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For a more detailed review of these results see Y. Choquet-Bruhat, “Positive Energy Theorems”, 1983 Les Houches Lectures.
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Although Eq. (20) looks identical toy (6) there is a slight difference. On an asymptotically flat surface, dgα A is defined to be a spinor satisfying ∂m °α A = 0 where ∂m is the derivative operator of the flat metric that the physical metric approaches. On an asymptotic null cone one does not have a flat three metric. Instead one defines °α A to a spinor whose components in the standard spinor basis are just spin ½ spherical harmonics.
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Horowitz, G.T. (1984). The positive energy theorem and its extensions. In: Flaherty, F.J. (eds) Asymptotic Behavior of Mass and Spacetime Geometry. Lecture Notes in Physics, vol 202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0048063
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