# Time dilation measurement and events

1. Jul 16, 2014

### fog37

Hello Forum,

An event is a point in spacetime with spatial coordinates and a time coordinate: (x,y,z,t). An event does not have a duration since it only lasts for an instant t. We can talk about time duration to mean the temporal separation between two different events, correct?

In the time dilation phenomenon of special relativity, one inertial observer see the time interval separating two events in another inertial frame of reference to be stretched (and vice versa since time delation is symmetric). Is the time interval called "proper time" the amount of time measured by the observer that is stationary with respect to the two events? What does it mean that an observer is stationary with respect to the two events? Events don't move or do they?

How does the moving observer manage to remotely measure the time dilation that occurs in the other reference frame which appears to be moving relative to his? Does it use light signals?

thanks,
fog37

2. Jul 16, 2014

### Staff: Mentor

Correct.

Yes.

"Stretched" may not be the best term. It's a good idea in these cases to give a specific scenario, using mathematical descriptions if at all possible (i.e., give the events of interest and their coordinates in some inertial frame), to avoid the ambiguities that arise when trying to describe physics using ordinary language.

Well, you picked the word "stationary"--what did you mean by it?

The proper time between two events is the time that would be measured by an observer for whom both events occur at the same point in space; i.e., the observer's worldline (the curve he follows through spacetime) passes through both events. (We are assuming that the observer is inertial, i.e., he is weightless, in free fall, feeling no acceleration.) That means there will be some inertial coordinate system in which the observer's spatial coordinates (x, y, z) are constant (and we can, without loss of generality, assume that x = y = z = 0, so the observer is at the spatial origin of the coordinates). In this coordinate system, the proper time between the two events is the same as the coordinate time t between them.

They don't.

If you mean "measure" as in "actually observe the time registered by a clock in motion relative to him", then of course he is going to have to get that information via light signals, since that's how he will see the time registered by the moving clock. He will also have to have some way of measuring how far away the moving clock is from him when it is showing a particular reading. He can use this information to determine what time, in his own coordinates, to assign to any event; by comparing the times between events by his coordinates, with the times actually shown by the moving clock at those same events, he can calculate the time dilation factor.

If you mean "measure" as in "calculate the coordinate values of particular events in the moving frame", then he can do that using the Lorentz transformation equations for any events whose coordinates he knows in his own frame, if he knows the relative velocity between the two frames (i.e., how fast an object at rest in the moving frame is moving, relative to him).

3. Jul 16, 2014

### bcrowell

Staff Emeritus
Call the events A and B. Then the proper time interval between A and B is the time measured on an inertially moving clock that was *present* at both events.

When we talk about a frame of reference described by coordinates (t,x), you can think of this as implying that someone has carried out a vast surveying project, placing a network of clocks throughout all of space and synchronizing them using some technique. One possible technique for synchronization is Einstein synchronization, which involves sending signals back and forth using light. Another technique is simply to transport the clocks slowly enough so that they don't get out of sync.

4. Jul 16, 2014

### Staff: Mentor

To be precise, the time duration between the two events is the difference between the $t$ coordinates of the two events. Because different observers assign different $(x,y,z,t)$ values to the same events, they can easily come up with different $t$ coordinates for the two events, and hence different times between the two events.

Events do not move. The proper time between two events is defined to be $\tau=\sqrt{\Delta{t}^2-Delta{x}^2-Delta{y}^2-Delta{z}^2}$ (where $\Delta{x}$ is the difference between the values for the $x$ coordinate, and so forth). Remarkably, this value will be the same for all observers even though they assign different coordinates to the two events.

The proper time is equal to what you're calling the "time duration" between between the two events only if $\Delta{x}=\Delta{y}=\Delta{z}=0$. That's true only if the spatial coordinates of the two events are the same, which is to say they happen at different times but at the same place relative to the observer.

In principle, you just look at the other clock with a powerful telescope and then correct for light travel time. For example, if at 12:37 PM you look through your telescope and see the moving clock reading 2:45 PM at a distance of five light-minutes... You know that you're seeing an image of the moving clock formed by light left the moving clock five minutes ago, so the moving clock read 2:45 PM at the same time that your clock read 12:32. It takes two such measurements to observe time dilation.

In practice, such clocks and telescopes are completely impossible, so we look for some system that changes at known rate (for example, the decay of short-lived particles - google for "time dilation muon"), see if it appears to take longer when the system is in motion relative to us.

Last edited: Jul 17, 2014
5. Jul 17, 2014

### ghwellsjr

I think it would be more meaningful to limit our discussion to events that happen to a clock and then we can talk about the events associated with the "ticking" (regular intervals) of the clock.

We have to be clear here: observers cannot actually see Time Dilation. What they can see is what is known as Relativistic Doppler and for observers moving away from each other it is a bigger factor than Time Dilation but it is also symmetric.

For example, let's suppose that two observers are moving away from each other at 0.6c. Here is a spacetime diagram showing the rest frame of the blue observer as he sees the one-nanosecond ticks of the red observer's clock. The ticks are indicated by the dots and the images of them are transmitted along the thin red lines:

The Time Dilation factor at 0.6c is 1.25 and you can see that in this frame, the dots for the red observer are spaced farther apart than the Coordinate Time by the factor of 1.25. However, the blue observer cannot see the distant coordinates, he can only see when the images of each tick reach his eyes and they are spaced by a factor of 2. In other words, 2 nanoseconds go by on his clock for every one that he sees on the red observers clock.

Now the same thing is happening for the red observer as he watches the blue observer's clock. Here is a spacetime diagram in the same frame but showing the images of the one-nanosecond ticks as they are transmitted from the blue observer to the red observer:

Even though this is the rest frame for the blue observer, it still correctly shows the symmetry of Relativistic Doppler as the red observer sees the blue observer's clock tick once for every two of his own.

Note that in all of these diagrams, the speed of light is 1 foot per nanosecond and travels along the 45-degree diagonals.

This is why I suggested that we limit our discussion to events that happen to a clock because even though the events cannot be said to be stationary, we can meaningfully talk about the blue observer's clock being stationary in the blue observer's rest frame and the Proper Time intervals on the blue clock are coincident with the Coordinate Time intervals. We can also meaningfully talk about the Time Dilation of the red observer's clock because it is moving in the above frames.

Yes, either observer can use light signals and the assumption that they travel at c (1 foot per nanosecond) to determine the Time Dilation of the other observer's clock. Here's how they do it:

Each observer sends a light signal to the other observer which reflects off of him and returns. The observer sending and receiving the light keeps track of when he sent it according to his own clock and when he received the reflection as well as the observed time on the other ones clock. He then assumes that the light traveled at c both going and coming and averages the two times to get the time at which the image of the other observer's clock was sent according to his rest frame.

Finally, he has to repeat the process which gives him a pair of differences for his own clock and for the other observer's clock. Dividing these gives him the Time Dilation factor based on his own reference frame.

Here is a spacetime diagram showing the light signals that the blue observer reflects off of the red observer:

At the blue observer's time of zero, he sends the first light signal to the red observer which reflects back to him at his time of 5 nanoseconds along with the image of the red observer's clock at 2 nanoseconds. The average of 0 and 5 is 2.5 nanoseconds at which time he assumes the red observer's clock displayed 2 nanoseconds.

Next, the blue observer sends a second light signal at his time of 1 nanosecond and receives the reflection at his time of 9 nanoseconds which averages to 5 nanoseconds. He sees the blue observer's clock at 4 nanoseconds.

Now he takes his own calculated time difference, 5-2.5=2.5 and divides that by the observed difference in the red observer's clock, 4-2=2, to get 2.5/2=1.25 as the Time Dilation Factor.

Now since this is symmetrical, here is a spacetime diagram showing how the red observer determines the Time Dilation of the blue observer's clock according to his own rest frame, even though the diagram is for the rest frame of the blue observer:

For the first light signal, the red observer takes the average of 0 and 4 which is 2 corresponding to the blue observer's time of 1. Then he takes the average of 1 and 8 which is 4.5 corresponding to the blue observer's time of 3. Finally, he takes the differences of 4.5-2=2.5 and 3-1=2 and dividing these yields 2.5/2=1.25 as the Time Dilation Factor.
You're welcome, assuming that you still want to offer me thanks. Hopefully it all makes sense to you. If not, just ask.

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6. Jul 17, 2014

### fog37

Thanks Ben.

1) Like you say, a frame of reference is a network of synchronized clocks positioned at specific locations (x,y,z) away from a location designated as the origin O (0,0,0). We can place the origin anywhere.
If two reference frames are both inertial but their origins O are not the same or if the directions of the coordinate axes x,y,z are not the same, then the two reference frames constitute two different frames, correct?

A frame of reference is also a different thing from a coordinate system: we can pick the origin O and Cartesian, spherical, cylindrical coordinates to specify the locations of the various clocks. Clearly, a frame of reference without a coordinate system is not useful but the two do not coincide.

2) A clock records the time t associated to that event. To do so, the clock must be instantaneously present at the location of the event. For example, an observer measures the duration of soccer game as the difference between the end (event B) and the beginning (event A). The two events happen at the same location in space but different times. Proper time is that time duration. The observer can be moving or not. What matters, as you say, is that his clock is instantaneously there at the location of the event to measure it.

The motion of an object in spacetime is not an event but I guess we can see the object as a collection of points. The position and time of each point mass represents an event. Is that correct?

Thanks
Fog37

7. Jul 17, 2014

### Staff: Mentor

Yes. There's even a word for this: If you have a point particle, the set of all events "the particle was at the point (x,y,z) at time t" forms a line called a "world line", which will be straight if the particle is moving inertially and more or less interestingly curved if it is undergoing acceleration.

8. Jul 17, 2014

### fog37

Thanks Nugatory.

So the motion of a particle is described by its worldline in spacetime. An extended object is a collection of point particles but if we are not interest in changes of orientation of the object, etc. we can describe the motion of the extended object in terms of the motion of one of its particles (or center of mass even if a physical particle may not exist there)....