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 will be sought in the form (3.16). Before considering the specific constructions themselves, we summarize the most important properties of the general expression of Eq. (3.16). Since the most detailed discussion of Eq. (3.16) is probably given in [494, 496], 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 . Obviously, transforms like a symmetric second-rank spinor under constant transformations of the dyad . Second, suppose that the spacetime is flat, and let be constant. Then the corresponding one-form basis is the constant Cartesian one, which consists of exact one-forms. Then, since the Bramson superpotential is the anti-self-dual part (in the name indices) of , which is also exact, for such spinor bases, Eq. (3.16) gives zero. Therefore, the integral of Bramson’s superpotential (3.16) measures the nonintegrability of the one-form basis , i.e., 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 to be able to interpret 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 as the ‘spinor constituents of the quasi-translations of ’, 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 , 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 , we refer to that.
Another suggestion for the quasi-local spatial angular momentum, proposed by Liu and Yau , will be introduced in Section 10.4.1.
Living Rev. Relativity 12, (2009), 4
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