Collapsing and Expanding Cylindrically Symmetric Fields with Ligh-tlike   Wave-Fronts in General Relativity
  The dynamics of collapsing and expanding cylindrically symmetric gravitational and matter fields with lightlike wave-fronts is studied in General Relativity, using the Barrabes-Israel method. As an application of the general formulae developed, the collapse of a matter field that satisfies the condition R_{AB}g^{AB} = 0, (A, B = z, phi), in an otherwise flat spacetime background is studied. In particular, it is found that the gravitational collapse of a purely gravitational wave or a null dust fluid cannot be realized in a flat spacetime background. The studies are further specified to the collapse of purely gravitational waves and the general conditions for such collapse are found. It is shown that after the waves arrive at the axis, in general, part of them is reflected to spacelike infinity along the future light cone, and part of it is focused to form spacetime singularities on the symmetry axis. The cases where the collapse does not result in the formation of spacetime singularities are also identified. 
Paulo R.C.T. Pereira and A.Z. Wang