Positivity of quasi-local mass II

Liu, Chiu-Chu Melissa; Yau, Shing-Tung

doi:10.1090/S0894-0347-05-00497-2

Positivity of quasi-local mass II
HTML articles powered by AMS MathViewer

by Chiu-Chu Melissa Liu and Shing-Tung Yau;

J. Amer. Math. Soc. 19 (2006), 181-204

DOI: https://doi.org/10.1090/S0894-0347-05-00497-2

Published electronically: July 12, 2005

PDF

Abstract:

We prove the following stronger version of the positivity of quasi-local energy (mass) stated by Liu and Yau: the quasi-local energy of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface and the boundary of the spacelike hypersurface is connected.

References

Robert Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989), no. 20, 2346–2348. MR 996396, DOI 10.1103/PhysRevLett.62.2346
Robert Bartnik, Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), no. 1, 31–71. MR 1198599
J. David Brown and James W. York Jr., Quasilocal energy in general relativity, Mathematical aspects of classical field theory (Seattle, WA, 1991) Contemp. Math., vol. 132, Amer. Math. Soc., Providence, RI, 1992, pp. 129–142. MR 1188439, DOI 10.1090/conm/132/1188439
J. David Brown and James W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D (3) 47 (1993), no. 4, 1407–1419. MR 1211109, DOI 10.1103/PhysRevD.47.1407
J. David Brown, Stephen R. Lau, and James W. York Jr., Energy of isolated systems at retarded times as the null limit of quasilocal energy, Phys. Rev. D (3) 55 (1997), no. 4, 1977–1984. MR 1438691, DOI 10.1103/PhysRevD.55.1977
Bernhelm Booß-Bavnbek and Krzysztof P. Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233386, DOI 10.1007/978-1-4612-0337-7
Michael E. Cahill and G. C. McVittie, Spherical symmetry and mass-energy in general relativity. I. General theory, J. Mathematical Phys. 11 (1970), 1382–1391. MR 260376, DOI 10.1063/1.1665273
D. Christodoulou and S.-T. Yau, Some remarks on the quasi-local mass, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 9–14. MR 954405, DOI 10.1090/conm/071/954405
A. J. Dougan and L. J. Mason, Quasilocal mass constructions with positive energy, Phys. Rev. Lett. 67 (1991), no. 16, 2119–2122. MR 1129012, DOI 10.1103/PhysRevLett.67.2119
Larry L. Smarr (ed.), Sources of gravitational radiation, Cambridge University Press, Cambridge-New York, 1979. MR 592635
Richard J. Epp, Angular momentum and an invariant quasilocal energy in general relativity, Phys. Rev. D (3) 62 (2000), no. 12, 124108, 30. MR 1814367, DOI 10.1103/PhysRevD.62.124018

Energy extraction

224

G. W. Gibbons, Collapsing shells and the isoperimetric inequality for black holes, Classical Quantum Gravity 14 (1997), no. 10, 2905–2915. MR 1476553, DOI 10.1088/0264-9381/14/10/016
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0

Gravitational radiation in an expanding universe

9

S. W. Hawking and Gary T. Horowitz, The gravitational Hamiltonian, action, entropy and surface terms, Classical Quantum Gravity 13 (1996), no. 6, 1487–1498. MR 1397130, DOI 10.1088/0264-9381/13/6/017
Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
Oussama Hijazi, Sebastián Montiel, and Xiao Zhang, Eigenvalues of the Dirac operator on manifolds with boundary, Comm. Math. Phys. 221 (2001), no. 2, 255–265. MR 1845323, DOI 10.1007/s002200100475
Pong Soo Jang, On the positivity of energy in general relativity, J. Math. Phys. 19 (1978), no. 5, 1152–1155. MR 488515, DOI 10.1063/1.523776
Jerzy Kijowski, A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity, Gen. Relativity Gravitation 29 (1997), no. 3, 307–343. MR 1439857, DOI 10.1023/A:1010268818255
Stephen R. Lau, New variables, the gravitational action and boosted quasilocal stress-energy-momentum, Classical Quantum Gravity 13 (1996), no. 6, 1509–1540. MR 1397132, DOI 10.1088/0264-9381/13/6/019
Chiu-Chu Melissa Liu and Shing-Tung Yau, Positivity of quasilocal mass, Phys. Rev. Lett. 90 (2003), no. 23, 231102, 4. MR 2000281, DOI 10.1103/PhysRevLett.90.231102
M. Ludvigsen and J. A. G. Vickers, Momentum, angular momentum and their quasilocal null surface extensions, J. Phys. A 16 (1983), no. 6, 1155–1168. MR 706691, DOI 10.1088/0305-4470/16/6/010

Volumen und Oberfläche

57

Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 202511, DOI 10.1007/978-3-540-69952-1
N. Ó Murchadha, L. B. Szabados, and K. P. Tod, Comment on: “Positivity of quasilocal mass” [Phys. Rev. Lett. 90 (2003), no. 23, 231102, 4 pp.; MR2000281] by C.-C. M. Liu and S.-T. Yau, Phys. Rev. Lett. 92 (2004), no. 25, 259001, 1. MR 2114434, DOI 10.1103/PhysRevLett.92.259001
Thomas Parker and Clifford Henry Taubes, On Witten’s proof of the positive energy theorem, Comm. Math. Phys. 84 (1982), no. 2, 223–238. MR 661134, DOI 10.1007/BF01208569
R. Penrose, Quasilocal mass and angular momentum in general relativity, Proc. Roy. Soc. London Ser. A 381 (1982), no. 1780, 53–63. MR 661716, DOI 10.1098/rspa.1982.0058
Roger Penrose and Wolfgang Rindler, Spinors and space-time. Vol. 2, 2nd ed., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1988. Spinor and twistor methods in space-time geometry. MR 944085
A. V. Pogorelov, Some results on surface theory in the large, Advances in Math. 1 (1964), no. fasc. 2, 191–264. MR 178438, DOI 10.1016/0001-8708(65)90039-3
Yuguang Shi and Luen-Fai Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), no. 1, 79–125. MR 1987378
Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. MR 612249, DOI 10.1007/BF01942062

Quasi-local energy-momentum and angular momentum in GR: a review article

7

K. P. Tod, More on Penrose’s quasilocal mass, Classical Quantum Gravity 3 (1986), no. 6, 1169–1189. MR 868724, DOI 10.1088/0264-9381/3/6/016
Edward Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. MR 626707, DOI 10.1007/BF01208277
Shing Tung Yau, Geometry of three manifolds and existence of black hole due to boundary effect, Adv. Theor. Math. Phys. 5 (2001), no. 4, 755–767. MR 1926294, DOI 10.4310/ATMP.2001.v5.n4.a4