All solutions of Einstein's field equations representing irrotational dust and possessing a metric of the form ds 2 = dt 2 -e 2a dr 2 -e 2β {dy 2 + dz 2 ) are found. The new metrics generalize the earlier Bondi-Tolman, Eardley-Liang-Sachs, and Kantowski-Sachs cosmological models.

A purely covariant approach to general relativity, using the equation of geodesic deviation, is adopted. The physical interpretation is essentially that due to Pirani, but instead of using clouds of particles to analyze the gravitational field, a ``gravitational compass'' is proposed which fulfills the same purpose. Particular attention is focussed on the different roles played by the matter and the free gravitational field. The latter splits up conveniently into a super-position of a transverse wave component, a longitudinal component, and a ``Coulomb'' field, all of which introduce ``shearing'' forces on the gravitational compass, while the matter contributes a general contraction. Applications to the Friedmann cosmological models and the problem of interacting gravitational waves are discussed.

A previously derived class of solutions representing gravitationally collapsing dust clouds is investigated in detail. The class includes all spherically symmetric collapses but is more general and is termed quasispherical. Most features of spherical collapse appear to go over in their essentials to the more general class. Under reasonable initial conditions the singularity is hidden behind an event horizon, but with slightly irregular initial density distributions portions of the singularity become naked. The latter feature is not restricted to the nonspherical case and will in general appear in situations such as collapsing spherical shells of matter.

The equations governing the collision of two plane gravitational waves are derived. The general exact solution representing this situation when both waves are linearly polarized are found, and some special solutions of possible physical interest are discussed in detail.