Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates

Abstract

Impulsive gravitational waves are (weak) solutions to the Einstein vacuum equations such that the Riemann curvature tensor admits a delta singularity along a null hypersurface. The interaction of impulsive gravitational waves is then represented by the transversal intersection of these singular null hypersurfaces. This is the first of a series of two papers in which we prove that for all suitable U(1)-symmetric initial data representing three "small amplitude" impulsive gravitational waves propagating towards each other transversally, there exists a local solution to the Einstein vacuum equations featuring the interaction of these waves. Moreover, we show that the solution remains Lipschitz everywhere and is H 2 loc ∩ C1, 1 4 − loc away from the impulsive gravitational waves. This is the first construction of solutions to the Einstein vacuum equations featuring the interaction of three impulsive gravitational waves.In this paper, we focus on the geometric estimates, i.e. we control the metric and the null hypersurfaces assuming the wave estimates. The geometric estimates rely crucially on the features of the spacetime with three interacting impulsive gravitational waves, particularly that each wave is highly localized and that the waves are transversal to each other. In the second paper of the series, we will prove the wave estimates and complete the proof.

High-frequency solutions to the Einstein equations

“…See section 2 for a discussion of the low-regularity results. We refer the reader also to [4,90,91] for more recent results regarding impulsive gravitational waves.…”

“…The result was later extended by Touati [119] to the general nonpolarized case with smallness only imposed on the W 1,4 norm of the wave part. The elliptic gauge in U(1) symmetry is useful in other low-regularity problems, see for instance [90,91].…”

Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in $U(1)$ symmetry

“…so noting that each term in the sum in the second part of (6.7. 19) is some multiple of a term of the form…”