Initial Value Problem in General Relativity

TL;DR

This review explains how to formulate and solve the initial value problem in general relativity, including methods for generating initial data, and discusses the stability and long-term behavior of solutions.

Contribution

It provides an accessible overview of the initial value problem in Einstein's theory, including new insights into data construction and stability analysis.

Findings

Initial value problem is well-posed with solutions existing for given initial data.

Methods for constructing initial data satisfying Einstein constraints are detailed.

Discussion on stability and long-term evolution of Einstein's solutions.

Abstract

This article, written to appear as a chapter in "The Springer Handbook of Spacetime", is a review of the initial value problem for Einstein's gravitational field theory in general relativity. Designed to be accessible to graduate students who have taken a first course in general relativity, the article first discusses how to reformulate the spacetime fields and spacetime covariant field equations of Einstein's theory in terms of fields and field equations compatible with a 3+1 foliation of spacetime with spacelike hypersurfaces. It proceeds to discuss the arguments which show that the initial value problem for Einstein's theory is well-posed, in the sense that for any given set of initial data satisfying the Einstein constraint equations, there is a (maximal) spacetime solution of the full set of Einstein equations, compatible with the given set of data. The article then describes how to generate initial data sets which satisfy the Einstein constraints, using the conformal (and conformal thin sandwich) method, and using gluing techniques. The article concludes with comments regarding stability and long term behavior of solutions of Einstein's equations generated via the initial value problem.