Stability of the static-fluid cylindrical regular spacetime

Abstract

In this paper, we study the mechanical stability of a static cylindrically symmetric regular solution to Einstein’s field equations that is supported by a static uniform perfect fluid with the equation of state p=−13ɛ in which p and ɛ are the pressure and the energy density of the uniform and isotropic perfect fluid. With the line-element ds2=−dt2+sech2ξρdρ2+dz2+1ξ2sinh2ξρdϕ2, this spacetime contains only one parameter, namely, ξ=8πɛ3 such that for small ρ or ξ→0 limit, the spacetime becomes flat. This solution belongs to a class of solutions found long ago by Bronnikov in the context of a static cylindrically symmetric perfect fluid, however, its interesting properties have not been noticed previously. Particularly, we show that this special cylindrically symmetric solution to the Einstein equations describes a regular distribution of gas of disordered cosmic strings. Moreover, with detailed calculations, we prove that the spacetime is unstable against a radial linear perturbation of its perfect fluid source.