Unpolarized radiative cylindrical spacetimes: trapped surfaces and quasilocal energy

We consider the most general vacuum cylindrical spacetimes, which are defined by two global, spacelike, commuting, non-hypersurface-orthogonal Killing vector fields. The cylindrical waves in such spacetimes contain both + and × polarizations, and are thus said to be unpolarized. We show that there are no trapped cylinders in the spacetime, and present a formal derivation of Thorneʹs C-energy, based on a Hamiltonian reduction approach. Using the Brown–York quasilocal energy prescription, we compute the actual physical energy (per unit Killing length) of the system, which corresponds to the value of the Hamiltonian that generates unit proper-time translations orthogonal to a given fixed spatial boundary. The C-energy turns out to be a monotonic non-polynomial function of the Brown–York quasilocal energy. Finally, we show that the Brown–York energy at spatial infinity is related to an asymptotic deficit angle in exactly the same manner as the specific mass of a straight cosmic string is to the former.