Effect of radiative gravitational modes on the dynamics of a cylindrical shell of counterrotating particles

Abstract

In this paper we consider some aspects of the relativistic dynamics of a cylindrical shell of counter rotating particles. In some sense these are the simplest systems with a physically acceptable matter content that display in a well defined sense an interaction with the radiative modes of the gravitational field. These systems have been analyzed previously, but in most cases resorting to approximations, or considering a particular form for the initial value data. Here we show that there exists a family of solutions where the space time inside the shell is flat and the equation of motion of the shell decouples completely from the gravitational modes. The motion of the shell is governed by an equation of the same form as that of a particle in a time independent one dimensional potential. We find that under appropriate initial conditions one can have collapsing, bounded periodic, and unbounded motions. We analyze and solve also the linearized equations that describe the dynamics of the system near a stable static solutions, keeping a regular interior. The surprising result here is that the motion of the shell is completely determined by the configuration of the radiative modes of the gravitational field. In particular, there are oscillating solutions for any chosen period, in contrast with the "approximately Newtonian plus small radiative corrections" motion expectation. We comment on the physical meaning of these results and provide some explicit examples. We also discuss the relation of our results to the initial value problem for the linearized dynamics of the shell.

Perturbative evolution of the static configurations, quasinormal modes and quasinormal ringing in the Apostolatos–Thorne cylindrical shell model

“…We notice, however, that the approximations introduced in the explicit examples considered in [3] are such that the radiative modes are effectively neglected, and, therefore, the results obtained are possibly relevant only close to the Newtonian limit. A detailed analysis of the full relativistic equations in the linearized approximation was carried out by the present authors in [4], where we obtained the general solution for the harmonic modes with real frequencies, and considered the solutions obtained by general linear combinations of these modes. Although in all the examples considered in [4] we found a stable evolution under perturbations of (essentially) compact support, at least two questions remained unanswered.…”

“…A detailed analysis of the full relativistic equations in the linearized approximation was carried out by the present authors in [4], where we obtained the general solution for the harmonic modes with real frequencies, and considered the solutions obtained by general linear combinations of these modes. Although in all the examples considered in [4] we found a stable evolution under perturbations of (essentially) compact support, at least two questions remained unanswered. The first was the possible completeness of the mode expansion, and its relation to the initial value problem for the system.…”

“…The examples chosen display clearly the quasi normal ringing associated to one of the QNM. The reason why this QNR was not seen in [4] is discussed in Section 12, using the Price-Husain toy model as an illustration.…”

On the evolution of the momentarily static radiation free data in the Apostolatos–Thorne cylindrical shell model

“…Equation (2) is the integrability condition for Eqs. (3). The coordinates (z, φ, r) and the metric function ψ are continuous across the shell Σ, while t and the metric function γ are discontinuous.…”

“…They also make use of some simplifying assumptions that reduce the full set of evolution equations to a particular ordinary differential equation for the motion of the shell. A different approach, where the evolution of the system is analyzed in the linear approximation was considered by Gleiser and Ramirez in [3]. The stability of the static configurations of the system under arbitrary perturbations was proved in an extended analysis given in [7], where the existence of quasi normal modes was also considered.…”