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Mean-curvature flow and quasilocal mass for 2-surfaces in Hamiltonian general relativity

### Abstract

A family of quasilocal mass definitions that includes as special cases the Hawking mass and the Brown-York “rest mass” energy is derived for spacelike 2-surfaces in space-time. The definitions involve an integral of powers of the norm of the space-time mean-curvature vector of the 2-surface, whose properties are connected with apparent horizons. In particular, for any spacelike 2-surface, the direction of mean curvature is orthogonal (dual in the normal space) to a unique normal direction in which the 2-surface has vanishing expansion in space-time. The quasilocal mass definitions are obtained by an analysis of boundary terms arising in the gravitational ADM Hamiltonian on hypersurfaces with a spacelike 2-surface boundary, using a geometric time flow chosen proportional to the dualized mean-curvature vector field at the boundary surface. A similar analysis is made choosing a geometric rotational flow given in terms of the twist covector of the dual pair of mean-curvature vector fields, which leads to a family of quasilocal angular momentum definitions involving the squared norm of the twist. The large sphere limit of these definitions is shown to yield the ADM mass and angular momentum in asymptotically flat space-times, while at apparent horizons a quasilocal version of the Gibbons-Penrose inequality is derived. Finally, some results concerning positivity are proved for the quasilocal masses, motivated by consideration of spacelike mean-curvature flow of 2-surfaces in space-time.

© 2007 American Institute of Physics

Received 12 August 2006
Accepted 21 March 2007
Published online 29 May 2007

Acknowledgments:
Roh Tung is thanked for contributions and fruitful discussions in the course of this research, particularly for an initial outline of the Noether charge analysis.

Article outline:

I. INTRODUCTION
II. GEOMETRY OF SPACELIKE 2-SURFACES IN SPACE-TIME
III. HAMILTONIAN ANALYSIS WITH SPATIAL BOUNDARY CONDITIONS RELATED TO MEAN-CURVATURE FLOW
A. Mean-curvature Hamiltonians and flows
B. Noether charge analysis for Hamiltonians with mean-curvature time flow
C. Analysis of null flows and Hamiltonians
IV. QUASILOCAL MEAN-CURVATURE QUANTITIES
A. Mean-curvature mass
B. Mean-curvature angular momentum
V. PROPERTIES OF MEAN-CURVATURE MASS AND ANGULAR MOMENTUM
A. Positivity
B. Large sphere limit and apparent horizons
C. Killing horizons and static 2-surfaces
D. Flat space-time
VI. CONCLUDING REMARKS

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2007-05-29

2016-09-01

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