Critical collapse of a cylindrically symmetric scalar field in four-dimensional Einstein's theory of gravity

Four-dimensional cylindrically symmetric spacetimes with homothetic self-similarity are studied in the context of Einstein's theory of gravity, and a class of exact solutions to the Einstein-massless scalar field equations is found. Their local and global properties are investigated and it is found that they represent gravitational collapse of a massless scalar field. In some cases the collapse forms black holes with cylindrical symmetry, while in the other cases it does not. The linear perturbations of these solutions are also studied and given in closed form. From the spectra of the unstable eigenmodes, it is found that there exists one solution that has precisely one unstable mode, which may represent a critical solution, sitting on a boundary that separates two different basins of attraction in the phase space.