Criticality in singularity formation without gravitational collapse

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 [26

] 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 [23 , 151 ] and in 3 dimensions [149 ] show that n = 0 is a global attractor and the n = 1 solution is the critical solution (see also [67 , 68 ] 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 ]. Update

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.