 ξ2 1 x − − and apply the Maclaurin expansions to the numerator,

2 4 p (6.2) x x ln x2 + 1 + x = ln 1 + x + + (x6) 2 − 8 O 2 4 2 4 x x 6 2 (6.3) x x x + + (x ) = x + + (x6) 2 − 8 O 2 − 8 O − 2 1Mathematics software package from Waterloo Maple Inc.

Hansevi, 2008. 23 24 Chapter 6. Results

2 3 2 4 x + x + (x4) x + x + (x4) + 2 O 2 O 3 − 4 5 x + (x2) + O + (x6) 5 O x3 3x5 = x + + (x6). (6.5) − 6 40 O Divide through by x, and change back to ξ,

2 2 2 arccosh ξ ξ 1 3(ξ 1) 2 5/2 p = 1 − + − + (ξ 1) . (6.6) ξ2 1 − 6 40 O − − Since ξ > 1, the ordo-term is actually

(ξ2 1)5/2 = (ξ + 1)5/2(ξ 1)5/2 = (ξ 1)5/2. (6.7) O − O − O − Thus, let ξ2 1 3(ξ2 1)2 f(ξ) = 1 − + − , (6.8) − 6 40 and calculate the second degree Taylor expansion of f about ξ = 1 . From f(1) = 1, f 0(1) = 1/3, and f 00(1) = 4/15 it follows that − 2 arccosh ξ ξ 1 2(ξ 1) 5/2 p = 1 − + − + (ξ 1) . (6.9) ξ2 1 − 3 15 O − −

6.1 Hawking mass in Minkowski spacetime

If we let an ellipse rotate around its minor axis we get the surface of a rotationally symmetric ellipsoid called an oblate spheriod, see ﬁg 6.1 (page 25). In Cartesian coordinates, the surface is given by the equation

x2 + y2 z2 + = 1. (6.10) A2 B2 Let A and B depend on a variable, say r > 0, in the way given by A2 = ξ2 r2 and B2 = r2. Then (6.10) is equivalent to

x2 + y2 + z2 = r2, (6.11) ξ2 which is the equation for an oblate spheroid, where 2r is the length of its minor axis. In this section, a closed-form expression for the Hawking mass within such an ellipsoid is given as a theorem with a proof. The result is also displayed as a graph in ﬁgure 6.2 (page 27). Furthermore, two corollaries regarding limits of the Hawking mass are proved. 6.1. Hawking mass in Minkowski spacetime 25

Figure 6.1: Oblate spheroid.

6.1.1 Closed-form expression of the Hawking mass

Theorem 6.1. Let ( , ηab) be Minkowski spacetime. For ξ > 1, let r be the spacelike oblate 2-spheroids,M in , given by S M = (x2 + y2)/ξ2 + z2 = r2 : r > 0 . Sr Then the Hawking mass within is given by Sr s 2 ! √2 r arccosh ξ 2 ξ 5 arccosh ξ mH r = − ξ + p − ξ + p . S 16√ξ ξ2 1 3 ξ2 1 − − Proof. Use the method provided in section 5.3. Start by introducing a coordinate system such that for a given r, and t held ﬁxed, θ and ϕ trace out the ellipsoidal 2-surface r. This is accomplished by using the parametrized form of , S Sr    x = ξ r sin θ cos ϕ  0 < r y = ξ r sin θ sin ϕ , 0 θ < π , (6.12)  z = r cos θ  0 ≤ ϕ < 2π ≤ as new variables (t, r, θ, φ). Using the tensor transformation law (2.19) yields the Minkowski metric (3.2) in the new coordinates,  1 0 0 0   0 (cos2 θ + ξ2 sin2 θ) (ξ2 1)r cos θ sin θ 0  (gab) =  − − −  .  0 (ξ2 1)r cos θ sin θ r2(ξ2 cos2 θ + sin2 θ) 0  − − − 0 0 0 ξ2r2 sin2 θ − (6.13) Ansatz (5.3). Solve equations (5.4), and pick a solution, say • p X = 1 / (√2 r ξ2 cos2 θ + sin2 θ) . (6.14) Y = 1 / (√2 ξ r sin θ)

Solve equations (5.5), and pick the solution given by A = 1/√2 and • p cos2 θ + ξ2 sin2 θ 2ξ2 r2 C2 √2(ξ2 1) r cos θ sin θ C B = − − − . √2(cos2 θ + ξ2 sin2 θ) (6.15) 26 Chapter 6. Results

Solve equation (5.6). The solution is given by • √2(ξ2 1) cos θ sin θ C = p − . (6.16) − 2 ξ r ξ2 cos2 θ + sin2 θ

The result of this procedure is the null tetrad (la, na, ma, m¯ a), given by

p 2 2 2 2 a √2 ξ cos θ + sin θ (ξ 1) cos θ sin θ l = 1 p − 0 , 2 ξ − ξ r ξ2 cos2 θ + sin2 θ p 2 2 2 2 a √2 ξ cos θ + sin θ (ξ 1) cos θ sin θ n = 1 p − 0 , 2 − ξ ξ r ξ2 cos2 θ + sin2 θ a √2 1 i m = 0 0 p , 2 r ξ2 cos2 θ + sin2 θ ξ r sin θ a √2 1 i m¯ = 0 0 p . (6.17) 2 r ξ2 cos2 θ + sin2 θ − ξ r sin θ Calculate the spin coefficients with the use of the null tetrad by first calculating the required λijk given by (5.7), λ = (∂ l ∂ l )ma m¯ b = 0, 314 b a − a b 2 2 2 a b ξ (1 + cos θ) + sin θ λ431 = (∂b ma ∂a mb)m ¯ l = , − − 2√2 ξ r (ξ2 cos2 θ + sin2 θ)3/2 2 2 2 a b ξ (1 + cos θ) + sin θ λ143 = (∂b m¯ a ∂a m¯ b)l m = , − 2√2 ξ r (ξ2 cos2 θ + sin2 θ)3/2 λ = (∂ n ∂ n )m ¯ a mb = 0, 423 b a − a b 2 2 2 a b ξ (1 + cos θ) + sin θ λ342 = (∂b m¯ a ∂a m¯ b)m n = , − 2√2 ξ r (ξ2 cos2 θ + sin2 θ)3/2 2 2 2 a b ξ (1 + cos θ) + sin θ λ234 = (∂b ma ∂a mb)n m¯ = , − − 2√2 ξ r (ξ2 cos2 θ + sin2 θ)3/2 (6.18) then the spin coefficients (5.8) follow easily as,

ξ2(1 + cos2 θ) + sin2 θ ρ = γ314 = 3/2 , − 2√2 ξ r ξ2 cos2 θ + sin2 θ ξ2(1 + cos2 θ) + sin2 θ ρ 0 = γ423 = 3/2 . (6.19) 2√2 ξ r ξ2 cos2 θ + sin2 θ

The surface area of r is part of the expression for the Hawking mass. From the surface element d S= pg g (g )2 dθ dϕ it follows that S θθ ϕϕ − θϕ 2π π Z Z q Area( ) = ξ r2 sin θ ξ2 cos2 θ + sin2 θ dθ dϕ Sr 0 0 p p 2π ξ r2 ξ ξ2 1 + ln( ξ2 1 + ξ) = −p − ξ2 1 − 6.1. Hawking mass in Minkowski spacetime 27

2 arccosh ξ = 2π ξ r ξ + p , (6.20) ξ2 1 − Integration over yields Sr 2π π 2π π 2 Z Z Z Z sin θ ξ2(1 + cos2 θ) + sin2 θ ρρ 0 d = 5/2 dθ dϕ S ξ ξ2 cos2 θ + sin2 θ 0 0 0 0 p p π (2 ξ3 + 7ξ) ξ2 1 + 3 arcsinh ( ξ2 1) = −p − − 6 ξ ξ2 1 − π 2 3 arccosh ξ = 2 ξ + 7 + p . (6.21) − 6 ξ ξ2 1 − Finally, from (6.20) and (6.21), the Hawking mass in follows as Sr r Z2πZπ Area( r) 1 m ( ) = S 1 + ρρ 0 d H Sr 16 π 2 π S 0 0 s ! √2 r arccosh ξ 2 ξ2 5 arccosh ξ = − ξ + p − ξ + p . (6.22) 16√ξ ξ2 1 3 ξ2 1 − −

As can be seen in ﬁgure 6.2, the Hawking mass becomes negative in Minkowski, even for a convex 2-surface. 1

1 − 2 − 3 − 0 1 2 3 4

Figure 6.2: m ( ) plotted against parameter ξ. H S1

6.1.2 Limit when approaching a metric sphere

Corollary 6.1. Let ( , ηab) be Minkowski spacetime. Given an r > 0, the M + Hawking mass vanishes in the limit when ξ 1 , that is, when r tends to a 2-sphere. → S

Proof. It follows from lemma 6.1 that

2 arccosh ξ ξ 1 2(ξ 1) 5/2 p = 1 − + − + (ξ 1) ξ2 1 − 3 15 O − − = 1 + (ξ 1), (6.23) O − 28 Chapter 6. Results and furthermore that 2 ξ2 5 arccosh ξ 2 ξ3 5 ξ − ξ + p = + 1 + (ξ 1) 3 ξ2 1 3 − 3 O − − 2 ξ(ξ + 1) 3 = − ξ 1 + (ξ 1) 3 − O − = (ξ 1). (6.24) O − Thus, it follows from (6.23), (6.24) and theorem 6.1 that s ! √2 r arccosh ξ 2 ξ2 5 arccosh ξ mH( r) = − ξ + p − ξ + p S 16√ξ ξ2 1 3 ξ2 1 − − r p = 1 + ξ + (ξ 1) (ξ 1) 0 as ξ 1+. (6.25) √ξ O − O − → →

6.1.3 Limit along a foliation

Corollary 6.2. Let ( , ηab) be Minkowski spacetime. For ω > 0, let r r>0 be foliations of a spacelikeM 3-surface Ω . Suppose that the leaves are{L given} by ⊂ M = (x2 + y2)/(1 + ω/r)2 + z2 = r2 : r > 0 . Lr Then the Hawking mass vanishes in the limit along all foliations . {Lr}r>0 Proof. Observe that

r = r , (6.26) L S ξ=1+ω/r and let ξ = 1 + ω/r in lemma 6.1. Then ξ 1+ as r , and it follows that → → ∞ arccosh (1 + ω/r) ω ω 2 ω2 1 1 + ω/r + p = 1 + + 1 + + (1 + ω/r)2 1 r − 3r 15r2 O r5/2 − 2ω 2 ω2 1 = 2 + + + 3r 15r2 O r5/2 1 = 2 + , (6.27) O r and furthermore that 2(1 + ω/r)2 5 arccosh (1 + ω/r) 32ω2 2ω3 1 − (1 + ω/r) + p = + + 3 (1 + ω/r)2 1 15r2 3r3 O r5/2 − 1 = . (6.28) O r2 From (6.27) and theorem 6.1, it follows that r √2 r 1 1 mH( r) = −p ω 2 + 2 L 16 1 + r O r O r 1 = 0 as r . (6.29) O r → → ∞ 6.2. Hawking mass in Schwarzschild spacetime 29

6.2 Hawking mass in Schwarzschild spacetime

In this section we will study the Hawking mass in the curved Schwarzschild spacetime. A consequence of the spacetime being curved is that the required calculations tend to be more complicated. Considering this, we will calculate the Hawking mass for an approximately ellipsoidal 2-surface, exterior to the event horizon (r > 2M), that has a simple expression in spherical polar coordinates. For small ε > 0, let the curve given by p r = 1 + ε sin2 θ, (6.30) rotate around the axis θ = 0. See ﬁgure 6.3. The surface of revolution, ˜, is approximately ellipsoidal. For ε = 0, it is a perfect sphere. S

θ = 0

r θ

p Figure 6.3: A curve given by r = 1 + ε sin2 θ.

Let us introduce a coordinate system such that for a given r, and t held ﬁxed, θ and ϕ trace out the spacelike 2-surface ˜r. This is accomplished by the following change of variables, S p r r 1 + ε sin2 θ, r > 2M. (6.31) 7→ In the new coordinates, the Schwarzschild metric (3.3) is given by

 β  rα 0 0 0 rα3 εr2α cos θ sin θ  0 β β 0  (g ) =  2− 2 −3 2 2 2  , ab  εr α cos θ sin θ r (α β+ε r cos θ sin θ)   0 0  − β − αβ 0 0 0 r2α2 sin2 θ − (6.32) where p p α = 1 + ε sin2 θ > 1 and β = r 1 + ε sin2 θ 2M > 0. (6.33) − By making the ansatz (5.3), solving equations (5.4), (5.5) and (5.6), we obtain a null tetrad (la, na, ma, m¯ a) given by, p √2 √rα α3β + ε2r cos2 θ sin2 θ 2C la = 0 , 2 √β √rα3 √2 30 Chapter 6. Results

p √2 √rα α3β + ε2r cos2 θ sin2 θ 2C na = 0 , (6.34) 2 √β − √rα3 − √2 a √2 √αβ i m = 0 0 p , 2 r α3β + ε2r cos2 θ sin2 θ − rα sin θ where √2 ε cos θ sin θ C = p . (6.35) − 2 √rα α3β + ε2 r cos2 θ sin2 θ

We calculate the spin coeﬃcients ρ and ρ0. Unfortunately, their expressions in Schwarzschild spacetime are very long, so we omit writing them out. The surface element of ˜ is given by Sr q d = g g (g )2 dθ dϕ S θθ ϕϕ − θϕ s ε2 r cos2 θ sin2 θ = r2α2 sin θ + 1 dθ dϕ, (6.36) αβ and the surface area by Z2πZπ Area( ˜ ) = d . (6.37) Sr S 0 0 Since the expression of the Hawking mass for ˜ , Sr s Z2πZπ Area( ˜r) 1 m ( ˜ ) = S 1 + ρρ0 d , (6.38) H Sr 16 π 2 π S 0 0 is rather complicated, we will rely on methods like Taylor expansion and numerical integration for further investigations.

6.2.1 Limit along a foliation Analogously to corollary 6.2, let ˜ be foliations of a spacelike 3-surface {Lr}r>2M Ω (now in Schwarzschild spacetime). Suppose that the leaves, ˜r, are given⊂ M by substituting L ω ε = , r > 2M, (6.39) r into ˜ . It is easy to see that ˜ becomes more spherical the larger r gets. Sr Lr In this section, we will show that the limit of the Hawking mass of ˜r is M along all foliations ˜ , that is, L {Lr}r>2M

lim mH( ˜r) = M. (6.40) r →∞ L

Since both d and ρρ0d are independent of ϕ, we make the substitution (6.39), and let S S

Z π 2 ˜ I1(ε) = ε Area( ω/ε) =: 2π Φ(ε, θ) dθ (6.41) L 0 6.2. Hawking mass in Schwarzschild spacetime 31 and Z π Z π I2(ε) = 2π ρρ0 dθ =: 2π Ψ(ε, θ) dθ. (6.42) 0 0

Under the reasonable assumption that Φ(ε, θ), Φε0 (ε, θ), Ψ(ε, θ), Ψε0 (ε, θ) and Ψεε00 (ε, θ) are continuous in some neighbourhood ε < 1, it follows by Maclaurin expansion that | | Z π 2 I1(ε) = 2π Φ(0, θ) dθ + (ε) = 4πω + (ε), ε < 1, (6.43) 0 O O | | and Z π Z π 2 I2(ε) = 2π Ψ(0, θ) dθ + 2π Ψε0 (0, θ) dθ ε + (ε ) 0 0 O 4πM = 2π + ε + (ε2), ε < 1. (6.44) − ω O | | Thus 4πω2 1 Area( ˜ ) = I (ε)/ε2 = + = 4πr2 + (r), r > ω , (6.45) Lr 1 ε2 O ε O | | and Z π 4πM 1 2π ρρ0 d = 2π + + 2 , r > ω . (6.46) 0 S − r O r | | By using the results (6.45) and (6.46), it follows that r 4πr2 + (r) 1 h 4πM 1 i m ( ˜ ) = O 1 + 2π + + H Lr 16π 2π − r O r2 r r 1 2M 1 = 1 + + 2 O r r O r2 r 1 1 = 1 + M + M as r . O r O r → → ∞ (6.47)

6.2.2 Numerical evaluations We evaluate the Hawking mass numerically with an adaptive Gaussian2 quadrature method. In numerical quadrature, an integral Z b I(f) = f(x)dx, (6.48) a is approximated by an n-point quadrature rule that has the form n X Qn(f) = wif(xi), (6.49) i=1 where a x < x < < x b. The points x are called nodes, and the ≤ 1 2 ··· n ≤ i multipliers wi are called weights. 2Carl Friedrich Gauss (1777-1855), German mathematician known as ‘princeps mathemati- corum’ (‘prince of mathematicians’). 32 Chapter 6. Results

In Gaussian quadrature, both the nodes and the weights are optimally cho- sen, hence Gaussian quadrature has the highest possible accuracy for the number of nodes used. Furthermore, it is stable and Qn(f) I(f) as n [6]. In adaptive quadrature, the interval of integration→ is selectively→ ∞ refined to reflect the behavior of the particular integrand. We set M = 1, and let Maple calculate the Hawking mass numerically for some values of ε. The results are presented as graphs. See figure 6.4, 6.5 and 6.6.

1.0 e = 0. e = 0.05

e = 0.10 0.5

e = 0.15 0.0

K0.5 e = 0.20

K1.0 e = 0.25 4 6 8 10 20 40 60 80 100 200 400 600

Figure 6.4: m ( ˜ ) plotted against 4 r 620 for some values of ε. H Sr ≤ ≤

1.0 e = 0. e = 0.2 0.9 e = 0.4 0.8

0.7 e = 0.6

0.6

0.5 e = 0.8

0.4

0.3 e = 1.0

2 4 6 8 10 12 14 16

Figure 6.5: m ( ˜ ) plotted against 2.3 r 16 for some values of ε. H Sr ≤ ≤

1.00

0.98

0.96

0.94

0.92

0.90

0.88

0.86

0.84

4 6 8 10 12 14 16

Figure 6.6: m ( ˜ ) plotted against 2.3 r 16 for ε = ω/r. H Sr ≤ ≤ Chapter 7

Discussion

7.1 Conclusions

In this thesis, we have derived a closed-form expression for the Hawking mass of a spacelike oblate 2-spheroid r, that is, a rotationally symmetric 2-ellipsoid, in Minkowski spacetime. If isS given by, Sr = (x2 + y2)/ξ2 + z2 = r2 : r > 0 . Sr Then the Hawking mass within is given by Sr s 2 ! √2 r arccosh ξ 2 ξ 5 arccosh ξ mH r = − ξ + p − ξ + p . S 16√ξ ξ2 1 3 ξ2 1 − − From this result, we can see that the Hawking mass can be negative even for convex 2-surfaces in Minkowski spacetime. However, the limits along particular foliations, were shown to vanish. Furthermore, we studied the Hawking mass in Schwarzschild spacetime. Nu- merical calculations for approximately ellipsoidal 2-surfaces were done, and the results were presented with diagrams that show that the Hawking mass can be negative in Schwarzschild. It this case, the limits along particular foliations, were shown to be M, that is, equal to the Hawking mass for a centered 2-sphere.

7.2 Future work

The calculations performed in this thesis can serve as a basis for similar studies of the Hawking mass in other spacetimes, for example in Reissner-Nordstr¨om which is the generalization of Schwarzschild that includes electric charge.

Hansevi, 2008. 33 34 Chapter 7. Discussion Bibliography

[1] Bergqvist, G., Quasilocal mass for event horisons, Class. Quantum Grav. 9, 1753-1768, (1992) [2] Chandrasekhar, S., The Mathematical Theory of Black Holes, Oxford Uni- versity Press, New York, 1983

[3] Einstein, A., Zur Elektrodynamik bewegter K¨orper, Annalen der Physik 18, 639-643, (1905) [4] Einstein, A., Die Grundlage der allgemeinen Relativit¨atstheorie, Annalen der Physik 49, 769-822, (1916)

[5] Hawking, S. W., Gravitational Radiation in an Expanding Universe, J. Math. Phys. 9, 598-604, (1968) [6] Heath, M. T., Scientiﬁc Computing: An Introductory Survey, McGraw-Hill, 2002 [7] Lawson, H. B., Jr., Foliations, Bull. Amer. Math. Soc. 80, 369-418, (1974)

[8] Misner, C. W, Thorne, K. S., Wheeler, J. A, Gravitation, W. H. Freeman and Company, 1973 [9] Newman, E. T. and Penrose, R., An approach to Gravitational Radiation by a Method of Spin Coeﬃcients, J. Math. Phys. 3, 566-578, (1962)

[10] Penrose, R. and Rindler, W., Spinors and space-time volume 1, Cambridge University Press, 1984 [11] R˚ade,L., Westergren, B., Mathematics Handbook for Science and Engi- neering, Studentlitteratur, Lund, 1995

[12] Sachs, R. K., Gravitational Waves in General Relativity. VI. The Outgoing Radiation Condition, Proc. Roy. Soc. (London) A 264, 309-337, (1961) [13] Szabados, L. B., ”Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article”, Living Rev. Relativity 7, (2004), 4. [Online Article] (cited on 2008-03-05): http://www.livingreviews.org/lrr-2004-4

[14] Wald, R. M., General Relativity, The University of Chicago Press, Chicago and London, 1984

Hansevi, 2008. 35 36 Bibliography Appendix A

Maple Worksheets

A.1 Null tetrad

# Determine a null-tetrad # Let m1=m and m2=conjugate(m), # m1*m1 = m2*m2 = 0 (null vectors) and m1*m2 = -1 (normalization) m1:=create([1],array(1..4,[0,0,X, I*Y])): m2:=create([1],array(1..4,[0,0,X,-I*Y])): sY:=solve((get_compts(prod(m1,lower(g,m1,1),[1,1])))=0,Y)[1]: m1:=create([1],array(1..4,[0,0,X, I*sY])): m2:=create([1],array(1..4,[0,0,X,-I*sY])): sX:=solve((get_compts(prod(m1,lower(g,m2,1),[1,1])))=-1,X)[1]: m1:=create([1],array(1..4,[0,0,sX, I*eval(sY,X=sX)])): m2:=create([1],array(1..4,[0,0,sX,-I*eval(sY,X=sX)])): # Let l and m, such that l*l = n*n = 0 (null vectors) # and l*n = 1 (normalization) l:=create([1],array(1..4,[A, B, C,0])): n:=create([1],array(1..4,[A,-B,-C,0])): sB:=solve((get_compts(prod(l,lower(g,l,1),[1,1])))=0,B)[1]: l:=create([1],array(1..4,[A, sB, C,0])): n:=create([1],array(1..4,[A,-sB,-C,0])): sA:=solve((get_compts(prod(l,lower(g,n,1),[1,1])))=1,A)[1]: sB:=eval(sB,A=sA): l:=create([1],array(1..4,[sA, sB, C,0])): n:=create([1],array(1..4,[sA,-sB,-C,0])): # l*m1 = l*m2 = n*m1 = n*m2 = 0 (orthogonality) sC:=solve((get_compts(prod(l,lower(g,m1,1),[1,1])))=0,C)[1]: sA:=simplify(eval(sA,C=sC)): sB:=simplify(eval(sB,C=sC)): l:=create([1],array(1..4,[sA, sB, sC,0])): n:=create([1],array(1..4,[sA,-sB,-sC,0])): # Control (result should be: 0,0,0,0,0,0,0,0,1,-1) simplify(get_compts(prod(l, lower(g,l, 1),[1,1]))), simplify(get_compts(prod(n, lower(g,n, 1),[1,1]))), simplify(get_compts(prod(m1,lower(g,m1,1),[1,1]))), simplify(get_compts(prod(m2,lower(g,m2,1),[1,1]))), simplify(get_compts(prod(l, lower(g,m1,1),[1,1]))), simplify(get_compts(prod(n, lower(g,m1,1),[1,1]))), simplify(get_compts(prod(l, lower(g,m2,1),[1,1]))), simplify(get_compts(prod(n, lower(g,m2,1),[1,1]))), simplify(get_compts(prod(l, lower(g,n, 1),[1,1]))), simplify(get_compts(prod(m1,lower(g,m2,1),[1,1])));

Hansevi, 2008. 37 38 Appendix A. Maple Worksheets

A.2 Minkowski restart:with(tensor): assume(xi::real,t::real,r::real,theta::real,phi::real): additionally(xi>=1,r>0,theta>=0,theta<=Pi,phi>=0,phi<=2*Pi): # Metric of Minkowski spacetime ("cartesian" coordinates) coords:=[t,r,theta,phi]: h_c:=array(sparse,1..4,1..4): h_c[1,1]:=1: h_c[2,2]:=-1: h_c[3,3]:=-1: h_c[4,4]:=-1: # Change of variables (theta and phi parametrize ellipsoids) x[1]:=t: x[2]:=xi*r*sin(theta)*cos(phi): x[3]:=xi*r*sin(theta)*sin(phi): x[4]:=r*cos(theta): g_c:=array(sparse,1..4,1..4): for i_ from 1 to 4 do for j_ from 1 to 4 do for i from 1 to 4 do for j from 1 to 4 do g_c[i_,j_]:=g_c[i_,j_] +h_c[i,j]*diff(x[i],coords[i_])*diff(x[j],coords[j_]): end do; end do; end do; end do; # Create (covariant) metric tensor g g:=create([-1,-1],eval(g_c)): # Null tetrad l:=create([1],array(1..4,[sqrt(2)/2, sqrt(2)/2/xi*sqrt(xi**2*cos(theta)**2+sin(theta)**2), -sqrt(2)/2/xi/r*(xi**2-1)*cos(theta)*sin(theta) /sqrt(xi**2*cos(theta)**2+sin(theta)**2), 0])): n:=create([1],array(1..4,[sqrt(2)/2, -sqrt(2)/2/xi*sqrt(xi**2*cos(theta)**2+sin(theta)**2), sqrt(2)/2/xi/r*(xi**2-1)*cos(theta)*sin(theta) /sqrt(xi**2*cos(theta)**2+sin(theta)**2), 0])): m1:=create([1],array(1..4,[0,0, sqrt(2)/2/r/sqrt(xi**2*cos(theta)**2+sin(theta)**2), sqrt(2)/2*I/xi/r/sin(theta)])): m2:=create([1],array(1..4,[0,0, sqrt(2)/2/r/sqrt(xi**2*cos(theta)**2+sin(theta)**2), -sqrt(2)/2*I/xi/r/sin(theta)])): # Calculate spin coefficients e[1]:=get_compts(l): e[2]:=get_compts(n): e[3]:=get_compts(m1): e[4]:=get_compts(m2): f[1]:=get_compts(lower(g,l, 1)): f[2]:=get_compts(lower(g,n, 1)): f[3]:=get_compts(lower(g,m1,1)): f[4]:=get_compts(lower(g,m2,1)): L314:=0: L431:=0: L143:=0: L423:=0: L342:=0: L234:=0: for i from 1 to 4 do for j from 1 to 4 do L314:=L314 +(diff(f[1][i],coords[j])-diff(f[1][j],coords[i]))*e[3][i]*e[4][j]: L431:=L431 +(diff(f[3][i],coords[j])-diff(f[3][j],coords[i]))*e[4][i]*e[1][j]: L143:=L143 +(diff(f[4][i],coords[j])-diff(f[4][j],coords[i]))*e[1][i]*e[3][j]: A.3. Schwarszchild 39

L423:=L423 +(diff(f[2][i],coords[j])-diff(f[2][j],coords[i]))*e[4][i]*e[3][j]: L342:=L342 +(diff(f[4][i],coords[j])-diff(f[4][j],coords[i]))*e[3][i]*e[2][j]: L234:=L234 +(diff(f[3][i],coords[j])-diff(f[3][j],coords[i]))*e[2][i]*e[4][j]: end do; end do; # rho = gamma[314], rho’ = gamma[423] rho_ :=simplify(1/2*(L314+L431-L143)); rho_2:=simplify(1/2*(L423+L342-L234)); # Calculate surface area of ellipsoid area_element:=simplify(sqrt(g_c[3,3]*g_c[4,4]-g_c[3,4]**2)); area:=simplify(int(int(area_element,theta=0..Pi),phi=0..2*Pi)); # Calculate integral of spin coefficients spin_integrand:=(rho_*rho_2*area_element); spin_integral:=simplify( int(eval(int(spin_integrand, theta ),theta=Pi)- eval(int(spin_integrand, theta ),theta=0),phi=0..2*Pi)); # Calculate Hawking mass mH_:=simplify(sqrt(area/16/Pi)*(1+1/2/Pi*spin_integral)); # Simplification by hand mH:=-sqrt(2)/(16*sqrt(xi))*r*sqrt(xi+arccosh(xi)/sqrt(xi**2-1)) *(xi*(2*xi**2-5)/3+arccosh(xi)/sqrt(xi**2-1)); # Limit along foliation limit(eval(mH,xi=1+omega/r),r=infinity); plot(eval(mH,r=1),xi=1..4,-3..1, thickness=2,axes=frame,gridlines=true,font=[TIMES,1,12]);

A.3 Schwarszchild restart:with(tensor):with(plots): assume(t::real,r::real,theta::real,phi::real,epsilon::real,omega::real,k::real,p::real): additionally(M>=0,r>2*M,epsilon>=0,theta>=0,theta<=Pi/2,alpha>=1,beta>0,omega>0): # Metric of Schwarzschild spacetime coords:=[t,r,theta,phi]: h_c:=array(sparse,1..4,1..4): h_c[1,1]:=1-2*M/r: h_c[2,2]:=-1/(1-2*M/r): h_c[3,3]:=-r**2: h_c[4,4]:=-r**2*sin(theta)**2: # Change of variables (theta and phi parametrize ellipsoids) F:=sqrt(1+epsilon*sin(theta)**2): x[1]:=t: x[2]:=F*r: x[3]:=theta: x[4]:=phi: gt_c:=array(sparse,1..4,1..4): for i_ from 1 to 4 do for j_ from 1 to 4 do for i from 1 to 4 do for j from 1 to 4 do gt_c[i_,j_]:=gt_c[i_,j_] +subs({t=x[1],r=x[2],theta=x[3],phi=x[4]},h_c[i,j]) *diff(x[i],coords[i_])*diff(x[j],coords[j_]): end do; end do; end do; end do; # Create (covariant) metric tensor g gt:=create([-1,-1],eval(gt_c)): AB:={alpha=sqrt(1+epsilon*sin(theta)**2), beta=r*sqrt(1+epsilon*sin(theta)**2)-2*M}: # Simplification by hand g_c:=array(sparse,1..4,1..4): g_c[1,1]:=beta/alpha/r: 40 Appendix A. Maple Worksheets g_c[2,2]:=-alpha**3*r/beta: g_c[3,3]:=-r**2*(alpha**3*beta+epsilon**2*r*sin(theta)**2*cos(theta)**2) /(alpha*beta): g_c[4,4]:=-alpha**2*r**2*sin(theta)**2: g_c[2,3]:=-epsilon*alpha*r**2*sin(theta)*cos(theta)/beta: g_c[3,2]:=-epsilon*alpha*r**2*sin(theta)*cos(theta)/beta: g:=create([-1,-1],eval(g_c)): # Null tetrad l:=create([1],array(1..4,[ sqrt(2)/2*sqrt(r*alpha/beta), sqrt(2)/2*sqrt(alpha**3*beta+r*epsilon**2*cos(theta)**2*sin(theta)**2) /(sqrt(r)*alpha**3), -sqrt(2)/2*epsilon*sin(theta)*cos(theta) /(alpha*sqrt(r)*sqrt(alpha**3*beta +r*epsilon**2*cos(theta)**2*sin(theta)**2)), 0])): n:=create([1],array(1..4,[ sqrt(2)/2*sqrt(r*alpha/beta), -sqrt(2)/2*sqrt(alpha**3*beta+r*epsilon**2*cos(theta)**2*sin(theta)**2) /(sqrt(r)*alpha**3), sqrt(2)/2*epsilon*sin(theta)*cos(theta) /(alpha*sqrt(r)*sqrt(alpha**3*beta +r*epsilon**2*cos(theta)**2*sin(theta)**2)), 0])): m1:=create([1],array(1..4,[ 0,0, sqrt(2)/2*sqrt(beta)*sqrt(alpha) /(r*sqrt(alpha**3*beta+r*epsilon**2*cos(theta)**2*sin(theta)**2)), -sqrt(2)/2*I/(alpha*r*sin(theta))])): m2:=create([1],array(1..4,[ 0,0, sqrt(2)/2*sqrt(beta)*sqrt(alpha) /(r*sqrt(alpha**3*beta+r*epsilon**2*cos(theta)**2*sin(theta)**2)), sqrt(2)/2*I/(alpha*r*sin(theta))])): # Calculate spin coefficients e[1]:=simplify(eval(get_compts(l), AB)): e[2]:=simplify(eval(get_compts(n), AB)): e[3]:=simplify(eval(get_compts(m1),AB)): e[4]:=simplify(eval(get_compts(m2),AB)): f[1]:=simplify(eval(get_compts(lower(g,l, 1)),AB)): f[2]:=simplify(eval(get_compts(lower(g,n, 1)),AB)): f[3]:=simplify(eval(get_compts(lower(g,m1,1)),AB)): f[4]:=simplify(eval(get_compts(lower(g,m2,1)),AB)): L314:=0: L431:=0: L143:=0: L423:=0: L342:=0: L234:=0: for i from 1 to 4 do for j from 1 to 4 do L314:=L314 +(diff(f[1][i],coords[j])-diff(f[1][j],coords[i]))*e[3][i]*e[4][j]: L431:=L431 +(diff(f[3][i],coords[j])-diff(f[3][j],coords[i]))*e[4][i]*e[1][j]: L143:=L143 +(diff(f[4][i],coords[j])-diff(f[4][j],coords[i]))*e[1][i]*e[3][j]: L423:=L423 +(diff(f[2][i],coords[j])-diff(f[2][j],coords[i]))*e[4][i]*e[3][j]: L342:=L342 +(diff(f[4][i],coords[j])-diff(f[4][j],coords[i]))*e[3][i]*e[2][j]: L234:=L234 +(diff(f[3][i],coords[j])-diff(f[3][j],coords[i]))*e[2][i]*e[4][j]: end do; end do; # rho = gamma[314], rho’ = gamma[423] rho_ :=simplify(1/2*(L314+L431-L143)); A.3. Schwarszchild 41 rho_2:=simplify(1/2*(L423+L342-L234)); # Hawking energy mH:=sqrt(area/16/Pi)*(1+1/2/Pi*spin_integral): # Surface area of ellipsoid area_element:=simplify(eval(sqrt(g_c[3,3]*g_c[4,4]-g_c[3,4]**2),AB)): area:=simplify(2*Pi*Int(area_element,theta=0..Pi, method=_Gquad)): # Integral of spin coefficients spin_integral:=2*Pi*Int(simplify(rho_*rho_2)*area_element,theta=0..Pi, method=_Gquad): # Limit along foliation Phi:=simplify(epsilon**2*eval(area_element,r=omega/epsilon)): 2*Pi*int(simplify(eval(Phi,epsilon=0)),theta=0..Pi)/epsilon**2: Psi_:=simplify(eval(simplify(rho_*rho_2*area_element),r=omega/epsilon)): 2*Pi*int(simplify(eval(Psi_,epsilon=0)),theta=0..Pi) +2*Pi*int(simplify(eval(diff(Psi_,epsilon),epsilon=0)),theta=0..Pi)*epsilon: eval(series(eval(mH,r=1/epsilon),epsilon=0,2),epsilon=1/r): # Plot eps1:=[0,1/5,2/5,3/5,4/5,1]: start1:=2.3: base1:=1.1: for i from 1 to 6 do R:=start1*base1**0: points1[i]:=[R,evalf(eval(mH,{M=1,r=R,epsilon=eps1[i]}))]: for a from 1 to 20 do R:=start1*base1**a: points1[i]:=points1[i],[R,evalf(eval(mH,{M=1,r=R,epsilon=eps1[i]}))]: end do: end do: plots[display]({ setoptions(symbol=diamond,symbolsize=12,color=black), plot([points1[1]]),pointplot([points1[1]]), plot([points1[2]]),pointplot([points1[2]]), plot([points1[3]]),pointplot([points1[3]]), plot([points1[4]]),pointplot([points1[4]]), plot([points1[5]]),pointplot([points1[5]]), plot([points1[6]]),pointplot([points1[6]]) }, textplot({[16,1.0,epsilon_=0.0],[16,0.95,epsilon_=0.2], [16,0.84,epsilon_=0.4],[16,0.69,epsilon_=0.6], [16,0.5,epsilon_=0.8],[16,0.29,epsilon_=1.0]}, align={right},font=[TIMES,ROMAN,12]), axes=boxed, axis[1]=[mode=log,gridlines=[8,thickness=1,subticks=false,color=grey]], axis[2]=[gridlines=[color=grey]], view=[2..16,0.25..1.03] ); eps2:=[0,0.05,0.1,0.15,0.2,0.25]: start2:=4: base2:=1.4: for i from 1 to 6 do R:=start2*base2**0: points2[i]:=[R,evalf(eval(mH,{M=1,r=R,epsilon=eps2[i]}))]: for a from 1 to 15 do R:=start2*base2**a: points2[i]:=points2[i],[R,evalf(eval(mH,{M=1,r=R,epsilon=eps2[i]}))]: end do: end do: plots[display]({ setoptions(symbol=diamond,symbolsize=12,color=black), plot([points2[1]]),pointplot([points2[1]]), plot([points2[2]]),pointplot([points2[2]]), 42 Appendix A. Maple Worksheets

plot([points2[3]]),pointplot([points2[3]]), plot([points2[4]]),pointplot([points2[4]]), plot([points2[5]]),pointplot([points2[5]]), plot([points2[6]]),pointplot([points2[6]]) }, textplot({[700,1.0,epsilon_=0.00],[700,0.9,epsilon_=0.05], [700,0.6,epsilon_=0.10],[700,0.17,epsilon_=0.15], [700,-0.45,epsilon_=0.20],[700,-1.18,epsilon_=0.25]}, align={right},font=[TIMES,ROMAN,12]), axes=boxed, axis[1]=[mode=log,gridlines=[12,thickness=1,subticks=false,color=grey]], axis[2]=[gridlines=[color=grey]], view=[4..700,-1.25..1.1] ); om:=[0.0,0.2,0.4,0.6,0.8,1]: start3:=2.3: base3:=1.1: for i from 1 to 6 do R:=start3*base3**0: points3[i]:=[R,evalf(eval(mH,{M=1,r=R,epsilon=om[i]/R}))]: for a from 1 to 20 do R:=start3*base3**a: points3[i]:=points3[i],[R,evalf(eval(mH,{M=1,r=R,epsilon=om[i]/R}))]: end do: end do: plots[display]({ setoptions(symbol=diamond,symbolsize=12,color=black), plot([points3[1]]),pointplot([points3[1]]), plot([points3[2]]),pointplot([points3[2]]), plot([points3[3]]),pointplot([points3[3]]), plot([points3[4]]),pointplot([points3[4]]), plot([points3[5]]),pointplot([points3[5]]), plot([points3[6]]),pointplot([points3[6]]) }, axes=boxed, axis[1]=[mode=log,gridlines=[8,thickness=1,subticks=false,color=grey]], axis[2]=[gridlines=[color=grey]], view=[2.2..16,0.83..1.01] ); Copyright The publishers will keep this document online on the Internet - or its possible replacement - for a period of 25 years from the date of publication barring exceptional circumstances. The online availability of the document implies a permanent permission for anyone to read, to download, to print out single copies for your own use and to use it unchanged for any non-commercial research and educational purpose. Subsequent transfers of copyright cannot revoke this permission. All other uses of the document are conditional on the consent of the copyright owner. The publisher has taken technical and administrative mea- sures to assure authenticity, security and accessibility. According to intellectual property law the author has the right to be mentioned when his/her work is accessed as described above and to be protected against infringement. For additional information about the Link¨opingUniversity Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: http://www.ep.liu.se/

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Hansevi, 2008. 43