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I'm having a difficulty in understanding Einstein's version of gravity as "space time curvature" (I could be a total physics idiot, please point out !).

  1. How does an object on the surface of earth stay fixed ? Since there is no concept of "attraction" and only space time why cannot an object keep moving around on the surface of earth.

  2. How is an object "attracted" towards earth when there is no concept of attraction ? Say an alien space ship hovers over the earths atmosphere, drops an apple and the apple falls to the ground. What causes this ? Or in other words, when there is only curvature in space time, what causes an object to move ?

  3. How does moon stay in a particular 2D plane ? In most depictions the space time curvature is presented in 2D but in reality isnt the curvature in 3D ? If so, why is the rotation of moon around earth restricted to a particular 2D plane ? For eg, lets consider earth in (0,0,0) of a 3D space and moon rotates the earth on the X-Y plane [ie (x,y,0), x&y being the variables, z fixed]. Why can it not move on the z plane too [ie (x,y,z)] ?

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    $\begingroup$ I believe it's because all geodesics near the surface of Earth lead towards it. $\endgroup$ – HDE 226868 Nov 22 '14 at 1:36
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    $\begingroup$ youtube.com/watch?v=DdC0QN6f3G4 $\endgroup$ – Schrödinger's Cat Nov 22 '14 at 1:41
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    $\begingroup$ I think it's useful sometimes to have visualizations which depict curved surfaces that are supposed to represent curved spacetime, not curved space, with geodesics drawn on--see here for a simple example, and here is a more sophisticated example for those who already have a good familiarity with special relativity (and ideally some conceptual familiarity with the Kruskal-Szekeres coordinate representation of a black hole). $\endgroup$ – Hypnosifl Nov 22 '14 at 3:24
  • $\begingroup$ Oh, and the parable of the apple from the textbook Gravitation is helpful for thinking about the basic concepts. $\endgroup$ – Hypnosifl Nov 22 '14 at 3:26
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The language of General Relativity is that of tensor calculus, which is the beautiful marriage of geometry and multivariable calculus. It equates symbolic representations of geometric objects--tensors--to analytical expressions representing the same object in terms of known operations like addition, multiplication, differentiation, and so forth. Most of what follows here will be the same as what HDE has said but with the attempt at providing you some geometric intuition which might feel a little more natural. All the notions I mention have mathematical counterparts but unless stated otherwise, I'm referencing them geometrically

One of the more vital notions of GR is that of the equivalent of straight lines, but embedded on curved manifolds(spaces), in particular the spacetime manifold of four dimensions and its curvature. We can make this easy to visualize geometrically by taking a snapshot ( time slice with constant time coordinate ), and look at the shape of space in all 3 spatial dimensions, a cross section, or what have you. In other words we can look at what the whole shape and geometry of a local spacetime by analyzing the whole picture or the projections against certain curves or planes. These aforementioned analogues of straight lines have a name. Geodesics. I won't go into the derivation via the calculus of variations, but know that the geometric description of it is backed analytically. A geodesic, mathematically, is a curve of a parameter such that $ \gamma(t): R \rightarrow R^{n} $ that at fixed endpoints--initial and final t values--the length of the path the curve takes is minimal. This should seem oddly familiar. What's the shortest distance between any two points in a flat, Euclidean space? A straight line! A straight line has the shortest distance between two points, but if your space isn't quite flat, you're going to have to compensate for that. That's where tensor calc, Euler, and Lagrange come in and save the day. Now as for your questions using the aforementioned:

Set the surface of the planet to be the termination of a geodesic path starting an infinitesimally small distance above the surface of the planet. Geodesic lines near this body are going to be such that all paths taken will be the shortest possible. Coincidentally, this is straight down/radially inward towards the source of spacetime curvature. Take the limit as we start closer and closer to the surface until we're standing there and you'll find the only path that satisfies the conditions mentioned is not moving at all.

The notion of attraction is encoded in the shape and curvature of a local system. The classical analogy of a bowling ball in the center of a sheet held above the ground at the corners does very well here. If we remove the bowling ball, the sheet will remain flat ( for the most part. this model is slightly flawed because the embedding space is already curved ). Put a marble on the sheet anywhere, and you'll find that it won't move at all, or if the initial conditions dictate that the marble was in motion to start, you get straight lines ( assuming perfect flatness ). Put the bowling ball in and do the same. If you put the marble at the surface of the ball, you witness no motion ( answer to question 1 again ). Start the marble in orbit ( alien spaceship ) and release it and you'll find it travels right to the surface in a "straight" path ( answer to question 3 ). If you start the marble moving with sufficient speed travelling at right angles to the radial lines to the bowling ball, however, you'll get orbit. In the absence of our own gravity, this would present a marble whose path is just a great circle that is traversed an infinite number of times as the parameter value gets large. If we look at the individual points of the path of the marble and try to find what they have in common, its that they all line in a plane (answer to question 3). That's not to say that we can't make a mathematical situation such that the curvature of the space makes the orbital circles break the planar shape of the trajectory, but we have yet to see something like that in nature.

Let me know if you want a slightly more rigorous clarification of any of the ideas here!

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I'm not a GR expert (and there are some on this site), but I'll try to answer this.

How does an object on the surface of earth stay fixed? Since there is no concept of "attraction" and only space time why cannot an object keep moving around on the surface of earth.

The answer here is that there generally aren't any forces moving the object to the side. The only force of any importance here is the gravity of the Earth. The reason it doesn't make the object move from side to side is that the geodesics in spacetime lead straight towards the Earth. A geodesic can be thought of as the preferred path for a particle to take.

2.How is an object "attracted" towards earth when there is no concept of attraction ? Say an alien space ship hovers over the earths atmosphere, drops an apple and the apple falls to the ground. What causes this ? Or in other words, when there is only curvature in space time, what causes an object to move ?

This goes back to geodesics.

3.How does moon stay in a particular 2D plane ? In most depictions the space time curvature is presented in 2D but in reality isnt the curvature in 3D ? If so, why is the rotation of moon around earth restricted to a particular 2D plane ? For eg, lets consider earth in (0,0,0) of a 3D space and moon rotates the earth on the X-Y plane [ie (x,y,0), x&y being the variables, z fixed]. Why can it not move on the z plane too [ie (x,y,z)] ?

This has been asked many, many times before on Astronomy and Physics (e.g. here, here, here, here, here, here, etc.). I highly encourage you to read all those posts. To summarize: The solar system formed out of a protoplanetary disk, which was disk-shaped and in a single plane.

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  1. Staying fixed? A: No force = no acceleration => stays put

  2. How is an object "attracted" towards earth? A: Gravity is not a force. Objects do not accelerate in their own frame of reference. Like in "freefall" you feel no acceleration, but observers may impune acceleration to you because they may observe you speeding away/toward etc.

  3. The moon in it's plane? A: See question 1) above. The moon is not accelerating, it's in freefall. I'm sitting here feeling the force of chair and the weight of my arms as I am not in freefall, that pesky chair. The moon will move from its freefall if you apply a force, but only "Gamera" can do that :-)

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