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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 2-surface S will be sought in the form (17View Equation). Before considering the specific constructions themselves we summarize the most important properties of the general expression of Equation (17View Equation). Since the most detailed discussion of Equation (17View Equation) is given probably in [360Jump To The Next Citation Point363Jump 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 {cA-}. Second, suppose that the spacetime is flat, and let {cA--} A be constant. Then the corresponding 1-form basis {haa } is the constant Cartesian one, which consists of exact 1-forms. Then since the Bramson superpotential w(cA-,cB-) ab is the anti-self-dual part (in the name indices) of a b a b hahb - h b ha, which is also exact, for such spinor bases Equation (17View Equation) gives zero. Therefore, the integral of Bramson’s superpotential (17View Equation) measures the non-integrability of the 1-form basis ' hAa-A' = cA-cA' A A, i.e. J AB- S is a measure of how much the actual 1-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 J S-- as angular momentum. It seems natural to choose those spinor fields that were used in the definition of the quasi-local energy-momenta in the previous 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 ,eAB-) 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-,eA B) ----, 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-,e ) A-B- we refer to that.

Another suggestion for the quasi-local spatial angular momentum, proposed by Liu and Yau [253Jump To The Next Citation Point], will be introduced in Section 10.4.1.

 9.1 The Ludvigsen-Vickers angular momentum
 9.2 Holomorphic/anti-holomorphic spin-angular momenta
 9.3 A specific construction for the Kerr spacetime

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