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研究生:巫明帆
研究生(外文):Ming-Fan Wu
論文名稱:球對稱時空的準局域能量
論文名稱(外文):Quasi-local Energy for Spherically Symmetric Spacetimes
指導教授:聶斯特
指導教授(外文):James M. Nester
學位類別:博士
校院名稱:國立中央大學
系所名稱:物理研究所
學門:自然科學學門
學類:物理學類
論文出版年:2011
畢業學年度:99
語文別:英文
論文頁數:78
中文關鍵詞:參考系位移向量準局域能量哈米爾頓
外文關鍵詞:referencedisplacement vectorquasi-local energyHamiltonian
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我們提出兩種互補的方法來決定哈米爾頓邊界項準局域能量表達式中的參考系,
並且實際應用於球對稱時空。我們一方面對能量取極值以決定參考系,稱之為能
量極值法;另一方面,我們也可以直接在邊界上要求兩組標架相等進而決定參考
系,稱之為標架法,兩種方法都給出合理的結果。如果我們直接對能量取極值而
不要求任何額外的限制條件,稱之為能量極值法一,得到的能量可以是正數也可
以是零,甚至是負數。如果在某些額外的限制條件之下對能量取極值,稱之為能
量極值法二,得到的能量則都是非負數的。另外,利用標架法且要求位移向量是
參考空間的Killing 向量,我們得到和能量極值法一相同的結果;如果要求邊界上
的二維面積元沿著位移向量的方向變化量相等,則得到的結果和能量極值法二相
同,並且符合準局域能量一般性的要求,特別是能量必須是非負的,若且唯若時
空是閔氏時空的時候能量是零。由能量極值法,我們了解使得能量得到極值的參
考系在邊界上是四維等度規的;而由標架法我們發現,在邊界上四維等度規的參
考系給出能量極值,因此,我們稱這兩種方法為互補的。
We present two complementary approaches for determining the reference for the co-
variant Hamiltonian boundary term quasi-local energy and test them on spherically
symmetric spacetimes. On the one hand, we extremize the energy in two ways, which
we call energy-extremization programs A and B. Both programs produce reasonable
results that allow us to discuss energies measured by diRerent observers. We show
that the energies produced by program A can be positive, zero, or even negative,
while in program B they are always non-negative. On the other hand, we match the
orthonormal frames of the dynamic and the reference spacetimes right on the two-
sphere boundary. If we further require that the reference displacement vector to be
the timelike Killing vector, the result is the same as program A. If, instead, we require
that the Lie derivatives of the two-area along the displacement vector in both the dy-
namic and reference spacetimes are the same, the result is the same as program B,
which satis¯es the usual criteria. In particular, the energies are non-negative and van-
ish only for Minkowski (or anti-de Sitter) spacetime. So by studying the spherically
symmetric spacetimes, both static and dynamic, we learn that the references deter-
mined by our energy extremization programs are those which isometrically match
the dynamic spacetimes on the boundary. And the energies determined by isometric
matching approach are actually the extremum measured by the associated observers.
1 Introduction 1
2 The Hamiltonian Formulation 5
2.1 First Order Langrangian 5
2.2 Local Translation Invariance 6
2.3 The Hamiltonian Formulation 7
2.4 Re¯ned Boundary Terms 9
2.5 Application to Einstein''s Gravity Theory 12
3 Quasi-local Energy for Static Spherically Symmetric Spacetimes 14
3.1 Energy Extremization Program A 14
3.2 Energy Measured by Various Observers 21
3.3 Observer Adapted Coordinates 25
3.4 Energy Extremization Program B 27
3.5 An Alternative Approach 31
3.6 Conclusion 35
4 Quasi-local Energy for Dynamic Spherically Symmetric Spacetimes 37
4.1 Energy Extremization Program A 37
4.2 Energy Extremization Program B 44
4.3 An Alternative Approach 48
4.4 Conclusion 51
5 Concluding Discussion 52
A Review of DiRerential Forms 55
B Variational Principles with DiRerential Forms 60
C Geometry in Exterior Covariant DiRerential Forms 66
D Explicit Calculations of Quasi-local Energies 69
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