Black Hole Formation from a Complete Regular Past for Collisionless Matter

Abstract

Initial data for the spherically symmetric Einstein-Vlasov system is constructed whose past evolution is regular and whose future evolution contains a black hole. This is the first example of initial data with these properties for the Einstein-matter system with a "realistic" matter model. One consequence of the result is that there exists a class of initial data for which the ratio of the Hawking mass • m = • m(r) and the area radius r is arbitrarily small everywhere, such that a black hole forms in the evolution. This result is in a sense analogous to the result [15] for a scalar field. Another consequence is that there exist black hole initial data such that the solutions exist for all Schwarzschild time t ∈ (−∞, ∞).

Cosmic Censorship Upheld in Spheroidal Collapse of Collisionless Matter

“…(See Ref. [14] for an analytic proof of black hole formation in a spherical symmetric collisionless matter configuration. )…”

The formation of trapped surfaces in spherically-symmetric Einstein–Euler spacetimes with bounded variation

“…Since a (0) is positive, so is a 0 for r < r * − δ. Since M (0) , a (0) , b (0) > 0 it is immediate from (6.9) that a 0 (r) = a (0) (r) + a (1) 0 (r) (6.10) Since b (0) is increasing and µ (0) = a (0) M (0) is monotonically decreasing and positive by Theorem 4.3, we conclude from (6.11) that for r ≥ r * − δ,…”

“…Step 1 made it obvious that a 0 is positive on [r * − ∆, r * + ∆] and a (1) 0 is negative on (r * − δ, +∞). Hence, we find…”

“…As introduced in Section 6.1, the effect of the perturbation M (1) , V (1) , a (1) , b (1) then takes place in the region…”

“…The collapse problem in spherical symmetry was extensively investigated by Christodoulou and followers in the past twenty years, under the assumption that the matter is represented by a scalar field [4,5] or is driven by a kinetic equation like Vlasov equation; cf. Andreasson [1], Andreasson and Rein [2], and Rendall [24,25] and the references cited therein. Furthermore, the problem of the generic formation of trapped surfaces in vacuum spacetimes without symmetry was solved by Christodoulou in the pioneering work [6].…”