A proof of the instability of AdS for the Einstein-null dust system with an inner mirror

Abstract

In 2006, Dafermos and Holzegel formulated the so-called AdS instability conjecture, stating that there exist arbitrarily small perturbations to AdS initial data which, under evolution by the Einstein vacuum equations for $\mathrm{\Lambda }<0$ with reflecting boundary conditions on conformal infinity $\mathcal{ℐ}$, lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically symmetric Einstein-scalar field system was initiated by Bizón and Rostworowski (Phys. Rev. Lett. 107:3 (2011), art. id. 031102), followed by a vast number of numerical and heuristic works by several authors.

In this paper, we provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein-massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein-null dust system, allowing for both ingoing and outgoing dust. In order to overcome the breakdown of this system occurring once the null dust reaches the center $r=0$, we place an inner mirror at $r=r_0>0$ and study the evolution of this system on the exterior domain $\left\{r\ge r_0\right\}$. The structure of the maximal development and the Cauchy stability properties of general initial data in this setting are studied in our companion paper (2017, arXiv: 1704.08685).

The statement of the main theorem is as follows: We construct a family of mirror radii $r_{0𝜀}>0$ and initial data ${\mathcal{𝒮}}_{𝜀}$, $𝜀\in \left(0,1\right]$, converging, as $𝜀\to 0$, to the AdS initial data ${\mathcal{𝒮}}_0$ in a suitable norm, such that, for any $𝜀\in \left(0,1\right]$, the maximal development $\left({\mathcal{ℳ}}_{𝜀},g_{𝜀}\right)$ of ${\mathcal{𝒮}}_{𝜀}$ contains a black hole region. Our proof is based on purely physical space arguments and involves the arrangement of the null dust into a large number of beams which are successively reflected off $\left\{r=r_{0𝜀}\right\}$ and $\mathcal{ℐ}$, in a configuration that forces the energy of a certain beam to increase after each successive pair of reflections. As $𝜀\to 0$, the number of reflections before a black hole is formed necessarily goes to $+\infty$. We expect that this instability mechanism can be applied to the case of more general matter fields.