Local Propagation of Impulsive GravitationalWaves

Abstract

Abstract. In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of non-regular characteristic data.

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

“…Hence, μ will vanish at a time exp |O(˚ )| −1 for data such 48 The work [60] showed that for equations of type (1.1a), the proof closes for small data verifying…”

“…Our strategy of propagating the smallness of some quantities while simultaneously allowing derivatives transversal to the characteristics to be large has roots in the similar approach taken by Christodoulou [16] in his celebrated proof of the formation of trapped surfaces in solutions to the Einstein-vacuum equations and in the related works [3,33,37,38,[47][48][49]. Similar strategies have been used [51,61,62,65] to prove global existence results for semilinear wave equations verifying the null condition in regimes that allow for large transversal derivatives.…”

Trapped surfaces in vacuum arising dynamically from mild incoming radiation

“…By the estimates forχ,χ, η and the commutation principle, σ obeys the pointwise bound |σ| δa |u| 3 , which is only consistent with (1.14) in the region |u| ≥ δa, but cannot be proved in the full region |u| ≥ δa Instead, we perform renormalized energy estimates as in [18], [19] and [17]. We make the observation that the following renormalized curvature components:…”

“…We now move to the terms where i 2 ≥ 1. It turns out that the ψ factors provide extra smallness and we can use the bootstrap assumption (4.2) together with Proposition 6.1 to deal with the first two terms 18 to get…”

“…We thus have to introduce weighted estimates to carefully track the growth of various geometric quantities as we approach the vertex. Moreover, in order to obtain the existence result in the region |u| ≥ δa 1 2 b, these weighted estimates are coupled with renormalized energy estimates introduced in [18], [19] and [17]. We will discuss the main ideas of the proof in Section 1.3.…”