11.1 The power and need of adaptivity

Physical problems governed by nonlinear equations can develop small scale structures, which are rarely easy to predict. The canonical example in hydrodynamics is turbulence, which results in short wavelength features up to the viscous scale. An example in general relativity of arbitrarily small scales is the one of critical phenomena [125, 223], where the solution develops a self-similar behavior revealing a universal approach to a singular one describing a naked singularity. Uncovering this phenomena is crucially required to dynamically adjust the grid structure to respond to the (exponentially) ever-shrinking features of the solution. Recently, such need was also demonstrated quite clearly by the resolution of the final fate of unstable black strings [279, 280]. This work followed the dynamics, in five-dimensional spacetimes, of an unstable black string: a black hole with topology S2 × S1, with the asymptotic length of S1 over the black hole mass per unit length above the critical value for linearized stability [216, 217]. As the evolution unfolds, pieces of the string shrink and elongate (so that the area increases), yielding another unstable stage; see Figure 10View Image. This behavior repeats in a self-similar manner, the black string developing a fractal structure of thin black strings joining spherical black holes. This behavior was followed through four generations and the numerical grid refined in some regions up to a factor of 17 2 compared to the initial one. This allowed the authors to extrapolate the observed behavior and conclude that the spacetime will develop naked singularities in finite time from generic conditions, thereby providing a counterexample to the cosmic censorship conjecture in five dimensions.
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Figure 10: Early (left) and late (right) stages of the apparent horizon describing the evolution of an unstable black string. Courtesy: Luis Lehner and Frans Pretorius.

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