只要提供一個合適的演化向量和適當的背景幾何，哈氏量的邊界項能給出引力場準局
域的值。本文的目標在建構一套最佳化的辦法來確定適當的背景值。首先將封閉二維面
上度規的十個分量，從動力學時空等度規地嵌入到在背景幾何上， 然後透過要求準局域
能量取極值的方法來確定適當的背景值。這套辦法也同時決定了演化向量的選取。我們
以軸對稱的動力學時空，針對克爾度規的情況明確地計算了準局域的能量和角動量。

The boundary term of the gravitational Hamiltonian can be used to give the values
of the quasi-local quantities as long as one can provide a suitable evolution vector field and an
appropriate reference geometry. On the two-surface boundary of a region we have proposed using
four dimensional isometric matching between the dynamic spacetime and the reference geometry
along with energy extremization to find both the optimal reference matching and the appropriate
quasi-Killing vectors. Here we consider the axisymmetric spacetime case. For the Kerr metric
in particular we can explicitly solve the equations to find the best matched reference and quasi-
Killing vectors. This leads to the exact expression for the quasi-local boundary term and the values
of our optimal quasi-local energy and angular momentum.

[1] P. G. Bergmann, R. Thomson, Spin and Angular Momentum in General Relativity, Phys.
Rev., 89, 400, (1953).
[2] I. S. Booth, R. B. Mann, Moving observers, nonorthogonal boundaries, and quasilocal energy, Phys. Rev. D, 59, 064021, (1999).
[3] I. S. Booth, J. D. E. Creighton, Quasilocal calculation of tidal heating, Phys. Rev. D, 62,
067503, (2000).
[4] J. D. Brown, J.W. York, Quasilocal energy and conserved charges derived from the gravita-
tional action, Phys. Rev. D, 47, 1407–1419, (1993).
[5] J. D. Brown, S. R. Lau, J. W. York, Canonical quasilocal energy and small spheres, Phys.
Rev. D, 59, 064028, (1999).
[6] P. T. Chru´sciel, On the relation between the einstein and the komar expression for the energy
of the gravitational field, Ann. Inst. Henri. Poincaré, 42, 267, (1985).
[7] S. Deser, J. S. Franklin, D. Seminara, Graviton-Graviton Scattering, Bel-Robinson and En-
ergy (Pseudo)-Tensors, Class.Quant.Grav., 16, 2815, (1999).
[8] M. Favata, Energy localization invariance of tidal work in general relativity, Phys. Rev. D,
63, 064013, (2001).
[9] Ph. Freud, Über die Ausdrücke der Geasmtenergie und des Gesamtimpulses eines Ma-
teriellen Systems in der allgemeinen Relativitätstheorie, Ann. Math., 40, 417, (1939).
[10] G. W. Gibbons, The isoperimetric and Bogomolny inequalities for black holes, in Willmore,
T.J., and Hitchin, N.J., eds., Global Riemannian Geometry, 194–202, (Ellis Horwood; Halsted Press, Chichester; New York, 1984).
[11] F.-H. Ho, Quasilocal center-of-mass for GR∥, Ms. Thesis, (National Central University,
Jung-Li, Taiwan, R.O.C.), (2003), unpublished.
[12] J. Katz, J. Biˇcák, D. Lynden-Bell, Relativistic conservation laws and integral constraints for
large cosmological perturbations, Phys. Rev. D, 55, 5957, (1997).
[13] J. Katz, J. Biˇcák, D. Lynden-Bell, Gravitational energy in stationary spacetimes, Class.
Quant. Grav., 23, 7111–7127, (2006).
[14] J. Kijowski, W. M. Tulczyjew, A Symplectic Framework for Field Theories, (Lecture Notes
in Physics 107, Springer-Verlag, Berlin, 1979).
[15] J. Kijowski, A Simple Derivation of Canonical Structure and Quasi-local Hamiltonians in
General Relativity, Gen. Relativ. Grav., 29, 307–43, (1996).
[16] A. Komar, Covariant conservation laws in general relativity, Phys. Rev., 113, 934–936,
(1959).
[17] C. Lanczos, The variational principles of mechanics (University of Toronto Press, Toronto,
1949).
[18] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Oxford: Pergamon, 1975).
[19] S. K. Chakrabarti, R. P. Geroch, C. Liang, Timelike Curves of Limted Acceleration in general
Relativity, J. Math. Phys., 24, 597–598, (1983).
[20] E. A. Martinez, Quasilocal energy for a Kerr black hole, Phys. Rev. D, 50, 4920, (1994).
[21] C. Møller, On the Localization of the Energy of a Physical System in the General Relativity,
Ann. Phys., 4, 347, (1958).
[22] J. M. Nester, F.-F. Meng, C.-M. Chen, Quasilocal Center-of-Mass, J.Korean Phys.Soc., 45,
S22–S25, (2004).
[23] C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (San Francisco: Freeman, 1973).
[24] J. M. Nester, A new gravitation energy expression with simpler positive proof, Phys. Lett. A,
83, 6, 241–242, (1981).
[25] W. Israel, J. M. Nester, Positivity of the Bondi gravitational mass, Phys. Lett. A, 85, 259–260.
(1981).
[26] C.-C. Chang, The Localization of Gravitational Energy: Pseudotensors and Quasilocal Expressions, MSc. Thesis (National Central University, Chung-li), 1999, unpublished.
[27] C.-M. Chen, J. M. Nester, R.-S. Tung, Quasilocal energy momentum for gravity theories,
Phys. Lett. A 203, 5 (1995)
[28] C.-C. Chang, J. M. Nester, C.-M. Chen, Pseudotensors and Quasilocal Energy-Momentum,
Phys. Rev. Lett., 83, 1897–901, (1999).
[29] C.-M. Chen and J. M. Nester, Quasilocal quantities for GR and other gravity theories, Class.
Quant.Grav., 16, 1279–1304, (1999)
[30] C.-M. Chen and J. M. Nester, A Symplectic Hamiltonian Derivation of Quasilocal Energy-
Momentum for GR, Grav. Cosmol., 6, 257, (2000).
[31] J. M. Nester, General pseudotensors and quasilocal quantities, Class. Quant. Grav., 21,
S261–S280, (2004).
[32] C.-M. Chen, J.-L. Liu, J. M. Nester, Quasi-local energy for cosmological models, Mod. Phys.
Lett. A, 22, 2039–2046, (2007).
[33] J. M. Nester, L.-L. So, T. Vargas, On the energy of homogeneous cosmologies, Phys. Rev. D,
78, 044035, (2008).
[34] J. M. Nester, L.-L. So, H. Chen, Energy-momentum density in small regions – the classical
pseudotensors Class. Quant. Grav., 26, 085004, (2009).
[35] R. Beig, N. Ó Murchadha, The Poincaré group as the symmetry group of canonical general
relativity, Ann. Phys., 174, 463–498, (1987).
[36] N. Ó Murchadha, L. B. Szabados, K. P. Tod, Comment on “Positivity of Quasilocal Mass”,
Phys. Rev. Lett., 92, 259001, (2004).
[37] A. Papapetrou, Einstein’s Theory of Gravitation and Flat Space, Proc. Roy. Irish. Acad. A52,
11, (1948).
[38] A. Papapetrou, Lectures on general relativity, (Springer, 1974)
[39] R. Penrose, Naked singularities, Ann. N.Y. Acad. Sci., 224, 125–134, (1973).
[40] R. Penrose, Sigularities and Time-Asmmetry in “General Relativity: An Einstein Centenary
Survey”, ed. S. W. Hawking, andW. Israel, (Cambridge: Cambridge University Press, 1979)
[41] R. Penrose, Quasilocal mass and angular momentum in general relativity, Proc. Roy. Soc.
Lond. A, 381, 53, (1982).
[42] N. P. Knopleva, V. N. Popov, Gauge fields, (San Francisco, 1973)
[43] P. Purdue, Gauge invariance of general relativistic tidal heating, Phys. Rev. D, 60, 104054,
(1999).
[44] T. Regge, C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general
relativity, Ann. Phys., 88, 286–319, (1974).
[45] G. Sun, Quasilocal conserved quantities for general relativity in small regions, Ms. Thesis,
(National Central University, Jung-Li, Taiwan, R.O.C.), (2005), unpublished.
[46] L.B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in General Relativity, Living Rev Relativity, 12, 4, (2009).
[47] J. Frauendiener, L. B. Szabados, A note on the post-Newtonian limit of quasi-local energy
expressions, Class. Quant. Grav., 28, 235009, (2011).
[48] R. D. Sorkin, Conserved Quantities as Action Variations, Contemporary Mathematics, 71,
23, (1988).
[49] M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 5, 2nd edition.
[50] Y. Shi, L.-F. Tam, Positive mass theorem and the boundary behaviors of compact manifolds
with nonnegative scalar curvature, J. Differ. Geom., 62, 79–125, (2002).
[51] R. C. Tolman, Relativity Thermodynamics and Cosmology, (London: Oxford University
Press, 1934).
[52] A. Trautman, Conservation Laws in General Relativity in “Gravitation: An Introduction to
Current Research”, ed. L. Witten (Wiley, New York, 1962)
[53] K. H. Vu, Quasilocal Energy-Momentum and Angular Momentum for Teleparallel Gravity,
MSc. Thesis (National Central University, Chung-li), (2000), unpublished.
[54] R. M. Wald, General Relativity, (The University of Chicago Press, 1984).
[55] S. Weinberg, Gravitation and Cosmology, (New York: Wiley, 1972).
[56] E. Witten, A new proof of the positive energy theorem, Commun. Math. Phys., 80, 381–402,
(1981).
[57] X. Wu, C.-M. Chen, J. M. Nester, Quasilocal energy-momentum and energy flux at null
infinity, Phys. Rev. D, 71, 124010 (2005).
[58] R. Schoen, S.-T. Yau, On the Proof of the Positive Mass Conjecture in General Relativity,
Commun. math. Phys., 65, 45–76 (1979).
[59] R. Schoen, S.-T. Yau, Proof that the Bondi mass is positive, Phys. Rev. Lett., 48, 369-371,
(1982).
[60] C.-C. M. Liu, S.-T. Yau, Positivity of quasilocal mass, Phys. Rev. Lett., 90, 231102-1-4,
(2003).
[61] M.-T. Wang, S.-T. Yau, A generalization of Liu-Yau’s quasi-local mass, Commun. Anal.
Geom., 15, 249, (2007).
[62] M.-T.Wang, S.-T. Yau, Quasilocal Mass in General Relativity, Phys. Rev. Lett., 102, 021101,
(2009).
[63] M.-T. Wang, S.-T. Yau, Commun. Math. Phys., 288, 919–942, (2009).
[64] N. Nadirashvili, Y. Yuan, Counterexamples for Local Isometric Embedding, arXiv e-print,
(2002). [math.DG/0208127].