9 Quasi-Local Spin Angular Momentum

In this section we review three specific quasi-local spin–angular-momentum constructions that are (more or less) ‘quasi-localizations’ of Bramson’s expression at null infinity. Thus, the quasi-local spin angular momentum for the closed, orientable spacelike two-surface 𝒮 will be sought in the form (3.16View Equation). Before considering the specific constructions themselves, we summarize the most important properties of the general expression of Eq. (3.16View Equation). Since the most detailed discussion of Eq. (3.16View Equation) is probably given in [494Jump To The Next Citation Point, 496Jump To The Next Citation Point], the subsequent discussions will be based on them.

First, observe that the integral depends on the spinor dyad algebraically, thus it is enough to specify the dyad only at the points of 𝒮. Obviously, AB- J 𝒮 transforms like a symmetric second-rank spinor under constant SL (2,ℂ ) transformations of the dyad A {λA}. Second, suppose that the spacetime is flat, and let {λA-} A be constant. Then the corresponding one-form basis {𝜗aa} is the constant Cartesian one, which consists of exact one-forms. Then, since the Bramson superpotential w(λA, λB) ab is the anti-self-dual part (in the name indices) of a b a b 𝜗a𝜗b − 𝜗b𝜗a, which is also exact, for such spinor bases, Eq. (3.16View Equation) gives zero. Therefore, the integral of Bramson’s superpotential (3.16View Equation) measures the nonintegrability of the one-form basis ′ 𝜗AaA′ = λA-¯λA′ A A, i.e., JAB- 𝒮 is a measure of how much the actual one-form basis is ‘distorted’ by the curvature relative to the constant basis of Minkowski spacetime.

Thus, the only question is how to specify a spin frame on 𝒮 to be able to interpret AB J𝒮--- as angular momentum. It seems natural to choose those spinor fields that were used in the definition of the quasi-local energy-momenta in Section 8. At first sight this may appear to be only an ad hoc idea, but, recalling that in Section 8 we interpreted the elements of the spin spaces A- (S ,𝜀AB) as the ‘spinor constituents of the quasi-translations of 𝒮’, we can justify such a choice. Based on our experience with the superpotentials for the various conserved quantities, the quasi-local angular momentum can be expected to be the integral of something like ‘superpotential’ × ‘quasi-rotation generator’, and the ‘superpotential’ is some expression in the first derivative of the basic variables, actually the tetrad or spinor basis. Since, however, Bramson’s superpotential is an algebraic expression of the basic variables, and the number of the derivatives in the expression for the angular momentum should be one, the angular momentum expressions based on Bramson’s superpotential must contain the derivative of the ‘quasi-rotations’, i.e., (possibly a combination of) the ‘quasi-translations’. Since, however, such an expression cannot be sensitive to the ‘change of the origin’, they can be expected to yield only the spin part of the angular momentum.

The following two specific constructions differ from each other only in the choice for the spin space (SA,-𝜀AB ) ---, and correspond to the energy-momentum constructions of the previous Section 8. The third construction (valid only in the Kerr spacetimes) is based on the sum of two terms, where one is Bramson’s expression, and uses the spinor fields of Section 8.3. Thus, the present section is not independent of Section 8, and, for the discussion of the choice of the spin spaces (SA, 𝜀 ) AB-, we refer to that.

Another suggestion for the quasi-local spatial angular momentum, proposed by Liu and Yau [338Jump To The Next Citation Point], will be introduced in Section 10.4.1.

 9.1 The Ludvigsen–Vickers angular momentum
 9.2 Holomorphic/antiholomorphic spin angular momenta
 9.3 A specific construction for the Kerr spacetime

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