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Conformal wave equations for the Einstein-tracefree matter system. (English) Zbl 1429.83005

Summary: Inspired by a similar analysis for the vacuum conformal Einstein field equations by T.-T. Paetz [Ann. Henri Poincaré 16, No. 9, 2059–2129 (2015; Zbl 1325.83006)], 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 than 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.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35L05 Wave equation
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
83C22 Einstein-Maxwell equations
53Z05 Applications of differential geometry to physics
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)

Citations:

Zbl 1325.83006
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References:

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