  We present results from a detailed study of spherically symmetric
Einstein-massless-scalar field dynamics with a negative cosmological constant
in four to nine spacetime dimensions. This study is the first to present a
detailed examination of the dynamics in AdS beyond five dimensions, including a
detailed comparison with numerical solutions of perturbative methods and their
gauge dependence. Using these perturbative methods, we provide evidence that
the oscillatory divergence of the first derivative used to argue for
instability of anti-de Sitter space by Bizon et al. is a gauge-dependent effect
in five spacetime dimensions but the divergence of the second derivative is
gauge-independent. We find that the divergence of the first derivative appears
to be gauge-independent in higher dimensions; however, understanding how this
divergence depends on the initial data is more difficult. We also find that
four dimensions is more difficult to study than higher dimensions. The results
we present show that while much progress has been made in understanding the
rich dynamics and stability of anti-de Sitter space, much work is still to be
done. The recent work of Moschidis is encouraging that it is possible to
understand the problem analytically.
