3.6 Self-similar solutions

The main reason that the question of global existence in certain time coordinates discussed in the previous Section 3.5 is of great importance is its relation to the cosmic censorship conjectures. Now there is, in fact, no theorem in the literature, which guarantees that weak cosmic censorship follows from such a global existence result, but there are strong reasons to believe that this is the case, cf. [116] and [18Jump To The Next Citation Point]. Hence, if initial data can be constructed, which lead to naked singularities, then either the conjecture that global existence holds generally is false or the viewpoint that global existence implies the absence of naked singularities is wrong. In view of a recent result by Rendall and Velazquez [161Jump To The Next Citation Point] on self similar dust-like solutions for the massless Einstein–Vlasov system, this issue has much current interest. Let us mention here that there is a previous study on self-similar solutions in the massless case by Martín-García and Gundlach [115]. However, this result is based on a scaling of the density function itself and therefore makes the result less related to the Cauchy problem. Also, their proof is, in part, based on numerics, which makes it harder to judge the relevance of the result.

The main aim of the work [161Jump To The Next Citation Point] is to establish self-similar solutions of the massive Einstein–Vlasov system and the present result can be viewed as a first step to achieving this. In the set-up, two simplifications are made. First, the authors study the massless case in order to find a scaling group, which leaves the system invariant. More precisely, the massless system is invariant under the scaling

-1-- r → 𝜃r,t → 𝜃t,w → √ -w, L → 𝜃L. 𝜃

The massless assumption seems not very restrictive since, if a singularity forms, the momenta will be large and therefore the influence of the rest mass of the particles will be negligible, so that asymptotically the solution can be self-similar also in the massive case, cf. [111], for the relativistic Vlasov–Poisson system. The second simplification is that the possible radial momenta are restricted to two values, which means that the density function is a distribution in this variable. Thus, the solutions can be thought of as intermediate between smooth solutions of the Einstein–Vlasov system and dust.

For this simplified system it turns out that the existence question of self-similar solutions can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. The proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The reason that an ODE system is obtained is due to the assumption on the radial momenta, and if regular initial data is considered, an ODE system is not sufficient and a system of partial differential equations results.

The self-similar solution obtained by Rendall and Velazquez has some interesting properties. The solution is not asymptotically flat but there are ideas outlined in [161] of how this can be overcome. It should be pointed out here that a similar problem occurs in the work by Christodoulou [59] for a scalar field, where the naked singularity solutions are obtained by truncating self-similar data. The singularity of the self-similar solution by Rendall and Velazquez is real in the sense that the Kretschmann scalar curvature blows up. The asymptotic structure of the solution is striking in view of the conditional global existence result in [12Jump To The Next Citation Point]. Indeed, the self similar solution is such that j ≤ 0, and 3m (t,r) → r asymptotically, but for any T, 3m (t,r) > r for some t > T. In [12] global existence follows if j ≤ 0 and if 3m (t,r) ≤ r for all t. It is also the case that if m ∕r is close to 1∕2, then global existence holds in certain situations, cf. [20Jump To The Next Citation Point]. Hence, the asymptotic structure of the self-similar solution has properties, which have been shown to be difficult to treat in the search for a proof of global existence.

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