How exactly does curved space-time describe the force of gravity?

Question

I understand that people explain (in layman's terms at least) that the presence of mass "warps" space-time geometry, and this causes gravity. I have also of course heard the analogy of a blanket or trampoline bending under an object, which causes other objects to come together, but I always thought this was a hopelessly circular explanation because the blanket only bends because of "real" gravity pulling the object down and then pulling the other objects down the sloped blanket. In other words, to me, it seems that curved space wouldn't have any actual effect on objects unless there's already another force present. So how is curved space-time itself actually capable of exerting a force (without some source of a fourth-dimensional force)? I apologize for my ignorance in advance, and a purely mathematical explanation will probably go over my head, but if it's required I'll do my best to understand.

Answer 1

No need to apologize. Your questions are astute. Truly understanding this concept requires skills and knowledge that are very sophisticated and beyond the scope of introductory or even intermediate level physics courses. It's actually a little beyond my capabilities. But I can explain a few details that might illuminate the concept to you and clear some of the confusion.

You are correct that the reason the membrane deforms on Earth is because the mass is being pulled down by gravity, so the logic is a bit circular. The part of the analogy that is relevant is the fact that the deformation happens. When a mass deforms a membrane on Earth, the reason the mass is being pulled down is gravitational attraction. The membrane deforms because it is a stretchy material. When space-time deforms in the presence of a mass, it deforms in all directions; not just beneath the mass. In space, there isn't even a defined concept of up or down. In reality, space-time is not a "fabric." Space-time is a just a mathematical model for something that is used to define directions and distances. Physicists call this a metric or a geodesic. In free space, far away from any mass, the metric is flat. In this sort of geometry, the inertial path is a straight line. In the absence of a force, mass travels in a straight line at a constant speed. In the presence of a mass, this metric is altered and the inertial path can become curved causing an acceleration in a non-acceleraed reference frame. Another relevant analogy is the shortest distance between two points. In empty space, the shortest distance is a straight line. But in the presence of some constraint, it may be some other path. For example, on the surface of the Earth, the shortest distance between two points is a portion of a circular arc. The presence of mass in space changes the metric so that the shortest distance between two points is something non-linear.

Why does mass do this? That is a little beyond my ability to explain. However, General Relativity does conform to conservation laws, so it might have something to do with conservation of momentum and energy. This is very important in Special Relativity so it stands to reason it could be relevant here as well. In any case, physics is about describing the Universe; not necessarily explaining why the Universe is they way it is.