Quasi-local energy for spherically symmetric spacetimes

Ming Fan Wu, Chiang Mei Chen, Jian Liang Liu, James M. Nester

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime.

Original languageEnglish
Pages (from-to)2401-2417
Number of pages17
JournalGeneral Relativity and Gravitation
Issue number9
StatePublished - Sep 2012


  • Hamiltonian boundary term
  • Quasi-local energy
  • Spherically symmetric spacetimes


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