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Student Number 91222014 Author Gang Sun(孫綱) Author's Email Address No Public. Statistics This thesis had been viewed 1691 times. Download 672 times. Department Physics Year 2004 Semester 2 Degree Master Type of Document Master's Thesis Language English Title Quasilocal Conserved Quantities For General Relativity In Small Regions Date of Defense 2005-06-21 Page Count 60 Keyword quasilocal small region limit Abstract Gravitational energy has been a concern for a long time. There are several ways to deal with the problem, but the best way is the quasilocal approach. The NCU group has been developing their quasilocal approach – the covariant Hamiltonian formalism, and has obtained good results for spatial and null infinity. In addition to these infinite cases; there is another limit case – the small region limit. The small region vacuum limit provides an important test of the quasilocal expression. Whereas the large scale asymptotic limit tests only the weak field linearized part of the expression, the small scale vacuum limit probes the next order non-linear part.

In this thesis the purpose is to test the covariant Hamiltonian formalism in the small region limit. In the first chapter, we will introduce the basic ideas of the quasilocal method and some related ideas. In chapter two, we will show the readers what the covariant Hamiltonian formulism is and how to derive it. In the chapter three, we will introduce some general concepts about energy-momentum, angular momentum and center-of-mass moment, and the relation between these physical quantities and conservation. In the next chapter, the detailed procedure on how to get quasilocal values in the small region limit, including the vacuum case and matter case, using covariant Hamiltonian formulism will follow. In the final chapter, we will discuss the meaning of our results and conclude that only one of the four covariant Hamiltonian expressions gives positive energy in the first non-linear order. Finally we will comment some deficiencies.Table of Content Contents

Chapter 1

Introduction 1

1.1 Fundamental concepts of quasilocal 1

1.2 Some applications of quasilocal 2

1.2.1 Tidal heating 2

1.2.2 Positivity of the gravitational energy in the finite region 3

1.2.3 The ADM mass and the (weak) cosmic censorship hypothesis 3

Chapter 2 4

Covariant symplectic quasilocal expression 4

2.1 From Lagrangian to Hamiltonian 4

2.2 Control over the Hamiltonian boundary term 6

2.3 The choice of evolving vector 9

2.3.1 Asymptotically 9

2.3.2 In the small region 12

Chapter 3 14

Physical quantities and conservation 14

3.1 Physical quantities in special relativity 14

3.1.1 Single particle 14

3.1.2 Matter field 15

3.2 Conservation Law and Noether theorem 16

3.2.1 Killing vector field 16

3.2.2 Noether theorem 17

3.2.3 Conservation quantities in Minkowski space 18

3.3 Total conservation for a gravitational field 19

3.3.1 Pseudotensor 19

3.3.2 Our quasilocal approach 20

Chapter 4 21

Quasilocal quantities in small region 21

4.1 Riemann normal coordinates 21

4.2 The expansion of the Hamiltonian in the small region 23

4.3 The quasilocal values expressed by the different unit volume 32

4.4 The values of the four quasilocal cases 36

4.4.1 The four cases expressed by the physical volume element 36

4.4.2 The four cases expressed by the flat volume element 38

Chapter 5 41

Discussion 41

5.1 Discussion to the results including a matter field 41

5.2 Discussion to those results concerning the vacuum 41

5.2.1 The physical quantities in the small region 41

5.2.2 Some comments on our quasilocal values in vacuum 46

5.3 Conclusion 50

Reference 53

Appendix A 55

Bel-Robinson tensor and quadratic forms in Riemann curvature 55

A.1 The definition of the Bel-Robinson tensor 55

A.2 The relationship between the Bel-Robinson tensor and energy in a small region 58

A.3 Quadratic form in Riemann curvature 59Reference References

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[19] S. Deser, J.S. Franklin, D. Seminara, “Graviton-Graviton Scattering, Bel-Robinson and Energy (Pseudo)-Tensors”, Class.Quant.Grav. 16 (1999) 2815-2821, or gr-qc/9905021

[20] Shi, Y., and Tam, L.-F., “Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature”, J. Differ. Geom., 62, 79-125, (2002) or http://arxiv.org/abs/math.DG/0301047v1

[21] Janusz Garecki, “Some remarks on the Bel-Robinson Tensor”, Annalen Phys. 10 (2001) 911-919, or gr-qc/0003006.Advisor J. Nester(聶斯特)

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91222014.pdf Date of Submission 2005-07-21

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