The construction of Penrose is based on twistor-theoretical ideas, and motivated by the linearized gravity integrals for energy-momentum and angular momentum. Since, however, twistor-theoretical ideas and basic notions are still considered ‘special knowledge’, the review here of the basic idea behind the Penrose construction is slightly more detailed than that of the others. The main introductory references of the field are the volumes [425, 426] by Penrose and Rindler on ‘Spinors and Spacetime’, especially volume 2, the very readable book by Hugget and Tod [277] and the comprehensive review article [516] by Tod.

7.1 Motivations

7.1.1 How do the twistors emerge?

7.1.2 Twistor space and the kinematical twistor

7.2 The original construction for curved spacetimes

7.2.1 Two-surface twistors and the kinematical twistor

7.2.2 The Hamiltonian interpretation of the kinematical twistor

7.2.3 The Hermitian scalar product and the infinity twistor

7.2.4 The various limits

7.2.5 The quasi-local mass of specific two-surfaces

7.2.6 Small surfaces

7.3 The modified constructions

7.3.1 The ‘improved’ construction with the determinant

7.3.2 Modification through Tod’s expression

7.3.3 Mason’s suggestions

7.1.1 How do the twistors emerge?

7.1.2 Twistor space and the kinematical twistor

7.2 The original construction for curved spacetimes

7.2.1 Two-surface twistors and the kinematical twistor

7.2.2 The Hamiltonian interpretation of the kinematical twistor

7.2.3 The Hermitian scalar product and the infinity twistor

7.2.4 The various limits

7.2.5 The quasi-local mass of specific two-surfaces

7.2.6 Small surfaces

7.3 The modified constructions

7.3.1 The ‘improved’ construction with the determinant

7.3.2 Modification through Tod’s expression

7.3.3 Mason’s suggestions

Living Rev. Relativity 12, (2009), 4
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