Here is something Steve Carlip (gravitational physicist) said over in sci.physic.relativity

"Remember, it's spacetime that's curved, not just space.
While the two partticles may initially be at rest in space,
they aren't ``at rest'' in time.

(When you're confused about relativity, a good starting
point is almost always to draw a spacetime diagram. That
won't necessarily give you an answer, but it will often
point you in a useful direction. If you do that in this case,
you'll immediately notice that even if the two particles
are initially at rest relative to each other, they still have
trajectories in spacetime.)

Steve Carlip"

I'm more inclined to go for the diagram since relativity is geometry.