The global stability of Minkowski space-time in harmonic gauge

Abstract

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations

“…Again, we stress that the difficult part of his argument was his proof of an analog of Theorem 21.1.1, and that the remaining aspects his work are easier to derive. We also stress that although Alinhac had already proved his small-data shock formation results (summarized in Theorem 1.11.1) for a larger class of equations, 26 there was great novelty in Christodoulou's thoroughness of his description of the dynamics and in particular, in his description of the solution along the boundary of the maximal development of the data. A particularly attractive feature of Christodoulou's detailed description is that it is suitable as a starting point for trying to extend the solution, in a generalized sense, beyond the shock.…”

“…Moreover, in the shock-formation proofs, the eikonal function played a more essential role than it did in the proof of the stability of Minkowski spacetime. Specifically, Lindblad and Rodnianski gave a second proof [26] of the stability of Minkowski spacetime, relative to wave coordinates, that did not rely upon a true eikonal function. Instead, they closed their small-data global existence proof by deriving estimates relative to the background Minkowskian geometry 13 with the help of a Minkowski eikonal function u (F lat) = t − r. In contrast, as we will see, for the solutions studied in this monograph, small-data shock formation exactly corresponds to the intersection of the level sets of a true eikonal function verifying (1.5.2).…”

Topological Censorship for Kaluza–Klein Space-Times

“…In order to proceed, we recall some facts about the space-times constructed in [24,25]. In Minkowski space-time R 1,n = (R n+1 , η) let q = r − t , and let H µν := g µν − η µν , where g µν is a small data vacuum metric on R n+1 as constructed in [24,25].…”

“…In this appendix we wish to point out that sufficiently small data vacuum spacetimes constructed using the Lindblad-Rodnianski method [24], as generalised by Loizelet to higher dimensions [25,26] (compare [5]), contain past inwards trapped, closed to the future (in a sense which should be made clear by what is said below), timelike hypersurfaces. This is irrelevant as far as the topological implications of our analysis are concerned, as in this case the space-time manifold is R n+1 anyway, but it illustrates the fact that such hypersurfaces can arise in vacuum space-times which are not necessarily stationary.…”

“…In Minkowski space-time R 1,n = (R n+1 , η) let q = r − t , and let H µν := g µν − η µν , where g µν is a small data vacuum metric on R n+1 as constructed in [24,25]. By [24,Corollary 9.3] for n = 3, and by [25,Corollary 5.1] for n ≥ 3, (3) there exist constants C, 0 < δ < δ ′ < 1 such that (A.1) |∂H| ≤ Cε(1 + t + |q|) 1−n 2 +δ (1 + |q|) −1−δ ′ , q ≥ 0 , Cε(1 + t + |q|) 1−n 2 +δ (1 + |q|) −1/2 , q < 0 , Here ǫ and δ are small constants determined by the initial data, and δ can be chosen as small as desired by choosing the data close enough to the Minkowskian ones. Next,∂ denotes partial coordinate derivatives ∂ µ to which a projection operator in directions tangent to the outgoing coordinate cones {t − r = const} has been applied; e.g., in spherical coordinates,∂ ∈ Span{L := ∂ t + ∂ r , 1 r ∂ θ , 1 r sin θ ∂ ϕ }.…”