How can I create a 3D spacetime grid animation?

A similar (though not physically realistic) effect can be done using array modifiers and a wireframe modifier to create the 3 dimensional space-time lattice, and then a cast modifier to create the gravitational effect.

1. Start with a cube, and then add three array modifiers, one for each dimension, and then a wire frame modifier, as indicated:

This gives you the space time lattice. Apply these modifiers and then center the lattice on the origin, just to make everything else a little easier.

2. Add a sphere. Then add to the lattice a cast modifier, shape sphere, with a negative factor. The lower this factor is, the more gravitation is simulated.

Moving the sphere will distort the lattice as one might expect:

Note well however that this effect is applied linearly over space, while real gravity would fall off proportionally to the inverse square. A more physically realistic effect I believe is possible using vertex groups and the hook modifier, but would be more complicated than this solution, which should be acceptable for most aesthetic uses.