Hayward Quasilocal Energy of Tori

Abstract

In this paper, the authors show that one cannot dream of the positivity of the Hayward energy in the general situation. They consider a scenario of a spherically symmetric constant density star matched to the Schwarzschild solution, representing momentarily static initial data. It is proved that any topological tori within the star, distorted or not, have strictly positive Hayward energy. Surprisingly we find analytic examples of ‘thin’ tori with negative Hayward energy in the outer neighborhood of the Schwarzschild horizon. These tori are swept out by rotating the standard round circles in the static coordinates but they are distorted in the isotropic coordinates. Numerical results also indicate that there exist horizontally dragged tori with strictly negative Hayward energy in the region between the boundary of the star and the Schwarzschild horizon.

Keywords

Quasilocal energy / Positivity / Toroidal topology

References

[1] Arnowitt R, Deser S, Misner S. Coordinate invariance and energy expressions in general relativity. Phys. Rev., 1961, 122(3): 997-1006

[2] Bengtsson I. The Hawking energy on photo surfaces. Gen. Rel. Grav., 2020, 52(5): 52

[3] Brown J D, York J W. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D, 1993, 47(4): 1407-1419

[4] Christodoulou D, Yau S -T. Isenberg J. Some remarks on the quasi-local mass. Mathematics and general relativity (Santa Cruz, CA, 1986), 1988, Providence, RI: Amer. Math. Soc.

[5] Geroch R. Energy extraction. Ann. N.Y. Acad. Sci., 1973, 224(1): 108-117

[6] Hawking S W. Gravitational radiation in an expanding universe. J. Math. Phys., 1968, 9(4): 598-604

[7] Hayward G. Gravitational action for spacetimes with nonsmooth boundaries. Phys. Rev. D, 1993, 47(8): 3275-3280

[8] Hayward S. Quasilocal gravitational energy. Phys. Rev. D, 1994, 49(2): 831-839

[9] He X, Xie N. Quasi-local energy and Oppenheimer-Snyder collapse. Class. Quantum Grav., 2020, 37(18): 185016

[10] Huisken G, Ilmanen T. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom., 2001, 59(3): 353-437

[11] Husa S. Initial data for general relativity containing a marginally outer trapped torus. Phys. Rev. D, 1996, 54(12): 7311-7321

[12] Karkowski J, Mach P, Malec E Toroidal trapped surfaces and isoperimetric inequalities. Phys. Rev. D, 2017, 95(6): 064037

[13] Kijowski J. A simple derivation of the canonical structure and the quasi-local Hamiltonians in general relativity. Gen. Rel. Grav., 1997, 29(3): 307-343

[14] Mach P, Xie N. Toroidal marginally outer trapped surfaces in closed Friedmann-Lemaître-Robertson-Walker spacetimes: stability and isoperimetric inequalities. Phys. Rev. D, 2017, 96(8): 084050

[15] Murchadha O, N How large can a star be?. Phys. Rev. Lett., 1986, 57(19): 2466-2469

[16] Penrose R. Yau S -T. Some unsolved problems in classical general relativity. Seminar on Differential Geometry, 1982, Press, NJ: Princeton Univ.

[17] Schoen R, Yau S -T. On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys., 1979, 65(1): 45-76

[18] Schoen R, Yau S -T. Proof of the positive mass theorem. II. Commun. Math. Phys., 1981, 79(2): 231-260

[19] Shi Y, Tam L -F. Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom., 2002, 62(1): 79-125

[20] Szabados L. Quasi-local energy-momentum and angular momentum in general relativity. Liv. Rev. Relativ., 2009, 12: 4

[21] Wang M -T, Yau S -T. Quasilocal mass in general relativity. Phys. Rev. Lett., 2009, 102(2): 021101

[22] Witten E. A new proof of the positive energy theorem. Commun. Math. Phys., 1981, 80(3): 381-402