Examensarbete
The Hawking mass for ellipsoidal 2-surfaces in
Minkowski and Schwarzschild spacetimes
Daniel Hansevi
LiTH - MAT - EX - - 08/14 - - SEThe 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¨oran Bergqvist,
Applied Mathematics, Link¨opings Universitet Examiner: G¨oran Bergqvist,
Applied Mathematics, Link¨opings Universitet Link¨oping: June 2008
Matematiska Institutionen 581 83 LINK ¨OPING SWEDEN
June 2008
x x LiTH - MAT - EX - - 08/14 - - SE
The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild space-times
Daniel Hansevi
In general relativity, the nature of mass is non-local. However, an appropriate def-inition 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.
Hawking mass, Quasi-local mass, General relativity, Ellipsoidal surface.
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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 characteri-zation 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.
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 . . . 82.5.5 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
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
List of Figures
2.1 A manifold M and two overlapping coordinate patches. . . 4
2.2 A foliation of a manifold M. . . 4
2.3 Intuitive picture of a tangent 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 2.7 Deviation vector ya between two nearby geodesics λsand λs0. . . 13
2.8 Spin coefficients. . . 14
6.1 Oblate spheroid. . . 25
6.2 mH(S1) plotted against parameter ξ. . . 27
6.3 A curve given by r =p1 + ε sin2θ. . . 29
6.4 mH( ˜Sr) plotted against 4 ≤ r ≤ 620 for some values of ε. . . 32
6.5 mH( ˜Sr) plotted against 2.3 ≤ r ≤ 16 for some values of ε. . . 32
6.6 mH( ˜Sr) plotted against 2.3 ≤ r ≤ 16 for ε = ω/r. . . 32
Chapter 1
Introduction
1.1
Background
In general relativity, the nature of the gravitational field is non-local, and there-fore 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 space-like 2-sphere, it can be calculated for a general spacespace-like 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.
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 oper-ator 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 formal-ism 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 interpreta-tion. 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 space-time. 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 M is a set M together with a maximal Ck
atlas{Uαφα}, that is, the collection of all charts (Uα, φα) where φαare bijective maps from subsets Uα⊂ M to open subsets φα(Uα) ⊂ Rn such that:
(i) {Uα} cover M, that is, each element in M lies in at least one Uα;
(ii) if Uα∩ Uβ is non-empty, then the transition map (see figure 2.1, page 4)
φβ◦ φ−1α : φα(Uα∩ Uβ) → φβ(Uα∩ Uβ), (2.1)
is a Ck map of an open subset of Rn to an open subset of Rn.
Each (Uα, φα) is a local coordinate patch with coordinates xα (α = 1, . . . , n)
defined by φα. In the overlap Uα∩ Uβ of two coordinate patches (Uα, φα) and
(Uβ, φβ), the coordinates xα are Ck functions of the coordinates xβ, and vice
versa.
M is said to be Hausdorff1 if for every distinct p, q ∈ M (p 6= q) there exist
two subsets Uα⊂ M and Uβ⊂ M such that p ∈ Uα, q ∈ Uβ and Uα∩ Uβ = ∅.
1Felix Hausdorff (1868-1942), German mathematician
4 Chapter 2. Mathematical preliminaries Rn M Uα Uβ φα φβ φβ◦ φ−1α p xα xβ
Figure 2.1: 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 (real-valued) function f on a Ck manifold M is a map f: M → R. It is
said to be of class Cr (r ≤ k) at p ∈ M if the map f ◦ φ−1
α : Rn → R in any
coordinate patch (Uα, φα) holding p is a Cr function of the coordinates at p.
A Ck curve in a manifold M is a map λ of an open interval I ∈ R → M
such that for any coordinate patch (Uα, φα), the map φα◦ λ: I → Rn is Ck.
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 M is a decomposition of M into a union of disjoint connected subsets {La}a∈A,
called the leaves of the foliation, with the property that for every point p ∈ M there is a coordinate patch (Uα, φα) holding p such that for each leaf La,
φα: (Uα∩ La) → (x1, . . . , xm, xm+1, . . . , xn) ∈ Rn, (2.2)
where xm+1, . . . , xn are constants. See figure 2.2.
M Rm Uα p Rn−m φα
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 M be an n-dimensional manifold and let F be the collection of all smooth, real-valued 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) (linear); (ii) X(fg) = f(p)X(g) + g(p)X(f) (Leibniz’ rule).
Let (U, φ) be a coordinate patch, with coordinates xµ, holding p. For µ =
1, . . . , n define the map Xµ: F → R by
Xµ(f) := ∂x∂µ(f ◦ φ−1) φ(p). (2.3)
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 Tp(M), or just Tp if the manifold is given 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 X = n X µ=1 XµX µ =: n X µ=1 Xµ ∂ ∂xµ, (2.4)
where (X1, . . . , Xn) ∈ Rnare the components of X with respect to the
coordi-nate 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.
M p
X Tp(M)
6 Chapter 2. Mathematical preliminaries
2.4
1-forms
Let M be an n-dimensional manifold. Given a point p ∈ M, let Tp be the
tangent space at p. Let T∗
p be the space of all linear maps
ω: Tp→ R. (2.5)
T∗
p is called the dual space of Tp, or the cotangent space at p, and is a vector
space of dimension n. Elements of T∗
p 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 hω, Xi.
If {e1, . . . , en} is a basis in Tp, then there exists an associated dual basis
{e1, . . . , en} of T∗
p consisting of 1-forms e1, . . . , en defined by the property
heµ, e
νi = δνµ :=
1, µ = ν
0, µ 6= ν . (2.6)
To see this, for every ω ∈ T∗
p, we define ωµ:= hω, eµi for µ = 1, . . . , n. Let X
be an arbitrary vector in Tp. Then
hω, Xi = hω,X µ Xµe µi = X α Xµhω, e µi = X µ ωµXµ = X µ,ν ωµXνδνµ (2.6)= X µ,ν ωµXνheµ, eνi = hX µ ωµeµ,X ν Xνe νi = hX µ ωµeµ, Xi. (2.7)
Since X was arbitrary, it follows that
ω = X
µ
ωµeµ. (2.8)
Thus every 1-form can be written as a linear combination of e1, . . . , en.
Given a coordinate basis {∂/∂x1, . . . , ∂/∂xn} for T
p, the associated dual
basis for T∗
p is the basis {dx1, . . . , dxn} of the so called coordinate differentials.
For a given 1-form ω, there is a subspace of Tp defined by all vectors X for
which hω, Xi is constant. Therefore, a 1-form can be pictured as planes, where hω, Xi is the number of planes that X is ‘piercing’, see figure 2.4.
X
p Y
hω, Xi = 3.5
hω, Yi = 0 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 M be an n-dimensional manifold and let p be a point in M. The multilinear map S: T∗ p × · · · × Tp∗ | {z } r factors × Tp× · · · × Tp | {z } s factors → R (2.9)
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 T∗∗
p is isomorphic to Tp. Thus, we can identify every
element in T∗∗
p 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 (U, φ) and (U0, φ0) be two overlapping coordinate patches holding a point
p ∈ M, with coordinates related by xµ0
= xµ0
(x1, . . . , xn). (2.10)
An object with components Sµ1...µr
ν1...νs in (U, φ) and Sµ 0 1...µ0rν0
1...ν0s in (U
0, φ0)
is called an (r, s)-tensor at p under the transformation xµ7→ xµ0
, if Sµ0 1...µ0r ν0 1...νs0 = n X µ1,...,µr,ν1,...νs=1 Sµ1...µr ν1...νs ∂xµ0 1 ∂xµ1 . . . ∂xµ0 r ∂xµr ∂xν1 ∂xν0 1. . . ∂xνs ∂xν0 s . (2.11)
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 by its components Sµ1···µr
ν1···νs with respect to a basis. Suppose that S is a
(1, 2)-tensor and that {eµ}, {eν} are dual bases. Define the basis component
Sµ
νρ := S(eµ, eν, eρ) ∈ R. (2.12)
Let ω = Pµωµeµ ∈ Tp∗ and X = PνXνeν, Y = PρYρeρ ∈ Tp. Then it
follows that S(ω, X, Y) = S(X µ ωµeµ, X ν Xνe ν, X ρ Yρe ρ) = X µ,ν,ρ S(eµ, e ν, eρ) ωµXνYρ = X µ,ν,ρ Sµ νρωµXνYρ. (2.13)
8 Chapter 2. Mathematical preliminaries
2.5.3
Tensor algebra
Let M be an n-dimensional manifold and let p be a point in M. Assume that S and T are (r, s)-tensors at p and that S0 is an (r0, s0)-tensor at p. Furthermore,
assume that ωi∈ T∗
p and Xj ∈ Tpfor i = 1, . . . , r + r0 and j = 1, . . . , s + s0.
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)(ω1, . . . , ωr, X 1, . . . , Xs) = S(ω1, . . . , ωr, X 1, . . . , Xs) + T(ω1, . . . , ωr, X1, . . . , Xs); (2.14) (αS)(ω1, . . . , ωr, X 1, . . . , Xs) = α S(ω1, . . . , ωr, X1, . . . , Xs). (2.15)
With addition and scalar multiplication defined as above, the space of all (r, s)-tensors at p ∈ M, constitutes a vector space of dimension nr+s.
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
, X1, . . . , Xs+s0) =
S(ω1, . . . , ωr, X
1, . . . , Xs) S0(ωr+1, . . . , ωr+r0, 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,
(Ci jS)(ω1, . . . , ωi−1, ωi+1, . . . , ωr; X1, . . . , Xj−1, Xj+1, . . . , Xs) = n X k=1 S(. . . , ek |{z} ith slot , . . . ; . . . , ek |{z} jth slot , . . .), (2.17)
where {ek} and {ek} are dual bases of Tp and Tp∗, respectively.
2.5.4
Tensor fields
It is natural to define a Ck 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 Ck 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]. In this notation, an (r, s)-tensor S is written as Sa1···ar
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
the outer product of S and T as Sa
bcTdefgh, and the contraction of S with
respect to the first slots as Saab.
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, Sµνρ
denotes a basis component of the (1, 2)-tensor Sabc.
Given a tensor equation written in the abstract index notation, the corre-sponding 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 M is a non-singular symmetric tensor field. Thus,
for every tangent space Tp of M:
(i) gabuavb = gbauavb for every ua, vb∈ Tp (symmetric);
(ii) gabuavb = 0 for every vb∈ Tp implies that ua = 0 (non-singular).
The metric has the structure of a (not necessarily positive definite) inner product on every tangent space of the manifold. If gabuavb = 0, then the
vectors ua and vb are said to be orthogonal.
For any vector va, the metric can be viewed as a linear map g
abvb: Tp→ R,
that is, a 1-form. Since gabis non-singular, there is a one-to-one correspondence
between elements of Tpand Tp∗. Given a vector va, we can apply the metric and
get the corresponding 1-form gabvb, 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
gabuavb = uava. (2.18)
Assume that we have two coordinate patches overlapping a neighbourhood of a point p ∈ M and that their coordinates at p are related by xµ0
= xµ0
(xµ).
Then the basis components of gabare related by (2.19), that is,
gµ0ν0 = X µ,ν gµν∂x µ ∂xµ0 ∂xν ∂xν0. (2.19)
It is always possible to find an orthonormal basis {v1a, . . . , vna} such that
viavja = ±δij. (2.20)
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-Lifshitz3 ‘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),
10 Chapter 2. Mathematical preliminaries
With a Lorentzian metric on M, all non-zero vectors in Tp can be divided
into three classes. With our particular choice of signature, a vector va is said
to be
timelike if vava > 0,
null if vava = 0,
spacelike if vava < 0.
(2.21) 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.
future cone past cone p spacelike vector timelike vector null vector
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.
u p
v q
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 M
assigns to every vector field xa on M, a differential operator xa∇a that maps
an arbitrary vector field ya on M into a vector field xa∇
ayb such that for all
vector fields xa, ya, za and functions f on M:
(i) xa∇a(yb+ zb) = xa∇ayb + xa∇azb (linear);
(ii) xa∇
a(fyb) = (xa∇af)yb + f(xa∇ayb) (Leibniz’ rule);
(iii) xa∇
af = x(f) (consistency with the notion of tangent vectors).
The vector field xa∇
ayb is called the covariant derivative of ya with respect to
xa, and the (1, 1)-tensor ∇
ayb mapping xa to xa∇ayb the covariant derivative
of yb.
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.
2.7.2
Metric connection
A connection ∇a is not uniquely defined by the above conditions. In [14],
how-ever, it is shown that if M is endowed with a metric gab, then there exists a
unique connection ∇awith the properties that for all smooth functions f on M:
(i) ∇agbc = 0 (compatible with metric);
(ii) ∇a∇bf − ∇b∇af = 0 (torsion-free).
This particular ∇a is called the metric connection, or the Levi-Civita4
connec-tion, on M, and has the properties that for all smooth vector fields ya on M: ∇ayb = ∂ayb + Γbacyc and ∇ayb = ∂ayb − Γcabyc, (2.22)
where ∂a is an ordinary derivative, and the Christoffel5 symbol Γcabis given by Γc
ab = 12gcd ∂agbd + ∂bgad − ∂dgab. (2.23)
Thus, in a given coordinate patch (U, φ) with coordinates xµ, the coordinate
basis components of ∇ayb are given by
∂yρ ∂xµ + 1 2 X σ,ν gρσ∂gνσ ∂xµ + ∂gµσ ∂xν − ∂gµν ∂xσ yν. (2.24)
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∇
ayb= 0 (2.25)
is satisfied along the curve.
The metric connection has the property that the inner product f = yaza
of any two smooth vector fields ya and za remains unchanged when parallelly
transported along any curve with tangent vector xa, since
xa(f) = xa∇ a(gbcybzc) = (xa∇ agbc | {z } =0 )ybzc + (xa∇ ayb | {z } =0 )gbczc + (xa∇azc | {z } =0 )gbcyb. (2.26)
2.7.4
Curvature
Let ωabe any smooth 1-form field on M. The Riemann6curvature tensorRabcd
is the tensor field on M defined by,
(∇a∇b − ∇b∇a) ωc =: Rabcdωd, (2.27)
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 M, the corresponding expression is
(∇a∇b − ∇b∇a) tc = −Rabdctd. (2.28)
The Ricci8 tensor is defined by contraction with respect to the second and
fourth slot, Rac := Rabcb, and the Ricci scalar curvature is defined by further
contraction, R := Raa. The Ricci tensor and scalar curvature are used to define
the Einstein tensor,
Gab := Rab − 12R gab, (2.29)
which is a fundamental tensor in general relativity.
2.7.5
Geodesics
Let xa 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 xa. Such
a curve is called an integral curve.
Let xa be a smooth vector field such that xa∇
axb = 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 ∈ M in a given direction xa∈ Tp.
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∇
axb = 0, (2.30)
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. ya xa xa λs λs0
Figure 2.7: Deviation vector ya between two nearby geodesics λ
sand λs0.
Let f be any smooth function on M. Since (xa∇ ayb − ya∇axb)∇bf = xa∇a(yb∇bf) − ya∇a(xb∇bf) = x(y(f)) − y(x(f)) = ∂t∂s∂2f − ∂s∂t∂2f = 0, (2.31) it follows that xa∇ ayb = ya∇axb. (2.32)
The relative acceleration za, in the direction of ya, of a nearby geodesic when
moving along the direction of xa, is given by
za = xc∇ c(xb∇bya)(2.32)= xc∇c(yb∇bxa) = (xc∇ cyb)∇bxa + yb(xc∇c∇bxa) (2.28)= (yc∇ cxb)∇bxa + yb(xc∇b∇cxa) − Rcbdaybxcxd = yc∇ c(xb∇bxa) − Rcbdaybxcxd (2.25)= − R cbdaybxcxd. (2.33)
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 Rabcd= 0.
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.
Let eia, i = 1, 2, 3, 4, be smooth vector fields that are linearly independent
at each point in spacetime (M, gab). Then the Ricci rotation coefficients are
defined by
γkij := ekceja∇aeic. (2.34)
It is shown in [2] that γijk = 1
2(λijk + λkij − λjki), where λijk = (∂beja − ∂aejb)eiaekb, (2.35)
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, intro-duced by Ezra Newman and Roger Penrose [9] in 1962, is usually called the Newman-Penrose formalism. The basis {la, na, ma, ¯ma} consists of null vec-tors, where la and na are real, ma and ¯ma are complex conjugates, satisfying
lala = nana = mama = ¯mam¯a = 0 (null);
lama = lam¯a = nama = nam¯a = 0 (orthogonal);
lana = −mam¯a = 1 (normalized).
(2.36)
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 = γ311 κ0= −ν = γ422 2α = γ214 + γ344
ρ = γ314 ρ0= −µ = γ423 2β = γ213 + γ343
σ = γ313 σ0 = −λ = γ424 2γ = γ212 + γ342
τ = γ312 τ0= −π = γ421 2ε = γ211 + γ341
Figure 2.8: Spin coefficients.
The spin coefficients ρ and ρ0are of particular interest to us, since they are used
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 (M, gab), which is represented by a smooth four-dimensional
Hausdorff manifold M endowed with a Lorentzian metric gaband a metric
con-nection ∇a. The presence of matter ‘warps’ spacetime according to the Einstein
field equation,
Gab := Rab−12R gab = 8π Tab, (3.1)
where Gab is the Einstein tensor that describes the curvature of M and Tab
is the stress-energy-momentum 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.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
the vacuum field equation Gab= 0. It is the flat4 spacetime (M, ηab) given by
the constant metric
(ηab) = 1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1 . (3.2)
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.
4R abcd= 0.
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 (M, gab)
given, in Schwarzschild coordinates, by the metric
(gab) = 1 − 2M r 0 0 0 0 −1 − 2M r −1 0 0 0 0 − r2 0 0 0 0 − r2sin2θ . (3.3)
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.
Chapter 4
Mass in general relativity
The stress-energy-momentum tensor, Tab, is the tensor that describes the
den-sity 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 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 Tabis 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].
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 Kerr4black 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
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 laand na be, respectively, the outgoing and the ingoing null vectors
orthog-onal to a spacelike 2-sphere S and let ma and ¯mabe tangent vectors to S. Then
the Hawking mass is defined by
mH(S) := r Area(S)16π 1 + 2π1 I Sρρ 0dS. (5.1)
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]. Both la and na 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.
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 expres-sion 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 S, as mais required to do. The other two
vectors, la and na, are parallel to the outgoing and ingoing null directions,
respectively, thus it is convenient to set up a null tetrad (la, na, ma, ¯ma),
by making the ansatz la = ( A B C 0 ), na = ( A −B −C 0 ), ma = ( 0 0 X iY ). (5.3)
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
ama = 0 (null vector)
mam¯a = −1 (normalization) , (5.4)
and pick one solution {X, Y }. 2. Solve the system
l
ala = 0 (null vector)
lana = 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
lama = 0 (orthogonality) . (5.6)
The result of the procedure above is a null tetrad (la, na, ma, ¯ma)
satis-fying 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),
λ314 = (∂bla − ∂alb) mam¯b λ431 = (∂bma − ∂amb) ¯malb λ143 = (∂bm¯a − ∂am¯b) lamb λ423 = (∂bna − ∂anb) ¯mamb λ342 = (∂bm¯a − ∂am¯b) manb λ234 = (∂bma − ∂amb) nam¯b , (5.7)
then the spin coefficients follow easily as,
ρ = γ314 = (λ314 + λ431 − λ143)/2
ρ0 = γ
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 cal-culating 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 spher-ical polar coordinates, the Minkowski metric (3.2) can be written as,
(gab) = 1 0 0 0 0 −1 0 0 0 0 −r2 0 0 0 0 −r2sin2θ . (5.9)
By solving the equations (5.4), (5.5) and (5.6) in three steps, we obtain the null tetrad (la, na, ma, ¯ma), given by la = √ 2 2 1 1 0 0 , na = √ 2 2 1 −1 0 0 , (5.10) ma = √ 2 2 0 0 1 r −i r sin θ , and then it follows easily from (5.7) and (5.8) that
ρρ0 = − 1
2r2. (5.11)
The area of the 2-sphere is the familiar Area(Sr) = 2π Z 0 π Z 0 r2sin θ dθ dϕ = 4πr2, (5.12)
and the integral of the spin coefficients I Sr ρρ0dS = 2π Z 0 π Z 0 −12sin θ dθ dϕ = −2π. (5.13) Finally, we can confirm the well-known result that the Hawking mass vanish for all 2-spheres in Minkowski spacetime, since
mH(Sr) = r 4πr 2 16π 1 + 1 2π(−2π) = r 2 1 − 1 = 0. (5.14)
5.4.2
In Schwarzschild spacetime
We repeat the calculations of the previous section, but this time for a cen-tered 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 S of a 2-sphere, thus we do not have to change to another coordinate system. We obtain a null tetrad (la, na, ma, ¯ma) given by
la = √ 2 2 √r √ r − 2M √ r − 2M√ r 0 0 , na = √ 2 2 √r √ r − 2M − √ r − 2M√ r 0 0 , (5.15) ma = √ 2 2 0 0 1 r −i r sin θ , from which it follows that
ρρ0 = −r − 2M
2r3 . (5.16)
The area of the 2-sphere is 4πr2, and the integral of the spin coefficients
I Sr ρρ0dS = 2π Z 0 π Z 0 −r − 2M2r sin θ dθ dϕ = −2π(r − 2M)r . (5.17) Finally, we get the expected result for the Hawking mass of a centered 2-sphere in Schwarzschild spacetime, mH(Sr) = r 4πr 2 16π 1 +2π1 −2π(r − 2M)r = 2r1 − 1 +2Mr = M. (5.18)
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 pre-sented 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
arccosh ξ p ξ2 − 1 = 1 − ξ − 1 3 + 2(ξ − 1)2 15 + O (ξ − 1)5/2 . Proof. Consider the equivalence,
ξ =px2+ 1, x > 0 ⇐⇒ x =pξ2− 1, ξ > 1, (6.1)
and the following Maclaurin expansions [11],
px2+ 1 = 1 + x2/2 − x4/8 + O(x6), (6.2)
ln(y + 1) = y − y2/2 + y3/3 − y4/4 + y5/5 + O(y6). (6.3)
Perform a change of variable, arccosh ξ p ξ2 − 1 (def.) = ln(ξ + pξp 2 − 1) ξ2 − 1 (6.1) = ln √ x2 + 1 + x x , (6.4)
and apply the Maclaurin expansions to the numerator, ln px2 + 1 + x (6.2)= ln 1 + x +x2 2 − x4 8 + O(x6) (6.3)= x + x2 2 − x4 8 + O(x6) − x + x2 2 −x 4 8 + O(x6) 2 2 1Mathematics software package from Waterloo Maple Inc.
24 Chapter 6. Results + x +x 2 2 + O(x4)3 3 − x + x2 2 + O(x4)4 4 + x + O(x2) 5 5 + O(x6) = x − x3 6 + 3x5 40 + O(x6). (6.5)
Divide through by x, and change back to ξ, arccosh ξ p ξ2 − 1 = 1 − ξ2− 1 6 + 3(ξ2− 1)2 40 + O (ξ2− 1)5/2 . (6.6)
Since ξ > 1, the ordo-term is actually O (ξ2− 1)5/2 = O (ξ + 1)5/2(ξ − 1)5/2 = O (ξ − 1)5/2 . (6.7) Thus, let f(ξ) = 1 −ξ2− 1 6 + 3(ξ2− 1)2 40 , (6.8)
and calculate the second degree Taylor expansion of f about ξ = 1 . From f(1) = 1, f0(1) = −1/3, and f00(1) = 4/15 it follows that
arccosh ξ p ξ2 − 1 = 1 − ξ − 1 3 + 2(ξ − 1)2 15 + O (ξ − 1)5/2 . (6.9)
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
A2 +
z2
B2 = 1. (6.10)
Let A and B depend on a variable, say r > 0, in the way given by A2 = ξ2r2
and B2= r2. Then (6.10) is equivalent to
x2 + y2
ξ2 + z2 = r2, (6.11)
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 (M, ηab) be Minkowski spacetime. For ξ > 1, let Sr be the
spacelike oblate 2-spheroids, inM, given by
Sr = (x2 + y2)/ξ2 + z2 = r2: r > 0 .
Then the Hawking mass withinSr is given by mH Sr = − √ 2 r 16√ξ s ξ + arccosh ξp ξ2− 1 2 ξ2− 5 3 ξ + arccosh ξ p ξ2− 1 ! .
Proof. Use the method provided in section 5.3. Start by introducing a coordi-nate system such that for a given r, and t held fixed, θ and ϕ trace out the ellipsoidal 2-surface Sr. This is accomplished by using the parametrized form
of Sr, 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 < 2π , (6.12) as new variables (t, r, θ, φ).
Using the tensor transformation law (2.19) yields the Minkowski metric (3.2) in the new coordinates,
(gab) = 1 0 0 0
0 −(cos2θ + ξ2sin2θ) −(ξ2− 1)r cos θ sin θ 0
0 −(ξ2− 1)r cos θ sin θ −r2(ξ2cos2θ + sin2θ) 0
0 0 0 −ξ2r2sin2θ . (6.13) Ansatz (5.3).
• Solve equations (5.4), and pick a solution, say
X = 1 / (√2 rpξ2cos2θ + sin2θ)
Y = 1 / (√2 ξ r sin θ) . (6.14)
• Solve equations (5.5), and pick the solution given by A = 1/√2 and B = pcos2θ + ξ2sin2θ − 2ξ2r2C2 − √ 2(ξ2 − 1) r cos θ sin θ C √ 2(cos2θ + ξ2 sin2θ) . (6.15)
26 Chapter 6. Results
• Solve equation (5.6). The solution is given by C = −
√
2(ξ2 − 1) cos θ sin θ
2 ξ rpξ2 cos2θ + sin2θ. (6.16)
The result of this procedure is the null tetrad (la, na, ma, ¯ma), given by
la =
√ 2 2
1 pξ2cos2ξθ + sin2θ − (ξ2 − 1) cos θ sin θ ξ rpξ2 cos2θ + sin2θ 0 , na = √ 2 2 1 − pξ2 cos2θ + sin2θ ξ (ξ2 − 1) cos θ sin θ ξ rpξ2 cos2θ + sin2θ 0 , ma = √ 2 2 0 0 1 rpξ2cos2θ + sin2θ i ξ r sin θ , ¯ ma = √ 2 2 0 0 1 rpξ2cos2θ + sin2θ − i ξ r sin θ . (6.17)
Calculate the spin coefficients with the use of the null tetrad by first calcu-lating the required λijk given by (5.7),
λ314 = (∂bla− ∂alb)mam¯b = 0, λ431 = (∂bma− ∂amb) ¯malb = − ξ 2(1 + cos2θ) + sin2θ 2√2 ξ r (ξ2 cos2θ + sin2θ)3/2, λ143 = (∂bm¯a− ∂am¯b)lamb = ξ 2(1 + cos2θ) + sin2θ 2√2 ξ r (ξ2cos2θ + sin2θ)3/2, λ423 = (∂bna− ∂anb) ¯mamb = 0, λ342 = (∂bm¯a− ∂am¯b)manb = ξ 2(1 + cos2θ) + sin2θ 2√2 ξ r (ξ2cos2θ + sin2θ)3/2, λ234 = (∂bma− ∂amb)nam¯b = − ξ 2(1 + cos2θ) + sin2θ 2√2 ξ r (ξ2 cos2θ + sin2θ)3/2, (6.18) then the spin coefficients (5.8) follow easily as,
ρ = γ314 = − ξ 2(1 + cos2θ) + sin2θ 2√2 ξ r ξ2cos2θ + sin2θ3/2, ρ0 = γ 423 = ξ 2(1 + cos2θ) + sin2θ 2√2 ξ r ξ2cos2θ + sin2θ3/2. (6.19)
The surface area of Sris part of the expression for the Hawking mass. From
the surface element dS = pgθθgϕϕ − (gθϕ)2dθ dϕ it follows that
Area(Sr) = 2π Z 0 π Z 0
ξ r2sin θqξ2 cos2θ + sin2θ dθ dϕ
= 2π ξ r2 ξ pξ2 − 1 + ln(pξp 2 − 1 + ξ) ξ2 − 1
6.1. Hawking mass in Minkowski spacetime 27 = 2π ξ r2ξ + arccosh ξ p ξ2− 1 , (6.20)
Integration over Sr yields 2π Z 0 π Z 0 ρρ0dS = 2π Z 0 π Z 0
sin θ ξ2(1 + cos2θ) + sin2θ2
ξ ξ2cos2θ + sin2θ5/2 dθ dϕ = −π6 (2 ξ3+ 7ξ) pξ2− 1 + 3 arcsinh (pξ2− 1) ξ pξ2− 1 = −π6 2 ξ2 + 7 + 3 ξ arccosh ξ p ξ2 − 1 . (6.21)
Finally, from (6.20) and (6.21), the Hawking mass in Srfollows as
mH(Sr) = r Area(S16 πr) 1 + 2 π1 2π Z 0 π Z 0 ρρ0dS = − √ 2 r 16√ξ s ξ +arccosh ξp ξ2− 1 2 ξ2− 5 3 ξ + arccosh ξ p ξ2− 1 ! . (6.22)
As can be seen in figure 6.2, the Hawking mass becomes negative in Minkowski, even for a convex 2-surface.
1 2 3 4 0 −3 −2 −1 0 1
Figure 6.2: mH(S1) plotted against parameter ξ.
6.1.2
Limit when approaching a metric sphere
Corollary 6.1. Let (M, ηab) be Minkowski spacetime. Given an r > 0, the
Hawking mass vanishes in the limit whenξ → 1+, that is, when Sr tends to a 2-sphere.
Proof. It follows from lemma 6.1 that arccosh ξ p ξ2 − 1 = 1 − ξ − 1 3 + 2(ξ − 1)2 15 + O (ξ − 1)5/2 = 1 + O(ξ − 1), (6.23)
28 Chapter 6. Results
and furthermore that 2 ξ2− 5 3 ξ + arccosh ξ p ξ2− 1 = 2 ξ3 3 − 5 ξ 3 + 1 + O(ξ − 1) = 2 ξ(ξ + 1) − 33 ξ − 1 + O(ξ − 1) = O(ξ − 1). (6.24)
Thus, it follows from (6.23), (6.24) and theorem 6.1 that mH(Sr) = − √ 2 r 16√ξ s ξ + arccosh ξp ξ2− 1 2 ξ2− 5 3 ξ + arccosh ξ p ξ2− 1 ! = √r ξ p 1 + ξ + O(ξ − 1) O(ξ − 1) → 0 as ξ → 1+. (6.25)
6.1.3
Limit along a foliation
Corollary 6.2. Let (M, ηab) be Minkowski spacetime. For ω > 0, let {Lr}r>0
be foliations of a spacelike 3-surfaceΩ ⊂ M. Suppose that the leaves are given by
Lr = (x2 + y2)/(1 + ω/r)2 + z2 = r2: r > 0 .
Then the Hawking mass vanishes in the limit along all foliations {Lr}r>0. Proof. Observe that
Lr = Sr
ξ=1+ω/r, (6.26) and let ξ = 1 + ω/r in lemma 6.1. Then ξ → 1+ as r → ∞, and it follows that
1 + ω/r +parccosh (1 + ω/r) (1 + ω/r)2 − 1 = 1 + ω r + 1 − ω 3r+ 2 ω2 15r2 + O 1 r5/2 = 2 +2ω3r +15r2 ω22+ Or5/21 = 2 + O1r, (6.27)
and furthermore that 2(1 + ω/r)2− 5 3 (1 + ω/r) + arccosh (1 + ω/r) p (1 + ω/r)2 − 1 = 32ω2 15r2 + 2ω3 3r3 + O 1 r5/2 = Or12. (6.28) From (6.27) and theorem 6.1, it follows that
mH(Lr) = − √ 2 r 16p1 +ω r r 2 + O1rO 1r2 = O1r → 0 as r → ∞. (6.29)
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
r = p1 + ε sin2θ, (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
r θ
Figure 6.3: A curve given by r =p1 + ε sin2θ.
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 ˜Sr. This is accomplished by the
following change of variables,
r 7→ rp1 + ε sin2θ, r > 2M. (6.31)
In the new coordinates, the Schwarzschild metric (3.3) is given by
(gab) = β rα 0 0 0 0 −rα3 β −εr 2α cos θ sin θ β 0 0 −εr2α cos θ sin θ β −r 2(α3β+ε2r cos2θ sin2θ) αβ 0 0 0 0 −r2α2sin2θ , (6.32) where
α = p1 + ε sin2θ > 1 and β = rp1 + ε sin2θ − 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, ¯ma) given by,
la = √ 2 2 √rα √ β pα3β + ε2r cos2θ sin2θ √rα3 2C√ 2 0 ,
30 Chapter 6. Results na = √ 2 2 √rα √ β − pα3β + ε2r cos2θ sin2θ √ rα3 − 2C √ 2 0 , (6.34) ma = √ 2 2 0 0 √ αβ rpα3β + ε2r cos2θ sin2θ − i rα sin θ , where C = − √ 2 2 ε cos θ sin θ √ rαpα3β + ε2r cos2θ sin2θ. (6.35)
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 ˜Sris given by
dS = qgθθgϕϕ − (gθϕ)2dθ dϕ
= r2α2sin θ
s
ε2r cos2θ sin2θ
αβ + 1 dθ dϕ, (6.36)
and the surface area by
Area( ˜Sr) = 2π Z 0 π Z 0 dS. (6.37)
Since the expression of the Hawking mass for ˜Sr,
mH( ˜Sr) = s Area( ˜Sr) 16 π 1 + 1 2 π 2π Z 0 π Z 0 ρρ0dS, (6.38)
is rather complicated, we will rely on methods like Taylor expansion and nu-merical integration for further investigations.
6.2.1
Limit along a foliation
Analogously to corollary 6.2, let { ˜Lr}r>2Mbe foliations of a spacelike 3-surface
Ω ⊂ M (now in Schwarzschild spacetime). Suppose that the leaves, ˜Lr, are
given by substituting
ε = ωr, r > 2M, (6.39)
into ˜Sr. It is easy to see that ˜Lr becomes more spherical the larger r gets.
In this section, we will show that the limit of the Hawking mass of ˜Lr is M
along all foliations { ˜Lr}r>2M, that is,
lim
r→∞mH( ˜Lr) = M. (6.40)
Since both dS and ρρ0dS are independent of ϕ, we make the substitution
(6.39), and let
I1(ε) = ε2Area( ˜Lω/ε) =: 2π
Z π
6.2. Hawking mass in Schwarzschild spacetime 31 and I2(ε) = 2π Z π 0 ρρ 0dθ =: 2πZ π 0 Ψ(ε, θ) dθ. (6.42)
Under the reasonable assumption that Φ(ε, θ), Φ0
ε(ε, θ), Ψ(ε, θ), Ψ0ε(ε, θ) and
Ψ00
εε(ε, θ) are continuous in some neighbourhood |ε| < 1, it follows by Maclaurin
expansion that I1(ε) = 2π Z π 0 Φ(0, θ) dθ + O(ε) = 4πω 2 + O(ε), |ε| < 1, (6.43) and I2(ε) = 2π Z π 0 Ψ(0, θ) dθ + 2π Z π 0 Ψ 0 ε(0, θ) dθ ε + O(ε2) = −2π + 4πMω ε + O(ε2), |ε| < 1. (6.44) Thus Area( ˜Lr) = I1(ε)/ε2 = 4πω 2 ε2 + O 1 ε = 4πr2 + O(r), r > |ω|, (6.45) and 2πZ π 0 ρρ 0dS = −2π + 4πM r + O 1 r2, r > |ω|. (6.46)
By using the results (6.45) and (6.46), it follows that mH( ˜Lr) = r 4πr 2+ O(r) 16π 1 +2π1 h − 2π + 4πMr + Or12i = r 2 r 1 + O1 r 2M r + O 1 r2 = r 1 + O1 r M + O1 r → M as r → ∞. (6.47)
6.2.2
Numerical evaluations
We evaluate the Hawking mass numerically with an adaptive Gaussian2
quadra-ture method.
In numerical quadrature, an integral I(f) = Z b
a f(x)dx, (6.48)
is approximated by an n-point quadrature rule that has the form Qn(f) =
n
X
i=1
wif(xi), (6.49)
where a ≤ x1 < x2 < · · · < xn ≤ b. The points xi are called nodes, and the
multipliers wi are called weights.
2Carl Friedrich Gauss (1777-1855), German mathematician known as ‘princeps
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. e = 0.25 e = 0.05 e = 0.10 e = 0.15 e = 0.20 e = 0. 4 6 8 10 20 40 60 80 100 200 400 600 K1.0 K0.5 0.0 0.5 1.0
Figure 6.4: mH( ˜Sr) plotted against 4 ≤ r ≤ 620 for some values of ε.
e = 0.8 e = 1.0 e = 0.4 e = 0.6 e = 0. e = 0.2 2 4 6 8 10 12 14 16 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 6.5: mH( ˜Sr) plotted against 2.3 ≤ r ≤ 16 for some values of ε.
4 6 8 10 12 14 16 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00
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 Sr, that is, a rotationally symmetric 2-ellipsoid,
in Minkowski spacetime. If Sr is given by,
Sr = (x2 + y2)/ξ2 + z2 = r2: r > 0 .
Then the Hawking mass within Sris given by
mH Sr = − √ 2 r 16√ξ s ξ + arccosh ξp ξ2− 1 2 ξ2− 5 3 ξ + arccosh ξ p 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.
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