# Why is the time interval between two events measured by two synchronised clocks seperated by a distance not proper?

Suppose that in an inertial frame two events A and B occur at two different places. If all the clocks are synchronised then why is the time interval (difference between the time of event A registered by clock A and the time of event B registered by clock B) not proper time interval? Is it only because the readings are taken from two different clocks at different position?

• What do you take to be the meaning of a proper time interval? In SR, I'd take it to be the time between events as measured in a frame of reference in which the events occur in the same place. So your proposal is wrong by definition, with no subtlety involved. You should now be able to inhale once more. – Philip Wood Oct 5 '17 at 9:01
• I note that you are using Resnick 's $Introduction\ to\ SR$. He defines (on p 63 in my edition) the proper time interval as the time interval recorded by a clock attached to the observed body. This is equivalent to the definition I gave above, provided that the body is not accelerating. So I repeat my claim that your proposal is wrong because it goes against the very definition of proper time. John Rennie's answer below rightly points out that the $significance$ of proper time goes beyond this simple definition (much as the constant $c$ is about much more than the speed of light). – Philip Wood Oct 5 '17 at 14:55
• yes,I did read that on Professor Resnick's Introduction to SR. But I didn't understand why we need to define proper time in this way, and that is because I didn't have very clear idea about space-time interval. But now I got very satisfactory answers, specially from Professor John Rennie, and Safesphere's answer was also very helpfull. – sid Oct 5 '17 at 17:20

The proper time is the length of a world line between two points in spacetime. It is calculated using the metric, which in special relativity is the Minkowski metric:

$$c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2$$

We often define a proper distance instead and this is related to the proper time by $ds^2 = -c^2d\tau^2$. Typically we use the proper time for timelike paths and the proper length for spacelike paths to avoid ending up having to square root a negative number.

The proper time depends on the world line. For any two points $A$ and $B$ there are an infinite number of paths that connect those two points, and those paths will in general have different proper times. However in introductory SR courses we frequently only consider straight lines joining the two points, and in that case for the two points:

\begin{align} A &= (t, x, y, z) \\ B &= (t+\Delta t, x+\Delta x, y+\Delta y, z+\Delta z) \end{align}

the proper time for a straight line joining $A$ and $B$ is simply:

$$c^2\Delta\tau^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 \tag{1}$$

Suppose in your example the two event are in the same place i.e. $A$ and $B$ are the same point. Then $\Delta x = \Delta y = \Delta z = 0$ so the proper time calculated using equation (1) is:

$$\Delta\tau = \Delta t$$

So in this case the proper time is equal to the time interval between the events i.e. the proper time $\Delta \tau$ and the coordinate time $\Delta t$ are the same. But if the two events have a spatial separation the proper time and the coordinate time will be different.

The significance of the proper time is that it is an invariant i.e. all observers in all frames of reference will agree on the value of $\Delta \tau$. However different observers will disagree about the value of $\Delta t$.

If you're interested in learning more bout this an excellent place to start would be right here on this site. A search will find you lots of related questions and answers. This is a particular hobby horse of mine, so starting with my posts on the subject would be an option.

• Sir, is there any difference between proper time interval and space-time interval? I think both are the same thing with different name. Please, Correct me if I'm wrong. – sid Oct 7 '17 at 14:18
• @sid: the terms proper time, $d\tau$, and proper distance, $ds$, tend to be used interchangeably. They are related by $ds^2=-c^2d\tau^2$. The proper distance is what I would usually call the interval, but as you say they are basically all the same thing. – John Rennie Oct 9 '17 at 4:33
• Hmmm...Got it Sir :) – sid Oct 9 '17 at 10:05

Proper time is your time measured by your clock. It is the time passed in your frame of reference regardless of how you moved. So the proper time is local to the observer. There is no such thing as "proper time between you and me" or "proper time at a distance" or "proper time between two separate objects".

A proper time interval is the distance between two events on a trajectory of an object in spacetime. If you have two objects, each would have its own trajectory and they would be separated. If you measure the distance between one event on one trajectory and another event on another trajectory, it is easy to imagine that the line connecting these points would not represent an actual spacetime trajectory of any object.

• Yes,though your answer is very precise clear and highly appreciable, but one thing you need to understand that I'm a beginner in SR, and my study material is only a book of Professor Robert Resnick. The understanding of space-time, invariace of the interval etc. are not very clear to me.So, One more thing I want to ask(may be that is irrelevent in this discussion). Can you please suggest me some study material/books where I can get such arranged topic wise discussion about space-time interval? – sid Oct 5 '17 at 9:44
• @sid Special Relativity can be explained at very different levels, so I cannot pick a reference, sorry. Just google and read what you understand. Instead, let me explain the interval. Take a sheet is paper and draw coordinates, time and distance. A straight line parallel to the time axis is for a non moving object that only moves in time. An angled straight line is a constant speed movement. A curved line is a movement with acceleration. All these lines are spacetime trajectories. Yet none of these objects move in their own frame, except in time. So every trajectory is the time axis for [...] – safesphere Oct 5 '17 at 15:24
• @sid [...] that observer. These lines are called world lines. Because in your own frame you move only in time, the distance measured along your world line is simply time measured by your wristwatch. It is called your proper time. Essentially, your spacetime trajectory is the time axis in your frame of reference. Straight lines - special relativity; curved lines (acceleration) - general relativity. Interval in general is the distance on this diagram between any two events, but it is only "proper", if both are on the same world line. Good luck and please check the answer that helps you most :) – safesphere Oct 6 '17 at 4:38