My textbook says roughly the following in then context of the invariant spacetime interval:
The proper time $\Delta \tau$ between two events is the time interval as observed in a coordinate system where the two events take place at the same position. This is not always possible. The invariant interval between two events equals c times the proper time, or $c\Delta \tau= \Delta S^2$. The proper time is the time interval measured by an observer who doesn't move relative to this position. The proper time is the time as for the observer himself; it is his own time. If the interval is positive, there will always be a coordinate system where the position of the two events is the same. This is true because a positive interval implies $|c\Delta t|>|\Delta r|$, so a reference frame that moves with velocity $\vec{v} = (\Delta \vec{r})/(\Delta t)$ will transform the events to the same position. And this speed is always smaller than the speed of light.
Alright, I understand that a positive spacetime interval implies that we could find an observer that measures two events at the same position, because even thought $\Delta x= 0$, we still have a positive time-interval $c\Delta t$. It makes sense that a time-interval is positive, so this seems to be about right to me.
However, I don't see how they come up with $\vec{v} = (\Delta \vec{r})/(\Delta t)$. It kind of makes sense intuitively, because I imagine following travelling from the first event to the second, so that they happen where you're at... but I can't connect the mathematics behind this intuition.
Can someone help me? Thanks!
