4 The Good-Cut Equation and ℋ-Space

In Section 3, we discussed NGCs in Minkowski spacetime that were shear-free. In this section we consider asymptotically shear-free NGCs in asymptotically-flat spacetimes. That is to say, we consider NGCs that have nonvanishing shear in the interior of the spacetime but where, as null infinity is approached, the shear vanishes. Whereas fully shear-free NGCs almost never occur in general asymptotically flat spacetimes, asymptotically shear-free congruences always exist. The case of algebraically-special spacetimes is the exception; they do allow one or two shear-free congruences.

We begin by reviewing the shear-free condition and follow with its generalization to the asymptotically shear-free case. From this we derive the generalization of the homogeneous good-cut equation to the inhomogeneous good-cut equation. Almost all the properties of the shear-free and asymptotically shear-free NGCs come from the study of these equations and virtually all the attributes of shear-free congruences are shared by the asymptotically shear-free congruences. It is from the use of these shared attributes that we will be able to extract physical identifications and information (e.g., complex center of mass/charge, Bondi mass, linear and angular momentum, equations of motion, etc.) from the asymptotic gravitational fields.

Though again the use of the formal complexification of + ℑ, i.e., + ℑℂ, is essential for our analysis, it is the extraction of the real structures that is important.

 4.1 Asymptotically shear-free NGCs and the good-cut equation
 4.2 ℋ-space and the good-cut equation
  4.2.1 Solutions to the shear-free equation
 4.3 Real cuts from the complex good cuts, II
 4.4 Summary of Real Structures
  Example: the (charged) Kerr metric

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