# Twin paradox: who decided who is the younger one

#### bobc2

I'm not understanding what these calibration curves are for or even how you derived them. On the bottom one, you show it going through the 5 year Proper Time for the traveling twin and through the 5 year Proper Time of the stationary twin.

Later you said:

Are you saying that you have a calibration scheme that allows you to determine that the twins have synchronized clocks or that it provides some means by which to "calibrate" them? It seems rather trivial to me if all you want to do is draw a line from a particular Proper Time for one twin to the same Proper Time for the other twin but then I don't understand why you would need a calibration curve.

I'll add some more commentary a little later. For now, let me provide these sketches with a little bit of commentary.

Equation 1) is derived from the upper left sketch. We have sketched the space-time diagram for a red guy moving to the left and a blue guy moving to the right. Both red and blue are moving at the same speed with respect to the rest black system. This is necessary in order that line lengths on the screen for red and blue will have the same scaling (one inch along a red coordinate has the same physical value as one inch on the corresponding blue coordinate). If you don’t use symmetric coordinate systems in this way, you must use hyperbolic calibration curves to compare physical distances among coordinate systems. This is what we wish to make clear with the hyperbolic curve derivation accompanying the sketches.

These slanted coordinate systems arise from special relativity theory. These unusual looking coordinates are selected as the only coordinates that always yield the same speed for light: c. That’s because the world line of a photon of light always bisects the angle between the X4 coordinate and the X1 axis.

Note that the X4 axis for a moving observer is rotated with respect to the rest system X4 axis (the slope is proportional to the speed). Then, the moving observer’s X1 axis is rotated so as to always maintain symmetric rotation with respect to a photon world line (which is always rotated to a 45-degree angle in the rest system).

Now, we see that the blue X1 axis is perpendicular to the red X4 axis. You will find this is the situation for any pair of symmetric coordinate systems. Further you can always find a rest system for which observers moving relative to each other will move in opposite directions at the same speed. So, contrary to some objections, this derivation is not a special case—it has completely general application. This allows us to write the Pythagorean Theorem equation involving the red and blue coordinates. The time dilation Lorentz transformation equation can be derived directly from the Pythagorean Theorem.

Here, we just want to derive the Proper Time hyperbola equation, i.e., equation 2) above. This equation may be modified for time scaling, using X4 = ct (we use units of years for time and use the compatible units of light-years distance along the X1 axis as shown when plotting the graphs for equation 3).

Equation 4) is shown for a constant value of 10 for the Proper Time. See the corresponding plot. This plot shows the points along a hyperbolic curve in the black rest system that correspond to a fixed Proper Time value of 10 years. The red slanted lines terminating on the hyperbola represent example world lines (time axes) associated with possible observers moving at various speeds.

So, even though the line lengths on the computer screen are different in the rest black rectangular coordinate system, the Proper Times are all the same.

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#### ghwellsjr

Gold Member
Bob, I appreciate the time you are spending on this. I can see now what your curve is.

But you can get the same curve using the Lorentz Transform by plugging in t=10 and x=0 and then sweeping β from +infinity to -infinity and plotting the locus of [t',x'] points. You are showing how a single event can transform into all possible frames.

But why? What has that got to do with anything?

Did you discover this on your own or can you point me to an on-line reference that explains this calibration curve and what its purpose is?

#### bobc2

Did you discover this on your own or can you point me to an on-line reference that explains this calibration curve and what its purpose is?

ghwellsjr, let me get back to you later this afternoon with more complete response to your questions. I did a quick google search and did not come across discussions of proper time that included the space-time diagram with hyperbolic curves. I'll look some more. In the meantime here is a figure from the Naber special relativity text book. But, no--this stuff is definitely not original with me by any means. My first encounter with the proper time calibration curves was in an udergraduate course on Modern Physics. They were also used by my special relativity prof in grad school.

#### ghwellsjr

Gold Member
But can't you give me a quick idea of why you do it? What are you calibrating? How do you use the curves once they are drawn?

#### yoron

Bob, you wrote "And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority."

Well, I think of it this way too, but when I think of 'time' I see it as a very 'local definition', radiation and gravity then becoming what gives us the 'whole unified experience' of SpaceTime. Doing the later one might assume that 'locality' solves it all, but if it is so that 'time', or better expressed, your local 'clock' defines all other frames of reference then there still will exist all those other 'frames of reference' defining you relative their 'clocks'. So even if 'times arrow' can be defined locally SpaceTime is very much like a jello to me.

Which makes it very understandable that some want 'time' to be anything than what it is :)

Eh, the last one was a slight joke relative entropy.

#### bobc2

But can't you give me a quick idea of why you do it? What are you calibrating? How do you use the curves once they are drawn?
Hi, ghwellsjr. Here is the short story. Example a) is a spacetime diagram with black rest frame and blue frame moving relative to rest frame. But you cannot compare times between the black frame and the blue frame. Example b) uses the hyperbolic calibration curves which allow you to compare times between t and t'. And you can see how much time dilation there is for the blue guy looking at a clock along black's world line (t axis). When blue's calendar says 30 years, he "sees" (correcting for light travel time, etc.) black's calendar showing about 26 years. You can measure the slope of blue's time axis to see how fast he is moving with respect to the black rest system.

By the way, notice that the X1 axis of blue is tangent to the hyperbolic curve at the time point of interest.

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#### bobc2

Bob, you wrote "And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority."

Well, I think of it this way too, but when I think of 'time' I see it as a very 'local definition', radiation and gravity then becoming what gives us the 'whole unified experience' of SpaceTime. Doing the later one might assume that 'locality' solves it all, but if it is so that 'time', or better expressed, your local 'clock' defines all other frames of reference then there still will exist all those other 'frames of reference' defining you relative their 'clocks'. So even if 'times arrow' can be defined locally SpaceTime is very much like a jello to me.

Which makes it very understandable that some want 'time' to be anything than what it is :)

Eh, the last one was a slight joke relative entropy.

Thanks for the comments and insight, yoron. You've given me something I'll have to reflect on for a while. You have hit on a point that has to be considered. I think of special relativity locally, but at the same time can envision a continuous sequence of light cones along world lines curving through curved space-time--the cones tipping more and more as they approach massive objects. In that sense I favor a more global application of special relativity.

#### yoron

Well, I think you can see it both ways, you start from a whole 'perspective', I start from a local. But as long as we both agree on that SpaceTime existing for all observers we should meet at some, eh :) 'point'. To me it feels simpler to define 'times arrow' from locality but the 'Jello' won't go away because of that. It just makes me look at 'frames of reference' and 'time' from another angle.

As I see it this was the way Einstein defined SpaceTime too, as a 'whole', using 'c' as the constant defining it, together with Gravity/acceleration relative motion. Maybe a little simplistic, but?

#### harrylin

Why would feeling acceleration in "totally empty universe" be mysterious?
Easy: if one accelerates relative to nothing, there would also be nothing to cause an effect from it.

#### harrylin

[..] I think, most of people when read about the twin paradox, take it at face value and then they think they've got the relativity. But I've got confused about the complex trajectories of both twins, in the field of gravity of the Earth and the Sun.
In the usual (SR) discussion the time dilation due to gravity fields are neglected, and indeed they just add unnecessary complexity for the understanding of SR time dilation.
The most important point I take from this discussion is that the frame of reference is irrelevant (the aging difference holds disregarding the FoR), and what serves here as the definition of the speed is "that what happens to an object that just felt acceleration".
Sorry but that is wrong: it has nothing to do with "feeling". Please read again the discussion by Langevin: he uses the orbit around the far away planet for the turn-around, so that the acceleration is not felt. What matters is the change of velocity.
However, the fact that there would be an acceleration to be felt even in totally empty universe, that escapes my deep understanding, feels like mystery. But I am happy to learn that Einstein and Mach were not that clear on that subject either :)
Although he was always a bit foggy about such topics, Einstein admitted (at least around 1918-1924) that "empty space" can't be truly empty. Indeed, such a view is inconsistent with field theory. See for example: http://en.wikisource.org/wiki/Ether_and_the_Theory_of_Relativity

Harald

#### tom.stoer

Easy: if one accelerates relative to nothing, there would also be nothing to cause an effect from it.
In that sense an empty universe has still the geometrical property to define geodesics. So you feel acceleration w.r.t. these geodesics (I guess this is not what Mach had in mind).

#### ghwellsjr

Gold Member
Hi, ghwellsjr. Here is the short story. Example a) is a spacetime diagram with black rest frame and blue frame moving relative to rest frame. But you cannot compare times between the black frame and the blue frame. Example b) uses the hyperbolic calibration curves which allow you to compare times between t and t'. And you can see how much time dilation there is for the blue guy looking at a clock along black's world line (t axis). When blue's calendar says 30 years, he "sees" (correcting for light travel time, etc.) black's calendar showing about 26 years. You can measure the slope of blue's time axis to see how fast he is moving with respect to the black rest system.

By the way, notice that the X1 axis of blue is tangent to the hyperbolic curve at the time point of interest.
Thanks again, Bob, for putting your time into making these graphics. I now understand what the calibration curve is for and how it is used.

I get the impression that back in the "old" days, before computers or even calculators, there must have been preprinted Minkowski diagrams available with the calibration curves already in place so that the user could label the black axes, draw in his sloping blue axis for whatever β he was interested in, and then he could easily use the calibration curves to label his blue axis--all without doing any calculation except determining the slope of the blue axis, which he could get from a lookup table (along with γ and its reciprocal).

But we have computers now which I'm sure you used to calculate and plot the calibration curves which is more work than simply plotting the blue axis with its appropriate labels.

You point out that the tangent of the calibration curve at the blue axis allows you to easily see the time dilation on the black axis but it is even easier to see if you start at the 30-year point on the black axis and just look at the horizontal (tangent) line going over to the blue axis and see the time dilation there 26 years. And once you know that, you also know that at 30 years for blue, he will "see" 26 years for black.

If the whole purpose of this is to graphically show on a Minkowski the reciprocal nature of time dilation, then why didn't you point this out?

However, all the same things can be shown using just the Lorentz Transform (which is the source of the information that gets drawn on a Minkowski diagram) so why not just stick with the exact numbers that you get from the Lorentz Transform, now that we all have computers and calculators? They work even when the values of β are close to zero or close to one where the Minkowski diagram becomes very difficult to evaluate.

#### bobc2

Thanks again, Bob, for putting your time into making these graphics. I now understand what the calibration curve is for and how it is used.
And thanks for your ideas on this subject.

I get the impression that back in the "old" days, before computers or even calculators, there must have been preprinted Minkowski diagrams available with the calibration curves already in place so that the user could label the black axes, draw in his sloping blue axis for whatever β he was interested in, and then he could easily use the calibration curves to label his blue axis--all without doing any calculation except determining the slope of the blue axis, which he could get from a lookup table (along with γ and its reciprocal).

But we have computers now which I'm sure you used to calculate and plot the calibration curves which is more work than simply plotting the blue axis with its appropriate labels.
I'll bet you are right about that. And yes, I used MatLab to do the math, then copied and pasted into Microsoft Paint to add a couple of things.

You point out that the tangent of the calibration curve at the blue axis allows you to easily see the time dilation on the black axis but it is even easier to see if you start at the 30-year point on the black axis and just look at the horizontal (tangent) line going over to the blue axis and see the time dilation there 26 years. And once you know that, you also know that at 30 years for blue, he will "see" 26 years for black.
A really good point. Thanks for pointing that out.

If the whole purpose of this is to graphically show on a Minkowski the reciprocal nature of time dilation, then why didn't you point this out?
I don't know. I guess I was originally more focused on using the diagram to emphasize it's representation of the 4-dimensional continuum and possibility of viewing objects as 4-dimensional as well.

However, all the same things can be shown using just the Lorentz Transform (which is the source of the information that gets drawn on a Minkowski diagram) so why not just stick with the exact numbers that you get from the Lorentz Transform, now that we all have computers and calculators? They work even when the values of β are close to zero or close to one where the Minkowski diagram becomes very difficult to evaluate.
You have a good point there. If space-time diagrams don't do anything for folks, then just stick to the calculations as you say.

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