Local linearization stability

Abstract

It is shown that Einstein’s equations are always linearization stable around any finite region of space-time. Let (Ω,g0ab ) be any region of space-time, admitting a compact Cauchy surface with nonempty smooth boundary, and with g0ab a sufficiently smooth solution of the vacuum Einstein equation. It is shown that for any solution g1ab of the linearized equation and any open region U⊆Ω, there exists a smooth one-parameter family gλab of solutions on U such that (gλab‖λ=0 =g0ab ) ‖U and ((d/dλ)gλab =g1ab )‖U. By using a result of Choquet-Bruhat and York [The Cauchy Problem, General Relativity and Gravitation, edited by A. Held (Plenum, New York, 1980), Vol. 1] asserting the smoothness of the map that sends initial data into solutions of Einstein’s evolution equations the proof of the above theorem is reduced to the proof of a similar theorem for Einstein’s constraint equations. The proof of this latter theorem involves the use of the implicit function theorem in Hilbert spaces. This local result on linearization stability asserts, in contrast to the general global case, that linearization about any vacuum solution is locally physically meaningful.