Abstract
Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz (Ann Henri Poincaré 16:2059, 2015), in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energy-momentum tensor has vanishing trace. In this case, the equation of conservation for the energy-momentum tensor is conformally invariant. Our analysis includes the construction of a subsidiary evolution which allows to prove the propagation of the constraints. We discuss how the underlying structure behind these systems of equations is the set of integrability conditions satisfied by the conformal field equations. The main result of our analysis is that both the evolution and subsidiary equations for the geometric part of the conformal Einstein-tracefree matter field equations close without the need of any further assumption on the matter models other that the vanishing of the trace of the energy-momentum tensor. Our work is supplemented by an analysis of the evolution and subsidiary equations associated to three basic tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang–Mills field. As an application we provide a global existence and stability result for de Sitter-like spacetimes. In particular, the result for the conformally invariant scalar field is new in the literature.
Highlights
The conformal Einstein field equations are a conformal representation of the Einstein field equations which permit us to study the global properties of the solutions to equations of General Relativity by means of Penrose’s procedure of conformal compactification—see e.g. [11,15] for an entry point to the literature on the subject
Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz (Ann Henri Poincaré 16:2059, 2015), in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energy-momentum tensor has vanishing trace
We supplement our general analysis of the metric conformal Einstein field equations with an analysis of the evolution and subsidiary evolution equations of some of the tracefree matter models more commonly used in the literature: the Maxwell field, the Yang–Mills field and the conformally invariant scalar field
Summary
The conformal Einstein field equations are a conformal representation of the Einstein field equations which permit us to study the global properties of the solutions to equations of General Relativity by means of Penrose’s procedure of conformal compactification—see e.g. [11,15] for an entry point to the literature on the subject. In [17] Paetz has obtained a satisfactory hyperbolic procedure for the metric vacuum Einstein field equations which is based on the construction of second order wave equations To round up his analysis, Paetz proceeds to construct a system of subsidiary wave equations for tensorial fields encoding the conformal Einstein field equations (the so-called geometric zero-quantities) showing, in this way, the propagation of the constraints. The case of tracefree matter is of particular interest since the equation of conservation satisfied by the energy-momentum is conformally invariant; the associated equations of motion for the matter fields can, usually, be shown to possess good conformal properties—see [15], Chapter 9 It clarifies the inner structure of Paetz’s original construction by identifying the integrability conditions underlying the mechanism of the propagation of the constraints. To the best of our knowledge, these integrability conditions have not
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