The rhythm of geometry
Distortions of geometry: what does that mean? First of all, distances shrink and expand in a certain coordinated way. That's the main mechanism by which gravitational waves act on the rest of the world: they rhythmically distort distances between freely falling objects.
For the simplest case of a gravitational wave, the consequences are shown in the animation below. Imagine that we are, once more, in empty space, far away from all sources of gravitation. On the floor of our spaceship, we create a mandala, painstakingly constructed by arranging coloured grains of sand:
Note that all the sand particles are free, so they float weightlessly near the cabin floor.
A simple gravitational wave traversing this mandala would change the distances between the sand particles as shown in the following animation. The wave moves from behind the computer screen towards the reader.
[Animation size 118kB; please allow time for loading]
The coordinated dance of stretching and shrinking - stretching in one direction, while distances shrink in the perpendicular direction - is a general property of gravitational waves, as is the fact that all distortion takes place in a plane perpendicular to the direction in which the wave travels.