Remark on the absence of self-similar solutions in a 4 + 1 vacuum gravitational collapse

Piotr Bizo (1) , Arthur Wasserman (2)

1 M. Smoluchowski Institute of Physics
2 Department of Mathematics

We give a very short proof that the vacuum Einstein equations in 4 + 1 dimensions have no cohomogeneity-two Bianchi IX continuously self-similar solutions. Introduction. In a recent paper [ 1 ] it was shown that in five spacetime dimensions one can perform a consistent cohomogeneity-two symmetry reduction of the vacuum Einstein equations which - in contrast to the spherically symmetric reduction - admits time dependent asymptotically flat solutions. The key idea was to modify the standard spherically symmetric ansatz by replacing the round metric on the three-sphere with the homogeneously squashed metric, thereby breaking the SO(4) isometry to SO(3) × U (1). In this way the squashing parameter becomes a dynamical degree of freedom and Birkhoff's theorem is evaded. This model (which we shall refer to as the BCS model) provides a simple theoretical setting for studying the dynamics of gravitational collapse in vacuum. Numerical simulations indicate that the spherically symmetric solutions, Minkowski and Schwarzschild, play the role of attractors in the evolution of generic regular initial data (small and large ones, respectively) and the transition between these two outcomes of evolution exhibits a discretely self-similar critical behavior [ 1 ]. In this respect the BCS model is very similar to the Einstein-massless scalar field system [ 2-4 ]. However, there is one interesting difference between these two models which we want to point out here. The difference is concerned with the existence of continuously self-similar (CSS) solutions. In [ 5 ] Christodoulou proved that the Einstein-massless scalar field system possesses CSS solutions. These solutions, suitably truncated, provide examples of naked singularities developing from regular initial data (however, being unstable [ 6 ], they do not contradict the weak cosmic censorship conjecture). We will show below that the BCS model has no CSS solutions. This result indicates that the CSS naked singularities found by Christodoulou for the self-gravitating massless scalar field are, in a sense, matter generated (mathematically, they are due to the fact that only derivatives of the scalar field appear in the equations).

Mots clés: Physical Sciences

Piotr Bizo, Arthur Wasserman. Remark on the absence of self-similar solutions in a 4 + 1 vacuum gravitational collapse. Classical and Quantum Gravity, 2010, 27 (12), pp.122001. ⟨10.1088/0264-9381/27/12/122001⟩. ⟨hal-00629996⟩