We saw in Section 3.1.1 that in the Newtonian theory of gravity the mass of the source in can be expressed as the flux integral of the gravitational field strength on the boundary . Similarly, in the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the gravitational field (i.e., the linearized energy-momentum tensor) is still analogous to charge. In fact, the total energy-momentum and angular momentum of the source can be expressed as appropriate two-surface integrals of the curvature at infinity . Thus, it is natural to expect that the energy-momentum and angular momentum of the source in a finite three-volume , given by Eq. (2.5), can also be expressed as the charge integral of the curvature on the two-surface . However, the curvature tensor can be integrated on only if at least one pair of its indices is annihilated by some tensor via contraction, i.e., according to Eq. (3.14) if some is chosen and . To simplify the subsequent analysis, will be chosen to be anti-self-dual: with .11 Thus, our goal is to find an appropriate spinor field on such thatvalence 2 twistor equation, a specific example for the general twistor equation for . Thus, as could be expected, depends on the Killing vector , and, in fact, can be recovered from as . Thus, plays the role of a potential for the Killing vector . However, as a consequence of Eq. (7.4), is a self-dual Killing 1-form in the sense that its derivative is a self-dual (s.d.) 2-form: In fact, the general solution of Eq. (7.4) and the corresponding Killing vector are real s, and the spatial angular momentum and center-of-mass are combined into the three complex components of the self-dual angular momentum , generated by .
Recall that the space of the contravariant valence-one twistors of Minkowski spacetime is the set of the pairs of spinor fields, which solve the valence-one–twistor equation . If is a solution of this equation, then is a solution of the corresponding equation in the conformally-rescaled spacetime, where and is the conformal factor. In general, the twistor equation has only the trivial solution, but in the (conformal) Minkowski spacetime it has a four complex-parameter family of solutions. Its general solution in the Minkowski spacetime is , where and are constant spinors. Thus, the space of valence-one twistors, called the twistor space, is four–complex-dimensional, and hence, has the structure . admits a natural Hermitian scalar product: if is another twistor, then . Its signature is , it is conformally invariant, , and it is constant on Minkowski spacetime. The metric defines a natural isomorphism between the complex conjugate twistor space, , and the dual twistor space, , by . This makes it possible to use only twistors with unprimed indices. In particular, the complex conjugate of the covariant valence-two twistor can be represented by the conjugate twistor . We should mention two special, higher-valence twistors. The first is the infinity twistor. This and its conjugate are given explicitly by
The solution of the valence-two twistor equation, given by Eq. (7.5), can always be written as a linear combination of the symmetrized product of the solutions and of the valence-one twistor equation. uniquely defines a symmetric twistor (see, e.g., ). Its spinor parts are
However, Eq. (7.1) can be interpreted as a -linear mapping of into , i.e., Eq. (7.1) defines a dual twistor, the (symmetric) kinematical twistor , which therefore has the structurethe quasi-local energy-momentum and self-dual angular momentum of the source are certain spinor parts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor, it has only ten real components as a consequence of its structure (its spinor part is identically zero) and the reality of . These properties can be reformulated by the infinity twistor and the Hermitian metric as conditions on : the vanishing of the spinor part is equivalent to and the energy momentum is the part of the kinematical twistor, while the whole reality condition (ensuring both and the reality of the energy-momentum) is equivalent to  or as its determinant : , the various components of the Pauli–Lubanski spin vector can also be expressed by the kinematic and infinity twistors and by certain special null twistors: if and are two different (null) twistors such that and , then
Thus, to summarize, the various spinor parts of the kinematical twistor are the energy-momentum and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian scalar product, are needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and, in particular, to define the mass and express the Pauli–Lubanski spin. Furthermore, the Hermiticity condition ensuring that has the correct number of components (ten reals) is also formulated in terms of these additional structures.
Living Rev. Relativity 12, (2009), 4
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