Examensarbete The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild spacetimes Daniel Hansevi LiTH - MAT - EX - - 08/14 - - SE The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild spacetimes Applied Mathematics, Link¨opings Universitet Daniel Hansevi LiTH - MAT - EX - - 08/14 - - SE Examensarbete: 30 hp Level: D Supervisor: G¨oranBergqvist, Applied Mathematics, Link¨opings Universitet Examiner: G¨oranBergqvist, Applied Mathematics, Link¨opings Universitet Link¨oping: June 2008 Datum Avdelning, Institution Date Division, Department Matematiska Institutionen June 2008 581 83 LINKOPING¨ SWEDEN Spr˚ak Rapporttyp ISBN Language Report category ISRN Svenska/Swedish Licentiatavhandling LiTH - MAT - EX - - 08/14 - - SE x Engelska/English x Examensarbete C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering 0348-2960 Ovrig¨ rapport URL f¨orelektronisk version Titel The Hawking mass for ellipsoidal 2-surfaces in Minkowski and Schwarzschild space- Title times F¨orfattare Daniel Hansevi Author Sammanfattning Abstract In general relativity, the nature of mass is non-local. However, an appropriate 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. Nyckelord Keyword Hawking mass, Quasi-local mass, General relativity, Ellipsoidal surface. vi Abstract In general relativity, the nature of mass is non-local. However, an appropriate definition of mass at a quasi-local level could give a more detailed 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. Hansevi, 2008. vii viii Contents 1 Introduction 1 1.1 Background . 1 1.2 Purpose . 1 1.3 Chapter outline . 2 2 Mathematical preliminaries 3 2.1 Manifolds . 3 2.2 Foliations . 4 2.3 Tangent vectors . 5 2.4 1-forms . 6 2.5 Tensors . 7 2.5.1 Abstract notation . 7 2.5.2 Component notation . 7 2.5.3 Tensor algebra . 8 2.5.4 Tensor fields . 8 2.5.5 Abstract index notation . 8 2.6 Metric . 9 2.7 Curvature . 10 2.7.1 Covariant derivative . 11 2.7.2 Metric connection . 11 2.7.3 Parallel transportation . 12 2.7.4 Curvature . 12 2.7.5 Geodesics . 12 2.8 Tetrad formalism . 14 2.9 Newman-Penrose formalism . 14 2.10 Spin coefficients . 14 3 General relativity 15 3.1 Solutions to the Einstein field equation . 15 3.1.1 Minkowski spacetime . 15 3.1.2 Schwarzschild spacetime . 16 4 Mass in general relativity 17 4.1 Gravitational energy/mass . 17 4.1.1 Non-locality of mass . 17 4.2 Total mass of an isolated system . 18 4.2.1 Asymptotically flat spacetimes . 18 4.2.2 ADM and Bondi-Sachs mass . 18 Hansevi, 2008. ix x Contents 4.3 Quasi-local mass . 18 5 The Hawking mass 19 5.1 Definition . 19 5.2 Interpretation . 19 5.3 Method for calculating spin coefficients . 20 5.4 Hawking mass for a 2-sphere . 21 5.4.1 In Minkowski spacetime . 21 5.4.2 In Schwarzschild spacetime . 21 6 Results 23 6.1 Hawking mass in Minkowski spacetime . 24 6.1.1 Closed-form expression of the Hawking mass . 25 6.1.2 Limit when approaching a metric sphere . 27 6.1.3 Limit along a foliation . 28 6.2 Hawking mass in Schwarzschild spacetime . 29 6.2.1 Limit along a foliation . 30 6.2.2 Numerical evaluations . 31 7 Discussion 33 7.1 Conclusions . 33 7.2 Future work . 33 A Maple Worksheets 37 A.1 Null tetrad . 37 A.2 Minkowski . 38 A.3 Schwarszchild . 39 List of Figures 2.1 A manifold and two overlapping coordinate patches. 4 2.2 A foliation ofM a manifold . .................... 4 2.3 Intuitive picture of a tangentM space. 5 2.4 Picture of a 1-form. 6 2.5 Null cone. 10 2.6 Parallel transportation in a plane and on the surface of a sphere. 10 a 2.7 Deviation vector y between two nearby geodesics λs and λs0 . 13 2.8 Spin coefficients. 14 6.1 Oblate spheroid. 25 6.2 mH( 1) plotted against parameter ξ. 27 S p 6.3 A curve given by r = 1 + " sin2 θ. 29 6.4 m ( ~ ) plotted against 4 r 620 for some values of ". 32 H Sr ≤ ≤ 6.5 mH( ~r) plotted against 2:3 r 16 for some values of ". 32 6.6 m (S~ ) plotted against 2:3 ≤ r ≤ 16 for " = !=r. 32 H Sr ≤ ≤ Hansevi, 2008. xi xii List of Figures Chapter 1 Introduction 1.1 Background In general relativity, the nature of the gravitational field is non-local, and 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 spacelike 2-surface. In this thesis we will calculate the Hawking mass for spacelike ellipsoidal 2-surfaces, both in flat Minkowski spacetime and curved Schwarzschild spacetime. Hansevi, 2008. 1 2 Chapter 1. Introduction 1.3 Chapter outline Chapter 2 This chapter provides a short introduction of the mathematics needed. We introduce the notion of a manifold that is used for the model of curved spacetime in general relativity. Structure is imposed on the manifold in the form of a covariant derivative 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.