Construction of Cauchy data of vacuum Einstein field equations evolving to black holes

Abstract

Abstract. We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside.The proof makes use of the full strength of Christodoulou's work on the dynamical formation of black holes and Corvino-Schoen's work on the constructions of initial data set.

Trapped surfaces in vacuum arising dynamically from mild incoming radiation

“…While all of the aforementioned works posed initial data on null hypersurfaces, Li-Yu [15] combined Christodoulou's result with the Corvino-Schoen gluing method to construct a class of Cauchy data such that a trapped surface is guaranteed to form in the future. Finally, we refer the interested readers to the beautiful exposition [8] for more background on the problem and a further discussion on the original work of Christodoulou.…”

“…15 To derive estimates from the schematic null structure equations, we first assume that ψ has small u-integral. This implies that the growth rate of ψ is determined by the non-integrable coefficient Substituting this bound for ψ into the equation (1.11), we obtain the following estimates:…”

On the local extension of the future null infinity

“…We should remark here that because χ = 0 when u = δ, then by null Codazzi equation we can obtain an improved estimate β ∼ 1 when u = δ (originally β ∼ δ −1/2 ). We remark that the trivial extension of the initial data on C u 0 to finite length is also used in [13] and such improvements are essential in the work [13]. In fact, we have obtained an even better improvement under an addition condition.…”

“…They used some specific approximately Killing and conformal Killing vectorfields as multipliers and commutators in order to capture the decay of different geometric quantities towards to the null infinity. The more recent works such as [10], [13], [11], [12], [7] used instead the null Bianchi equations which are written in components and integration by parts. When dealing with the null infinity, such as in [13] and [7] and the current work, since the decay of the geometry is well understood after the work of [6] and [8], we can use suitable weights to generate the weighted energy estimate using integration by parts.…”

Untitled

“…Further, see Yu [77], and Li and Yu [53]. Miao and Yu [56] applied the short pulse method successfully to study shock formation in quasilinear wave equations.…”