Having accepted that the gravitational energy-momentum and angular momentum should be introduced at the quasi-local level, we next need to discuss the special tools and concepts that are needed in practice to construct (or even to understand) the various special quasi-local expressions. Thus, first, in Section 4.1 we review the geometry of closed spacelike two-surfaces, with special emphasis on two-surface data. Then, in Sections 4.2 and 4.3, we discuss the special situations where there is a more-or-less generally accepted ‘standard’ definition for the energy-momentum (or at least for the mass) and angular momentum. In these situations any reasonable quasi-local quantity should reduce to them.

4.1 The geometry of spacelike two-surfaces

4.1.1 The Lorentzian vector bundle

4.1.2 Connections

4.1.3 Embeddings and convexity conditions

4.1.4 The spinor bundle

4.1.5 Curvature identities

4.1.6 The GHP formalism

4.1.7 Irreducible parts of the derivative operators

4.1.8 -connection one-form versus anholonomicity

4.2 Standard situations to evaluate the quasi-local quantities

4.2.1 Round spheres

4.2.2 Small surfaces

4.2.3 Large spheres near spatial infinity

4.2.4 Large spheres near null infinity

4.2.5 Other special situations

4.3 On lists of criteria of reasonableness of the quasi-local quantities

4.3.1 General expectations

4.3.2 Pragmatic criteria

4.3.3 Incompatibility of certain ‘natural’ expectations

4.1.1 The Lorentzian vector bundle

4.1.2 Connections

4.1.3 Embeddings and convexity conditions

4.1.4 The spinor bundle

4.1.5 Curvature identities

4.1.6 The GHP formalism

4.1.7 Irreducible parts of the derivative operators

4.1.8 -connection one-form versus anholonomicity

4.2 Standard situations to evaluate the quasi-local quantities

4.2.1 Round spheres

4.2.2 Small surfaces

4.2.3 Large spheres near spatial infinity

4.2.4 Large spheres near null infinity

4.2.5 Other special situations

4.3 On lists of criteria of reasonableness of the quasi-local quantities

4.3.1 General expectations

4.3.2 Pragmatic criteria

4.3.3 Incompatibility of certain ‘natural’ expectations

Living Rev. Relativity 12, (2009), 4
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