Living Reviews in Relativity

"Quasi-Local Energy-Momentum and Angular Momentum in General Relativity"
László B. Szabados 

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1 Introduction
2 Energy-Momentum and Angular Momentum of Matter Fields
2.1 Energy-momentum and angular-momentum density of matter fields
2.2 Quasi-local energy-momentum and angular momentum of the matter fields
3 On the Energy-Momentum and Angular Momentum of Gravitating Systems
3.1 On the gravitational energy-momentum and angular momentum density: The difficulties
3.2 On the global energy-momentum and angular momentum of gravitating systems: The successes
3.3 The necessity of quasi-locality for observables in general relativity
4 Tools to Construct and Analyze Quasi-Local Quantities
4.1 The geometry of spacelike two-surfaces
4.2 Standard situations to evaluate the quasi-local quantities
4.3 On lists of criteria of reasonableness of the quasi-local quantities
5 The Bartnik Mass and its Modifications
5.1 The Bartnik mass
5.2 Bray’s modifications
6 The Hawking Energy and its Modifications
6.1 The Hawking energy
6.2 The Geroch energy
6.3 The Hayward energy
7 Penrose’s Quasi-Local Energy-Momentum and Angular Momentum
7.1 Motivations
7.2 The original construction for curved spacetimes
7.3 The modified constructions
8 Approaches Based on the Nester–Witten 2-Form
8.1 The Ludvigsen–Vickers construction
8.2 The Dougan–Mason constructions
8.3 A specific construction for the Kerr spacetime
9 Quasi-Local Spin Angular Momentum
9.1 The Ludvigsen–Vickers angular momentum
9.2 Holomorphic/antiholomorphic spin angular momenta
9.3 A specific construction for the Kerr spacetime
10 The Hamilton–Jacobi Method
10.1 The Brown–York expression
10.2 Kijowski’s approach
10.3 Epp’s expression
10.4 The expression of Liu and Yau
10.5 The expression of Wang and Yau
11 Towards a Full Hamiltonian Approach
11.1 The 3 + 1 approaches
11.2 Approaches based on the double-null foliations
11.3 The covariant approach
12 Constructions for Special Spacetimes
12.1 The Komar integral for spacetimes with Killing vectors
12.2 The effective mass of Kulkarni, Chellathurai, and Dadhich for the Kerr spacetime
12.3 Expressions in static spacetimes
13 Applications in General Relativity
13.1 Calculation of tidal heating
13.2 Geometric inequalities for black holes
13.3 Quasi-local laws of black hole dynamics
13.4 Entropy bounds
13.5 Quasi-local radiative modes of general relativity
13.6 Potential applications in cosmology
14 Summary: Achievements, Difficulties, and Open Issues
14.1 On the Bartnik mass and Hawking energy
14.2 On the Penrose mass
14.3 On the Dougan–Mason energy-momenta and the holomorphic/antiholomorphic spin angular momenta
14.4 On the Brown–York–type expressions
15 Acknowledgments
Open References References