### Abstract

Given a spacelike 2-surface σ in a spacetime N and a constant future timelike unit vector T_{0} in R_{3},1, we derive upper and lower estimates of Wang-Yau quasilocal energy E(σ,X, T_{0}) for a given isometric embedding X of σ into a flat 3-slice in R3,1. The quantity E(σ,X, T_{0}) itself depends on the choice of X; however, the infimum of E(σ,X, T_{0}) over T_{0} does not. In particular, when σ bounds a compact domain ω in a time symmetric 3-slice in N and has nonnegative Brown-York quasilocal mass mBY(σ, ω), our estimates show that infT0 E(σ,X, T_{0}) equals mBY(σ, ω). We also study the spatial limit of infT0 E(Sr,Xr, T0), where Sr is a large coordinate sphere in a fixed end of an asymptotically flat initial data set (M, g, p) and X_{r} is an isometric embedding of S_{r} into R^{3} ∪ R^{3,1}. We show that if (M, g, p) has future timelike ADM energy-momentum, then lim r→∞ infT_{0} E(S_{r},X_{r}, T_{0}) equals the ADM mass of (M, g, p).

Original language | English (US) |
---|---|

Article number | 245017 |

Journal | Classical and Quantum Gravity |

Volume | 26 |

Issue number | 24 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Classical and Quantum Gravity*,

*26*(24), [245017]. https://doi.org/10.1088/0264-9381/26/24/245017

**Some estimates of Wang-Yau quasilocal energy.** / Miao, Pengzi; Tam, Luen Fai; Xie, Naqing.

Research output: Contribution to journal › Article

*Classical and Quantum Gravity*, vol. 26, no. 24, 245017. https://doi.org/10.1088/0264-9381/26/24/245017

}

TY - JOUR

T1 - Some estimates of Wang-Yau quasilocal energy

AU - Miao, Pengzi

AU - Tam, Luen Fai

AU - Xie, Naqing

PY - 2009

Y1 - 2009

N2 - Given a spacelike 2-surface σ in a spacetime N and a constant future timelike unit vector T0 in R3,1, we derive upper and lower estimates of Wang-Yau quasilocal energy E(σ,X, T0) for a given isometric embedding X of σ into a flat 3-slice in R3,1. The quantity E(σ,X, T0) itself depends on the choice of X; however, the infimum of E(σ,X, T0) over T0 does not. In particular, when σ bounds a compact domain ω in a time symmetric 3-slice in N and has nonnegative Brown-York quasilocal mass mBY(σ, ω), our estimates show that infT0 E(σ,X, T0) equals mBY(σ, ω). We also study the spatial limit of infT0 E(Sr,Xr, T0), where Sr is a large coordinate sphere in a fixed end of an asymptotically flat initial data set (M, g, p) and Xr is an isometric embedding of Sr into R3 ∪ R3,1. We show that if (M, g, p) has future timelike ADM energy-momentum, then lim r→∞ infT0 E(Sr,Xr, T0) equals the ADM mass of (M, g, p).

AB - Given a spacelike 2-surface σ in a spacetime N and a constant future timelike unit vector T0 in R3,1, we derive upper and lower estimates of Wang-Yau quasilocal energy E(σ,X, T0) for a given isometric embedding X of σ into a flat 3-slice in R3,1. The quantity E(σ,X, T0) itself depends on the choice of X; however, the infimum of E(σ,X, T0) over T0 does not. In particular, when σ bounds a compact domain ω in a time symmetric 3-slice in N and has nonnegative Brown-York quasilocal mass mBY(σ, ω), our estimates show that infT0 E(σ,X, T0) equals mBY(σ, ω). We also study the spatial limit of infT0 E(Sr,Xr, T0), where Sr is a large coordinate sphere in a fixed end of an asymptotically flat initial data set (M, g, p) and Xr is an isometric embedding of Sr into R3 ∪ R3,1. We show that if (M, g, p) has future timelike ADM energy-momentum, then lim r→∞ infT0 E(Sr,Xr, T0) equals the ADM mass of (M, g, p).

UR - http://www.scopus.com/inward/record.url?scp=84908139935&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908139935&partnerID=8YFLogxK

U2 - 10.1088/0264-9381/26/24/245017

DO - 10.1088/0264-9381/26/24/245017

M3 - Article

VL - 26

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 24

M1 - 245017

ER -