In the n+1 formalism of general relativity, the (n+1)-dimensional space-time is decomposed into n-dimensional space-like slices that are parametrized by a time function. This is the basis for formulating Einstein’s equation as an initial value problem. In an effort to understand which space-times are constructible, an important question is, “What is the admissible class of initial data for this problem?” This question is addressed by analyzing the so-called Einstein constraint equations, which are an undetermined system of equations to be solved for a metric and an extrinsic curvature tensor.
The conformal method of Lichnerowicz and York transforms the constraint equations into a determined nonlinear system of elliptic equations by allowing certain pieces of the initial data to be freely specified. Among the freely specified data are the conformal class of the metric, the trace of the extrinsic curvature, and a traceless, divergence free portion of the extrinsic curvature. The specified data act as parameters for solutions to the conformal equations, which consist of a conformal factor and a vector field. Once a solution is obtained, it is used along with the specified data to build solutions to the constraints.
In our article in Classical and Quantum Gravity we show that the coupled conformal system is solvable on asymptotically Euclidean manifolds with only moderate regularity assumptions on the mean curvature and smallness conditions on the other parameters. This result is a development in the solution theory for the constraints because previously, the only existence results for the conformal equations on asymptotically Euclidean manifolds relied either on the mean curvature being constant or nearly constant with small derivative. To obtain our result, we show that solutions to the coupled conformal system must be fixed points of a certain nonlinear operator.
Using sub-and supersolutions, we build a set on which the associated operator is invariant and then apply the Schauder fixed point Theorem. There are incentives for obtaining solutions to the conformal equations with minimal assumptions on the mean curvature beyond developing the solution theory for coupled elliptic systems with critical nonlinearities on asymptotically Euclidean manifolds.
A well known result of Isenberg, Pollack, and Chrusciel is that not all space-times can be foliated by maximal slices. From a physical point of view this work is relevant because it provides a method for constructing initial data for an isolated system that evolves into a space-time foliated by far from maximal slices. Therefore our results extend the class of constructible space-times that can be modeled using the n+1 formalism.
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