When the action of curvature of spacetime happen, does that mean that spacetime is moving? But if it is, How can space be curved in space? That means that space needs space to be curved, which I can't understand.
The curvature of spacetime is distances and times that do not match Euclidian geometry (sometimes called flat space.) In flat space, if you walk around a rectangle, you end up back where you started. Or equivalently, if you walk around two sides of the rectangle, you get to the same place as if you walked around the other two sides.
The surface of the Earth is an example of a curved space. If you start on the equator and walk east 1/4 of the around the world, turn right and walk 1/4 of the way around the world, you are at the north pole. Turn right again and you are back where you started. Turn right again, and you are 1/4 of the way around the world from where you started.
Curved spacetime is similar, but the sides of the rectangle can be a time interval as well as a space interval.
For an example of curved spacetime, consider a result from general relativity: time travels slower in a gravitational well. That is clocks run slower on the surface of the earth than in outer space above the earth. The effect is small, but measurable.
So consider a rectangle that starts in outer space. One side is a distance of 10 miles toward the earth. Another side is a wait of 10 seconds. To travel around the rectangle one way, you wait 10 seconds, and then drop 10 miles. To travel the other way, you drop 10 miles and then wait 10 seconds. But time travels slower if you drop first. So you wind up at the same place, but at different times. This does not happen in flat space.
General relativity relies on four dimensions to model nature. Three of them are the usual space dimensions and the fourth is time. The verb "moves" implies a difference in three dimensional space with respect to time so it is not good to use in the four dimensional space time.
Curvature though is defined mathematically and can be drawn for any variables, as the change in one variable with respect to another. That is why in general relativity one talks of curvature of space time , and not of motion. It is a general term that includes both space and time.
General relativity's equation correlates space-time with the energy/masses included in that space time in the famous equation seen here.
The equation means that if there are no masses/energy space time is completely flat. Assume there are only two space dimensions and a stretched membrane describes them:
Einstein's theory of general relativity predicts that massive objects warp the space-time around them. NASA's Gravity Probe B found that the space-time around Earth is indeed curved by our planet, and twisted by its rotation. (Image: © NASA)
So in projected dimensions, one can define the usual "motion" as dx/dt, but the functional map of spacetime would "move" if the masses/energy change, changing the surface's curvature.
This is an easy to understand video.
Yes spacetime can move. Distant Patches of the universe, ( spacetime events) are moving away from us. Space itself can move when for example when it's being dragged by a massive rotating body which also causes a wavlike movement in space time which can be detected (gravitational waves). Space itself, not just spacetime can be curved in space. For example the Earth produces very little space curvature, only when you include the dimension of time do you get curvature. But around more massive bodies space can be curved so much so that you see you back yust by looking in front. If you measured the angles of a triangle in such a space you wouldn't get 180 degrees but if you moved further away in space space would straighten out again.
Spacetime does move and Einstein recognized this. As stars and planets move, the spacetime surrounding them also moves. In the simplest form, think of boats moving along in the water. And then think about eddies in the water, which are nothing more than water moving inside water. The movement of stars and planets creates moving wells in spacetime where the planet is actually located, along with larger waves and smaller ripples. You can consider that gravitational waves from colliding black holes as moving spacetime.
I was just reading about this in The Fabric of the Cosmos just a few days ago.
Edit: To quote directly from the book above: "An essential lesson of general relativity is that mass and energy cause the fabric of spacetime to warp. We illustrated this in figure 3.10 by showing the curved environment surrounding the sun.... General relativity predicts that, just as a trampoline assumes a fixed, warped shape if you stand perfectly still, but heaves when you jump up and down, space can assume a fixed warped shape if matter is perfectly still, but ripples undulate through its fabric when matter moves to and fro. Einstein came to this realization between 1916 and 1918 when he used the newly fashioned equations of general relativity to show that - much as electric charges racing up and down a broadcast antenna produce electromagnetic waves,... matter racing this way and that (as in a supernova explosion) produces gravitational waves. Just as tossing a pebble in a pond produces outward spreading ripples, gyrating matter produces outward spreading spatial ripples. ... an important distinguishing feature of gravitational waves: unlike electromagnetic, sound and water waves - waves that travel through space - gravitational waves travel within space. They are traveling distortions in the geometry of space itself."
I hope this helps.