Dynamics of interacting multiple shells in confined spaces

Summary

The study of gravitational dynamics in anti-de Sitter (AdS) plays a central role in modern theoretical physics. Through the celebrated AdS/CFT correspondence it has found many applications to seemingly unrelated fields, ranging from fluid dynamics and superconductors to heavy ion collisions and plasma physics.

In stark contrast with Minkowski space, which was proved to be nonlinearly stable by Christodoulou and Klainerman in the 90's, AdS turns out to be nonlinearly unstable towards the formation of black holes. This so-called turbulent instability was discovered in 2011 by Bizon and Rostworowski through investigations of a spherically symmetric scalar field minimally coupled to gravity. It is now known to arise also in other confining spaces. The case of AdS is particularly interesting because under the AdS/CFT duality this instability gets mapped into an evolution toward equilibrium of a strongly coupled quantum system perturbed away from its vacuum.

In a pair of papers (this one and this other one), in collaboration with V. Cardoso and R. Brito, I investigated the turbulent instability of AdS with an extremely clean multiple-shell model, thus clarifying the physical mechanisms at play. More precisely, we studied the evolution of systems composed of two concentric thin shells, interacting only gravitationally, in both a spherical reflecting cavity or in AdS. The shells are endowed with some pressure to counteract their tendency to collapse (which would result in uninteresting dynamics), and the reflectiveness of the cavity or the confining nature of AdS space force the two membranes to interact repeatedly.

Figure: Example of an evolution of a two-shell system, crossing three times before one of the shells collapses to form a black hole. The upper panel shows the radial location of each shell as a function of time. The lower panel displays the value of the metric component g_tt evaluated at the inner shell, and its vanishing corresponds to the formation of a black hole.

The results are encouraging, showing that this very simple model captures the essence of the problem of scalar field collapse in AdS. In particular, the two-shells model exhibits critical phenomena and chaotic behaviour, both of which are observed with scalar fields in AdS. These studies confirm that the physical mechanism behind the instability toward black hole formation in AdS is the transfer of energy to shorter wavelength modes. In the context of the interacting shells model this results from energetic exchanges between the two shells when they cross: it is always the ingoing shell that gains energy from the outgoing one. Unless the initial configuration is finely tuned, this will lead to a cascading process, ultimately terminating in the formation of a black hole. Thus, our model also indicates how this cascade can be avoided, i.e., in what situations do small perturbations of AdS not lead to black hole formation.