Quasi-local energy from a Minkowski reference

Chiang Mei Chen, Jian Liang Liu, James M. Nester

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4 Scopus citations


The specification of energy for gravitating systems has been an unsettled issue since Einstein proposed his pseudotensor. It is now understood that energy-momentum is quasi-local (associated with a closed 2-surface). Here we consider quasi-local proposals (including pseudotensors) in the Lagrangian–Noether–Hamiltonian formulations. There are two ambiguities: (1) there are many possible expressions, (2) they depend on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. The Hamiltonian perspective helped us to make a remarkable discovery: with an isometric Minkowski reference a large class of expressions—namely all those that agree with the Einstein pseudotensor’s Freud superpotential to linear order—give a common quasi-local energy value. Moreover, with a best-matched reference on the boundary this is the Wang–Yau mass value.

Original languageEnglish
Article number158
JournalGeneral Relativity and Gravitation
Issue number12
StatePublished - 1 Dec 2018


  • Hamiltonian
  • Minkowski reference
  • Quasi-local energy


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