| 摘要: | 始於愛氏，能動量區域化是引力論重要課題。基於對等原理，引力能動量密度不存在。傳統解決方法是用不同標架膺張量，近代方法是膺區域能動量。陳江梅透過哈密頓方法，提出四組準局域能動量方程式。大部份古典膺張量方程式有相同宏觀能動量值，卻不同於小區域；在眾多表達式中，尋找適切描述引力能動量。我們研究古典膺張量(例如愛氏、柏氏或布氏、1958和1961年版本的毛氏與溫氏)和協變哈密頓準局域邊界方程式，物理條件在物質和真空。於小區域尺度，其計算結果可以篩選那種表達式能夠滿足正能量要求，沒有一個古典膺張量表達式滿足正能量特性。但有1個常數綫性組合能夠給出彪張量，即小區域正能量；還有3個常數綫性組合的柯座標架膺張量亦能給出彪張量。再者，應用正座標架，1961年的毛氏式子和陳江梅的其一表達式，與及在柯座標架修正版的陳聶表達式，這三個式子均在自然邊界條件下給出彪張量。 The localization of energy-momentum for gravitating systems has remained an important problem since the time of Einstein. Due to the equivalence principle there is no proper energy-momentum density. Traditional approaches led to a va- riety of reference frame dependent expressions, referred to as pseudotensors. A more modern idea is quasilocal energy-momentum. C.M. Chen, using a covariant Hamiltonian formalism, gave four preferred Hamiltonian boundary term quasilocal energy-momentum expressions. The classical pseudotenor expressions, as well as the quasilocal expressions generally agree for the total (i.e. global) values but give quite di®erent values locally. It is desirable to ?nd some way to choose which expression gives a better description of the energy-momentum for a gravitating system. Here we shall test both the well-known classical pseudotensors (in particular, Einstein, Papapetrou, Landau-Lifshits ' Bergmann-Thomson, M¿ller (1958), M¿ller (1961), Weinberg) and the covariant Hamiltonian quasilocal boundary expressions in a dif- ferent regime, namely the small region limit|both inside matter and in vacuum. All of the expressions|except for M¿ller's 1958 expression|give the correct mate- rial limit. In small vacuum regions we found some interesting results which allows us to choose which expressions satisfy an important physical property: positive en- ergy. None of the classical pseudotensors satis?es this positivity property, however there is a one-parameter set of linear combinations which, to lowest non-vanishing order is proportional to the Bel-Robinson tensor and hence is positive for small regions. Moreover, we have constructed an in?nite set (with 10 constant parame- ters) of additional new holonomic pseudotensors which, although rather contrived, satisfy this important positive energy requirement. On the other hand we found that M¿ller's 1961 teleparallel-tetrad energy-momentum expression naturally has this Bel-Robinson property. For C.M. Chen's covariant-symplectic quasilocal ex- pressions we found that one, corresponding to the natural boundary choices, gives this desired Bel-Robinson positivity result in orthonormal frames. Moreover within a two parameters modi?cation of the Chen-Nester four expressions, one gives an extra nice result in holonomic frames. |
