The thought experiment depicted geometrically
In general relativity, planets, stars, galaxies and other massive bodies warp spacetime. Warped spacetime in turn guides the motions of those massive bodies[1]. The late American theoretical physicist John Archibald Wheeler summed up the relevant principles by saying that spacetime (the equation's left hand side) tells matter and energy how to move; matter and energy (the right hand side) tell spacetime how to curve.
Spacetime tells matter how to move [2]
In other words, the Einstein equations describe the different geometries of space, but as Robert Crease points out, the fact that spacetime has a geometry does not mean that space is literally curved (something can only be curved with reference to something else taken to be straight), but rather has to do with the way that the measurements of a path of something travelling through that space, say a beam of light, add up. If light is bent by a gravitational field, the measurements of its trajectory add up in a way that corresponds to the mathematics of a certain geometry. So the notion of curvature is a mathematical construct, and it is the geometric nature of the theory which leads to the term 'curvature'. This involves treating gravitation not as a force, that is, something that exerts a tug, but as a property of space itself, as having a structure that must be followed by things passing through it[3]. The basic idea is that space itself has a "shape", and the shape of the space depends on the distribution of matter and energy within it.[4]
Objects moving in space therefore travel in straight lines (they only appear curved) and move along the shortest possible paths through spacetime called geodesics, which dictate how freely-falling matter moves through space-time.[5] This is the principle of extremal (minimal or least) action, the shortest distance between two points being a straight line. Gravity and the curvature of space-time do not make a body depart from a geodesic, so neither of them is strictly speaking a force. The paths of objects only seem curved because of the warping of spacetime, and when a massive (or even a massless) object's trajectory appears curved in response to gravity, it doesn't take any force to keep it moving along a straight line, as with light in the lower graphic:
Spacetime tells matter how to move [2]
In other words, the Einstein equations describe the different geometries of space, but as Robert Crease points out, the fact that spacetime has a geometry does not mean that space is literally curved (something can only be curved with reference to something else taken to be straight), but rather has to do with the way that the measurements of a path of something travelling through that space, say a beam of light, add up. If light is bent by a gravitational field, the measurements of its trajectory add up in a way that corresponds to the mathematics of a certain geometry. So the notion of curvature is a mathematical construct, and it is the geometric nature of the theory which leads to the term 'curvature'. This involves treating gravitation not as a force, that is, something that exerts a tug, but as a property of space itself, as having a structure that must be followed by things passing through it[3]. The basic idea is that space itself has a "shape", and the shape of the space depends on the distribution of matter and energy within it.[4]
Objects moving in space therefore travel in straight lines (they only appear curved) and move along the shortest possible paths through spacetime called geodesics, which dictate how freely-falling matter moves through space-time.[5] This is the principle of extremal (minimal or least) action, the shortest distance between two points being a straight line. Gravity and the curvature of space-time do not make a body depart from a geodesic, so neither of them is strictly speaking a force. The paths of objects only seem curved because of the warping of spacetime, and when a massive (or even a massless) object's trajectory appears curved in response to gravity, it doesn't take any force to keep it moving along a straight line, as with light in the lower graphic:
Source: http://www.askamathematician.com/2009/10/q-why-does-curved-space-time-cause-gravity/
[1] Tim Folger, "A brief history of time travel", Scientific American, Special Issue, 100 years of General Relativity, September 2015, 58 at 61
[2] How matter and energy tells spacetime how to curve, the second limb of Wheeler's analysis is considered on the previous page when dealing with the mathematical side of general relativity.
[3] Crease, op.cit, 196.
[4] http://www.physicsforums.com/showthread.php?t=295449
[5] David Zaslavsky and Luboš Motl in http://physics.stackexchange.com/questions/3009/how-exactly-does-curved-space-time-describe-the-force-of-gravity
Far from any gravity source, the shortest distance is a straight line in three-dimensional space, but near a massive object, the shortest distance is curved in three-dimensional space. Describing the circle of the equator on a flat surface is one of the straightest lines one can draw on the surface of Earth, and the same is true for all actual trajectories that objects choose in the spacetime of general relativity. Similarly, planets orbiting the sun follow straight lines, because they are the most natural - the straightest - lines they can find in curved spacetime. They are moving along a straight trajectory in a curved space that is bent around the Sun. In other words, the trajectory of an object (including light) moving along the straightest and shortest possible line in spacetime (a geodesic) appears to curve and this apparent curvature is governed (caused) by the presence of matter and energy (some sources greater, some smaller; some near, some far) in the architecture or fabric of spacetime.
Instead of the familiar but simplistic billiard ball on trampoline example, imagine that there is a hemisphere replacing the ball on the trampoline, so there exists an almost straight line on the hemisphere, namely the equator near the junction with the rest of the trampoline. This circle is one of the straightest lines you can draw on the surface of Earth, and the same is true for all actual trajectories that objects choose in the spacetime of general relativity. So in the hemisphere-above-trampoline example, particles can orbit around the equator of the attached hemisphere just like planets, because it is the straightest and most natural line they can choose. Particles thus choose the most natural - the straightest - line they can find in curved spacetime.
To understand this, it is sometimes necessary to call in aid what is commonly referred to as ant geometry. Imagine a sphere in two dimensions like the surface of a ball. An ant lives upon the surface and keeps walking straight, so that it returns to its starting point, his trajectory describing a circle. A second ant then walks from the same point as the first ant and at the same speed but in a different direction. He also traces a circle and the two circles intersect at two points. You can imagine those circles as meridians and the crossing points as respectively the north and south poles. From the ants' perspectives (who aren't aware that they are living in a curved space), their distance will be changing in time non-linearly, and this is one of the features of curved spacetime on the movement of particles[1].
[1] Tim Folger, "A brief history of time travel", Scientific American, Special Issue, 100 years of General Relativity, September 2015, 58 at 61
[2] How matter and energy tells spacetime how to curve, the second limb of Wheeler's analysis is considered on the previous page when dealing with the mathematical side of general relativity.
[3] Crease, op.cit, 196.
[4] http://www.physicsforums.com/showthread.php?t=295449
[5] David Zaslavsky and Luboš Motl in http://physics.stackexchange.com/questions/3009/how-exactly-does-curved-space-time-describe-the-force-of-gravity
Far from any gravity source, the shortest distance is a straight line in three-dimensional space, but near a massive object, the shortest distance is curved in three-dimensional space. Describing the circle of the equator on a flat surface is one of the straightest lines one can draw on the surface of Earth, and the same is true for all actual trajectories that objects choose in the spacetime of general relativity. Similarly, planets orbiting the sun follow straight lines, because they are the most natural - the straightest - lines they can find in curved spacetime. They are moving along a straight trajectory in a curved space that is bent around the Sun. In other words, the trajectory of an object (including light) moving along the straightest and shortest possible line in spacetime (a geodesic) appears to curve and this apparent curvature is governed (caused) by the presence of matter and energy (some sources greater, some smaller; some near, some far) in the architecture or fabric of spacetime.
Instead of the familiar but simplistic billiard ball on trampoline example, imagine that there is a hemisphere replacing the ball on the trampoline, so there exists an almost straight line on the hemisphere, namely the equator near the junction with the rest of the trampoline. This circle is one of the straightest lines you can draw on the surface of Earth, and the same is true for all actual trajectories that objects choose in the spacetime of general relativity. So in the hemisphere-above-trampoline example, particles can orbit around the equator of the attached hemisphere just like planets, because it is the straightest and most natural line they can choose. Particles thus choose the most natural - the straightest - line they can find in curved spacetime.
To understand this, it is sometimes necessary to call in aid what is commonly referred to as ant geometry. Imagine a sphere in two dimensions like the surface of a ball. An ant lives upon the surface and keeps walking straight, so that it returns to its starting point, his trajectory describing a circle. A second ant then walks from the same point as the first ant and at the same speed but in a different direction. He also traces a circle and the two circles intersect at two points. You can imagine those circles as meridians and the crossing points as respectively the north and south poles. From the ants' perspectives (who aren't aware that they are living in a curved space), their distance will be changing in time non-linearly, and this is one of the features of curved spacetime on the movement of particles[1].
“Ant geometry” explained: An example of curved 2-d space is the surface of a ball. If you draw two parallel lines on the ball, then eventually they will cross. The curvature forces the two lines to come together. A straight line on a sphere always traces out a "great circle", like the equator. The lines are parallel twice, but also intersect twice. In other words, parallel lines do not exist in spherical geometry, as all lines (which are great circles) eventually cross each other. This contrasts with Euclidean geometry, where parallel lines are a unique pair, and with hyperbolic space which is a type of non-Euclidean geometry with a negative curvature, (as to which see The flatness problem revisited following the section on string theory later herein).
Source: “Why does “curved space-time” cause gravity? http://www.askamathematician.com/2009/10/q-why-does-curved-space-time-cause-gravity/ Posted on 25 Oct 2009 by The Physicist.
[1] This is an abbreviated version of "Marek"’s explanation in http://physics.stackexchange.com/questions/3009/how-exactly-does-curved-space-time-describe-the-force-of-gravity. As to ant geometry, see also Dana Mackenzie, The Universe in Zero words – The Story of Mathematics, New South, 126-7 (city and date of publication not disclosed).
Source: “Why does “curved space-time” cause gravity? http://www.askamathematician.com/2009/10/q-why-does-curved-space-time-cause-gravity/ Posted on 25 Oct 2009 by The Physicist.
[1] This is an abbreviated version of "Marek"’s explanation in http://physics.stackexchange.com/questions/3009/how-exactly-does-curved-space-time-describe-the-force-of-gravity. As to ant geometry, see also Dana Mackenzie, The Universe in Zero words – The Story of Mathematics, New South, 126-7 (city and date of publication not disclosed).