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


 Philip Carter
 3 months ago
 Views:
Transcription
1 Examensarbete The Hawking mass for ellipsoidal surfaces in Minkowski and Schwarzschild spacetimes Daniel Hansevi LiTH  MAT  EX / SE
2
3 The Hawking mass for ellipsoidal surfaces in Minkowski and Schwarzschild spacetimes Applied Mathematics, Linköpings Universitet Daniel Hansevi LiTH  MAT  EX / SE Examensarbete: 30 hp Level: D Supervisor: Göran Bergqvist, Applied Mathematics, Linköpings Universitet Examiner: Göran Bergqvist, Applied Mathematics, Linköpings Universitet Linköping: June 008
4
5 Avdelning, Institution Division, Department Matematiska Institutionen LINKÖPING SWEDEN Datum Date June 008 Språk Language Svenska/Swedish Rapporttyp Report category Licentiatavhandling ISBN ISRN x x LiTH  MAT  EX / SE Engelska/English Examensarbete Cuppsats Serietitel och serienummer ISSN Duppsats Övrig rapport Title of series, numbering URL för elektronisk version Titel Title The Hawking mass for ellipsoidal surfaces in Minkowski and Schwarzschild spacetimes Författare Author Daniel Hansevi Sammanfattning Abstract In general relativity, the nature of mass is nonlocal. However, an appropriate definition of mass at a quasilocal 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 closedform expression for the Hawking mass of ellipsoidal 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 sphere. Numerical results for Schwarzschild are also presented. Nyckelord Keyword Hawking mass, Quasilocal mass, General relativity, Ellipsoidal surface.
6 vi
7 Abstract In general relativity, the nature of mass is nonlocal. However, an appropriate definition of mass at a quasilocal 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 closedform expression for the Hawking mass of ellipsoidal 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 sphere. Numerical results for Schwarzschild are also presented. Keywords: Hawking mass, Quasilocal mass, General relativity, Ellipsoidal surface. Hansevi, 008. vii
8 viii
9 Contents 1 Introduction Background Purpose Chapter outline Mathematical preliminaries 3.1 Manifolds Foliations Tangent vectors forms Tensors Abstract notation Component notation Tensor algebra Tensor fields Abstract index notation Metric Curvature Covariant derivative Metric connection Parallel transportation Curvature Geodesics Tetrad formalism NewmanPenrose formalism Spin coefficients General relativity Solutions to the Einstein field equation Minkowski spacetime Schwarzschild spacetime Mass in general relativity Gravitational energy/mass Nonlocality of mass Total mass of an isolated system Asymptotically flat spacetimes ADM and BondiSachs mass Hansevi, 008. ix
10 x Contents 4.3 Quasilocal mass The Hawking mass Definition Interpretation Method for calculating spin coefficients Hawking mass for a sphere In Minkowski spacetime In Schwarzschild spacetime Results Hawking mass in Minkowski spacetime Closedform expression of the Hawking mass Limit when approaching a metric sphere Limit along a foliation Hawking mass in Schwarzschild spacetime Limit along a foliation Numerical evaluations Discussion Conclusions Future work A Maple Worksheets 37 A.1 Null tetrad A. Minkowski A.3 Schwarszchild
11 List of Figures.1 A manifold M and two overlapping coordinate patches A foliation of a manifold M Intuitive picture of a tangent space Picture of a 1form Null cone Parallel transportation in a plane and on the surface of a sphere Deviation vector y a between two nearby geodesics λ s and λ s Spin coefficients Oblate spheroid m H (S 1 ) plotted against parameter ξ A curve given by r = 1 + ε sin θ m H ( S r ) plotted against 4 r 60 for some values of ε m H ( S r ) plotted against.3 r 16 for some values of ε m H ( S r ) plotted against.3 r 16 for ε = ω/r Hansevi, 008. xi
12 xii List of Figures
13 Chapter 1 Introduction 1.1 Background In general relativity, the nature of the gravitational field is nonlocal, 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 quasilocal level, that is, for the mass within a compact spacelike 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 quasilocal mass originates from a paper about gravitational radiation written by Stephen Hawking [5] in 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 sphere, and it has been shown to have various desirable properties [13]. To reach an appropriate definition for quasilocal 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. Purpose Even though Hawking s expression was given for the mass contained in a spacelike sphere, it can be calculated for a general spacelike surface. In this thesis we will calculate the Hawking mass for spacelike ellipsoidal surfaces, both in flat Minkowski spacetime and curved Schwarzschild spacetime. Hansevi,
14 Chapter 1. Introduction 1.3 Chapter outline Chapter 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 NewmanPenrose 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 localized. 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  surfaces in both Minkowski spacetime and Schwarzschild spacetime. The main result is the closedform expression for the Hawking mass of an 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.
15 Chapter Mathematical preliminaries This chapter provides a short introduction of the mathematics needed for a mathematical formulation of the general theory of relativity..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 twodimensional 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, noncompact 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 C k ndimensional manifold M is a set M together with a maximal C k atlas {U α φ α }, that is, the collection of all charts (U α, φ α ) where φ α are bijective maps from subsets U α M to open subsets φ α (U α ) R n such that: (i) {U α } cover M, that is, each element in M lies in at least one U α ; (ii) if U α U β is nonempty, then the transition map (see figure.1, page 4) φ β φ 1 α : φ α (U α U β ) φ β (U α U β ), (.1) is a C k map of an open subset of R n to an open subset of R n. Each (U α, φ α ) is a local coordinate patch with coordinates x α (α = 1,..., n) defined by φ α. In the overlap U α U β of two coordinate patches (U α, φ α ) and (U β, φ β ), the coordinates x α are C k functions of the coordinates x β, and vice versa. M is said to be Hausdorff 1 if for every distinct p, q M (p q) there exist two subsets U α M and U β M such that p U α, q U β and U α U β =. 1 Felix Hausdorff ( ), German mathematician Hansevi,
16 4 Chapter. Mathematical preliminaries M p Uα φ α x α φ β φ 1 α U β φ β x β R n Figure.1: A manifold M and two overlapping coordinate patches. Furthermore, it is natural to introduce the notion of a function on M and the notion of a curve in M as follows. A (realvalued) function f on a C k manifold M is a map f: M R. It is said to be of class C r (r k) at p M if the map f φ 1 α : R n R in any coordinate patch (U α, φ α ) holding p is a C r function of the coordinates at p. A C k curve in a manifold M is a map λ of an open interval I R M such that for any coordinate patch (U α, φ α ), the map φ α λ: I R n is C k. 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.. 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 ndimensional manifold M is a decomposition of M into a union of disjoint connected subsets {L a } a A, called the leaves of the foliation, with the property that for every point p M there is a coordinate patch (U α, φ α ) holding p such that for each leaf L a, φ α : (U α L a ) (x 1,..., x m, x m+1,..., x n ) R n, (.) where x m+1,..., x n are constants. See figure.. M φ α R n m U α p R m Figure.: A foliation of a manifold M.
17 .3. Tangent vectors 5.3 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 onetoone 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 M be an ndimensional manifold and let F be the collection of all smooth, realvalued functions on M. A tangent vector, or vector for short, X at a point p M, is a map X: F R such that for all f, g F and all α, β R: (i) X(αf + βg) = αx(f) + βx(g) (ii) X(fg) = f(p)x(g) + g(p)x(f) (linear); (Leibniz rule). Let (U, φ) be a coordinate patch, with coordinates x µ, holding p. For µ = 1,..., n define the map X µ : F R by X µ (f) := x µ (f φ 1 ). (.3) φ(p) It is shown in [14], that X 1,..., X n are linearly independent tangent vectors that span an ndimensional vector space at p. We call this vector space the tangent space at p and denote it by T p (M), or just T p if the manifold is given by the context. The basis {X 1,..., X n } is called a coordinate basis and is usually denoted by { / x 1,..., / x n }. Thus, an arbitrary tangent vector X can be expressed as X = n X µ X µ =: µ=1 n µ=1 X µ x µ, (.4) where (X 1,..., X n ) R n 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.3. T p (M) p X M Figure.3: Intuitive picture of a tangent space.
18 6 Chapter. Mathematical preliminaries.4 1forms Let M be an ndimensional manifold. Given a point p M, let T p be the tangent space at p. Let Tp be the space of all linear maps ω: T p R. (.5) Tp is called the dual space of T p, 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 1forms. The number which ω maps a vector X T p into, is often written as ω, X. If {e 1,..., e n } is a basis in T p, then there exists an associated dual basis {e 1,..., e n } of Tp consisting of 1forms e 1,..., e n defined by the property e µ, e ν = δ µ ν := { 1, µ = ν 0, µ ν. (.6) To see this, for every ω T p, we define ω µ := ω, e µ for µ = 1,..., n. Let X be an arbitrary vector in T p. Then ω, X = ω, µ X µ e µ = α X µ ω, e µ = µ ω µ X µ = µ,ν ω µ X ν δ µ ν (.6) = ω µ X ν e µ, e ν µ,ν = µ ω µ e µ, ν X ν e ν = µ ω µ e µ, X. (.7) Since X was arbitrary, it follows that ω = µ ω µ e µ. (.8) Thus every 1form can be written as a linear combination of e 1,..., e n. Given a coordinate basis { / x 1,..., / x n } for T p, the associated dual basis for T p is the basis {dx 1,..., dx n } of the so called coordinate differentials. For a given 1form ω, there is a subspace of T p defined by all vectors X for which ω, X is constant. Therefore, a 1form can be pictured as planes, where ω, X is the number of planes that X is piercing, see figure.4. X ω, X = 3.5 p Y ω, Y = 0 Figure.4: Picture of a 1form.
19 .5. Tensors 7.5 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..5.1 Abstract notation Let M be an ndimensional manifold and let p be a point in M. The multilinear map S: Tp Tp T p T p R (.9) }{{}}{{} 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 1form is a tensor of type (0, 1). Since T p is a finite dimensional vector space, it is (algebraicly) reflexive, and therefore the second (algebraic) dual space Tp is isomorphic to T p. Thus, we can identify every element in Tp with a unique element in T p and we consider a tangent vector as a tensor of type (1, 0)..5. Component notation Let (U, φ) and (U, φ ) be two overlapping coordinate patches holding a point p M, with coordinates related by x µ = x µ (x 1,..., x n ). (.10) An object with components S µ1...µr ν 1...ν s in (U, φ) and S µ 1...µ r ν 1...ν s in (U, φ ) is called an (r, s)tensor at p under the transformation x µ x µ, if S µ 1...µ r ν 1...ν s = n µ 1,...,µ r,ν 1,...ν s=1 S µ1...µr ν 1...ν s x µ 1 x µ1... xµ r x ν1... xνs. (.11) x µr x ν 1 x ν s A (0, 1)tensor (1form) 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 by its components S µ1 µr ν 1 ν s with respect to a basis. Suppose that S is a (1, )tensor and that {e µ }, {e ν } are dual bases. Define the basis component S µ νρ := S(e µ, e ν, e ρ ) R. (.1) Let ω = µ ω µe µ Tp and X = ν Xν e ν, Y = ρ Y ρ e ρ T p. Then it follows that S(ω, X, Y) = S( µ ω µ e µ, ν X ν e ν, ρ = S(e µ, e ν, e ρ ) ω µ X ν Y ρ µ,ν,ρ Y ρ e ρ ) = µ,ν,ρ S µ νρ ω µ X ν Y ρ. (.13) The components of S satisfy the tensor transformation law (.11).
20 8 Chapter. Mathematical preliminaries.5.3 Tensor algebra Let M be an ndimensional manifold and let p be a point in M. Assume that S and T are (r, s)tensors at p and that S is an (r, s )tensor at p. Furthermore, assume that ω i T p and X j T p for i = 1,..., r + r and j = 1,..., s + s. 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: (S + T)(ω 1,..., ω r, X 1,..., X s ) = S(ω 1,..., ω r, X 1,..., X s ) + T(ω 1,..., ω r, X 1,..., X s ); (.14) (αs)(ω 1,..., ω r, X 1,..., X s ) = α S(ω 1,..., ω r, X 1,..., X s ). (.15) With addition and scalar multiplication defined as above, the space of all (r, s)tensors at p M, constitutes a vector space of dimension n r+s. The outer product of S and S, denoted by S S, is the (r +r, s+s )tensor defined by, (S T)(ω 1,..., ω r+r, X 1,..., X s+s ) = S(ω 1,..., ω r, X 1,..., X s ) S (ω r+1,..., ω r+r, X s+1,..., X s+s ). (.16) The contraction with respect to the ith (1form) and j th (tangent vector) slots is a map from an (r, s)tensor to an (r 1, s 1)tensor defined by, (CjS)(ω i 1,..., ω i 1, ω i+1,..., ω r ; X 1,..., X j 1, X j+1,..., X s ) = n S(..., }{{} e k,... ;..., e k,...), (.17) k=1 ith slot }{{} jth slot where {e k } and {e k } are dual bases of T p and T p, respectively..5.4 Tensor fields It is natural to define a C k tensor field of type (r, s) on a manifold M as an assignment of an (r, s)tensor at each p M such that the components with respect to any coordinate basis are C k functions. We call a C tensor field a smooth tensor field. A vector field is a tensor field of type (1, 0), and a 1form field is a tensor field of type (0, 1)..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, basisindependent equations that will hold between tensors. It is convenient to introduce a notation called abstract index notation [10]. In this notation, an (r, s)tensor S is written as S a1 ar b 1 b s, where the indices are abstract markers telling us what type of tensor it is. Assume that S is a (1, )tensor and that T is a (3, )tensor. In abstract index notation, we write
21 .6. Metric 9 the outer product of S and T as S a bc T def gh, and the contraction of S with respect to the first slots as S a 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 letters and indices of the latter with lowercase greek letters, for example, S µ νρ denotes a basis component of the (1, )tensor S a 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..6 Metric A metric g ab on a manifold M is a nonsingular symmetric tensor field. Thus, for every tangent space T p of M: (i) g ab u a v b = g ba u a v b for every u a, v b T p (symmetric); (ii) g ab u a v b = 0 for every v b T p implies that u a = 0 (nonsingular). The metric has the structure of a (not necessarily positive definite) inner product on every tangent space of the manifold. If g ab u a v b = 0, then the vectors u a and v b are said to be orthogonal. For any vector v a, the metric can be viewed as a linear map g ab v b : T p R, that is, a 1form. Since g ab is nonsingular, there is a onetoone correspondence between elements of T p and T p. Given a vector v a, we can apply the metric and get the corresponding 1form g ab v b, usually denoted by v a in order to make the correspondence with v a 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 u a and v a as g ab u a v b = u a v a. (.18) Assume that we have two coordinate patches overlapping a neighbourhood of a point p M and that their coordinates at p are related by x µ = x µ (x µ ). Then the basis components of g ab are related by (.19), that is, g µ ν = µ,ν x µ x ν g µν. (.19) x µ x ν It is always possible to find an orthonormal basis {v 1 a,..., v n a } such that v ia v j a = ±δ i j. (.0) The number of basis vectors for which (.0) equals 1 and the number of basis vectors for which (.0) equals 1 are basis independent and called the signature of the metric. A metric of signature (+ ) or ( + +) is called Lorentzian. We will follow the LandauLifshitz 3 timelike convention [8] and use a metric of signature (+ ) for spacetime. Hendrik Antoon Lorentz ( ), Dutch physicist. 3 Lev Davidovich Landau ( ) and Evgeny Mikhailovich Lifshitz ( ), Russian physicists.
22 10 Chapter. Mathematical preliminaries With a Lorentzian metric on M, all nonzero vectors in T p can be divided into three classes. With our particular choice of signature, a vector v a is said to be timelike if v a v a > 0, null if v a v a = 0, (.1) spacelike if v a v a < 0. Thus, a Lorentzian metric defines a certain structure on each T p, called a null cone; the set of null vectors form what looks like a double cone if we suppress one spatial dimension, see figure.5. future cone null vector timelike vector spacelike vector p past cone Figure.5: Null cone..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 paralleltransports 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.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.6: Parallel transportation in a plane and on the surface of a sphere.
23 .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 paralleltransported 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..7.1 Covariant derivative A connection, or covariant derivative operator, a on a smooth manifold M assigns to every vector field x a on M, a differential operator x a a that maps an arbitrary vector field y a on M into a vector field x a a y b such that for all vector fields x a, y a, z a and functions f on M: (i) x a a (y b + z b ) = x a a y b + x a a z b (ii) x a a (fy b ) = (x a a f)y b + f(x a a y b ) (iii) x a a f = x(f) (linear); (Leibniz rule); (consistency with the notion of tangent vectors). The vector field x a a y b is called the covariant derivative of y a with respect to x a, and the (1, 1)tensor a y b mapping x a to x a a y b the covariant derivative of y b. The definition of a can be extended to apply to any tensor field on M by the additional requirement that a when acting on contracted products should satisfy Leibniz rule..7. Metric connection A connection a is not uniquely defined by the above conditions. In [14], however, it is shown that if M is endowed with a metric g ab, then there exists a unique connection a with the properties that for all smooth functions f on M: (i) a g bc = 0 (ii) a b f b a f = 0 (compatible with metric); (torsionfree). This particular a is called the metric connection, or the LeviCivita 4 connection, on M, and has the properties that for all smooth vector fields y a on M: a y b = a y b + Γ b acy c and a y b = a y b Γ c aby c, (.) where a is an ordinary derivative, and the Christoffel 5 symbol Γ c ab is given by Γ c ab = 1 gcd( a g bd + b g ad d g ab ). (.3) Thus, in a given coordinate patch (U, φ) with coordinates x µ, the coordinate basis components of a y b are given by y ρ x µ + 1 ( g ρσ gνσ x µ + g µσ x ν g ) µν x σ y ν. (.4) σ,ν 4 Tullio LeviCivita ( ), Italian mathematician. 5 Elwin Bruno Christoffel ( ), German mathematician.
24 1 Chapter. Mathematical preliminaries.7.3 Parallel transportation Given a derivative operator a we can define the notion of parallel transport. A smooth vector field y a is said to be parallelly transported around a curve with tangent vector x a if the equation x a a y b = 0 (.5) is satisfied along the curve. The metric connection has the property that the inner product f = y a z a of any two smooth vector fields y a and z a remains unchanged when parallelly transported along any curve with tangent vector x a, since x a (f) = x a a (g bc y b z c ).7.4 Curvature = (x a a g }{{ bc )y b z c + (x a } a y b )g }{{} bc z c + (x a a z c )g }{{} bc y b. (.6) =0 =0 =0 Let ω a be any smooth 1form field on M. The Riemann 6 curvature tensor R abc d is the tensor field on M defined by, ( a b b a ) ω c =: R abc d ω d, (.7) 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 t a on M, the corresponding expression is ( a b b a ) t c = R abd c t d. (.8) The Ricci 8 tensor is defined by contraction with respect to the second and fourth slot, R ac := R abc b, and the Ricci scalar curvature is defined by further contraction, R := R a a. The Ricci tensor and scalar curvature are used to define the Einstein tensor, which is a fundamental tensor in general relativity. G ab := R ab 1 R g ab, (.9).7.5 Geodesics Let x a be a smooth vector field on M. From the theory of ordinary differential equations, we know that given a point p M, there exists a unique curve that passes through p and has the property that for each point on the curve, the tangent vector of the curve coincides with the corresponding vector of x a. Such a curve is called an integral curve. Let x a be a smooth vector field such that x a a x b = 0. Then the integral curves of x a are called geodesics. 6 Georg Friedrich Bernhard Riemann ( ), German mathematician. 7 Some authors reverse the sign of the lefthand side in the definition. 8 Gregorio RicciCurbastro ( ), Italian mathematician.
25 .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 M in a given direction x a T p. Let λ s (t) be a oneparameter family of geodesics and consider the twodimensional surface, with coordinates (t, s), spanned by λ s (t). The vector field x a = ( / t) a is tangent to the family of geodesics, thus x a a x b = 0, (.30) and the vector field y a = ( / s) a represents the deviation vector, which is the displacement from the geodesic λ s to an infinitesimally nearby geodesic λ s, see figure.7. λ s λ s x a x a y a Figure.7: Deviation vector y a between two nearby geodesics λ s and λ s. Let f be any smooth function on M. Since it follows that (x a a y b y a a x b ) b f = x a a (y b b f) y a a (x b b f) = x(y(f)) y(x(f)) = f t s f s t = 0, (.31) x a a y b = y a a x b. (.3) The relative acceleration z a, in the direction of y a, of a nearby geodesic when moving along the direction of x a, is given by z a = x c c (x b b y a ) (.3) = x c c (y b b x a ) = (x c c y b ) b x a + y b (x c c b x a ) (.8) = (y c c x b ) b x a + y b (x c b c x a ) R cbd a y b x c x d = y c c (x b b x a ) R cbd a y b x c x d (.5) = R cbd a y b x c x d. (.33) Thus, the geodesic deviation, that is, the acceleration of geodesics toward or away from each other, which is a characterization of the curvature of M, is determined by the Riemann curvature tensor. M is flat if and only if R abc d = 0.
26 14 Chapter. Mathematical preliminaries.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 satisfied by them. Let e i a, i = 1,, 3, 4, be smooth vector fields that are linearly independent at each point in spacetime (M, g ab ). Then the Ricci rotation coefficients are defined by γ kij := e k c e j a a e ic. (.34) It is shown in [] that γ ijk = 1 (λ ijk + λ kij λ jki ), where λ ijk = ( b e ja a e jb )e i a e k b, (.35) which is an efficient way of calculating the Ricci rotation coefficients, since there is no need to calculate any covariant derivatives..9 NewmanPenrose formalism The tetrad formalism with the choice of a particular type of null basis, introduced by Ezra Newman and Roger Penrose [9] in 196, is usually called the NewmanPenrose formalism. The basis {l a, n a, m a, m a } consists of null vectors, where l a and n a are real, m a and m a are complex conjugates, satisfying l a l a = n a n a = m a m a = m a m a = 0 (null); l a m a = l a m a = n a m a = n a m a = 0 (orthogonal); l a n a = m a m a = 1 (normalized). (.36).10 Spin coefficients In the NewmanPenrose formalism, the Ricci rotation coefficients are called spin coefficients, and are given in figure.8. κ = γ 311 κ = ν = γ 4 α = γ 14 + γ 344 ρ = γ 314 ρ = µ = γ 43 β = γ 13 + γ 343 σ = γ 313 σ = λ = γ 44 γ = γ 1 + γ 34 τ = γ 31 τ = π = γ 41 ε = γ 11 + γ 341 Figure.8: Spin coefficients. The spin coefficients ρ and ρ are of particular interest to us, since they are used in the definition of the Hawking mass.
27 Chapter 3 General relativity General relativity is the modern geometric theory of space, time and gravitation published by Albert Einstein [4] in In the theory, space and time are unified into spacetime (M, g ab ), which is represented by a smooth fourdimensional Hausdorff manifold M endowed with a Lorentzian metric g ab and a metric connection a. The presence of matter warps spacetime according to the Einstein field equation, G ab := R ab 1 R g ab = 8π T ab, (3.1) where G ab is the Einstein tensor that describes the curvature of M and T ab is the stressenergymomentum tensor describing the distribution 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. 3.1 Solutions to the Einstein field equation The Einstein field equation might look simple when written with tensors. However, it constitutes a system of coupled, nonlinear partial differential equations Minkowski spacetime The spacetime of special relativity Minkowski 3 spacetime is a solution to the vacuum field equation G ab = 0. It is the flat 4 spacetime (M, η ab ) given by the constant metric (η ab ) = (3.) 1 Some authors use a different sign in the definition of the Ricci tensor resulting in a minus sign in front of the righthand side. Furthermore, we use geometrized units where the gravitational constant G, and the speed of light c, are set equal to one. Brief explanation of general relativity by the student portrayed in chapter one of [8]. 3 Hermann Minkowski ( ), Russianborn German mathematician. 4 R abc d = 0. Hansevi,
28 16 Chapter 3. General relativity In the coordinate system implied by η ab, all geodesics appear straight, that is, they take the form, x a (t) = y a + z a t Schwarzschild spacetime One of the solutions of the vacuum field equation was discovered by Karl Schwarzschild 5 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 (nonrotating) star, planet, or black hole, and it remains one of the most important exact solutions. Schwarzschild spacetime is the curved spacetime (M, g ab ) given, in Schwarzschild coordinates, by the metric 1 M r ( ) 1 (g ab ) = 0 1 M r r 0. (3.3) r sin θ In these coordinates, the metric becomes singular at the surface r = M, 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. 5 Karl Schwarzschild ( ), German physicist.
29 Chapter 4 Mass in general relativity The stressenergymomentum tensor, T ab, 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 energy is associated with mass, 1 therefore T ab is in the righthand 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, T ab, 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 nonlinear Nonlocality of mass The contribution of gravitational mass should be included in a description for total mass in a spacelike volume of spacetime. The stressenergymomentum tensor T ab 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 (Riemannnormal 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. 1 According to E = mc ; Einstein s famous equation [3]. Hansevi,
30 18 Chapter 4. Mass in general relativity 4. 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 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.. ADM and BondiSachs mass In asymptotically flat spacetimes, the total mass can be determined by the asymptotic form of the metric. The ADM mass is the total mass measured at spacelike infinity, whereas the BondiSachs 3 mass is measured at future null infinity [14]. 4.3 Quasilocal mass A meaningful definition of mass at a quasilocal level, that is, for the mass within a compact spacelike surface, should have certain properties. For example, the quasilocal 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 BondiSachs 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 quasilocal level. Several attempts have been made, for example, the DouganMason 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 Kerr 4 black hole [1]. There is still no generally accepted definition. To reach an appropriate definition for quasilocal 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]. R. Arnowitt, S. Deser and C. Misner. 3 H. Bondi and R. K. Sachs. 4 A solution to the Einstein field equation found in 1963 by Roy Kerr. It describes spacetime outside a rotating black hole.
31 Chapter 5 The Hawking mass One of many suggestions that have been made for a definition of quasilocal mass originates from a paper about gravitational radiation written by Stephen Hawking [5] in 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 BondiSachs mass, respectively [13]. Its advantage is its simplicity, calculability and monotonicity for special families of surfaces. In Minkowski spacetime, the Hawking mass vanish for spheres. However, it can give negative results for general surfaces, for example for nonconvex surfaces. 5.1 Definition Let l a and n a be, respectively, the outgoing and the ingoing null vectors orthogonal to a spacelike sphere S and let m a and m a be tangent vectors to S. Then the Hawking mass is defined by m H (S) := Area(S) 16π ( ) ρρ ds. (5.1) π S 5. Interpretation Consider a oneparameter family of null geodesics, that is light rays, that intersects a circle in a plane. Following the geodesics in the future direction, the optical scalars, θ and σ, introduced by Rainer Sachs [1] in 1961, can be defined as θ = Re ρ and σ = Im ρ, (5.) and interpreted as the expansion and rotation, respectively, of the circle []. Both l a and n a are orthogonal to the surface, thus both ρ and ρ are real [10]. Hence ρ and ρ 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 sphere. Hansevi,
32 0 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.10), introduce a coordinate system such that for a given r, and t held fixed, 0 θ < π and 0 ϕ < π trace out the surface S. In the new coordinates, every vector with the first two components vanished lies in the tangent plane of S, as m a is required to do. The other two vectors, l a and n a, are parallel to the outgoing and ingoing null directions, respectively, thus it is convenient to set up a null tetrad (l a, n a, m a, m a ), by making the ansatz l a = ( A B C 0 ), n a = ( A B C 0 ), m a = ( 0 0 X iy ). (5.3) By imposing the conditions (.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 { ma m a = 0 (null vector) m a m a = 1 (normalization), (5.4) and pick one solution {X, Y }.. Solve the system { la l a = 0 (null vector) l a n a = 1 (normalization), (5.5) 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 l a m a = 0 (orthogonality). (5.6) The result of the procedure above is a null tetrad (l a, n a, m a, m a ) satisfying the conditions (.36). Calculate the spin coefficients with the use of the null tetrad by first calculating the required λ ijk given by (.35), λ 314 = ( b l a a l b ) m a m b λ 431 = ( b m a a m b ) m a l b λ 143 = ( b m a a m b ) l a m b λ 43 = ( b n a a n b ) m a m b λ 34 = ( b m a a m b ) m a n b λ 34 = ( b m a a m b ) n a m b then the spin coefficients follow easily as,, (5.7) ρ = γ 314 = (λ λ 431 λ 143 )/ ρ = γ 43 = (λ 43 + λ 34 λ 34 )/. (5.8)
33 5.4. Hawking mass for a sphere Hawking mass for a sphere In the following subsections, we demonstrate the method of section 5.3 by calculating the Hawking mass for a sphere 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.) can be written as, (g ab ) = r 0. (5.9) r sin θ By solving the equations (5.4), (5.5) and (5.6) in three steps, we obtain the null tetrad (l a, n a, m a, m a ), given by ( ) l a = , ( ) n a = , (5.10) ( ) m a 1 i = 0 0, r r sin θ and then it follows easily from (5.7) and (5.8) that The area of the sphere is the familiar ρρ = 1 r. (5.11) Area(S r ) = π π 0 0 r sin θ dθ dϕ = 4πr, (5.1) and the integral of the spin coefficients S r ρρ ds = π π sin θ dθ dϕ = π. (5.13) Finally, we can confirm the wellknown result that the Hawking mass vanish for all spheres in Minkowski spacetime, since ( 4πr m H (S r ) = ) 16π π ( π) = ( r ) 1 1 = 0. (5.14) 5.4. In Schwarzschild spacetime We repeat the calculations of the previous section, but this time for a centered sphere in Schwarzschild spacetime. The coordinates (t, r, θ, ϕ) of the
34 Chapter 5. The Hawking mass Schwarzschild metric (3.3) have the property that 0 θ < π and 0 ϕ < π trace out the surface S of a sphere, thus we do not have to change to another coordinate system. We obtain a null tetrad (l a, n a, m a, m a ) given by l a = n a = m a = from which it follows that ( r r M ( r r M ( r ) r M 0 0, r ) r M 0 0, (5.15) r ) i, r sin θ ρρ = r M r 3. (5.16) The area of the sphere is 4πr, and the integral of the spin coefficients S r ρρ ds = π π 0 0 r M r sin θ dθ dϕ = π(r M). (5.17) r Finally, we get the expected result for the Hawking mass of a centered sphere in Schwarzschild spacetime, ( 4πr m H (S r ) = ) π(r M) = r ( M ) = M. (5.18) 16π π r r
35 Chapter 6 Results In this chapter, we calculate the Hawking mass, with the aid of Maple 1, for ellipsoidal 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 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 arccosh ξ ξ 1 = 1 ξ 1 (ξ 1) + + O ( (ξ 1) 5/) Proof. Consider the equivalence, ξ = x + 1, x > 0 x = ξ 1, ξ > 1, (6.1) and the following Maclaurin expansions [11], x + 1 = 1 + x / x 4 /8 + O(x 6 ), (6.) ln(y + 1) = y y / + y 3 /3 y 4 /4 + y 5 /5 + O(y 6 ). (6.3) Perform a change of variable, arccosh ξ ξ 1 (def.) = ln(ξ + ξ 1) ξ 1 (6.1) = ln ( x x ) x, (6.4) and apply the Maclaurin expansions to the numerator, ln ( x x ) (6.) = ln ( 1 + x + x x4 8 + O(x6 ) ) (6.3) = x + x x4 8 + O(x6 ) 1 Mathematics software package from Waterloo Maple Inc. ( x + x x4 8 + O(x6 ) ) Hansevi,
36 4 Chapter 6. Results Divide through by x, and change back to ξ, ( x + x + + O(x4 ) ) 3 ( x + x + O(x4 ) ) ( x + O(x ) ) O(x 6 ) 5 = x x x O(x6 ). (6.5) arccosh ξ ξ 1 = 1 ξ 1 + 3(ξ 1) + O ( (ξ 1) 5/). (6.6) 6 40 Since ξ > 1, the ordoterm is actually Thus, let O ( (ξ 1) 5/) = O ( (ξ + 1) 5/ (ξ 1) 5/) = O ( (ξ 1) 5/). (6.7) f(ξ) = 1 ξ 1 + 3(ξ 1), (6.8) 6 40 and calculate the second degree Taylor expansion of f about ξ = 1. From f(1) = 1, f (1) = 1/3, and f (1) = 4/15 it follows that arccosh ξ ξ 1 = 1 ξ 1 (ξ 1) + + O ( (ξ 1) 5/). (6.9) 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 5). In Cartesian coordinates, the surface is given by the equation x + y A + z = 1. (6.10) B Let A and B depend on a variable, say r > 0, in the way given by A = ξ r and B = r. Then (6.10) is equivalent to x + y ξ + z = r, (6.11) which is the equation for an oblate spheroid, where r is the length of its minor axis. In this section, a closedform 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. (page 7). Furthermore, two corollaries regarding limits of the Hawking mass are proved.
37 6.1. Hawking mass in Minkowski spacetime 5 Figure 6.1: Oblate spheroid Closedform expression of the Hawking mass Theorem 6.1. Let (M, η ab ) be Minkowski spacetime. For ξ > 1, let S r be the spacelike oblate spheroids, in M, given by S r = { (x + y )/ξ + z = r : r > 0 }. Then the Hawking mass within S r is given by ( ) ( ) r m H Sr = 16 ξ + arccosh ξ ξ 5 ξ + arccosh ξ. ξ ξ 1 3 ξ 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 surface S r. This is accomplished by using the parametrized form of S r, x = ξ r sin θ cos ϕ y = ξ r sin θ sin ϕ z = r cos θ, 0 < r 0 θ < π 0 ϕ < π, (6.1) as new variables (t, r, θ, φ). Using the tensor transformation law (.19) yields the Minkowski metric (3.) in the new coordinates, (g ab ) = Ansatz (5.3) (cos θ + ξ sin θ) (ξ 1)r cos θ sin θ 0 0 (ξ 1)r cos θ sin θ r (ξ cos θ + sin θ) ξ r sin θ Solve equations (5.4), and pick a solution, say { X = 1 / ( r ξ cos θ + sin θ) Y = 1 / ( ξ r sin θ). (6.13). (6.14) Solve equations (5.5), and pick the solution given by A = 1/ and B = cos θ + ξ sin θ ξ r C (ξ 1) r cos θ sin θ C (cos θ + ξ sin θ). (6.15)
38 6 Chapter 6. Results Solve equation (5.6). The solution is given by (ξ 1) cos θ sin θ C = ξ r ξ cos θ + sin θ. (6.16) The result of this procedure is the null tetrad (l a, n a, m a, m a ), given by ( l a ξ cos = 1 θ + sin ) θ (ξ 1) cos θ sin θ ξ ξ r 0, ξ cos θ + sin θ ( n a ξ cos = 1 θ + sin θ (ξ ) 1) cos θ sin θ ξ ξ r 0, ξ cos θ + sin θ ( ) m a 1 = 0 0 r i, ξ cos θ + sin θ ξ r sin θ ( ) m a 1 = 0 0 r i. (6.17) ξ cos θ + sin θ ξ r sin θ Calculate the spin coefficients with the use of the null tetrad by first calculating the required λ ijk given by (5.7), λ 314 = ( b l a a l b )m a m b = 0, λ 431 = ( b m a a m b ) m a l b = λ 143 = ( b m a a m b )l a m b = λ 43 = ( b n a a n b ) m a m b = 0, λ 34 = ( b m a a m b )m a n b = ξ (1 + cos θ) + sin θ ξ r (ξ cos θ + sin θ) 3/, ξ (1 + cos θ) + sin θ ξ r (ξ cos θ + sin θ) 3/, ξ (1 + cos θ) + sin θ ξ r (ξ cos θ + sin θ) 3/, λ 34 = ( b m a a m b )n a m b ξ (1 + cos θ) + sin θ = ξ r (ξ cos θ + sin θ), 3/ (6.18) then the spin coefficients (5.8) follow easily as, ρ = γ 314 = ρ = γ 43 = ξ (1 + cos θ) + sin θ ξ r ( ξ cos θ + sin θ ) 3/, ξ (1 + cos θ) + sin θ ξ r ( ξ cos θ + sin θ ) 3/. (6.19) The surface area of S r is part of the expression for the Hawking mass. From the surface element ds = g θθ g ϕϕ (g θϕ ) dθ dϕ it follows that Area(S r ) = π π 0 0 ξ r sin θ ξ cos θ + sin θ dθ dϕ = π ξ r ( ξ ξ 1 + ln( ξ 1 + ξ) ) ξ 1
39 6.1. Hawking mass in Minkowski spacetime 7 ( = π ξ r ξ + arccosh ξ ), (6.0) ξ 1 Integration over S r yields π π 0 0 ρρ ds = π π 0 0 = π 6 = π 6 sin θ ( ξ (1 + cos θ) + sin θ ) ξ ( ξ cos θ + sin θ ) dθ dϕ 5/ ( ξ 3 + 7ξ) ξ arcsinh ( ξ 1) ( ξ ξ ξ ξ 1 arccosh ξ ξ 1 ). (6.1) Finally, from (6.0) and (6.1), the Hawking mass in S r follows as m H (S r ) = Area(Sr ) = r 16 ξ 16 π ( π ξ + arccosh ξ ξ 1 π π 0 0 ) ρρ ds ( ξ 5 ξ + arccosh ξ 3 ξ 1 ). (6.) As can be seen in figure 6., the Hawking mass becomes negative in Minkowski, even for a convex surface Figure 6.: m H (S 1 ) plotted against parameter ξ Limit when approaching a metric sphere Corollary 6.1. Let (M, η ab ) be Minkowski spacetime. Given an r > 0, the Hawking mass vanishes in the limit when ξ 1 +, that is, when S r tends to a sphere. Proof. It follows from lemma 6.1 that arccosh ξ ξ 1 = 1 ξ (ξ 1) 15 + O ( (ξ 1) 5/) = 1 + O(ξ 1), (6.3)
40 8 Chapter 6. Results and furthermore that ξ 5 3 ξ + arccosh ξ ξ 1 = ξ3 3 5 ξ O(ξ 1) 3 = ξ(ξ + 1) 3( ) ξ 1 + O(ξ 1) 3 = O(ξ 1). (6.4) Thus, it follows from (6.3), (6.4) and theorem 6.1 that m H (S r ) = ( ) r 16 ξ + arccosh ξ ξ 5 ξ + arccosh ξ ξ ξ 1 3 ξ 1 = r 1 + ξ + O(ξ 1) O(ξ 1) 0 as ξ 1 +. (6.5) ξ Limit along a foliation Corollary 6.. Let (M, η ab ) be Minkowski spacetime. For ω > 0, let {L r } r>0 be foliations of a spacelike 3surface Ω M. Suppose that the leaves are given by L r = { (x + y )/(1 + ω/r) + z = r : r > 0 }. Then the Hawking mass vanishes in the limit along all foliations {L r } r>0. Proof. Observe that L r = S r ξ=1+ω/r, (6.6) and let ξ = 1 + ω/r in lemma 6.1. Then ξ 1 + as r, and it follows that 1 + ω/r + arccosh (1 + ω/r) (1 + ω/r) 1 = 1 + ω r + 1 ω 3r + ω 15r + O ( 1 r 5/ ) = + ω 3r + ( ω 1 ) 15r + O r ( 5/ 1 ) = + O, (6.7) r and furthermore that (1 + ω/r) 5 arccosh (1 + ω/r) (1 + ω/r) + 3 (1 + ω/r) 1 ( = 3ω 15r + ω3 1 ) 3r 3 + O r 5/ ( 1 ) = O r. (6.8) From (6.7) and theorem 6.1, it follows that r ( 1 ) ( 1 ) m H (L r ) = ω + O O r r r ( 1 ) = O 0 as r. (6.9) r
41 6.. Hawking mass in Schwarzschild spacetime 9 6. 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 surface, exterior to the event horizon (r > M), that has a simple expression in spherical polar coordinates. For small ε > 0, let the curve given by r = 1 + ε sin θ, (6.30) rotate around the axis θ = 0. See figure 6.3. The surface of revolution, S, is approximately ellipsoidal. For ε = 0, it is a perfect sphere. θ = 0 θ r Figure 6.3: A curve given by r = 1 + ε sin θ. Let us introduce a coordinate system such that for a given r, and t held fixed, θ and ϕ trace out the spacelike surface S r. This is accomplished by the following change of variables, r r 1 + ε sin θ, r > M. (6.31) In the new coordinates, the Schwarzschild metric (3.3) is given by β rα rα3 β εr α cos θ sin θ β 0 (g ab ) = 0 εr α cos θ sin θ β r (α 3 β+ε r cos θ sin θ) αβ 0, r α sin θ (6.3) where α = 1 + ε sin θ > 1 and β = r 1 + ε sin θ M > 0. (6.33) By making the ansatz (5.3), solving equations (5.4), (5.5) and (5.6), we obtain a null tetrad (l a, n a, m a, m a ) given by, l a = ( rα β α3 β + ε r cos θ sin θ rα 3 ) C 0,
42 30 Chapter 6. Results n a = m a = ( rα β ( 0 0 α3 β + ε r cos θ sin θ rα 3 αβ r α 3 β + ε r cos θ sin θ ) C 0, (6.34) ) i, rα sin θ where ε cos θ sin θ C = rα α3 β + ε r cos θ sin θ. (6.35) We calculate the spin coefficients ρ and ρ. Unfortunately, their expressions in Schwarzschild spacetime are very long, so we omit writing them out. The surface element of S r is given by ds = and the surface area by g θθ g ϕϕ (g θϕ ) dθ dϕ = r α sin θ ε r cos θ sin θ + 1 dθ dϕ, (6.36) αβ Area( S r ) = π π 0 0 Since the expression of the Hawking mass for S r, m H ( S r ) = Area( S ( r ) π π ds. (6.37) π π 0 0 ) ρρ ds, (6.38) is rather complicated, we will rely on methods like Taylor expansion and numerical integration for further investigations Limit along a foliation Analogously to corollary 6., let { L r } r>m be foliations of a spacelike 3surface Ω M (now in Schwarzschild spacetime). Suppose that the leaves, Lr, are given by substituting ε = ω, r > M, (6.39) r into S r. It is easy to see that L r becomes more spherical the larger r gets. In this section, we will show that the limit of the Hawking mass of L r is M along all foliations { L r } r>m, that is, lim m H( L r ) = M. (6.40) r Since both ds and ρρ ds are independent of ϕ, we make the substitution (6.39), and let I 1 (ε) = ε Area( L ω/ε ) =: π π 0 Φ(ε, θ) dθ (6.41)
43 6.. Hawking mass in Schwarzschild spacetime 31 and I (ε) = π π 0 ρρ dθ =: π π 0 Ψ(ε, θ) dθ. (6.4) Under the reasonable assumption that Φ(ε, θ), Φ ε(ε, θ), Ψ(ε, θ), Ψ ε(ε, θ) and Ψ εε(ε, θ) are continuous in some neighbourhood ε < 1, it follows by Maclaurin expansion that and I 1 (ε) = π π 0 I (ε) = π Φ(0, θ) dθ + O(ε) = 4πω + O(ε), ε < 1, (6.43) π 0 Ψ(0, θ) dθ + π π 0 Ψ ε(0, θ) dθ ε + O(ε ) = π + 4πM ω ε + O(ε ), ε < 1. (6.44) Thus Area( L r ) = I 1 (ε)/ε = 4πω 1 ) ε + O( ε and π π 0 ρρ ds = π + 4πM r = 4πr + O(r), r > ω, (6.45) ( 1 ) + O r, r > ω. (6.46) By using the results (6.45) and (6.46), it follows that ( m H ( L 4πr + O(r) r ) = [ π + 4πM ( 1 )] ) + O 16π π r r = r ( 1 ) ( M ( 1 ) ) 1 + O + O r r r ( 1 ) ( ( 1 ) ) = 1 + O M + O M as r. r r (6.47) 6.. Numerical evaluations We evaluate the Hawking mass numerically with an adaptive Gaussian quadrature method. In numerical quadrature, an integral I(f) = b a f(x)dx, (6.48) is approximated by an npoint quadrature rule that has the form n Q n (f) = w i f(x i ), (6.49) i=1 where a x 1 < x < < x n b. The points x i are called nodes, and the multipliers w i are called weights. Carl Friedrich Gauss ( ), German mathematician known as princeps mathematicorum ( prince of mathematicians ).
44 3 Chapter 6. Results In Gaussian quadrature, both the nodes and the weights are optimally chosen, hence Gaussian quadrature has the highest possible accuracy for the number of nodes used. Furthermore, it is stable and Q n (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 e = e = 0.10 e = 0.15 K0.5 e = 0.0 K e = 0.5 Figure 6.4: m H ( S r ) plotted against 4 r 60 for some values of ε e = 0. e = 0. e = 0.4 e = 0.6 e = e = Figure 6.5: m H ( S r ) plotted against.3 r 16 for some values of ε Figure 6.6: m H ( S r ) plotted against.3 r 16 for ε = ω/r.
45 Chapter 7 Discussion 7.1 Conclusions In this thesis, we have derived a closedform expression for the Hawking mass of a spacelike oblate spheroid S r, that is, a rotationally symmetric ellipsoid, in Minkowski spacetime. If S r is given by, S r = { (x + y )/ξ + z = r : r > 0 }. Then the Hawking mass within S r is given by ( ) ( ) r m H Sr = 16 ξ + arccosh ξ ξ 5 ξ + arccosh ξ. ξ ξ 1 3 ξ 1 From this result, we can see that the Hawking mass can be negative even for convex surfaces in Minkowski spacetime. However, the limits along particular foliations, were shown to vanish. Furthermore, we studied the Hawking mass in Schwarzschild spacetime. Numerical calculations for approximately ellipsoidal 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 sphere. 7. 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 ReissnerNordström which is the generalization of Schwarzschild that includes electric charge. Hansevi,
46 34 Chapter 7. Discussion
47 Bibliography [1] Bergqvist, G., Quasilocal mass for event horisons, Class. Quantum Grav. 9, , (199) [] Chandrasekhar, S., The Mathematical Theory of Black Holes, Oxford University Press, New York, 1983 [3] Einstein, A., Zur Elektrodynamik bewegter Körper, Annalen der Physik 18, , (1905) [4] Einstein, A., Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 49, 7698, (1916) [5] Hawking, S. W., Gravitational Radiation in an Expanding Universe, J. Math. Phys. 9, , (1968) [6] Heath, M. T., Scientific Computing: An Introductory Survey, McGrawHill, 00 [7] Lawson, H. B., Jr., Foliations, Bull. Amer. Math. Soc. 80, , (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, , (196) [10] Penrose, R. and Rindler, W., Spinors and spacetime volume 1, Cambridge University Press, 1984 [11] Råde, L., Westergren, B., Mathematics Handbook for Science and Engineering, Studentlitteratur, Lund, 1995 [1] Sachs, R. K., Gravitational Waves in General Relativity. VI. The Outgoing Radiation Condition, Proc. Roy. Soc. (London) A 64, , (1961) [13] Szabados, L. B., QuasiLocal EnergyMomentum and Angular Momentum in GR: A Review Article, Living Rev. Relativity 7, (004), 4. [Online Article] (cited on ): [14] Wald, R. M., General Relativity, The University of Chicago Press, Chicago and London, 1984 Hansevi,
48 36 Bibliography
49 Appendix A Maple Worksheets A.1 Null tetrad # Determine a nulltetrad # Let m1=m and m=conjugate(m), # m1*m1 = m*m = 0 (null vectors) and m1*m = 1 (normalization) m1:=create([1],array(1..4,[0,0,x, I*Y])): m:=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])): m:=create([1],array(1..4,[0,0,x,i*sy])): sx:=solve((get_compts(prod(m1,lower(g,m,1),[1,1])))=1,x)[1]: m1:=create([1],array(1..4,[0,0,sx, I*eval(sY,X=sX)])): m:=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*m = n*m1 = n*m = 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(m,lower(g,m,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,m,1),[1,1]))), simplify(get_compts(prod(n, lower(g,m,1),[1,1]))), simplify(get_compts(prod(l, lower(g,n, 1),[1,1]))), simplify(get_compts(prod(m1,lower(g,m,1),[1,1]))); Hansevi,
50 38 Appendix A. Maple Worksheets A. 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<=*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[,]:=1: h_c[3,3]:=1: h_c[4,4]:=1: # Change of variables (theta and phi parametrize ellipsoids) x[1]:=t: x[]:=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()/, sqrt()//xi*sqrt(xi***cos(theta)**+sin(theta)**), sqrt()//xi/r*(xi**1)*cos(theta)*sin(theta) /sqrt(xi***cos(theta)**+sin(theta)**), 0])): n:=create([1],array(1..4,[sqrt()/, sqrt()//xi*sqrt(xi***cos(theta)**+sin(theta)**), sqrt()//xi/r*(xi**1)*cos(theta)*sin(theta) /sqrt(xi***cos(theta)**+sin(theta)**), 0])): m1:=create([1],array(1..4,[0,0, sqrt()//r/sqrt(xi***cos(theta)**+sin(theta)**), sqrt()/*i/xi/r/sin(theta)])): m:=create([1],array(1..4,[0,0, sqrt()//r/sqrt(xi***cos(theta)**+sin(theta)**), sqrt()/*i/xi/r/sin(theta)])): # Calculate spin coefficients e[1]:=get_compts(l): e[]:=get_compts(n): e[3]:=get_compts(m1): e[4]:=get_compts(m): f[1]:=get_compts(lower(g,l, 1)): f[]:=get_compts(lower(g,n, 1)): f[3]:=get_compts(lower(g,m1,1)): f[4]:=get_compts(lower(g,m,1)): L314:=0: L431:=0: L143:=0: L43:=0: L34:=0: L34:=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]:
51 A.3. Schwarszchild 39 L43:=L43 +(diff(f[][i],coords[j])diff(f[][j],coords[i]))*e[4][i]*e[3][j]: L34:=L34 +(diff(f[4][i],coords[j])diff(f[4][j],coords[i]))*e[3][i]*e[][j]: L34:=L34 +(diff(f[3][i],coords[j])diff(f[3][j],coords[i]))*e[][i]*e[4][j]: end do; end do; # rho = gamma[314], rho = gamma[43] rho_ :=simplify(1/*(l314+l431l143)); rho_:=simplify(1/*(l43+l34l34)); # Calculate surface area of ellipsoid area_element:=simplify(sqrt(g_c[3,3]*g_c[4,4]g_c[3,4]**)); area:=simplify(int(int(area_element,theta=0..pi),phi=0..*pi)); # Calculate integral of spin coefficients spin_integrand:=(rho_*rho_*area_element); spin_integral:=simplify( int(eval(int(spin_integrand, theta ),theta=pi) eval(int(spin_integrand, theta ),theta=0),phi=0..*pi)); # Calculate Hawking mass mh_:=simplify(sqrt(area/16/pi)*(1+1//pi*spin_integral)); # Simplification by hand mh:=sqrt()/(16*sqrt(xi))*r*sqrt(xi+arccosh(xi)/sqrt(xi**1)) *(xi*(*xi**5)/3+arccosh(xi)/sqrt(xi**1)); # Limit along foliation limit(eval(mh,xi=1+omega/r),r=infinity); plot(eval(mh,r=1),xi=1..4,3..1, thickness=,axes=frame,gridlines=true,font=[times,1,1]); 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>*m,epsilon>=0,theta>=0,theta<=pi/,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*m/r: h_c[,]:=1/(1*m/r): h_c[3,3]:=r**: h_c[4,4]:=r***sin(theta)**: # Change of variables (theta and phi parametrize ellipsoids) F:=sqrt(1+epsilon*sin(theta)**): x[1]:=t: x[]:=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[],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)**), beta=r*sqrt(1+epsilon*sin(theta)**)*m}: # Simplification by hand g_c:=array(sparse,1..4,1..4): g_c[1,1]:=beta/alpha/r:
52 40 Appendix A. Maple Worksheets g_c[,]:=alpha**3*r/beta: g_c[3,3]:=r***(alpha**3*beta+epsilon***r*sin(theta)***cos(theta)**) /(alpha*beta): g_c[4,4]:=alpha***r***sin(theta)**: g_c[,3]:=epsilon*alpha*r***sin(theta)*cos(theta)/beta: g_c[3,]:=epsilon*alpha*r***sin(theta)*cos(theta)/beta: g:=create([1,1],eval(g_c)): # Null tetrad l:=create([1],array(1..4,[ sqrt()/*sqrt(r*alpha/beta), sqrt()/*sqrt(alpha**3*beta+r*epsilon***cos(theta)***sin(theta)**) /(sqrt(r)*alpha**3), sqrt()/*epsilon*sin(theta)*cos(theta) /(alpha*sqrt(r)*sqrt(alpha**3*beta +r*epsilon***cos(theta)***sin(theta)**)), 0])): n:=create([1],array(1..4,[ sqrt()/*sqrt(r*alpha/beta), sqrt()/*sqrt(alpha**3*beta+r*epsilon***cos(theta)***sin(theta)**) /(sqrt(r)*alpha**3), sqrt()/*epsilon*sin(theta)*cos(theta) /(alpha*sqrt(r)*sqrt(alpha**3*beta +r*epsilon***cos(theta)***sin(theta)**)), 0])): m1:=create([1],array(1..4,[ 0,0, sqrt()/*sqrt(beta)*sqrt(alpha) /(r*sqrt(alpha**3*beta+r*epsilon***cos(theta)***sin(theta)**)), sqrt()/*i/(alpha*r*sin(theta))])): m:=create([1],array(1..4,[ 0,0, sqrt()/*sqrt(beta)*sqrt(alpha) /(r*sqrt(alpha**3*beta+r*epsilon***cos(theta)***sin(theta)**)), sqrt()/*i/(alpha*r*sin(theta))])): # Calculate spin coefficients e[1]:=simplify(eval(get_compts(l), AB)): e[]:=simplify(eval(get_compts(n), AB)): e[3]:=simplify(eval(get_compts(m1),ab)): e[4]:=simplify(eval(get_compts(m),ab)): f[1]:=simplify(eval(get_compts(lower(g,l, 1)),AB)): f[]:=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,m,1)),ab)): L314:=0: L431:=0: L143:=0: L43:=0: L34:=0: L34:=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]: L43:=L43 +(diff(f[][i],coords[j])diff(f[][j],coords[i]))*e[4][i]*e[3][j]: L34:=L34 +(diff(f[4][i],coords[j])diff(f[4][j],coords[i]))*e[3][i]*e[][j]: L34:=L34 +(diff(f[3][i],coords[j])diff(f[3][j],coords[i]))*e[][i]*e[4][j]: end do; end do; # rho = gamma[314], rho = gamma[43] rho_ :=simplify(1/*(l314+l431l143));
53 A.3. Schwarszchild 41 rho_:=simplify(1/*(l43+l34l34)); # Hawking energy mh:=sqrt(area/16/pi)*(1+1//pi*spin_integral): # Surface area of ellipsoid area_element:=simplify(eval(sqrt(g_c[3,3]*g_c[4,4]g_c[3,4]**),ab)): area:=simplify(*pi*int(area_element,theta=0..pi, method=_gquad)): # Integral of spin coefficients spin_integral:=*pi*int(simplify(rho_*rho_)*area_element,theta=0..pi, method=_gquad): # Limit along foliation Phi:=simplify(epsilon***eval(area_element,r=omega/epsilon)): *Pi*int(simplify(eval(Phi,epsilon=0)),theta=0..Pi)/epsilon**: Psi_:=simplify(eval(simplify(rho_*rho_*area_element),r=omega/epsilon)): *Pi*int(simplify(eval(Psi_,epsilon=0)),theta=0..Pi) +*Pi*int(simplify(eval(diff(Psi_,epsilon),epsilon=0)),theta=0..Pi)*epsilon: eval(series(eval(mh,r=1/epsilon),epsilon=0,),epsilon=1/r): # Plot eps1:=[0,1/5,/5,3/5,4/5,1]: start1:=.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 0 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=1,color=black), plot([points1[1]]),pointplot([points1[1]]), plot([points1[]]),pointplot([points1[]]), 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.], [16,0.84,epsilon_=0.4],[16,0.69,epsilon_=0.6], [16,0.5,epsilon_=0.8],[16,0.9,epsilon_=1.0]}, align={right},font=[times,roman,1]), axes=boxed, axis[1]=[mode=log,gridlines=[8,thickness=1,subticks=false,color=grey]], axis[]=[gridlines=[color=grey]], view=[..16, ] ); eps:=[0,0.05,0.1,0.15,0.,0.5]: start:=4: base:=1.4: for i from 1 to 6 do R:=start*base**0: points[i]:=[r,evalf(eval(mh,{m=1,r=r,epsilon=eps[i]}))]: for a from 1 to 15 do R:=start*base**a: points[i]:=points[i],[r,evalf(eval(mh,{m=1,r=r,epsilon=eps[i]}))]: end do: end do: plots[display]({ setoptions(symbol=diamond,symbolsize=1,color=black), plot([points[1]]),pointplot([points[1]]), plot([points[]]),pointplot([points[]]),
54 4 Appendix A. Maple Worksheets plot([points[3]]),pointplot([points[3]]), plot([points[4]]),pointplot([points[4]]), plot([points[5]]),pointplot([points[5]]), plot([points[6]]),pointplot([points[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_=0.0],[700,1.18,epsilon_=0.5]}, align={right},font=[times,roman,1]), axes=boxed, axis[1]=[mode=log,gridlines=[1,thickness=1,subticks=false,color=grey]], axis[]=[gridlines=[color=grey]], view=[4..700, ] ); om:=[0.0,0.,0.4,0.6,0.8,1]: start3:=.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 0 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=1,color=black), plot([points3[1]]),pointplot([points3[1]]), plot([points3[]]),pointplot([points3[]]), 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[]=[gridlines=[color=grey]], view=[...16, ] );
55 Copyright The publishers will keep this document online on the Internet  or its possible replacement  for a period of 5 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 noncommercial 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 measures 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 Linköping University Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: Upphovsrätt Detta dokument hålls tillgängligt på Internet  eller dess framtida ersättare  under 5 år från publiceringsdatum under förutsättning att inga extraordinära omständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut enstaka kopior för enskilt bruk och att anvnda det oförändrat för ickekommersiell forskning och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart. För ytterligare information om Linköping University Electronic Press se förlagets hemsida c 008, Daniel Hansevi Hansevi,