Change Log

Fast-track revision published on 7 December 2012

One of the key features of Living Reviews is that its articles are kept up-to-date by their authors. Errata and fast-track revisions of important new research developments are added directly to the original review and marked throughout the article.

Global description:

Recent developments of the field are included. A few subsections and more than fifty new references are added, minor improvements and corrections of the text are made at several points, and the bibliography is updated. The major changes are as follows:

Changes in detail:

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Update A new Subsection 3.1.1, on the gravitational energy in Newtonian theory and in a relativistically corrected Newtonian theory is added.
Update A new paragraph on a candidate for the total mass of closed universes is added.
Update A discussion of the monotonicity properties of the quasi-local mass expressions near spatial infinity, motivated by the compatibility with the post Newtonian limit is added.
Update A new paragraph on the incompatibility of the monotonicity of the quasi-local mass expressions and two ‘standard’ requirements is added.
Update A displayed formula is corrected, and two references are added.
Update New references on the generalization of the Hawking energy and its monotonicity properties, and on the Hawking and Geroch energy operators in loop quantum gravity are added.
Update A new reference to a modified construction by Zhang, having better asymptotic behaviour at spatial infinity, is given.
Update New references on the initial boundary value problem and its potential connection with the quasi-local Hamiltonian approach are added.
Update The more detailed discussion of the role of the area 2-form (as a part of the boundary conditions) is given and new references are added.
Update A new Subsection 11.3.3 is added with the suggestion for the reference configuration and a definition for the quasi-local quantities by Nester, Chen, Liu and Sun. Several new references are added.
Update A more detailed discussion of the properties of the Komar integral is given.
Update Section 12.3 is reorganized by adding two more subsections. One is on Tolman’s energy expression, the other is on the gravitational energy in the post-Newtonian limit of GR.
Update Several new references on geometric inequalities for black heles are added.
Update A new paragraph on (and the reference to) a recent reformulation and its proof of Thorne’s hoop conjecture for spherically symmetric configurations is given.
Update A new Subsection 13.4.3 on a new entropy bound for uncollapsed bodies is added.