Could it be possible to bend space in more than one "way"? 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.
 A: General relativity describes gravity as it has been observed, and gravitation fields are always attractive, not repulsive which is what your pin analogy would mean.
Theorists do examine possibilities , that antiparticles have negative gravitational charge and are repulsed by particles. There are experiments at CERN that are aiming to measure the gravitational behavior of antihydrogen. At present, the mainstream model of the cosmos works with the assumption that gravitation is always attractive.
A: We already know many things that curve spacetime. And none of them actually are mass. Mass has energy associated with it and it's the energy you notice.


*

*Energy density curves spacetime (and things with mass have energy, and so if you have some mass in a region you have an energy density in that region, and that's the biggest effect for lots of systems).

*Pressure curves spacetime.

*Stress curves spacetime.

*Momentum density curves spacetime.

*Energy flux curves spacetime.

*Spacetime curvature curves spacetime.
And the last one is the biggest deal. What the first five do is just change how spacetime curves itself so that it causes itself to curve differently than it already would curve itself.
For instance when a gas collapses to form a star, the energy density changes the curvature of the spacetime as it passes through a region and turns the curvature there into a kind of curvature that keeps itself curved even after the gas passed through to settle into the newly formed star. That's how spacetime gets, and stays, curved. And as the star shines, energy is released and as that energy flows outwards it turns the curvature out there into a weaker curvature as it passes by, so you don't feel like you orbit the smaller star until after the energy it releases actually passes you by.
Now if energy density is the biggest effect you usually see, then to have a different effect, you'd need a different energy.
And matter with a negative energy is called exotic matter.
