If I bend a paper, it doesn't change the things written on it. If gravity is bent space(time), we shouldn't be able to detect it. (e.g. things shouldn't be dragged by it) So why are we talking about bending space, if it's not the space itself that is bending? it's only our mental concept of how things will be that is "bending".
What? If I curve a sheet of paper, of course what's written on it will change...I don't understand your argument. Draw a straight line on a piece of paper, and then curve that piece of paper into a cylinder, now your straight line turned into a circle...
If you are created by the molecules of paper, you can't tell if it's curved, or strechted. Do we exist beyond spacetime?
If I lived on the paper, I can't tell the paper is curved locally - this is a statement of the Einstein Equivalence principle. But globally, I can tell if the paper is curved. If me and my friend started on parallel paths, we find that eventually our paths either converge or diverge, depending on the curvature. This is the so-called "geodesic deviation".
You could only measure the distance between paths with counting molecules. The number of molecules is not changing, so the distance would be the same. It only works if bending means creating or removing space.
I don't necessarily need to measure distance by "counting molecules". The proper distance between me and my friend changes. A light pulse will take shorter and shorter time to bounce from me to my friend and back as I move along this piece of paper assuming our geodesics are converging. Even without regard to this, at some point, I'll collide with my friend, so obviously the distance between us changed.
That's what I don't understand. Why will it take shorter time? Light (or equivalent pulse) would allways pass the same number of molecules (the same amount of paper) regardless of how the paper is bent.
I'm getting closer to my friend. The proper distance between us decreases. The proper distance is not "number of molecules" between me and my friend. The distance between two points is not governed by some "counting" of molecules or intermediate particles.
molecules of paper are equivalent to fabric of space, not particles. If we assume that space is discrete, the distance between two points is the number of points of fabric of space (= molecules of paper) between them. If we bend the space/paper, we don't change the distance.
As far as I know, no successful quantization procedure of space-time is known. If you want to deal with causets or something, I am not well versed in those theories. In general relativity, the proper distance between two points is given by the metric, and this proper distance changes if the space-time changes. There is no way to count the events of space-time, as they are uncountably infinite. The events of space-time form a differentiable manifold. As such, we can make a differentiable mapping between some small region of space-time to an open set of R^4. Any open region of R^4 necessarily contains an uncountably infinite set of points, just as there is an uncountably infinite set of points between the numbers 0 and 1 on the real line.
You can bend a non-stretchy piece of paper into some shapes (like cylinders) without changing any distances, but you can't, for instance, make a flat non-stretching piece of paper smoothly cover a sphere, for the same reason that you can't make a flat, continuous map of the Earth's surface that preserves the size and shape of continents (i.e. doesn't stretch them or distort them). This is because a cylinder has no intrinsic curvature, though it has an extrinsic curvature. See for instance the Wikipedia: http://en.wikipedia.org/w/index.php?title=Curvature&oldid=460970826 While the article above, and your post, deals with the first sense of the ambiguous word, curvature, GR is most interested in the second - "intrinsic" curvature, not "extrinsic" curvature. The curvature of space-time that's of interest for gravity is intrisic curvature, not extrinsic curvature.
The following thread may be more than you bargained for and will take some time to plow through but there are some really good insights regarding 'curvature': https://www.physicsforums.com/showthread.php?t=548148
I think I get what OP is saying. I think he is saying that "distance" is just the number of points between two points, and that YOU an intrinsic are part of the "space" or "set of points". He thinks it doesn't matter what direction a point is from you.... it doesn't matter WHERE it is, but rather how many points are between you and that point. However, this ignores the concept of "distance" - which basically HAS to be used in physics for any meaningful theories to arise. And "distance" is also well-defined in maths - and I advice you to look up the concept of a "metric"/"metric space". Also, pervect's post is very good and may have already answered OP's question, but oh well. Edit: Also OP, you mentioned discrete space-time, and theories involving this always interest me. However, many theories based on discrete space-time apparently violates the Lorentz Invariance Principle (according to a lecturer).
Think about it this way: we exist in spacetime. You say things wouldn't move? Well, let's use your paper example. If I set a paper clip on the paper, and then curved the piece of paper to a convexed (hump) shape, would it slide of? Yes. The same occurs with spacetime in a general sense.
The point you are conveying here sounds similar to the hairy ball theorem and how all the hair can't be combed in on direction because there is always a point where the vector is 0.
Hmmm. I agree that ALL physics theories are just mental concepts. But everyone seems to find they have a useful relation with the real world. If you want to use General Relativity then you use its concepts and definitions. If you don't want to, fine. But I will mention that if you bend the paper then it affects the motion of grains of sand on the paper, right?
IMHO that model is misleading, because you need another external space (where grains could exist, where the observer is, and where you actually can bend a paper). As was said earlier (by pervect), spacetimes curvatures are different.
Yes, grains of sand rolling on a bend paper have nothing to do with General Relativity. This is a better analogy: http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html http://www.relativitet.se/spacetime1.html
Obviously there's some conflict between everyday usages of these words and their mathematical definitions. I agree with the OP about people living on a sheet of paper not being able to tell if it had been rolled or not. This caught me eye though: Please correct me if I'm wrong, but isn't this exactly the kind of thing the curvature of space-time effects? If you have a massive object, space-time curves around it. Think about if you were surveying distances by how long it took you to walk from place to place, moving at a steady pace. Of course since gravity sucks you in, you would get to that massive object much faster than you had expected, based upon how long it took you to get to that same spot when it was empty. Therefore (absent a proper understanding of higher physics) you would have to conclude that somehow the distance between the two spots had shrunk. So it seems like the curvature would produce results suggesting the distance between the points really had shrunk, even though maybe at first glance no "space" had evaporated?