Ever since the announcement of the discovery of gravitational waves, I have found myself pondering related notions of space. Recently, I was thinking about an analogy I've heard to explain how mass bends spacetime. Imagine holding a piece of paper with a stone placed in the middle, and then place a marble on the edge. Because of how the stone warps the paper, the marble will roll towards the stone.
This got me thinking about the "ways" in which space could potentially be bent. For example, imagine holding the same piece of paper, but this time poke the paper (without tearing it) from underneath with the eraser of a pencil. Placing a marble on the piece of paper not directly on the eraser will result in the marble rolling away from eraser.
Thus depending on how the paper is bent, the behavior of the marble changes. I wonder, has anyone considered the possible ways spacetime can be bent? If so, what are they? The only one I've ever heard of is through mass; it seems (at least in my ignorance) the paper/pencil analogy would represent a way to bend spacetime which does not involve mass. Perhaps this just illustrates the pitfall of using analogies.
Regardless, in case it isn't clear, I have a very limited understanding of physics. However, do not shy away from mathematics in your explanation if it is necessary, I am quite comfortable with mathematics.