Time Dilation and The Twin Paradox?

[JimiJams]

Hey everyone, I've searched the threads for an answer to my question but came up short. Basically what I want to know is how does time dilation relate to the twin paradox. I'm a newb to relativity and just started reading about it last night. I hit this bump and I'm trying to get over it.

The way I understand time dilation best is through the example of a light beam traveling straight up and down, and hence less distance, to an observer moving with the light emitting contraption. But to a still observer watching the "contraption pass by it forms a triangular type trajectory and hence travels more distance. Since light travels at a constant this implies that time is different in each frame.

So, can anyone explain to me how time dilation applies to the twin paradox using my understanding of time dilation. I have no clue why the person on the ship is experiencing a slow down of time.

Thanks a lot!

 

[ghwellsjr]

Scroll down to the bottom of this page and you will see a bunch of similar threads including some with almost the same title as yours. Read those threads and see if they answer your questions.

 

[JimiJams]

I'm still lost here. I think the OPs of those other threads had a bit more of a grasp than I do. Bear in mind my understanding of time dilation is defined by the example I gave in my original post.

Can anyone take the example and relate it to the twin paradox.

For example I'm guessing since the twin is near the speed of light that plays a factor. If he were traveling at a more realistic speed the time difference would be less, why?

I just need the basics broken down and explained so I can wrap my head around it and move on b/c I already find it fascinating.

I learn visually the best, so if you can explain in a visual manner that'll really help, but any help is appreciated.

 

[Matterwave]

If I have a digital watch which displayed my time, and you have an identical digital watch that displayed your time, and you go into a space-ship and fly away at high velocity, and I am here on Earth looking through a telescope at your digital watch and I looked at your watch tick, then I would find that even taking into account the fact that the light takes a finite amount of time to get from you and me, I would still find that your watch ticks slower than mine.

For example, if you were going at .5c, and I was looking at your watch, I would expect, without regard to time dilation, that to see your clock tick 1 second, I would need to wait 1.5 seconds simply because you are moving away from me and the light from your watch takes some time to get to me.

What I actually see is that I need to wait MORE than 1.5 seconds to see your clock tick 1 second. That is relativistic time dilation.

 

[JimiJams]

so are you saying that the delay that it takes for the twin on Earth to see the spaceship/watch has something to do with it? I understand time dilation I'm pretty sure. I don't think that needs to be explained in case that's what you were doing. I'm just having trouble applying it to this problem. I haven't reached that aha moment just yet.

And from a mathematical perspective I understand. It's basic plug and chug but I'm just trying to really get the concept. The book I've been reading gave a few examples and I understood them fine, then it threw this problem into the mix.

 

[Matterwave]

No, the time dilation effect is independent of the effect of the finite transmission speed of the signals, the time ACTUALLY dilates.

My point was only that there REALLY is a difference in time between the two observers, it's not some artifact of "seeing" being delayed due to the light. I thought that that might have been your confusion. Maybe I misinterpreted.

If you understand time-dilation, I'm not sure what you don't understand about the twin paradox. It's simply an application of the time-dilation.

 

[JimiJams]

Yeah I didn't think it was a simple delay in seeing the watch. If the twin paradox is a simple example of time dilation, though, is there any way you can explain it simply? I mean can you break it down into simple digestible pieces? Everything I read kind of says well the one experiences this time the other experiences his time, then the ship experiences an acceleration at the turning point creating a new inertial ref frame and then the tables turn as far as the time experience goes for each observer. I just want to know why. Like if a teacher during lecture, for instance, just finished teaching time dilation and moved into the twin paradox how would he tie them together, you know?

 

[Matterwave]

One twin stays on Earth, one twin goes off into space. From the point of view of the twin who stayed on Earth, his twin ages more slowly than he does. From the point of view of the twin who went off on the rocket, he sees his twin aging at a rate slower than him during his outgoing and returning journeys.

This shows the symmetric nature of relativity. I.e. there's no way to determine who is "actually moving" for inertial motion. There is only relative motion.

When the twin that was in the rocket returns to Earth, he finds that he is actually younger than his twin brother.

The "paradox" is asking why is the twin in the rocket wrong?

In other words, how can both twins see the OTHER twin as aging slower, but in the end the twin who is younger is the twin in the rocket ship?

The answer is that the twin in the rocket ship had to accelerate to stop and return. He is in 2 different inertial reference frames at the 2 different times, whereas the twin on Earth is in 1 inertial reference frame the whole time and so he is the "correct" one.

 

[JimiJams]

"From the point of view of the twin who stayed on Earth, his twin ages more slowly than he does. From the point of view of the twin who went off on the rocket, he sees his twin aging at a rate slower than him during his outgoing and returning journeys."

Yeah but why?

 

[jartsa]

We have found out that a light beam traveling up and down in a rocket that travels back and forth, makes a reduced number of up and down trips.

So we guess that a human being travelin in this same rocket makes a reduced number of visits to bathroom.

Or shall we make the human traveller to jump up and down the whole trip. The number of jumps during a trip to other galaxy can be for example two.

 

[Matterwave]

"From the point of view of the twin who stayed on Earth, his twin ages more slowly than he does. From the point of view of the twin who went off on the rocket, he sees his twin aging at a rate slower than him during his outgoing and returning journeys."

Yeah but why?

To one twin, it's the other twin which is moving. The situation, as long as nobody is accelerating, is symmetric.

 

[JimiJams]

Hey guys, I think you're assuming I now more than I do about all this. If you can explain using the utmost basics/fundamentals of time dilation or relativity that would help the most. Telling me "because he sees the clock moving slower" doesn't explain to me why he sees the clock moving slower.

Can anyone chime in and really explain how dilation is related to this and maybe explain dilation a bit more clearly to help aid in your explanation. Whatever it takes to really explain it because telling me he simply sees the clock moving slower fails to explain why. I mean I may be missing something about time dilation that's hindering me from grasping how it's applied to this twin paradox example.

 

[jartsa]

Hey guys, I think you're assuming I now more than I do about all this. If you can explain using the utmost basics/fundamentals of time dilation or relativity that would help the most. Telling me "because he sees the clock moving slower" doesn't explain to me why he sees the clock moving slower.

Can anyone chime in and really explain how dilation is related to this and maybe explain dilation a bit more clearly to help aid in your explanation. Whatever it takes to really explain it because telling me he simply sees the clock moving slower fails to explain why. I mean I may be missing something about time dilation that's hindering me from grasping how it's applied to this twin paradox example.

We all understand how Einstein light clock is dilated. We don't understand how Rolex quartz clock is dilated.

Or do we understand? Is it the increase of mass of the quartz crystall?

(a bouncing light beam is an Einstein light clock)

 

[harrylin]

Yeah I didn't think it was a simple delay in seeing the watch. If the twin paradox is a simple example of time dilation, though, is there any way you can explain it simply? I mean can you break it down into simple digestible pieces? Everything I read kind of says well the one experiences this time the other experiences his time, then the ship experiences an acceleration at the turning point creating a new inertial ref frame and then the tables turn as far as the time experience goes for each observer. I just want to know why. Like if a teacher during lecture, for instance, just finished teaching time dilation and moved into the twin paradox how would he tie them together, you know?

OK, I see... The original version of the twin paradox is a difficult topic, as it involves a discussion of the equivalence of gravitation and acceleration, based on a presumed understanding of special relativity. And even the modern version of the twin paradox is far beyond what you are asking, as it involves a discussion about inertial frames. It seems to me that what you are asking, is:

How does time dilation of clocks work, and how can it be mutual, so that each observer interprets what he sees as the other clock running slow.

The point of the light clock example is, that if the relativity principle holds (see for example https://simple.wikipedia.org/wiki/Principle_of_relativity), then someone who is at rest relative to the light clock should not see his watch tick faster than the light clock. Thus what is valid for the light clock must also be valid for his clock.
And the reason that the observed effect is mutual can be explained by the combined effects of relativity of simultaneity and length contraction. Most important is relativity of simultaneity, and this is often not understood or misunderstood. Thus you could search for discussions about that topic. Alternatively, if you know about clock synchronization you could work out for yourself how clocks will be synchronized in a "moving" system, and how that will result in a different perspective of observations.

PS: I see that the first, basic aspect is currently discussed in this thread:
https://www.physicsforums.com/showthread.php?t=610164

 

[JimiJams]

Hi Harrylin, thanks for the response. I suppose I kind of figured this out last night. It seems to me that this is a case where it's very difficult to describe with words. I think I was over-thinking things. Basically the only way it seems to tie the example of time dilation involving the light clock with the twin paradox is through mathematics.

In the book I was reading I understood the light clock example right away and then it pretty much jumped into the twin paradox. Without carefully looking at the derivation of the time dilation formula you end up asking why. But through the light clock example it must be realized that the derived time dilation formula can then be applied to just about any situation. I still can't relate the example visually to the twin paradox, but I have more faith in the formula and tonight I'll go through the problem and apply variables to each component of the twin paradox and then I may be able to relate it to the light clock example.

 

[harrylin]

Hi Harrylin, thanks for the response. I suppose I kind of figured this out last night. It seems to me that this is a case where it's very difficult to describe with words. I think I was over-thinking things. Basically the only way it seems to tie the example of time dilation involving the light clock with the twin paradox is through mathematics.

More or less so: I think that in order to understand it well, words are not enough. However, after you understand it with mathematics and/or sketches, it's possible to summarize your understanding with words. I have also seen nice pictures and even animations that explain it rather well, showing how the (de-)synchronization of the moving clocks affects the measurement of the stationary clocks by the moving system, but now I don't have a link ready.

In the book I was reading I understood the light clock example right away and then it pretty much jumped into the twin paradox. Without carefully looking at the derivation of the time dilation formula you end up asking why. But through the light clock example it must be realized that the derived time dilation formula can then be applied to just about any situation. I still can't relate the example visually to the twin paradox, but I have more faith in the formula and tonight I'll go through the problem and apply variables to each component of the twin paradox and then I may be able to relate it to the light clock example.

Good. The first thing to be sure to understand well is mutual time dilation.

 

[bobc2]

...In the book I was reading I understood the light clock example right away and then it pretty much jumped into the twin paradox. Without carefully looking at the derivation of the time dilation formula you end up asking why. But through the light clock example it must be realized that the derived time dilation formula can then be applied to just about any situation. I still can't relate the example visually to the twin paradox, but I have more faith in the formula and tonight I'll go through the problem and apply variables to each component of the twin paradox and then I may be able to relate it to the light clock example.

JimiJams, if you like visualizations, then google the subject of spacetime diagrams. If you think in terms of a four-dimensional space-time universe, then you can apply that concept to the twin paradox and the problem is resolved quite clearly (clear only after understanding the 4-dimensional space-time picture and how to understand observers moving along worldlines and distances in space-time).

Some physicists like to avoid focusing too much on the effects of deceleration and acceleration--they rather focus on the path length in the 4-dimensional universe taken by the stay-at-home twin versus the traveling twin. Each one is traveling at the speed of light along his own world line in the 4-dimensional universe. The traveling twin takes a shorter path through the four-dimensional universe.

The twin paradox has been discussed in the context of space-time diagrams in other posts here and someone can probably point you to them if you are interested in that approach. Here is one of the posts:

https://www.physicsforums.com/showthread.php?t=572581 (Look at post #16)

Edit 6-8-12: JDoolin shows space-time diagrams for three different rest frames in post #30 of this same . He then posts an interesting anim...nt rest frames: [ATTACH=full]153027[/ATTACH]

 

[JDoolin]

Ah, I see. Someone has since added captions to the image since I last saw it on Wikipedia. I did not use that diagram, myself, because the fourth image shows non-orthogonal space and time axes, which struck me as deeply misleading. Now someone (Loedel?) has put a caption under it, which clears up at least what was intended, if not what was actually conveyed. The time axis should, of course, be vertical in every picture.

Thanks, bobc for the note on my user-page.

 

[bobc2]

Ah, I see. Someone has since added captions to the image since I last saw it on Wikipedia. I did not use that diagram, myself, because the fourth image shows non-orthogonal space and time axes, which struck me as deeply misleading. Now someone (Loedel?) has put a caption under it, which clears up at least what was intended, if not what was actually conveyed. The time axis should, of course, be vertical in every picture.

Thanks, bobc for the note on my user-page.

Yes, I touched up the graphics from the Wiki sketches (boxes and captions). I thought the Loedel diagram for the twin paradox was interesting. But you are right, they should have put in the rest frame time axis for the Loedel diagram to make it a little more clear. Here are more graphics to help those who may not be familiar with Loedel diagrams.

Again, the Loedel diagram is constructed by finding a rest system (our black perpendicular coordinates below) in which two observers (blue and red twins) are moving in opposite directions at the same speed. The thing that makes it nice is that line distances on the screen have the same actual distance (or time) scaling for the two Lorentz boosted frames (the blue and red frames for the twins). Thus, in the rest frame corresponding to the black perpendicular reference coordinates (again, the vertical axis should have been shown), both of the twins travel the same distance (time) through the 4-D universe for the outgoing trip of the "traveling twin" (of course both twins are traveling with respect to the black perpendicular coordinates). To compare the distances between the twins for the return trip (the blue twin is doing the turn-around and return), you need to use hyberbolic calibration curves. We are using the black rest frame for making distance comparisons.

I have added the sketches below to help visualize the space-time scaling that results in the "traveling twin" taking a shorter path on the return trip. The hyperbolic curves needed to see the scaling of space-time distances are (hopefully) developed. The sketch on the left is a Loedel diagram. Two observers (the twins) are moving in opposite directions with the same speed with respect to rest frame represented by the perpendicular coordinates. Thus, the actual time and distance scaling on the diagram are the same for each twin during the out-going trip of the "traveling twin."

Again, for the return trip we must use hyperbolic calibration curves for distance (time) comparisons. But, the Loedel diagram allows us to use the Pythagorean theorem directly as shown, from which the metric and Lorentz transformations result. This result also gives us the hyperbolic curves that must be used for calibrating distances (times) in the black rest frame (perpendicular coordinate system). The upper right sketch below illustrates how a hyperbolic curve is used to calibrate points in the black frame located ten years from the origin. The locus of points, all at ten years distance, form a hyperbolic curve in accordance with the derived metric equation. The lower right sketch shows a collection of calibration curves for locating space-time distances (times).

Note that a photon worldline always bisects the time and space coordinate axes for all Lorentz coordinate systems (green lines rotated at a 45 degree angle in the rest frame of the black perpendicular coordinates).

 

[bobc2]

Here is an example of using the hyperbolic calibration curves. We have perpendicular black coordinates for the rest frame. The stay-at-home twin's worldline is along the black vertical time axis--the path is 13 years long (measured by stay-at-home's clock). The traveling twin's path through space-time is shown with the blue lines--the path is 10 years long (measured by traveling twin's clock).

 

[JDoolin]

The term "hyperbolic calibration curves" is apt. I kind of think of them being concentric. Just like concentric circles could be made as

$$x^2+y^2=r^2$$ as r={0,1,2,3,4,...} etc, all representing circles; the locus of positions equidistant from the origin.

you can have "concentric" hyperbolic curves, going

$$x^2-c^2 t^2=s^2$$ as s={0,1,2,3,4...}, all representing "concentric" space-like hyperbolas, the locus of events which, in some reference frame, are simultaneous with the origin event, all the same distance to the left or right of the origin in those respective frames,

and
$$c^2 t^2-x^2=\tau^2$$ as τ={0,1,2,3,4...} for the "concentric" time-like hyperbolas, the locus of events, which, in some reference frame, are all in the same position as the origin, all the same time since or before the origin in those respective frames.

In any case, whether you call it a "hyperbolic calibration curve" or a "concentric hyperbola" it's a helpful mental construct.

I especially like that you re-centered your hyperbolic calibration curve at the turn-around event.

The analogy with circles works so long as you re-center the circle with each measurement. You can't measure the path distance just by looking at the distance to the center to the end-point. You have to move the center of the circle every time the path turns. I'm probably just explaining something really obvious in a complicated way, but I remember myself thinking about it for a few days before it occurred to me, how to perfect the analogy; measuring distances with concentric circles vs measuring space-time intervals with "hyperbolic calibration curves."

 

[bobc2]

The term "hyperbolic calibration curves" is apt. I kind of think of them being concentric. Just like concentric circles could be made as

$$x^2+y^2=r^2$$ as r={0,1,2,3,4,...} etc, all representing circles; the locus of positions equidistant from the origin.

you can have "concentric" hyperbolic curves, going

$$x^2-c^2 t^2=s^2$$ as s={0,1,2,3,4...}, all representing "concentric" space-like hyperbolas, the locus of events which, in some reference frame, are simultaneous with the origin event, all the same distance to the left or right of the origin in those respective frames,

and
$$c^2 t^2-x^2=\tau^2$$ as τ={0,1,2,3,4...} for the "concentric" time-like hyperbolas, the locus of events, which, in some reference frame, are all in the same position as the origin, all the same time since or before the origin in those respective frames.

In any case, whether you call it a "hyperbolic calibration curve" or a "concentric hyperbola" it's a helpful mental construct.

I especially like that you re-centered your hyperbolic calibration curve at the turn-around event.

The analogy with circles works so long as you re-center the circle with each measurement. You can't measure the path distance just by looking at the distance to the center to the end-point. You have to move the center of the circle every time the path turns. I'm probably just explaining something really obvious in a complicated way, but I remember myself thinking about it for a few days before it occurred to me, how to perfect the analogy; measuring distances with concentric circles vs measuring space-time intervals with "hyperbolic calibration curves."

Excellent points, JDoolin. I'll adapt your "concentric hyperbola" teminology from now on.