Speaker
Walter Simon
(University of Vienna)
Description
We generate initial data for vacuum spacetimes (possibly with cosmological constant) via the conformal method, with a three sphere or a ”three- donut” (the product of a two-sphere with a circle) as "seed metrics". Our motivation comes from black-hole-lattice cosmology. However, in contrast to known schemes for constructing corresponding data, we do not allow for punctures, which prevents us from changing topology. On the other hand, we include the non-linear terms ("cosmological" and "momentum") in the Lichnerowicz equation. We discuss how theorems from differential geometry (Yamabe, Obata, Premoselli) guarantee or exclude the existence of suitable solutions of this equation. We consider two examples: As a warmup, we recall the maximal Kottler (="Schwarzschild de Sitter") data on the three-donut. We then focus on the round three sphere where we take as momentum a transverse, trace-free tensor constructed algebraically out of certain pairs of the sphere’s Killing vectors and their derivatives. This endows the sphere with some "structure" - but not necessarily with black holes. This is joint work with Robert Beig and Piotr Bizo\'n.
