Cylindrically symmetric models of gravitational collapse to black holes: A short review

Abstract

We survey results about exact cylindrically symmetric models of gravitational collapse in General Relativity. We focus on models which result from the matching of two spacetimes having collapsing interiors which develop trapped surfaces and vaccum exteriors containing gravitational waves. We collect some theorems from the literature which help to decide a priori about eventual spacetime matchings. We revise, in more detail, some toy models which include some of the main mathematical and physical issues that arise in this context, and compute the gravitational energy flux through the matching boundary of a particular collapsing region. Along the way, we point out several interesting open problems.

Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

“…Finally, we observe that the Schwarzschild limit κ → 0 of the conformal metric (19) and the wave equation (25) are slightly different from the expressions used in the numerical investigations in [12,13]. With our present approach, we interestingly obtain somewhat simpler equations.…”

“…For instance, reference [16] shows the separability of conformally coupled scalar field equations in general rotating black-hole spacetimes, opening the possibilities for studies in higher dimensions. Another interesting application could be the investigation of gravitational waves in a cylindrically-symmetric gravitational collapse [25].…”

“…The other additional function f 1 needs to satisfy equation ( 45) at order n = 1. Therefore, we numerically solve this equation together with the equation for F (ρ, θ, τ ) that results from substituting (88) into the wave equation (25).…”

Cylindrically symmetric inhomogeneous dust collapse with a zero expansion component

“…This means, in particular, that one requires that the matching hypersurface is tangent to the orbits of the symmetry group to be preserved (see [27], for more details). As a consequence, since the interior has θ 22 = 0, it is not expected that it can be matched to static or stationary exteriors (see [13]). We shall then explore non-stationary exteriors.…”

“…On the other hand, spatially homogeneous dust collapsing models have been proved to match Einstein-Rosen exteriors although with the dramatic consequence of containing trapped cylinders initially [11]. See also [12,13], for recent reviews.…”