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9.1 The Ludvigsen-Vickers angular momentum

Under the conditions that ensured the Ludvigsen-Vickers construction for the energy-momentum would work in Section 8.1, the definition of their angular momentum is straightforward [259Jump To The Next Citation Point]. Since in Minkowski spacetime the Ludvigsen-Vickers spinors are just the restriction to S of the constant spinor fields, by the general remark above the Ludvigsen-Vickers spin-angular momentum is zero in Minkowski spacetime.

Using the asymptotic solution of the Einstein-Maxwell equations in a Bondi-type coordinate system it has been shown in [259] that the Ludvigsen-Vickers spin-angular momentum tends to that of Bramson at future null infinity. For small spheres [360Jump To The Next Citation Point] in non-vacuum it reproduces precisely the expected result (29View Equation), and in vacuum it is

A B 4 ' ' '( ' A B ) ( ) JSr--= ----r5TAA'BB'CC'DD'tAA tBB tCC rtD EeDF E(EE F) + O r7 . (62) 45G
We stress that in both the vacuum and non-vacuum cases the factor ' rtDEeDF A B E(EE F), interpreted in Section 4.2.2 as an average of the boost-rotation Killing fields that vanish at p, emerges naturally. No (approximate) boost-rotation Killing field was put into the general formulae by hand.
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