準區域的膺張量和陳聶式子

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蘇樓來(Lau-Loi So) 查詢紙本館藏

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物理學系

論文名稱

準區域的膺張量和陳聶式子
(Quasi-local energy-momentum and pseudotensors for GR in small regions)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

關鍵字(英)

★ gravitation
★ pseodotensor

論文目次

1 Introduction 1
2 Gravitational energy-momentum and its localization 7
2.1 The classical pseudotensors . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Hamiltonian boundary and quasi-local expressions . . . . . . . . . . . 11
2.3 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Riemann Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . 15
3 Some properties of the Bel-Robinson and Weyl tensors 17
3.1 Some properties of the Bel-Robinson tensor . . . . . . . . . . . . . . 17
3.2 Some properties of the Weyl tensor . . . . . . . . . . . . . . . . . . . 20
4 The classical holonomic pseudotensors 24
4.1 Einstein pseudotensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Bergmann-Thomson pseudotensor . . . . . . . . . . . . . . . . . . . . 28
4.3 Papapetrou pseudotensor . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Weinberg pseudotensor . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 M¿ller 58 pseudotensor . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 The modi¯cation of the M¿ller 58 pseudotensor . . . . . . . . . . . . 34
4.7 The combination of the classical holonomic pseudotensors . . . . . . . 35
4.8 A large class of new pseudotensors which satisfy the positive require-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 The tetrad/teleparallel theory 43
5.1 The Tetrad theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Teleparallel theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 Orthonormal frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 The M¿ller 1961 classical energy-momentum density in tetrad-teleparallel
expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Covariant Hamiltonian formalism & quasilocal boundary terms for
metric-compatible gravity 49
6.1 Covariant Hamiltonian formalism . . . . . . . . . . . . . . . . . . . . 49
6.2 Metric-compatible gravity . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Einstein and Einstein-Cartan theory 54
7.1 Einstein GR theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.2 Einstein-Cartan theory . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8 The four orthonormal frame expressions: small region limit 60
8.1 Einstein's gravity theory . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.2 Expression for quasilocal quantities . . . . . . . . . . . . . . . . . . . 61
8.3 Application of the four expressions in small region limit . . . . . . . . 64
8.4 Comparison of our results with others . . . . . . . . . . . . . . . . . . 67
9 The modi¯cation of Chen-Nester's four quasilocal expressions 69
9.1 Chen-Nester's 4 boundary expressions . . . . . . . . . . . . . . . . . . 69
9.2 The modi¯cation of Chen-Nester's 4 boundary expressions . . . . . . 73
9.3 On the physical interpretation of the modi¯ed Chen-Nester's 4 bound-
ary expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.4 Gravitation application of the new quasilocal expressions . . . . . . . 76
10 Conclusion 82
Bibliography 86

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指導教授

聶斯特(James M. Nester)

審核日期

2006-6-16

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