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Recent work on the Bondi-Metzner-Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R3 and the properties of ð are derived from its relationship to an angular-momentum raising operator. The relationship of the sTlm(θ, φ) to the spherical harmonics of R4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

It is shown that a vacuum metric is algebraically special in the sense of the Petrov classification if and only if it contains a shear-free null geodesic congruence. By noting which field equations are used in the proof of the theorem, the theorem is extended to include an electromagnetic field with geodesic rays as well as a null field. From a theorem of Robinson's on the existence of a null electromagnetic "test field", one observes that such a field can be constructed in vacuum if and only if the Riemannian space is algebraically special. From the original theorem, one can easily show that the existence of two independent shear-free null congruences guarantees that the space is Petrov Type I-degenerate.

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