No abstract

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.

No abstract

No abstract

We use the stress-energy tensor and the associated energy-momentum conservation to study the interactions between two widely separated monopoles (or monopole and antimonopole). By defining a set of minimal conditions to represent the above systems, we show how the problem reduced mathematically to a known electrostatic problem. The force between the monopoles (or monopole and antimonopole) is then, to the leading order, the expected repulsive (attractive) Coulomb force. We also discuss how the Prasad-Sommerfield limit alters the problem, leading to twice the Coulomb force between a monopole and antimonopole and zero force between two monopoles.

Following Ashtekar's (1987) recently revised version of the standard canonical theory, the construction of a new variables canonical formalism for Einstein's theory of gravity is investigated when the time parameter has level sets which are null hypersurfaces. The configuration space variables are the components of a tetrad and the self-dual components of a connection. Because a null time parameter is used, the Hamiltonian formalism has second-class constraints as well as the first-class constraints which are associated with the invariance of the theory under diffeomorphisms and local gauge transformations. The first-class constraint algebra is discussed and reality conditions which relate the complex formalism to real general relativity are displayed.

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.