Thank you guys. That is very helpful. I still have some questions though. I guess I am gonna need more time to absorb the concept here :)
So whenever [tex]\Delta x=0[/tex] in some reference frame,
[tex](\Delta s)^2=(c\Delta \tau )^2=(c\Delta t)^2[/tex]
Is this always true? at least in the context of special relativity?
Also, in a problem I found it says suppose there are two fire crackers. And we put a clock in between the two firecrackers, but not half way. The two firecrackers explode.
So, my thinking is: in the reference frame of the clock, where the two firecrackers are stationery to the clock, the time the clock will measure between the two events (explosion of the two fire crackers) is the coordinate time and also proper time since [tex]\Delta x=0[/tex]
However, if we supposed the fire crackers along with the clocks were moving with a constant velocity with respect to another ref frame, then in that frame the time the previous clock will measure is the coordinate time but not the proper time because [tex]\Delta x\neq 0[/tex] between the two events (clock receives light from one explosion then after some time receives the light from the other firecracker)
Is my understanding of the problem correct? or no? Also, can we safely say proper time is a special case of coordinate time? And space time interval is a special case of proper time?
@pervect: We have not actually studied the spacelike and timelike intervals yet. But may I ask you, if you know or you know a good link, who came up first with the idea that if [tex]c\Delta t[/tex] is used, then a deep quantity such as spacetime will show up?
Thank you again