Abstract
We investigate the effects of an analytic boundary metric for smooth asymptotically anti-de Sitter gravitational solutions. The boundary dynamics is then completely determined by the initial data due to corner conditions that all smooth solutions must obey. We perturb a number of familiar static solutions and explore the boundary dynamics that results. We find evidence for a nonlinear asymptotic instability of the planar black hole in four and six dimensions. In four dimensions we find indications of at least exponential growth, while in six dimensions, it appears that a singularity may form in finite time on the boundary. This instability extends to pure AdS (at least in the Poincare patch). For the class of perturbations we consider, there is no sign of this instability in five dimensions.
Highlights
We investigate the effects of an analytic boundary metric for smooth asymptotically anti-de Sitter gravitational solutions
For both a2/r2 and a4/r4 perturbations in α(r) to the planar black hole, we find a rapid growth of the size of one circle relative to the other in the boundary metric
Assuming that the boundary metric is analytic, it is completely determined by the perturbed initial data due to corner conditions that all smooth solutions must obey
Summary
The six dimensional planar black hole is similar to (3.1) with the T 3 replaced by T 4 and M/r2 replaced by M/r3. The prefactors of a24t12 and a26t16 are both negative, and as a result these terms contribute negatively to the evolution regardless of the signs of a4 and a6 This is quite different from the four dimensional case: for these two perturbations we do not see evidence for an exponential growth. If F vanishes in finite time, we could choose another conformal frame in which the shrinking torus is kept fix but the remaining circles expand. In this frame, the evolution would look very similar to the four dimensional case, we would know that the expanding circles diverge in finite time. That might be the case for the four dimensional solutions if the Taylor series fails to converge at a finite time
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