Localizing solutions of the Einstein constraint equations

Abstract

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of N -body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large T we can engineer solutions where any two massive bodies do not interact at all for any time t ∈ (0, T ), in striking contrast with the Newtonian gravity scenario.

Shielding linearized gravity

“…For cone-like geometries, as considered in [5,9], and with h µν = O(1/r), the gravitational field in the screening region will fall-off again as O(1/r). This is rather surprising, as the gluing approach of [5] leads to a loss of decay even for the linear problem.…”

“…This is a simple consequence of the equation ∆φ = 4πGρ , where φ is the gravitational potential, G is Newton's constant and ρ is the matter density: The requirement that ρ ≥ 0, and the asymptotic behaviour −M/r of φ, where M is the total mass, implies that φ vanishes at large distances along a curve extending to infinity if and only if there is no matter whatsoever and φ ≡ 0. It is therefore extremely surprising that in general relativity, gravitational fields can be shielded away by gravitational fields, as proved recently in a remarkable paper by Carlotto and Schoen [5]. Since Newtonian gravity is part of the weak-field limit of general relativity (indeed, this is weak-field GR with small velocities), one wonders if a similar screening can occur for linearised relativity.…”

“…Since Newtonian gravity is part of the weak-field limit of general relativity (indeed, this is weak-field GR with small velocities), one wonders if a similar screening can occur for linearised relativity. As it turns out, the analysis of Carlotto and Schoen can be readily generalised to linearised gravitational fields on cone-like sets as considered in [5] (compare [9]). This, however, requires sophistical mathematical machinery which imposes restrictions to the sets considered and, as an intermediate step, uses solutions blowing-up at the relevant boundaries, which leads to difficulties when trying to implement the method numerically.…”

General relativity and gravitation: a centennial perspective

“…They show [71] that, just as one can construct a solution of the constraints by gluing an arbitrary interior region across an annular transition region to an exterior Schwarzschild or Kerr region, one can also do so by gluing an arbitrary solid conical region (stretching out to spatial infinity) across a transition region to a region of Minkowski initial data with a conical region removed. Moreover, one can do this without significantly changing the ADM mass.…”

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

“…It is certainly appropriate to mention here the article [CS14], which is joint work with R. Schoen, where we show that the rigidity Theorem 2 is essentially sharp by constructing asymptotically flat initial data sets that have large ADM energy and momentum and are exactly trivial outside of a solid cone (of given, yet arbitrarily small opening angle) so that they contain plenty of complete, stable MOTS of planar type. A posteriori, this strongly justifies our requirement that the metric g in the previous statement has some nice asymptotics at infinity.…”