Alexandru D. Ionescu and Benoît Pausader
PRINCETON UNIVERSITY PRESS 2022, 308 PAGES
PRICE (PAPERBACK) £70.00 ISBN 978-0-691-23304-8
The Einstein–Klein–Gordon system couples the classical Einstein equations of general relativity with the Klein–Gordon equation for a massive scalar field. Although one can write down many physically relevant spacetimes which solve the Einstein equations, demonstrating that these solutions are stable against perturbations is highly non-trivial. As an example, the global nonlinear stability of Minkowski spacetime (the spacetime analogue of flat Euclidean space) as a solution of the vacuum Einstein equations is a central result in general relativity. The analysis for the Einstein–Klein–Gordon system is significantly more complicated because of all the possible interactions, especially those between the metric tensor and the massive scalar field. The purpose of this monograph is to prove a set of general results for the global stability of Minkowski spacetime as a solution of the Einstein–Klein–Gordon equations.
The overall strategy is to choose a suitable system of local coordinates (known as wave coordinates) and transform the geometric system into a system of quasilinear wave and Klein–Gordon equations which is amenable to sophisticated techniques from PDE analysis. More specifically, the new system can be analysed using a combination of energy estimates and Fourier analysis inspired by existence theory for quasilinear dispersive PDEs.
Chapter 1 outlines the PDE formulation of the problem and also provides some background from mathematical relativity such as the definition of the Bondi and ADM energies. Chapter 2 introduces notation and definitions and describes a technical type of result known in the literature as a bootstrap result for a metric tensor-scalar field pair which satisfies the wave version of Einstein–Klein–Gordon on a large time interval given some other assumptions on the initial data. Chapter 3 analyses the major nonlinearities of the system, providing bounds and decomposition of the nonlinearities. Chapter 4 starts from this decomposition to prove the main energy estimates in Chapter 2 and establishes spacetime bounds. Chapter 5 proves the remaining bootstrap bounds from Chapter 2.
Chapter 6 pulls together the previous results to establish the main theorems. The first of these is a global regularity result for solutions of Einstein–Klein–Gordon which are small smooth perturbations of Minkowski spacetime. Roughly speaking, the result states that given a suitable initial data set which satisfies equations of constraint, the Einstein–Klein–Gordon system in wave coordinates has a unique global solution which satisfies suitable harmonic gauge conditions, bounds on norms, and decay bounds. The second set of results establishes rigorous asymptotics and information on growth rates and convergence bounds for the metric tensor and massive scalar field as the time goes to infinity.
The third set of results is concerned with classical estimates from mathematical relativity. These include a proof that future-directed causal geodesics in the spacetimes studied can be extended arbitrarily far into the future and that they become asymptotically parallel to geodesics in Minkowski spacetime. Weak peeling estimates are also proved for the Riemann curvature tensor. The terminology originates in the peeling property, where principal null directions in a gravitationally radiating system are visualised as being ’peeled away’ as one moves from the leading order r−1 term to the higher order terms. Identities and estimates are proved for the ADM and Bondi energies, including the fact that the ADM energy is well-defined and non-negative.
Physicists and researchers working in relativity will likely be most interested in this final set of theorems and the general reader who is only interested in the flavour of the results may wish to go immediately to Chapter 6 and avoid the large amount of preceding technical analysis. The book assumes knowledge of differential geometry, Fourier analysis and wave equations and is mostly suited to graduate students and experts in PDE analysis and mathematical physics.
Hollis Williams
Book review published directly onto IMA website
