Good thinking! This stuff is something that can really be a trick to get your head wrapped around it. Space time tends to act opposite in ways that matter energy do. Unfortunately, all we are and all we do, how we study, the equipment we use, the pencil and paper we take notes on, all is based in the existence of matter and energy. This fact makes it tuff to explain as well as grasp. To show how some of this stuff works by using this world of matter and energy we live in ,well, ...hmmm, very careful analogies maybe. I'm winging this so....OK....here is an example of something working rather opposite to how we experience things:
Consider the total energy in a spring. OK wait....think of a spring that is compressed. Never mind what is compressing it, just consider the existence of a spring in a compressed state. Now consider all the energy it encompasses. It has the energy contained in it's mass. Good ol' E=mc2. This is a lot of energy. It also has some potential energy in being compressed. The potential energy will pale in comparison to the energy in it's mass. This is how it will be in just about any such situation as we experience it. Now, how about something that can contain massive potential energy with very little 'substance' ? Welcome to the world of space time, except that there isn't any matter energy here to think about, to make it tangible. Come to think of it, it does go a long way towards making sense of dark energy being much more prominent compared to dark matter as calculated by the physics. (physics gives math purpose) Lets move to your question now, I have a sad but hopefully functional analogy.
Consider a room of people, crowded, standing around, casual, comfy, just crowded, shoulder to shoulder, OK with the situation, stable but rubbing right against each other....and a door opens. The door is not sucking people out, it's the people that naturally go out on their own. They approach the door, it's just what they do, they aren't under pressure like air in a balloon, they just approach the door, it's what they want to do. The crowd is thinner towards the door, thicker away from the door. Lets try something better because space time would rather be proactive on matter so.....how about the white corpuscles. Those guys are proactive after infestations. They are wandering around in a nice uniform distribution, then that disgusting matter shows up. The corpuscles go for it, push onto it. The corpuscles distribution gets thinner close to the matter by their own nature as they attack and remain denser further away. The disgusting matter isn't sucking them in, it's what the corpuscles do.
Now you could go into saying things about how the corpuscles would be dense on the surface of the matter or what about the people outside the room. Analogies aren't perfect. You'll start to get into digesting concepts of singularities, and infinite densities, and never ending pondering. Ok, once I get going things kind of settle in so here's one for you. Think of an infinitely long magnetic spring. The spring is relaxed just hanging around. Then a piece of iron shows up. The end of the spring latches out to the iron. The spring is less dense close to the iron and the effect travels out along the spring forever to infinity. ooooohhh, ahhhhhh, ok if you want to actively find failures in an analogy you can. I don't have a lot of time for this and it is a bit arduous to try to convey. Hope this has helped.