{'fraction_non_alphanumeric': 0.027280727486066296, 'fraction_numerical': 0.0038134350249339984, 'mean_word_length': 5.111111111111111, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A New Tradition between Discrete and Contiuous Self-Similarity in Critical Gravitational Collapse

Gravitational collapse at the threshold of black hole formation ties together many of the fundamental issues of general relativity, such as the global aspects of solutions, the structure of singularities arising from regular initial data and the cosmic censorship hypothesis [1]. By finetuning initial data for a gravitating massless scalar field to the boundary between eventual dispersal and complete collapse, Choptuik [2] found phenomena reminiscent of criticality associated with phase transitions in statistical physics, such as universality and scaling (e.g. of the black hole mass). Considerable qualitative understanding has been gained by explaining critical collapse in terms of a single unstable mode of the universal critical solution. This solution is understood as an intermediate attractor located in a codimension-one stable hypersurface in phase space, separating data which do or do not form black holes. Critical collapse has by now been studied in a number of matter models – in all of these the physics of the threshold of black hole formation was found to be governed by symmetry: the critical solution exhibits either continuous or discrete self-similarity, or staticity or periodicity in time [3]. In this paper, we study numerically a simple model with a single scalar field in spherical symmetry which exhibits CSS critical behavior at small coupling constants, and DSS critical behavior at large ones. At intermediate coupling constants we observe a competition between CSS and DSS solutions giving rise to a new phenomenon: within an approximately-DSS critical evolution we find several episodes of approximate CSS. Our main focus here is on the interpretation of this observation in terms of an analogy to a heteroclinic loop bifurcation in finite dimensional dynamical systems, at which the limit cycle (DSS) merges with two fixed points (the CSS solution and its negative). Apart from the bifurcation itself, the model also shows other interesting features, such as the existence of a stable (with respect to linear spherical perturbations) self-similar solution for some finite range of the coupling, and a “suppression” effect for the CSS solution in critical searches which we interpret as “shielding” by an apparent horizon. Our results are based on the direct numerical construction of the CSS and DSS solutions, a linear perturbation analysis of the CSS solutions, and comparison with critical evolutions. The latter are defined by considering a 1-parameter family of initial data � = �p(u0,r), such that (say) for small values of p the field eventually disperses (as determined by a numerical evolution [4]), while for large values of p it eventually forms a black hole (diagnosed by the appearance of an apparent horizon). We use a binary search in p to numerically approximate the critical solution at the threshold of black hole formation. We refer to such fine-tuned numerical solutions as near-critical evolutions, and our results are taken from initial data which are fine-tuned to the same tolerance �p/p < 10 −14 . The self-gravitating SU(2) �-model [5] under investigation is a wave map from spacetime to the target manifold S 3 with the standard metric. The so called Hedgehog ansatz of spherical symmetry leaves a single matter field �(u,r) coupled to gravity: