The basic question was inspired by some other recent threads on the "fabric of space". If we imagine a 2-d spatial rubber sheet, how closely can we make its vibrational modes compare to gravity waves (in the limit of non-relativistic velocities). It's well known that gravity waves locally stretch in one direction and compress in the other, I would assume this would place some constraints on Poisson's ratio of our hypothetical material. Is this even possible at all, or is the analogy unproductive if we try to take it too seriously? As I recall, there are no spherically symmetric gravity waves, but if we drop a pebble on a sheet, we expect as a primary mode spherically symmetric ripples.