After reading this thread I am again reminded of my own confusion of this gravity/spce-time warping thing.
My understanding of the argument - based partly on those balls on rubber demonstrations - is this:
Objects move in straight lines unless some force acts on them (Heh - even Newton knew that! ). The mass of objects has the effect of bending space around them so that the ‘straight’ lines close to that mass are actually bent, therefore an object which object passes within the region of said mass will travel along the ‘bent’ straight line and so be attracted towards the mass. I hope that is correct as I can picture all that.
The bit which I have difficulty picturing is this: suppose I suspend an apple several feet from the ground then let go. The apple falls towards the Earth as per Sir Issac. What I cannot picture, in the special relativity realm, is how the apple is moving along any path at all relative to the Earth, up until I let it fall. What is this path of the apple which is being bent by the mass of the Earth?
The center of gravity of the earth is at the literal center of the earth. The apple, you, the ground you’re standing on, everything, is orbiting around that center of gravity. So to an outside observer, that apple is moving at a thousand miles per hour in an orbit, just like a satellite. Unlike the satellite, that speed is not enough to keep the falling and the orbiting in balance. You therefore see the apple fall to earth, marking out a curving trail through space from the 1000 mph horizontal vector and the 32 feet per second squared vertical vector. (That needs to be nitpicked, but it gives the right picture.)
It helps to imagine that there is nothing between you and the earth’s center. Once you do that your movement through space is more obvious.
Remember, we’re dealing with space-time, not just space. Everything is “moving forward” in time, even if they’re not moving in space.
If you want to consider gravity, you need to ignore other forces, which confuse the picture. Imagine the two apples floating freely in outer space. They will slowly approach, and either collide or orbit each other. If they don’t do that (because they’re sitting on a table on Earth), it’s because other forces are also in play (typically electro-magnetism).
If you think of gravity as a force you don’t need the bending of space and vice versa. According to Einstei’s General (not Special) Theory of Relativity objects travel along geodesics. A geodesic is the shortest distance between any two points that lie on a surface. “Surface” here is not a plane surface but rather a multidimensional coordinate set. We think of things traveling in straight lines and so when they deviate from that we ascribe to the system some force that causes the deviation.
Isaac Asimov gave on illustrative example that I think is pretty good. Imagine a golf green with humps and valleys. As the ball rolls it curves around and we can see that the path is a result of the humps and valleys. But suppose we could see the ball but not the green’s surface. Now the ball follows a curving path and we could ascribe that path to the result of a series of different forces acting on the ball.
In this sense the force of gravity is similar to Coriolis force. That “force” is a consequence of the geometry of the situation. If we throw an object, say to the east it travels with the original velocity along a line tangent to the surface and in the original direction we were traveling at the time of the throw. However we are on the surface of the earth and are being carried in a circle away from the original path. The makes the object appear to curve away from us and we ascribe that curved path to a force called the Coriolis force.
Your question is a tad obscure. In order to be affected by gravity an object need not have a path relative to another object in the sense I think you are using “patn”. In the case of the apple held by a string Newton’s gravitation is amply sufficient. The apple’s future path will be straight, relative to the earth, toward the center of mass of the earth. Well, almost straight. There will be miniscule varitations resulting from local differences in the thickness of the earth’s crust. The tilt of a plumb line in the vicinity of mountains needs to be accounted for in super accurate surveys. (Google deflection of plumb line by mountains for further info). However, for purpose of this discussion it’s a straight line.
I’m not sure how the “path” of the apple previous to cutting the string enters into the problem.
Let me try and simplify what it is I don’t understand.
If we have 2 apples close to each other and stationary with respect to each other somewhere in space and isolated from all other influences, what causes the pull of gravity between them? I cannot picture how curvature of space/time causes this as each apple is stationary wrt the other, and so also stationary wrt the resulting region of curvature.
Well, imagine the two apples sitting on a rubber sheet, both making a depression in it. But instead of two spatial dimensions, let’s say we have one space and one time dimension. So tilt the entire sheet (as if on an inclined plane). The apples will roll down the incline. Their mutual depressions will affect their paths and they will deviate toward each other.
“at rest” “stationary” “motionless” etc. are meaningful only with a frame of reference.
For instance if there were only one object in the universe, it could not have any motion, since it occupies the only point in space. Given more objects, they can have velocities relative to each other. One possible velocity is zero. Two objects can have a zero velocity with respect to each other. However, given that all mass does exhibit the property of gravity, the two objects will gain velocity towards each other from the very moment that they are at first motionless, with respect to each other. If there are three objects, the possibilities become much more complex. More than three, even more complex.
Imagine two apples on a special rubber sheet with no friction, and on which there’s no point at which the bending caused by an object is completely canceled out. Your two apples may appear to be in completely separate depressions, but the area between them will be affected by both, and will be slightly lower than the rest, so they’ll start sliding towards each other.
Because the analogy is only correct if you do so. When you stick two apples on a waterbed, they only slide towards each other because of the Earth’s gravity pulling down on both of them, but when we’re talking about the sheet as the whole Universe, there isn’t any “external gravity” playing that role. That aspect of the rubber-sheet model doesn’t actually correspond to anything in the real world it’s trying to represent.