Singularities and self-similarity in gravitational collapse

Einstein’s field equations in general relativity admit a variety of solutions with spacetime singularities. Numerical relativity has recently revealed the properties of somewhat generic spacetime singularities. It has been found that in a variety of systems self-similar solutions can describe asymptotic or intermediate behaviour of more general solutions. The typical example is the convergence to an attractor self-similar solution in gravitational collapse. This is closely related to the cosmic censorship violation in the spherically symmetric collapse of a perfect fluid. The self-similar solution also plays an important role in critical phenomena in gravitational collapse. The critical phenomena are understood as the intermediate behaviour around a critical self-similar solution. We see that the convergence and critical phenomena are understood in a unified manner in terms of attractors of codimension zero and one, respectively, in renormalisation group flow. §1. The framework of general relativity The essential assumption of general relativity is that the spacetime is given by a curved manifold with a metric ds = gabdx dx of the Lorentzian signature. g denotes the inverse of gab. The curvature of the spacetime is given by the Riemann tensor Rabcd. The metric lifts and lowers the tensor indices. A vector is timelike, spacelike and null if vva < 0, v va > 0 and v va = 0, respectively. A hypersurface is called timelike, spacelike and null, if its normal vector is spacelike, timelike and null, respectively. We use the abstract index notation [1] in this article. The field equation for the metric is given by Einstein’s equations (1) Rab − 1 2 gabR = 8πTab, where Rab ≡ Rcacb is the Ricci tensor, R ≡ Ra is the scalar curvature and Tab is the stress-energy tensor of matter fields. We adopt the units in which G = c = 1. Einstein’s equations were proposed so that it has