A gluing construction for non-vacuum solutions of the Einstein-constraint equations.(English)Zbl 1101.83005

The glueing construction described in the present paper is applicable e.g. to the Einstein-Yang-Mills field equation and the Einstein-Vlasov equation, but is restricted to the case of spacetimes where the spatial part is a compact manifold of arbitrary dimension $$n$$. The calculations are done in the constant mean curvature (CMC) gauge for the initial hypersurface and applies to such types of matter fields where the Cauchy problem is well-posed. The abstract of the paper reads: We extend the conformal gluing construction of J. Isenberg, R. Mazzeo, D. Pollack [Commun. Math. Phys. 231, 529–568 (2002; Zbl 1013.83008)] by establishing an analogous gluing result for field theories obtained by minimally coupling Einstein’s gravitational theory with matter fields. We treat classical fields such as perfect fluids and the Yang-Mills equations as well as the Einstein-Vlasov system, which is an important example coming from kinetic theory. In carrying out these extensions, we extend the conformal gluing technique to higher dimensions and codify it in such a way as to make more transparent where it can, and can not, be applied. In particular, we show exactly what criteria need to be met in order to apply the construction, in its present form, to any other non-vacuum field theory.

MSC:

 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C15 Exact solutions to problems in general relativity and gravitational theory 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)

Keywords:

Yang-Mills field; Vlasov equation

Zbl 1013.83008
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