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
with reflecting boundary conditions on conformal infinity
, 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
, we place an
inner mirror at
and study the evolution of this system on the exterior domain
. 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
and initial
data
,
, converging, as
, to the AdS initial data
in a suitable norm,
such that, for any
, the
maximal development
of
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
and
, in a
configuration that forces the energy of a certain beam to increase after each successive pair of
reflections. As
,
the number of reflections before a black hole is formed necessarily goes to
. We
expect that this instability mechanism can be applied to the case of more general
matter fields.
