廣義相對論中以四維度規適配為參考的準局域能量

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劉建良(Jian-Liang Liu) 查詢紙本館藏

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物理學系

論文名稱

廣義相對論中以四維度規適配為參考的準局域能量
(4D-metric matching for the reference of quasi-local energy in general relativity)

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摘要(中)

哈密頓三形式扮演了沿著N向量演化方程的生成子的角色。它決定了哈密頓邊界表示式，也因而決定了準局域量。能量其意義實為能差，能差的概念總是涉及一個相對的參考值，因此無法唯一定義物理的能量。協變哈密頓法[PRD 72 (2005)
104020]指定了一個適當的邊界表示式，而近期的工作中[PRD 84 (2011) 084047;GRG 44 (2011) 2401]，考慮球對稱時空的情形，我們藉由四維度規在封閉二維面上的適配條件得到令人滿意的結果。本文分析了一般情形的四維度規在封閉二維面的適配條件。我們發現對於一個二維面，滿足等距嵌入到閔氏空間，在度規適配的條件下仍然具有兩個自由度可以決定參考系的選擇。準局域能量的值形成一個集，若它是這兩個自由函數的泛函，則臨界點為其一階變分的解，而準局域能量則為相應的臨界值。

摘要(英)

The Hamiltonian 3-form plays the role of the generator of the evolution w.r.t. the displacement vector. It is uniquely deﬁned up to a total diﬀerential term, the Hamiltonian boundary expression. The latter determines the quasi-local quantities. The meaningful concept of energy involves the diﬀerence of the dynamical values w.r.t. the reference values, so that we do not have a unique deﬁnition of the physical energies. For the covariant Hamiltonian approach a suitable boundary expression [PRD 72 (2005) 104020] was identiﬁed, and in recent works [PRD 84 (2011) 084047; GRG
44 (2011) 2401] we found satisfactory results obtained from matching the four metrics on a 2-sphere for spherically symmetric spacetimes. Here we analyze the general
4D-metric matching on a closed 2-surface. We ﬁnd that for a 2-surface which satisﬁes isometric embedding into Minkowski space there are still two degrees of freedom remaining to determine the choice of reference. The quasi-local energy values form a set, and, if it is a functional of the two free functions, the critical values could be determined by the solution of its variation.

關鍵字(中)

★ 四維度規適配
★ 準局域能量
★ 哈密頓量
★ 邊界表示式
★ 等距嵌入
★ 臨界值

關鍵字(英)

★ 4D-metric matching
★ quasi-local energy
★ Hamiltonian
★ boundary expression
★ isometric embedding
★ critical value

論文目次

1 Introduction 1
2 Covariant Hamiltonian approach 4
2.1 First-order Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 The boundary conditions . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Modiﬁcation of the boundary expression . . . . . . . . . . . . 10
2.2.4 Remark on the conserved quantities . . . . . . . . . . . . . . . 12
2.3 Application to general relativity . . . . . . . . . . . . . . . . . . . . . 13
3 Brief review of our previous works 17
3.1 The choice of reference . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Quasi-local energy for the strategy (i) . . . . . . . . . . . . . . . . . . 19
3.3 Quasi-local energy for the strategy (ii) . . . . . . . . . . . . . . . . . 21
3.3.1 Program I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 Program II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.3 Alternative approach . . . . . . . . . . . . . . . . . . . . . . . 25
4 Metric matching on a subspace 27
4.1 Notation settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 Dynamical spacetime (M, g) and a 2-surface S . . . . . . . . . 27
4.1.2 Reference space ( R
1,3 , ¯g) . . . . . . . . . . . . . . . . . . . . . 27
4.1.3 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.4 Extrinsic curvature of S . . . . . . . . . . . . . . . . . . . . . 31
i4.2 Matching the four spacetime metric on the closed space-like two surface 31
4.3 The existence of the metric matching . . . . . . . . . . . . . . . . . . 35
5 Application to quasi-local energy in general relativity 45
5.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.1 The holonomic expression in x µ . . . . . . . . . . . . . . . . . 47
5.1.2 The holonomic expression in y a . . . . . . . . . . . . . . . . . 48
5.1.3 The orthonormal frame expression in ϑ
α
. . . . . . . . . . . . 49
5.1.4 The covector basis expression in
¯
ϑ
¯α
. . . . . . . . . . . . . . . 51
5.2 Application to the Kerr like metric . . . . . . . . . . . . . . . . . . . 51
5.2.1 The choice of reference from the variation of the quasi-local
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Alternative choice of the controlled variables . . . . . . . . . . 61
6 Conclusion 66
Bibliography 68

參考文獻

[1] M. F. Wu, C. M. Chen, J. L. Liu and J. M. Nester, “Quasi-local Energy for Spher-
ically Symmetric Spacetimes,” Gen. Rel. Grav. 44 (2012) 2401 [arXiv:1206.0506
[gr-qc]].
[2] J. L. Liu, C. M. Chen and J. M. Nester, “Quasi-local energy and the choice of
reference,” Class. Quant. Grav. 28 (2011) 195019 [arXiv:1105.0502 [gr-qc]].
[3] M. F. Wu, C. M. Chen, J. L. Liu and J. M. Nester, “Optimal Choices of Reference
for a Quasi-local Energy: Spherically Symmetric Spacetimes,” Phys. Rev. D 84
(2011) 084047 [arXiv:1109.4738 [gr-qc]].
[4] C. M. Chen, J. L. Liu, J. M. Nester and M. F. Wu, “Optimal Choices of Reference
for Quasi-local Energy,” Phys. Lett. A 374 (2010) 3599 [arXiv:0909.2754 [gr-qc]].
[5] N.
´
O Murchadha, R. S. Tung, N. Xie and E. Malec, “The Brown-York
mass and the Thorne hoop conjecture,” Phys. Rev. Lett. 104 (2010) 041101
[arXiv:0912.4001 [gr-qc]].
[6] L. B. Szabados, “Quasi-Local Energy-Momentum and Angular Momentum in
GR: A Review Article,” Living Rev. Rel. 12 (2009) 4.
[7] M. T. Wang, S. T. Yau, “Isometric Embeddings into the Minkowski Space
and New Quasi-Local Mass,” Commun. Math. Phys. 288, 919 – 942 (2009).
[arXiv:0805.1370 [math-ph]].
[8] L. L. So and J. M. Nester, “New positive small vacuum region gravitational
energy expressions,” Phys. Rev. D 79 (2009) 084028 [arXiv:0901.2400 [gr-qc]].
68[9] C. M. Chen, J. M. Nester and R. S. Tung, “The Hamiltonian boundary term and
quasi-local energy ﬂux,” Phys. Rev. D 72 (2005) 104020 [arXiv:gr-qc/0508026].
[10] J. M. Nester, “General pseudotensors and quasilocal quantities,” Class. Quant.
Grav. 21, S261 (2004).
[11] C. C. M. Liu and S. T. Yau, “Positivity of Quasilocal Mass,” Phys. Rev. Lett.
90 (2003) 231102 [gr-qc/0303019].
[12] L. B. Szabados, “On the roots of the Poincare structure of asymptotically ﬂat
space-times,” Class. Quant. Grav. 20 (2003) 2627 [gr-qc/0302033].
[13] C. M. Chen and J. M. Nester, Grav. Cosmol. 6 (2000) 257 [gr-qc/0001088].
[14] C. M. Chen and J. M. Nester, “Quasilocal quantities for GR and other gravity
theories,” Class. Quant. Grav. 16 (1999) 1279 [gr-qc/9809020].
[15] C. C. Chang, J. M. Nester and C. M. Chen, “Pseudotensors and quasilocal grav-
itational energy momentum,” Phys. Rev. Lett. 83, 1897 (1999) [gr-qc/9809040].
[16] C. M. Chen, J. M. Nester and R. S. Tung, “Quasilocal energy momentum for
gravity theories,” Phys. Lett. A 203 (1995) 5 [gr-qc/9411048].
[17] L. B. Szabados, “Two-dimensional Sen connections in general relativity,” Class.
Quant. Grav. 11 (1994) 1833 [gr-qc/9402001].
[18] L. B. Szabados, “Two-dimensional Sen connections and quasilocal energy mo-
mentum,” Class. Quant. Grav. 11 (1994) 1847 [gr-qc/9402005].
[19] J. D. Brown and J. W. York, Jr., Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012].
[20] J. M. Nester, “Special orthonormal frames and energy localization”, Class.
Quant. Grav. 8, L19 (1991).
[21] J. M. Nester, “A covariant Hamiltonian for gravity theories,” Mod. Phys. Lett.
A 6 (1991) 2655.
[22] R. Beig and N.
´
O Murchadha, “The Poincar´e Group as the Symmetry Group of
Canonical General Relativity,” Annals Phys. 174 (1987) 463.
69[23] K. Kuchaˇr, “Dynamics of tensor ﬁelds in hyperspace. III,” J. Math. Phys. (N.Y.)
17, 801 (1976).
[24] T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian For-
mulation of General Relativity,” Annals Phys. 88 (1974) 286.
[25] L. Nirenberg, “The Weyl and Minkowski problems in diﬀerential geometry in the
large,” Comm. Pure Appl. Math. 6, 337-394, (1953).
[26] Ph. Freud, “Uber Die Ausdrucke Der Gesamtenergie Und Des Gesamtimpulses
Eines Materiellen Systems in Der Allgemeinen Relativitatstheorie”, Ann. Math.
40, 417-419, (1939).
Books
[27] 梁燦彬, 周彬, “微分幾何入門與廣義相對論”, 社會科學出版社, (2006).
[28] F. W. Hehl and Yuri N. Obukhov, “Foundations of Classical Electrodynamics,”
§B.1., (2003).
[29] S. S. Chern, W. H. Chen, K. S. Lam, “Lectures on Diﬀerential Geometry,” World
Scientiﬁc, (1999).
[30] M. Spivak, “A Comprehensive Introduction to Diﬀerential Geometry,” volume
2, third edition, Ch4, Part D. (1999).
[31] M. Spivak, “A Comprehensive Introduction to Diﬀerential Geometry,” volume
5, third edition, Ch11, (1999).
[32] Ray D’Inverno, “Introducing Einstein’s Relativity” Oxford: New York, Toronto,
(1993).
[33] 陳省身, “微分幾何講義,” 聯經出版, (1990).
[34] 伍鴻熙, 沈純理, 虞言林, “黎曼幾何初步,”北京大學出版, (1989).
[35] Phillip A. Griﬃths, Gary R. Jensen, “DIFFERENTIAL SYSTEMS AND ISOMETRIC EMBEDDINGS,” Princeton University Press, (1987).
70[36] C. Lancos, “The Variational Principles of Mechanics,” Dover (1986).
[37] R. M. Wald, “General Relativity,” The University of Chicago Press, (1984).
[38] H. Blaine Lawson, “Lectures on Minimal Submanifolds,” Volume I, (1980), §2.
[39] A. Papapetrou, “Lectures on general relativity”, (1974), §31–§35, eq. (31.14), eq.
(35.2).
[40] C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation,” Freeman, New
York, (1973), §5.8.

指導教授

聶斯特(James M. Nester)

審核日期

2013-8-28

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