# Quick time dilation question (Proper time)

This is super frustrating. I literally cannot grasp the concept of what proper time is and I have no idea why. There's something I'm missing here.

If someone is standing on earth and there's a clock next to him in the same reference frame, and then there's someone in a spaceship traveling at $0.5c$ away from the earth with another clock in the frame of the ship, then from the view of the earth the clock on the ship will run slow. I get that.

But who measures proper time? I don't get it at all. The definition given everywhere makes no sense to me. My book says that the proper time interval is "the time interval between two events measured by an observer who sees the events occur at the same point in space." At what same point in space? For whom? What does this even mean at all? Can both the observer on earth and the observer in the space ship both have a proper time? By the time dilation formula it's obvious that it refers to the minimum time that's possible I guess. Would one the proper time in the ship be the improper time on earth? Would the proper time on earth be the improper time on the ship?

I'm literally hitting my desk just not being able to understand this and not getting a clear answer from anywhere.

• One source of confusion is that the "proper" in proper time has the same root as "proprietary", but this was lost in the translation to English. It is the time that is "owned" by a specific clock (the clock that is at both events). It has nothing to do with the idea that "the Victorian age was a very proper time." – DMPalmer Apr 18 '17 at 14:37

In the scenario you present, only the observer holding the clock can measure the elapse of its proper time. The moving observer observes his own proper time elapse by looking at the clock that he is holding. Each can calculate the proper time observed by the other, but they observe the proper time of their own clocks only.

• So, let's say I'm on earth and I see the space ship travel 2 kilometers. I see that the time (that I measured on earth from my own clock) it did this was in 0.2 seconds. If I wanted to find how long the guy in the ship measured that he flew the 2 km (which would be LESS than what the guy on earth saw, I think), could I say that the improper time for the guy in the ship was that 2 km in 2 seconds that the guy on earth saw? – hhh Apr 18 '17 at 12:09
• @Locrian7 This is really a comment on your comment and not on the answer. Have never seen the term "improper time" and it doesn't make sense to use it. Proper time comes from a (not so good) direct translation of the french "temps propre" which actually translates to english as: Own time (I believe this would have avoided much confusion). – PML Apr 18 '17 at 13:31
• @Locrian7 please also keep in mind the relativity of simultaneity. When you try to compare two proper times measured by different observers the answer is: generally you can't do this because simultaneity is relative. – Prof. Legolasov Apr 22 '17 at 8:42

A geometric interpretation might be helpful.

On a spacetime diagram, proper-time is the [arc-]length of along a worldline between two timelike-related events (like the arc-length of a line or a curve in Euclidean space). Physically, it is measured by a wristwatch worn by the observer who experiences that worldline. In the reference frame of that observer, he is "in the same place"---that is, he will assign the same spatial coordinates to his location.

In the clock effect, between separation (event A) and reunion (event Z), two observers take different worldlines from A to Z. Each worldline has its own proper time... and in general, the proper-time measured along each worldline is different.

Between timelike-related events A and Z, the worldline with the longest proper-time is that of the inertial observer from A to Z. (I think some books mistakenly refer to this as "the proper time".)

With knowledge of Alice's worldline path from A to Z, another observer Bob (who might not have experienced events A or Z) can in principle indirectly measure the proper-time measured by Alice using measurements Bob makes, together with some calculation. However, this is not generally equal to the elapsed time between events A and Z according to Bob.

The proper time that elapses for a person is very simple: it's whatever the watch on their wrist says.

If you stay on Earth and I go on a rocket ship to the moon at $.9c$ and back, then I will have aged less than you. I will have experienced less time, had to sleep less, had to eat less, my watch didn't tick over as many times, etc. However much time my watch says has passed is the proper time that I experienced between setting out to the moon, and landing back on Earth. You, having stayed on earth, experienced a larger amount of proper time, waiting there by the launch pad. Whatever your watch says is the proper time that you experienced between me setting out to the moon and landing back on Earth. Each observer experiences their own amount of time (called proper time) between any pair of events.

Maybe this helps to clarify the meaning of proper time.

Imagine you want to measure the separation of points (where x and t are just numbers):

$(x_1,t_1), (x_2,t_2)$

on a 2D plane. With an appropriate choice of the coordinate system you can always achieve that e.g. $x_1 = x_2$. So neither $x_1-x_2$ nor $t_1-t_2$ tells you what the actual distance between the points is.

What has significance is the quantity $(x_1 - x_2)^2 + (t_1-t_2)^2$, which is invariant under translations and rotations of the coordinate system.

Going to SR the situation is similar (now x is a distance and t has units of time), neither the difference in $t$ nor the difference in $x$ between two points have physical significance. But the coordinate transformations are different than in the previous example, they are translations and Lorentz-Transformations.

It turns out that the quantity that's left invariant by them is $-(x_1 - x_2)^2 + (c t_1-c t_2)^2$, as can be checked easily. This looks already very similar to the proper time, which you get by taking the root and pulling out c.

So the intuition is that proper time is the analogue of distance between two points from classical mechanics. Since it's invariant under coordinate transformations the different definitions you mentioned all agree.