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 [346Jump 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 [346] that the Ludvigsen–Vickers spin angular momentum tends to that of Bramson at future null infinity. For small spheres [494Jump To The Next Citation Point] in nonvacuum it reproduces precisely the expected result (4.10View Equation), and in vacuum it is

4 ( ) ( ) J𝒮AB-= ----r5TAA′BB ′CC ′DD ′tAA ′tBB ′tCC ′ rtD′E𝜀DF ℰA(EℰFB) + 𝒪 r7 . (9.1 ) r 45G
We stress that in both the vacuum and nonvacuum cases, the factor rtD ′E 𝜀DF ℰ A(E-ℰBF), 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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