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The self-similarity hypothesis claims that in classical general relativity,spherically symmetric solutions may naturally evolve to a self-similar form incertain circumstances. In this context, the validity of the correspondinghypothesis in nonspherical geometry is very interesting as there may existgravitational waves. We investigate self-similar vacuum solutions to theEinstein equation in the so-called whole-cylinder symmetry. We find that thosesolutions are reduced to part of the Minkowski spacetime with a regular orconically singular axis and with trivial or nontrivial topology if thehomothetic vector is orthogonal to the cylinders of symmetry. These solutionsare analogous to the Milne universe, but only in the direction parallel to theaxis. Using these solutions, we discuss the nonuniqueness (and nonvanishingnature) of C energy and the existence of a cylindrical trapping horizon inMinkowski spacetime. Then, as we generalize the analysis, we find atwo-parameter family of self-similar vacuum solutions, where the homotheticvector is not orthogonal to the cylinders in general. The family includes theMinkowski, the Kasner and the cylindrical Milne solutions. The obtainedsolutions describe the interior to the exploding (imploding) shell ofgravitational waves or the collapse (explosion) of gravitational wavesinvolving singularities from nonsingular initial data in general. Since recentnumerical simulations strongly suggest that one of these solutions may describethe asymptotic behavior of gravitational waves from the collapse of a dustcylinder, this means that the self-similarity hypothesis is naturallygeneralized to cylindrical symmetry.Comment: 21 pages, 10 figure files, revised version, accepted for publication  in Physical Review D, minor correction, erratum adde

Einstein-Rosen waves and the self-similarity hypothesis in cylindrical symmetry