That explanation that "things are always moving - through time" is exactly correct. You need to think of the full four dimensional spacetime. It is that spacetime which is curved rather than just space.
The crucial concept is Newton's first law (objects continue on their trajectory if a force is not applied). The straight lines in the 4D spacetime (geodesics) - the lines that an object would follow if no force is applied - correspond to the paths that look as if a gravitational force is applied.
In order to help people think about curvature and gravity, look at the following examples:
- The surface of the earth is a 2 dimensional positive curved space. To see this, draw a triangle with corners on the north pole, on the equator near Somalia and on the equator in Equador. The resulting triangle has a sum of all corners > 180 degrees.
- In a negative curved space, the sum would be less than 180 degrees. In a flat space, it is equal to 180 degrees.
- Another way to see the curvature of the surface of the earth is to observe that it's impossible to draw 2 parallel lines that do not intersect.
- The 2D torus (e.q. the surface of a donut) is flat. Test it with triangles.
- The towers of the Verrazano–Narrows Bridge are wider at their top than at their base. This has nothing to do with the earth have a positive curvature. Test it with a torus.
- 3D space is nearly always flat in the universe, especially at the surface of our planet.
- 4D space-time is not remotely flat. If I throw up a ball, it will come down. This is due the mass of the earth curving its surrounding 4D space-time. The straight line for a ball in the curved space-time looks like the ball changes directions and comes down in our flat 3D space.
- If you try to find the triangle of a sphere with the biggest sum of corners, you'll discover that the outside and inside of a triangle are interchangeable. We've entered the field of topology now and this has nothing to do with its curvature.
>It is that spacetime which is curved rather than just space.
This is the crucial point that many discussions overlook. The concept of "curvature of space" is obvious nonsense because curvature is measured with respect to space. For space itself to curve implies the existence of meta-space (and as many nested metaspaces as you like). But "spacetime" does not work like "space", and there actually is no metaspace. Talking about "space" curving or expanding is unnecessarily confusing.
> The concept of "curvature of space" is obvious nonsense
This is emphatically not correct. Space can be curved without being embedded in a higher "meta-space". You can have a 2D surface that is curved as if it were the surface of a sphere without it actually being on an actual 3D sphere -- it just has to "connect up" the right way and have "parallel" lines bend towards each other and so on. Ditto in more dimensions.
The concept of "curvature of space" is obvious nonsense because curvature is measured with respect to space.
This is not really true. If you take a flat two dimensional space, a sheet of paper, this space has no intrinsic curvature. You surely now imagine this sheet of paper laying flat on a table in front of you embedded in our usual three dimensional space. Now pick it up and role it to form a cylinder. This gives extrinsic curvature to the space, curvature in the space the sheet is embedded in. But the sheet has still no intrinsic curvature, if you were a two dimensional creature living on the sheet you can not tell whether it is laying flat on the table or is rolled up to a cylinder, at least ignoring the fact that the cylinder connects two opposite edges of the space.
The surface of the earth on the other hand has intrinsic curvature, angles of triangles don't add up to 180 degrees for example. And this intrinsic curvature is a feature of the space not of the embedding into a higher dimensional space. It is not easy to visualize if possible at all, but spaces can have intrinsic curvature independent or even without an embedding, our three dimensional space can be curved without being embedded in a higher dimensional space and the same of course holds for space-time.
"Extrinsic curvature" is normally called just "curvature". "Intrinsic curvature" isn't normally called anything because it has minimal relevance to everyday life. If you talk about "curvature of space" the natural assumption is that you mean extrinsic curvature.
But in general relativity extrinsic curvature is the irrelevant thing, it's all about intrinsic curvature. We don't think of space-time being embedded in a higher dimensional space but we still talk about curvature of space-time, its intrinsic curvature.
These demonstrations with balls rolling on a stretchable rubber sheet to visualize gravity are really misleading in this regard because they use a good deal of extrinsic curvature to make things work but in reality mass doesn't deform space-time into a fifth dimension or at least it doesn't necessarily do so.
Curvature is not "measured with respect to [an embedding] space". If you have a triangle and the sum of the angles is not 180 degrees that's a non-Euclidian (curved) space.
But that's not how the word "curved" is used in everyday life. Call it non-Euclidian if it's non-Euclidian. Redefining common words only leads to confusion.
Participating in a discussion about an article titled "the quantum source of space-time" to say that basic concepts are nonsense because some words do not have the same meaning as in everyday life also leads to confusion.
Take a 50-kg object (110 pounds) and carry it in your arms for an hour on a flat road. You will have done no work against gravity. You may complain that this is not how the word "work" is used in everyday life - and that you will have done a lot of "work" against gravity (preventing the object from falling to the ground). However, if you wish to discuss Physics and Mathematics, using terms that have precise definitions in those fields, you cannot object on the basis that these words do not have the same meaning as non-Physicist and non-Mathematicians ascribe to them.
Why stop there? Field, black hole, wormhole, string, force, spin, colour, charm, strangeness. All of these terms and more are used by physicists and have very different meanings to those used in "everyday life".
The crucial concept is Newton's first law (objects continue on their trajectory if a force is not applied). The straight lines in the 4D spacetime (geodesics) - the lines that an object would follow if no force is applied - correspond to the paths that look as if a gravitational force is applied.