Space-Time Interval 

Consider two events that are close together in space-time. 

Just as we can compute the distance between two points in space, dl, we can also compute the distance between two nearby events in space-time. 

The important fact about the space-time distance, called the space-time interval, is that it is invariant; that is, it is measured to be the same by all observers and its value does not depend on how we choose to label the events. 

The space-time interval ds is defined by

where c is the speed of light and dt is the difference between the time labels at the two nearby events. 

The difference dt is called the coordinate time difference to remind us of the fact that its value depends on how we choose to label the events, in contrast to the interval ds which does not. 

Notice something peculiar about the interval. Its square can be positive, zero or negative! 
If ds2 < 0 the two nearby events are said to separated by a time-like interval. 

If ds2 > 0 the two nearby events are said to be separated by a space-like interval. 

If ds2 = 0 the two events are said to be separated by a light-like interval. 

Consider a space-time interval with dl = 0. In this case What does this mean? This means that the two nearby events are actually at the same spatial point, but are separated in time by an amount dt. 

The time measured at any given point is called the proper time.  The difference dt is the proper time that has elapsed between the two events. It is the time measured by a clock at that point. 

The elapsed proper time is just |ds2|1/2/c. 

Now consider setting dt = 0. In this case

This shows that when two events have the same time, that is, when they are simultaneous the spatial distance between them is invariant, that is, measured to be the same by all observers. This distance is called the proper distance between the two points.

What about the proper time of observers who are moving relative to the coordinate system?  How do we compute their proper times? Let's start with the formula for the space-time interval

After a little bit of algebra we get  The following 

dt = |ds2|1/2/c

is the expression for the proper time difference not only for observers at rest in the coordinate system but for all observers, however they move,  

provided that they move only along time-like world-lines.