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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 sphericallysymmetric spacetimes. On the one hand, we extremize the energy in two ways, whichwe call energy-extremization programs A and B. Both programs produce reasonableresults that allow us to discuss energies measured by diRerent observers. We showthat 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 theorthonormal frames of the dynamic and the reference spacetimes right on the two-sphere boundary. If we further require that the reference displacement vector to bethe timelike Killing vector, the result is the same as program A. If, instead, we requirethat 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 sphericallysymmetric spacetimes, both static and dynamic, we learn that the references deter-mined by our energy extremization programs are those which isometrically matchthe dynamic spacetimes on the boundary. And the energies determined by isometricmatching approach are actually the extremum measured by the associated observers.
 1 Introduction 12 The Hamiltonian Formulation 52.1 First Order Langrangian 52.2 Local Translation Invariance 62.3 The Hamiltonian Formulation 72.4 Re¯ned Boundary Terms 92.5 Application to Einstein''s Gravity Theory 123 Quasi-local Energy for Static Spherically Symmetric Spacetimes 143.1 Energy Extremization Program A 143.2 Energy Measured by Various Observers 213.3 Observer Adapted Coordinates 253.4 Energy Extremization Program B 273.5 An Alternative Approach 313.6 Conclusion 354 Quasi-local Energy for Dynamic Spherically Symmetric Spacetimes 374.1 Energy Extremization Program A 374.2 Energy Extremization Program B 444.3 An Alternative Approach 484.4 Conclusion 515 Concluding Discussion 52A Review of DiRerential Forms 55B Variational Principles with DiRerential Forms 60C Geometry in Exterior Covariant DiRerential Forms 66D Explicit Calculations of Quasi-local Energies 69Bibliography 74