Understanding Time Dilation in Special Relativity

[Sculptured]

Alright, I'm having a bit of trouble understanding time dilation. My biggest concern is in how an observer in a frame will see all other frames at different relative speeds moving slower. I've been trying to show it mathematically, but haven't had the luck yet.

 

[bernhard.rothenstein]

time dilation

Alright, I'm having a bit of trouble understanding time dilation. My biggest concern is in how an observer in a frame will see all other frames at different relative speeds moving slower. I've been trying to show it mathematically, but haven't had the luck yet.

To start off I'll go with the meaning of proper time first. Since it is the time measured from a frame (unprimed) in which the event occurs at the same position such as a ball thrown up returning to the same point. Therefore, only one frame can be of the proper time. Am I correct in this?

Next, I understand and can derive how time is longer for a moving (primed) frame looking at this event. What I'm lost with is when it comes to an observer in the primed frame looking at the unprimed frame in which the event occurred. To me it would look as though the unprimed frame should be moving faster. From what I've been told, the primed observer should view the unprimed observer as going slower as well. How is this derived? I'm pretty sure it's similar as going from the unprimed observer to the primed observer. Am I messing up a concept or is there a key thing I am missing in this?

I think the best thing is to start with a good understanding of Einstein's clock synchronization procedure which leads in the case of a single space dimension approach to
xx-cctt=x'x'-cct't' (1)
Consider that in I' a proper time interval is measured i.e. x'=0, x=Vt, with which (1) leads to
t=t'/sqrt(1-VV/cc). (2)
If you consider that the proper time interval is measured in I then x=0, x'=-Vt' and the final result is
t'=t/sqrt(1-VV/cc). (3)
I think that the use of "going slower" or of "going faster" is counterproductive.
Equations (1) and (2) establish a relationship between a proper time interval and a non-proper one (t-0) or (t'-0).
The formula which accounts for the Doppler shift establishes a relationship between two poper time intervals measured in I and in I' respectively.
The transformation equation for the time coordinates of the same event detected from I and I'
t=(t'+Vx')/sqrt(1-VV/cc)=t'(1+Vu')/sqrt(1-VV/cc) (4)
establishes a relationship between two non-proper (ccordinate) time intervals. Properly handled it accounts for time dilation and for the Doppler Effect as well.
If there are further questions please put them.

 

[Sculptured]

As usual, a mathematical representation has helped to see it from another angle. I've identified my troubles, stretching the concept out farther than it applies, and feel good about the concept. Thank you.

 

[country boy]

Here's another way to look at it: Make a clock by bouncing a light pulse back and forth along the y-axis between two mirrors separated by a distance D. The mirrors are at rest in your frame; their x, y positions are not changing. Say that each cycle of the light pulse takes one second of time in your frame.

Now consider another pair of mirrors, still separated by D in the y direction, but moving along x. Because the speed of light is the same and the diagonal distance is greater, you will measure a time longer than one second for each cycle between the moving mirrors.

However, an observer traveling with the moving clock will measure one second for each cycle, because he measures the same transverse distance D and the same speed of light. You therefore measure a dilation of time compared to what he measures. And it doesn't matter what direction he is moving (along x) or how fast. The diagonal distance gets longer when the clock is moving.

This applies equally well to the traveling observer. He measures the light pulse cycle time in his clock as one second. But looking back at your frame, which is moving with respect to him, he sees a diagonal distance traveled by the light pulse in your clock. That longer distance gives a time dilation in your frame as seen by him.

The use of a transverse length, along y in this example, is the key to comparing measurements among moving frames. Transverse lengths are agreed upon by all observers. [Note that the proper interval ds acts like a transverse length.]

 

[robphy]

The use of a transverse length, along y in this example, is the key to comparing measurements among moving frames. Transverse lengths are agreed upon by all observers.

While your method is fine [I've used it myself], I'd downgrade the use of the invariance of the transverse length from "the key" to merely "one key". If we restricted ourselves to a (1+1)-spacetime, we'd still have time-dilation [with the light clock]. We would have to appeal to some other invariances, e.g., the invariance of the square-interval [which at first glance may not be obvious to a new student] or the symmetry of the inertial observers and their doppler factors in a radar experiment (a la Bondi's k-calculus).

It's probably most accurate to say that it is the use of invariance that is key. In addition, it is the invariance of the square interval that is at the heart of all other invariances that one could have used instead.

[Note that the proper interval ds acts like a transverse length.]

As it stands [without more of an explanation why this may appear to be true], I'm not sure how useful this comment is in further developing more intuition for relativity.

 

[country boy]

robphy,
Thanks for the thoughtful response. Transverse length is not, of course the only key. Another is the universal speed of light. The example was intended to make time dilation easier to understand, and I hope it helps. Your use of (1+1)-spacetime makes it clear that the transformation properties derived from the transverse motion must apply also to the longitudinal motion. My example only treats t, but the Lorentz transformations affect x and t.

As for the proper interval, if you write the expression for ds in your (1+1) geometry and then replace ds with dy, you see that it is the expression for a light path in x, y space. If ds is regarded as a "hidden" transverse direction, much of the character of special relativity can be derived intuitively using light pulses. Energy, momentum and rest mass can also be studied in this faux space-time. The rest mass can even be quantized by closing the s-axis.

But, of course, this is just a thinking tool, right?

 

[bernhard.rothenstein]

time dilation

Here's another way to look at it: Make a clock by bouncing a light pulse back and forth along the y-axis between two mirrors separated by a distance D. The mirrors are at rest in your frame; their x, y positions are not changing. Say that each cycle of the light pulse takes one second of time in your frame.

Now consider another pair of mirrors, still separated by D in the y direction, but moving along x. Because the speed of light is the same and the diagonal distance is greater, you will measure a time longer than one second for each cycle between the moving mirrors.

However, an observer traveling with the moving clock will measure one second for each cycle, because he measures the same transverse distance D and the same speed of light. You therefore measure a dilation of time compared to what he measures. And it doesn't matter what direction he is moving (along x) or how fast. The diagonal distance gets longer when the clock is moving.

This applies equally well to the traveling observer. He measures the light pulse cycle time in his clock as one second. But looking back at your frame, which is moving with respect to him, he sees a diagonal distance traveled by the light pulse in your clock. That longer distance gives a time dilation in your frame as seen by him.

The use of a transverse length, along y in this example, is the key to comparing measurements among moving frames. Transverse lengths are agreed upon by all observers. [Note that the proper interval ds acts like a transverse length.]

The light clocks in relative motion are largely used in teaching time dilation. Even if it is simple, involving Pythagoras' theorem, it obscures the fact that after a period we compare the reading t' of an usual clock C'(0,0) located on the lower mirror of the moving clock with the reading t of a clock C of the stationary reference frame located on the overlapped axes OX(O'X'), where the reflected ray intersects it as detected from the stationary reference frame. IMHO it is worth to mention that fact for a better understanding of the relationship between light clocks and usual clocks say wrist watches we use in many thought experiments.

 

[country boy]

The light clocks in relative motion are largely used in teaching time dilation. Even if it is simple, involving Pythagoras' theorem, it obscures the fact that after a period we compare the reading t' of an usual clock C'(0,0) located on the lower mirror of the moving clock with the reading t of a clock C of the stationary reference frame located on the overlapped axes OX(O'X'), where the reflected ray intersects it as detected from the stationary reference frame. IMHO it is worth to mention that fact for a better understanding of the relationship between light clocks and usual clocks say wrist watches we use in many thought experiments.

All of the measurements can be made with light clocks, so we should have no problem. As you say, though, this is just a teaching aid. Of course we need to to generalize to all clocks, including wristwatches. But we do assume that all valid methods of measuring time will give the same results, isn't that correct?

 

[bernhard.rothenstein]

time measurement

All of the measurements can be made with light clocks, so we should have no problem. As you say, though, this is just a teaching aid. Of course we need to to generalize to all clocks, including wristwatches. But we do assume that all valid methods of measuring time will give the same results, isn't that correct?

Thank you for your answer. If I understand correctly your question my answer is that if in both the involved inertial reference frames we perform the same synchronization procedure compatible with the two postulates then yes. Einstein's clock synchronization procedure satisfies that condition.

 

[country boy]

Thank you for your answer. If I understand correctly your question my answer is that if in both the involved inertial reference frames we perform the same synchronization procedure compatible with the two postulates then yes. Einstein's clock synchronization procedure satisfies that condition.

Yes, that's what I meant. I did leave out the synchronization of clocks to try to keep the description simple, but you are quite right in implying that synchronization is an important element.