Critical Behavior in Vacuum Gravitational Collapse in4+1Dimensions

Abstract

We show that the (4 + 1)-dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the (t,r) plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar type II critical behavior at the threshold of black hole formation.

Codimension-Two Critical Behavior in Vacuum Gravitational Collapse

“…This is a continuation of our studies in [5] where we have shown the existence of a critical DSS solution with one unstable mode and the associated type II critical behavior in this model. On the basis of our result it is tempting to conjecture that the critical DSS solution is a ground state of a countable family of DSS solutions with increasing number of unstable modes.…”

“…Discrete self-similarity (DSS) means invariance under rescalings of space and time variables by a constant factor e ∆ where ∆ is a number, usually called the echoing period. Critical solutions possessing this curious symmetry (with different echoing periods) have been found for several selfgravitating matter models (massless scalar field [1], YangMills field [3], σ-model [4] and few others) and recently also in the vacuum gravitational collapse in higher dimensions [5,6].Our current understanding of DSS solutions is very limited in comparison with continuously self-similar (CSS) solutions. In the case of spherical symmetry the CSS ansatz leads to an ODE system which can be handled analytically and sometimes even rigorous proofs of existence are feasible.…”

“…Discrete self-similarity (DSS) means invariance under rescalings of space and time variables by a constant factor e ∆ where ∆ is a number, usually called the echoing period. Critical solutions possessing this curious symmetry (with different echoing periods) have been found for several selfgravitating matter models (massless scalar field [1], YangMills field [3], σ-model [4] and few others) and recently also in the vacuum gravitational collapse in higher dimensions [5,6].…”

Fractal Threshold Behavior in Vacuum Gravitational Collapse

“…Critical phenomena.-The study [9] of the critical phenomena in the model restricted to biaxial symmetry revealed type-II gravitational collapse with the critical solution being discretely self-similar (DSS). It was also shown [7] that in triaxial symmetry the phenomenology of critical behavior remains the same.…”

“…Namely, in order to evade Birkhoff's theorem and save radial symmetry, we take five dimensional vacuum Einstein equations and reduce the number of degrees of freedom by the BCS ansatz [9], [7] …”