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GRG Editor's Choice: Quasi-local energy from a Minkowski reference

Chen, CM., Liu, JL. & Nester, J.M., Quasi-local energy from a Minkowski reference, Gen Relativ Gravit (2018) 50: 158.

Editor's Choice (Research Article)

First Online: 19 November 2018

"How to measure the amount of energy carried by gravitational field? This was one of the most intriguing problems in General Relativity Theory since its advent in 1915 and it remains such hundred years later. There are dozens of different approaches and even more definitions of the "quasi-local mass". The paper by Chen, Liou and Nester is surely one of the best analysis of these issues which I have ever seen in the literature."


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.

The authors:  GERG Ed's Chen Liu Nester © SpringerChiang-Mei Chen (陳江梅) is Professor of Physics at the Department of Physics and Center for High Energy and High Field Physics (CHiP) of the National Central University, Chungli, Taiwan. His research areas are Gravitation, Cosmology, and String Theory.

Jian-Liang Liu (刘建良) is working at the Department of Mathematics and Data Science, Dongguan University of Technology and the Department of Mathematics, Shantou University, China.

James M. Nester is Retired Professor at the Department of Physics and Graduate Institute of Astronomy of the National Central University, Chungli, and visiting scholar at the Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei, Taiwan. His research areas are Gravitational Fields and Relativity.

GRG Editor's Choice:

In each volume of GRG, a few papers are marked as “Editor’s Choice”. The primary criteria is original, high quality research that is of wide interest within the community.