Construction of N-Body Initial Data Sets in General Relativity

Abstract

Given a collection of N solutions of the (3 + 1) vacuum Einstein constraint equations which are asymptotically Euclidean, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier work which handled the time-symmetric case, thus providing a construction of large classes of initial data for the many body problem in general relativity.2000 Mathematics Subject Classification. 53C21, 83C99. 1 Here and throughout this paper, we use the Einstein summation convention.

Black-hole lattices as cosmological models

“…• Electrovacuum: T ab = F ac F b c − 1 4 g ab F cd F cd • Free scalar field: T ab = ∇ a ϕ∇ b ϕ − 1 2 g ab ∇ c ϕ∇ c ϕ These stress-energy tensors can be decomposed into quantities measured by an observer following the integral curves of u a , and who is hence at rest with respect to S t , as follows: ρ = u a u b T ab energy density p = 1 3 h ab T ab pressure q a = −h a b u c T bc energy flux (or equivalently momentum density) π ab = T ab anisotropic pressure and tangential stresses (4) Here the angle brackets indicate the spatial tracefree part of any symmetric pair of indices, such that…”

“…Though the metric of an isolated gravitational system was one of the first solutions of Einstein's equations to be discovered, finding the gravitational field corresponding to many-body systems is still something of an open problem in GR (outside of specific regimes [4,5]). Unlike in Newton's gravitational theory, the superposition principle does not hold, and the nonlinear effects of combining several gravitating systems together have to be addressed on a case-by-case basis.…”

General relativity and gravitation: a centennial perspective

“…Moreover, one can do this without significantly changing the ADM mass. Both the Carlotto-Schoen conical gluing and the Corvino-Schoen annular gluing [72], can be used to produce N-body initial data sets of specific design which satisfy the constraints. The evolutions of such N-body initial data sets have not yet been studied.…”

Principe de recollement des équations des contraintes en relativité générale

“…Chruściel, J. Corvino et J. Isenberg [9]. À partir de N paires de données initiales asymptotiquement euclidiennes et solutions des contraintes, une nouvelle paire de données initiales y est construite, satisfaisant les mêmes propriétés et contenant des régions prescrites de chacune des N données initiales d'origine.…”