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For example, if an observer in the rest frame of a room measures an explosion on one side of the room and then at a time $\Delta t$ later observes another explosion on the opposite side of the room, is this time interval $\Delta t$ the proper time? I know that the length of the room measured by the observer in the room will be the proper length as they are measuring the length in the same frame but I am unsure of the time.

So far, proper time has been explained to me as the time interval between two events that happen at the same point. However, I am confused as to what this means. Does this mean the two events have to happen at the same point in the frame or does it mean that the observer recording the time interval between two events in the frame has to record them at the same point. I say this because the way I see it is that the events could be the observer observing the explosions and these observations happen at the same point in space.

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Suppose we have two events $(t_1, x_1)$ and $(t_2, x_2)$, where $t$ denotes the time the event happened and $x$ denotes the place the event happened, then the proper time, $\tau$, between the points is given by:

$$ c^2 \tau^2 = c^2 (t_1 - t_2)^2 - (x_1 - x_2)^2 $$

It is convenient to write $\Delta t = t_1 - t_2$ and $\Delta x = x_1 - x_2$ as this simplifies the equation to:

$$ c^2\tau^2 = c^2 \Delta t^2 - \Delta x^2 $$

This should immediately remind you of Pythagoras' theorem, and indeed the proper time is the length of the straight line joining the two points except that in special relativity the equation for the length has a minus sign where Pythagoras' theorem has a plus sign. This minus sign turns out to be responsible for the weird effects like time dilation and length contraction.

If you want a physical interpretation of the proper time then it is the time recorded by a clock that travels in a straight line at constant speed between the two events.

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As is often the case when questions about special relativity are at issue, you need to be really really clear to avoid any ambiguities.

Yes, if you have an observer who sees any two events A and B, the time between the two observations can be a proper time. But to be really really clear, the time between the two observations of A and B isn't necessarily the same as the time between the events A and B- you need to distinguish between an original event and the observation of it which is a separate event.

As John Rennie said, a proper time interval between two events is one measured by a single clock that's present at both of them. Another way of saying the same thing is that the proper time between two events is the time that passes in a frame in which the two events happen in the same place. So it is possible to have two events that are spatially separated- like two lightning strikes at either end of a platform (to use a classic Einsteinism)- so that the time between those two events in the frame of the platform is not a proper time. However, the arrival of the light from each flash at the eye of the observer are different events altogether, and the time between them can be a proper time.

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Three observers standing in a room will have their clocks synchronised, they will see that their private clocks run at the same rate as the other two. If two of them snap their fingers simultaneously then all three will agree that the snaps were simultaneous, after allowing for the speed of light (they don't hear but see the snaps). If one observer snaps his finger one second after another then all three will agree that there was a one second gap. But if they start tossing a clock at each other then that clock will show a slower time compared to the clocks that stay with them. Comoving clocks show each other's proper time. Hope it answers your question

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