How do I displace mesh like this effect

Hey guys, I am a noob when it comes to blender. I found this animation that this artist made @matthieubraccini and wondered if it would be possible in blender.

my first problem was on how would I be able to recreate the soft groove around the object especially when it is moving.

This is what mine looks like right now, I tried first by experimenting with a basic cube and plane. My idea was to use a displace modifier to get the soft groove and a vertex weight proximity to follow the cube.

As you can see there are 2 subdivision surfaces, the first one on top is to refine and sharpen the weight paint of the weight proximity modifier (as I have googled) and the bottom to smooth out the mesh, but as you can see the displacement looks really square and chunky.

Any help would be greatly appreciated.