Decay of the Weyl curvature in expanding black hole cosmologies

Abstract

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein's equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null infinity. More precisely we establish decay estimates for Weyl fields which are (i) uniform (with respect to a global time function) (ii) optimal (with respect to the rate) and (iii) consistent with a global existence proof (in terms of regularity). The proof relies on a geometric positivity property of compatible currents which is a manifestation of the global redshift effect capturing the expansion of the spacetime.

v2: 122 pages, 7 figures; the main theorem now establishes sharp r^{-3} decay under stronger assumptions than in v1 which are motivated by [arXiv:1910.05799]; also an extended version of the previous Section 2 of v1 now appears instead in [arXiv:1910.05799]; numerous improvements throughout the manuscript, in particular Section 1.5 revised, and Appendix B added