4.4 Criticality in singularity formation without gravitational collapse

It is well known that the Yang–Mills field does not form singularities from smooth initial conditions in 3+1 dimensions [70], but Bizoń and Tabor [26Jump To The Next Citation Point] have shown singularity formation in 4+1 (the critical dimension for this system from the point of view of energy scaling arguments) and 5+1 (the first supercritical dimension). In 5+1 there is the countable family W n of CSS solutions with n unstable modes, such that W 1 acts as a critical solution separating singularity (W 0) formation from dispersal to infinity. In 4+1 there are no self-similar solutions and the formation of singularities seems to proceed through adiabatic shrinking of a static solution.

Completely parallel results can be found for wave maps, for which the critical dimension is 2+1. For the wave map from 3+1 Minkowski to the 3-sphere, Bizoń [16] has shown that there is a countable family of regular (before the CH) CSS solutions labeled by a nodal number n ≥ 0, such that each solution has n unstable modes. Simulations of collapse in spherical symmetry [23151] and in 3 dimensions [149] show that n = 0 is a global attractor and the n = 1 solution is the critical solution (see also [6768] for computations of the largest perturbation-eigenvalues of W 0 and W 1). Again, for the wave map from 2+1 Minkowski to the 2-sphere generic singularity formation proceeds through adiabatic shrinking of a static solution [24].

These results have led to the suggestion in [26] that criticality (in the sense of the existence of a codimension-1 solution separating evolution towards qualitatively different end states) could be a generic and robust feature of evolutionary PDE systems in supercritical dimensions, and not an effect particular of gravity.

Garfinkle and Isenberg [88] examine the threshold between the round end state and pinching off in Ricci flow for a familiy of spherically symmetric geometries on S3. They have found intermediate approach to a special “javelin” geometry, but have not investigated whether this is universal. See also [89] and [135]. UpdateJump To The Next Update Information

A scaling of the shape of the event horizon at the moment of merger in binary black hole mergers is noted in [44], but this is really a kinematic effect.

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