What is the proper time interval in special relativity?

[Rasalhague]

In Tipler & Mosca: Physics for Scientists and Engineers, e5, extended edition (page R-14 of the supplementary section on special relativity), there is a question:

“You are standing on a corner and a friend is driving past in an automobile. Each of you is wearing a wrist watch. Both of you note the times when the car passes two different intersections and determine from your watch readings the time that elapses between the two events. Which of you has determined the proper time interval.”

My answer: the friend. The two events (the car passing each intersection) happen in the same place (at the same spatial coordinates) in a reference frame in which the car is at rest, namely the location of the car. And the friend's watch is at rest with respect to the friend’s car in this frame.

Book’s answer: “Neither of you has determined the proper time interval. By definition the proper time interval is measured by a clock in the rest frame of the car, that is by a clock in the car.”

But assuming that an automobile is a car, and a wrist watch is a clock, and the friend’s wrist is attached to the friend, why would that not count as a clock in the car? Also, could a clock still be said to measure the proper time interval between the two events even if it wasn’t in the car, so long as it was in the same inertial reference frame, and that the delay in information from the events reaching the clock was taken into account? Earlier (p. R-6), Tipler & Mosca define the proper time interval between two events as “the time between events 1 and 2 in a frame of reference in which the two events occur at the same location”.

 

[Nabeshin]

I agree with you on this one. The observer in the car should measure the proper time between the two events.

 

[Fredrik]

A clock measures the proper time of the curve in Minkowski space that represents its motion. (Note that there are many curves connecting any two points).

If it's attached to your friend, and your friend is in the car, then the clock is in the car too.

I'm not a big fan of the phrase "...it was in the same inertial reference frame". All objects are always present in all inertial frames, but they are only at rest in some of them, so you should say something like "at rest in the car's rest frame". If the clock isn't in the car, it's measuring the proper time of the wrong curve, but the result may or may not be the same, depending on what the clock's world line looks like.

 

[Rasalhague]

Thanks, both, for your replies. I see what you mean, Fredrik: "in the same inertial reference frame" was careless phrasing on my part. If the clock was not in the car, would it still measure the proper time interval between the two events provided that it was at rest in the car's rest frame? And is that equivalent to saying that the clock's world line would have to differ from that of the car only by a translation through space?

 

[Fredrik]

If the only difference between their world lines is a translation by some four-vector, then the proper time of both world lines is the same. In this case, saying that the clock's velocity is the same as the car's, is equivalent to saying that their world lines only differ by a translation.

 

[Rasalhague]

Again, thanks. That's very helpful.