Question

Can someone explain me carefully what Coordinate Time, Proper Time and Spacetime Interval are (eventual incl. an example)?

Answer 1

Let me give you a mere picture - an approximate view on these three.
Proper Time: It is just an interval of two moments. For example, water with 0 C and making it to boil upto 100C. you are trying to measure how long it took. This measurement is proper time - Here you are measuring a single event. In day to day life we are using this only.

Co-ordinate Time, let there are two events going on, one in US and another one is going on in Europe. So you want to measure the time here. so here you need to sync the clock first and then to measure the fact. so a co-ordinate time usually comes with where the place you are measuring it(co-ordinates) to assure simultaneity. This is mostly used in space with relativity.

Space-Time: Just go to a overview without thinking of Proper Time and Coordinate time. Eg. the big-bang is going in space. Snap shot of a moment where you are standing in a second. on the next second the all the things changed - expanded, so the next moment you are in another space-time. The difference between the two moments how you measure? Previous moment things are in one alignment and on the next moment everything in another alignment. The difference between the two moment is called space time. In a moment we will be in a point and on the next moment we are going to another scene. you can imagine a space-time only you see the world from outside of everything.

Answer 2

Thanks greatleo!