Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations (Q1750241)

The authors consider the initial value problem associated to the nonlinear interaction of two impulsive gravitational waves for the vacuum Einstein equations \(R_{\mu \nu }=0\). The initial data have delta function singularities supported on embedded 2-spheres \(S_{0,\underline{u}_{s}}\) and \( S_{u_{s},0}\) on the initial null hypersurfaces \(H_{0}\) and \(\underline{H}_{0} \) respectively. In their previous paper [Commun. Pure Appl. Math. 68, No. 4, 511-624 (2015; Zbl 1316.35281)], the authors proved the existence of a unique solution to the vacuum Einstein equations before the interaction of the two impulsive gravitational waves, i.e., for \(u<u_{s}\) or \(\underline{u}< \underline{u}_{s}\), and that the singularity is supported on the null hypersurfaces emanating from the initial singularities. In the present paper, the authors consider the behavior of the solution beyond the first interaction of the two impulsive gravitational waves. The authors suppose that the data are smooth on \(H_{0}\) (resp. \(\underline{H}_{0}\)), except across a 2-sphere \(S_{0,\underline{u}_{s}}\) (resp. \(S_{u_{s},0}\)), where the traceless part of the second fundamental form of \(H_{0}\) (resp. \(\underline{H }_{0}\)) has a jump discontinuity. The main result of the paper proves that for such initial data and \(\epsilon \) sufficiently small, there exists a unique spacetime \((\mathcal{M},g)\) endowed with a double null foliation \(u\), \(\underline{u}\) that solves the characteristic initial value problem for the vacuum Einstein equations in the region \(0\leq u\leq u^{\ast }\), \(0\leq \underline{u}\leq \underline{u}^{\ast }\), whenever \(u^{\ast }\leq \epsilon \) or \(\underline{u}^{\ast }\leq \epsilon \). If \(\underline{H}_{\underline{u} _{s}}\) (resp. \(H_{u_{s}}\)) is the incoming (resp. outgoing) null hypersurface emanating from \(S_{0,\underline{u}_{s}}\) (resp. \(S_{u_{s},0}\)), the curvature components \(\alpha _{AB}=R(e_{A},e_{4},e_{B},e_{4})\) and \( \underline{\alpha }_{AB}=R(e_{A},e_{3},e_{B},e_{3})\) are measures with singular atoms supported on \(\underline{H}_{\underline{u}_{s}}\) and \( H_{u_{s}}\) respectively. Finally, all other components of the curvature tensor can be defined in \(L^{2}\) and the solution is smooth away from \( \underline{H}_{\underline{u}_{s}}\cup H_{u_{s}}\). For the proof of their main result, the authors first rewrite the Einstein equations as a system for Ricci coefficients and curvature components. They improve renormalization tools they developed in their above-indicated previous paper to get energy estimates. They derive mixed norm estimates for the Ricci coefficients studying transport equations and estimates in an arbitrarily long \(u\) interval using Gronwall's inequality and a bootstrap argument. They introduce a notion of signature which is linked to the scaling properties of the Einstein equations. They analyze the nonlinear interaction of impulsive gravitational waves to prove that the spacetime is smooth away from the null hypersurfaces emanating from the initial singularities. Finally, they build a large class of spacetimes such that the initial data do not contain a trapped surface, and a trapped surface is formed in evolution. The authors also prove an existence and uniqueness result for spacetime \((\mathcal{M},g)\) solutions to the vacuum Einstein equations. The successive steps of the proofs are given with details along this long paper.