Measuring time with respect to reference frame

Question

Let us suppose a person $P$ is standing on earth. A car with velocity $v$(close to the speed of light) went past him. If the person's watch showed $t$ time and if the car measured $t'$ time,then which one will be the proper time?

I have two explanations and each one of them gives different result. First of all the watch was on the person's hand. So the watch was at rest in the person's referencr frame. So $t$ should be the proper time and the equation should be $t'=\frac{t}{\gamma}$.

Another one is that the car's clock was also at rest in the car's frame,so the time showed by the car will be the proper time and hence the equation should be $t=\frac{t'}{\gamma}$.

Maybe i haven't understood the concept of special relativity yet or maybe i haven't realised figuring out events which may be important in this case. I think the event is clocks measuring the time. Please correct me the place where i am making the mistake and fix my concept.

Answer 1

The general rule in special relativity is that you can use the time dilation formula to compare the times between two events where those events happen in one place in one inertial frame and in two places in another inertial frame. The time difference is always shorter in the frame in which the events happen in one place.

In your example, there is only one event, namely the car passing the person, so you can't apply the time dilation formula to it at all.

To use theme dilation formula, you could have the car passing two people in different places, in which case the elapsed time measured by the car between passing the first person and passing the second, will be less than the elapsed time according to the people.

The alternative would be to have two cars in a line passing the person. In that case the reverse would apply- the time between the first car passing and the second, as recorded by the person, would be less than the difference in time according to clocks on the cars.

The time dilation effect is entirely symmetrical between two interval reference frames.