Positivity of quasi-local mass II

Liu, Chiu-Chu Melissa^1, 2 ; Yau, Shing-Tung¹

¹ Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
² Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Journal of the American Mathematical Society, Tome 19 (2006) no. 1, pp. 181-204

Résumé

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.

Détail

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

Bibliographie

[1] Bartnik, Robert New definition of quasilocal mass Phys. Rev. Lett. 1989 2346 2348

[2] Bartnik, Robert Quasi-spherical metrics and prescribed scalar curvature J. Differential Geom. 1993 31 71

[3] Brown, J. David, York, James W., Jr. Quasilocal energy in general relativity 1992 129 142

[4] Brown, J. David, York, James W., Jr. Quasilocal energy and conserved charges derived from the gravitational action Phys. Rev. D (3) 1993 1407 1419

[5] Brown, J. David, Lau, Stephen R., York, James W., Jr. Energy of isolated systems at retarded times as the null limit of quasilocal energy Phys. Rev. D (3) 1997 1977 1984

[6] BooãŸ-Bavnbek, Bernhelm, Wojciechowski, Krzysztof P. Elliptic boundary problems for Dirac operators 1993

[7] Cahill, Michael E., Mcvittie, G. C. Spherical symmetry and mass-energy in general relativity. I. General theory J. Mathematical Phys. 1970 1382 1391

[8] Christodoulou, D., Yau, S.-T. Some remarks on the quasi-local mass 1988 9 14

[9] Dougan, A. J., Mason, L. J. Quasilocal mass constructions with positive energy Phys. Rev. Lett. 1991 2119 2122

[10] Sources of gravitational radiation 1979

[11] Epp, Richard J. Angular momentum and an invariant quasilocal energy in general relativity Phys. Rev. D (3) 2000

[12] Gibbons, G. W. Collapsing shells and the isoperimetric inequality for black holes Classical Quantum Gravity 1997 2905 2915

[13] Gilbarg, David, Trudinger, Neil S. Elliptic partial differential equations of second order 1983

[14] Hawking, S. W., Horowitz, Gary T. The gravitational Hamiltonian, action, entropy and surface terms Classical Quantum Gravity 1996 1487 1498

[15] Huisken, Gerhard, Ilmanen, Tom The inverse mean curvature flow and the Riemannian Penrose inequality J. Differential Geom. 2001 353 437

[16] Hijazi, Oussama, Montiel, Sebastiã¡N, Zhang, Xiao Eigenvalues of the Dirac operator on manifolds with boundary Comm. Math. Phys. 2001 255 265

[17] Jang, Pong Soo On the positivity of energy in general relativity J. Math. Phys. 1978 1152 1155

[18] Kijowski, Jerzy A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity Gen. Relativity Gravitation 1997 307 343

[19] Lau, Stephen R. New variables, the gravitational action and boosted quasilocal stress-energy-momentum Classical Quantum Gravity 1996 1509 1540

[20] Liu, Chiu-Chu Melissa, Yau, Shing-Tung Positivity of quasilocal mass Phys. Rev. Lett. 2003

[21] Ludvigsen, M., Vickers, J. A. G. Momentum, angular momentum and their quasilocal null surface extensions J. Phys. A 1983 1155 1168

[22] Morrey, Charles B., Jr. Multiple integrals in the calculus of variations 1966

[23] Ó Murchadha, N., Szabados, L. B., Tod, K. P. Comment on: “Positivity of quasilocal mass” [Phys. Rev. Lett. 90 (2003), no. 23, 231102, 4 pp. Phys. Rev. Lett. 2004

[24] Parker, Thomas, Taubes, Clifford Henry On Witten’s proof of the positive energy theorem Comm. Math. Phys. 1982 223 238

[25] Penrose, R. Quasilocal mass and angular momentum in general relativity Proc. Roy. Soc. London Ser. A 1982 53 63

[26] Penrose, Roger, Rindler, Wolfgang Spinors and space-time. Vol. 2 1988

[27] Pogorelov, A. V. Some results on surface theory in the large Advances in Math. 1964 191 264

[28] Shi, Yuguang, Tam, Luen-Fai Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature J. Differential Geom. 2002 79 125

[29] Schoen, Richard, Yau, Shing Tung Proof of the positive mass theorem. II Comm. Math. Phys. 1981 231 260

[30] Tod, K. P. More on Penrose’s quasilocal mass Classical Quantum Gravity 1986 1169 1189

[31] Witten, Edward A new proof of the positive energy theorem Comm. Math. Phys. 1981 381 402

[32] Yau, Shing Tung Geometry of three manifolds and existence of black hole due to boundary effect Adv. Theor. Math. Phys. 2001 755 767