Does the Twin Paradox Break Symmetry in the Zig-Zag Scenario?

[ash64449]

"Seeing" always includes a certain amount of interpretation of what it is that you think you are seeing.

What is that interpretation?

 

[harrylin]

What is that interpretation?

Ehm no. Just try to come up with an example of such a thing that you are looking for, and in which you do not interpret! (and compare for example post #7 of this thread).

 

[Dale]

What we see is Relativistic Doppler Effect which includes Time Dilation.

You may wish to read about "transverse Doppler".

 

[ghwellsjr]

I think some spacetime diagrams might help to illustrate the two scenarios specified in this thread. The first one comes from post #55:

Let the distance between the Earth and planet be 8 light years.(rest length between them)
Let speed of the traveling twin(TT) be 0.8c in both part's of the scenario.
...
In the frame HT uses, TT is moving and hence time runs slower for TT. So HT predicts TT would measure 20*0.6=12 years to reach earth. And hence concludes that in between the events 'TT leaves the earth' and 'TT reaches the earth', proper time elapsed for TT is 12 years and proper time elapsed for HT is 20 years and hence TT is 8 years younger than HT.

HT is shown in blue and elapses 20 years while TT is shown in red and elapses 12 years (the dots mark off 1-year increments of Proper Time for each twin):

The issue of relativistic Doppler was brought up as a way for each twin to observe and keep track of the other ones aging. Here is a diagram showing how HT views TT's aging:

Please note that HT cannot see the progress of TT in real time, he can only see the red signals coming to him from TT at the moment they hit him. But since he knows that he has been inertial the whole time, he can calculate the speed of TT and plot his trajectory as shown in the above diagram based just on what he sees and knows of Special Relativity. But this only works for him because TT starts out coincident with him so he can start integrating TT's speed to determine his distance from him. I did not show the planet in the diagrams but HT would not be able to tell how far away it was simply by looking at the relativistic Doppler coming from it. He could however determine the distance to the planet after he sees TT arrive there.

Similarly, TT can use relativistic Doppler to track the aging of HT and since he presumes that HT remained inertial while he did all the accelerating, he can construct a diagram of the scenario just like HT did:

Another method that was discussed is radar coordinates. In this scheme, each twin continually sends out coded signals earmarked with the time they were sent and later when the echo is received, the received time is averaged with the sent time to establish the time of the measurement and the difference of the received time and the sent time divided by two establishes the distance (assuming units where c=1). Here is a diagram showing just a few of the radar signals sent (in blue) and received (in red) by HT:

You should be able to confirm in the three radar examples that the coordinates that HT calculates match the coordinates in the diagram.

Now let's see what TT does:

I've only shown two radar examples which are enough, plus the start and stop data for TT to construct a diagram but since he is not inertial, it will look different than the one that HT makes or the one he made using relativistic Doppler:

This may look like a surprising result but what is significant about this is that it maintains all the same radar and Doppler signals that the previous diagrams enacted. In fact, after TT makes this diagram, he can do the same exercise over again from the point of view of HT and get the inertial reference frame that HT got.

One question that might come up is why would TT bother to use radar coordinates when the relativistic Doppler got him the inertial reference frame more directly. The most important answer is that this method works for distant objects such as the planet. Another answer is that it provides a method to show "the point of view" of a non-inertial observer. However, this particular POV (or any other) does not provide any more information to an observer than an inertial reference frame does but it does provide a mechanism for a non-inertial observer to first construct a non-inertial frame and then convert it to an inertial frame (provided that there is an inertial object that he can take data off of.

In my next post I will show Peter's zig-zag scenario.

 

[ghwellsjr]

Finally, we get to Peter's zig-zag scenario:

The scenario I am talking about, which I don't think had been discussed in this thread before I brought it up, would be described using the HT's rest frame as follows: TT leaves and travels outbound to a distant location (planet, space station, whatever); then TT turns around and heads back towards HT; halfway back, TT turns around again and heads back to the planet/space station/whatever; then TT turns around and heads back all the way to HT and they meet up again.

TT can use radar coordinates just like before to construct this diagram but here I show some of the significant radar signals that TT employs:

And from that data he will construct this non-inertial frame: