Understanding the Twin Paradox: A Quick Question Answered

[ghwellsjr]

I do, but what's bothering me is in fact the difference between simultaneity in two perspectives. The first one is the observer which is at rest with the moving twin and the events that the considers to be happening now over a period of time.

This would be the non-inertial rest frame of the black moving twin, the last diagram in the above link according to the radar method.

The second is the blue home twin.

This would be the inertial rest frame of the home twin, the first diagram in the above link.

I have redrawn the two diagrams into one drawing and added in additional radar signals. The moving black twin's non-inertial rest frame is on the left and the blue home twin's inertial rest frame is on the right:

Note that each twin establishes the distance to the other twin by sending a radar signal to the other twin, noting the time it was sent and waiting for the return signal, noting the time it was received, along with the time that he sees on the other twin's clock. He then takes the difference between the sent and received times and divides that by two and assumes (according to Einstein's second postulate) that the signal traveled at c to get to the other twin and that the echo traveled back at the same speed so this allows him to establish a distance to the other twin as simply the time multiplied by c. He also assumes that the time according to his own clock at which this distance applies is the average of the sent and received times.

You can look at either diagram and see how they support the radar signals for either twin but each twin draws his own diagram such that he is at rest. For example, the blue home twin sends a signal at his time of 2008 years (follow the thin blue line up and to the right) and receives the echo at his time of 2011 (follow the thin black line up and to the left) along with his observation that the other black moving twin's clock displayed 2009 at the point of reflection. So he takes the difference of the 2008 and 2011 which is 3 and divides that by 2 to get a distance of 1.5 light-years and since the average of 2008 and 2011 is 2009.5, he puts that black moving twin at 1.5 light-years away at his time of 2009.5 and marks the black moving twin's time at 2009.

In the same way, the black moving twin does a similar thing gets the same answers, except that his outgoing signal is a thin black line and the reflected signal is a thin blue line. But if you repeat the process for later years, you will see that they get different answers.

From the perspective of the home twin, the events on the worldtube of the moving twin that he considers present are past, or happened before, the events that the observers which is at rest with the moving twin consider to be present. Then after the turnaround, on the inbound trip, the events that the stationary twin considers to be present are really the future, or happened after the events that the co-moving observer considers to be the present. This is what confuses me. The description of two perspectives and why the twin, for instance doesn't consider the past of the moving twin to be his present all the time, but in fact, it's first past, then after the turnaround, the future.

I think you are a little mixed up here. The blue home twin always establishes that the black moving twin's clock is behind his own for the entire trip. In fact, the moving twin's clock is ticking at 80% of his own so that during the ten-year interval on the blue home twin's clock, he establishes by radar measurements, that the black moving twin's clock has ticked eight years and that's exactly what has happened when they get back together.

On the other hand, the black moving twin establishes that the blue home twin's clock is behind his for only three years. In fact, for the first two years and a half, their experiences are symmetrical, they both have established that the other ones clock has progressed through only two years but at that point the black moving twin establishes that the blue home twin quits moving away and his clock speeds up so that in just another half year, the blue home twin's clock matches his own at the year 2010. During the next two and a half years, the black moving twin establishes that the blue home twin's clock continues to tick away at twice the rate of his own so that it has reached the year 2015 while he has only progressed half way through 2012. At that point, the black moving twin establishes that the blue home twin starts moving towards him and his clock slows down to the 80% rate once again so that when they reunite, the time on the blue home twin's clock is 2017 compared to his own at 2015.

Does this make sense to you? Remember, they both are establishing the distance to the other one as a function of their own clock by making radar measurements. They both do the same thing but they get different answers.

 

[PeterDonis]

what happens from the stationary twin's perspective? I still don't understand it. How come his lines of simultaneity do not change orientation

Because he never changes his state of motion. Using the definition of "lines of simultaneity" that you are using, the only way they will change orientation is if your state of motion changes--i.e., if you experience proper acceleration. The stationary twin never experiences proper acceleration, so his lines of simultaneity never change orientation.

since we can consider him to move away during the first leg of the trip, and therefore having events that are past (or that preceded others) in his present, and after that he has future events of the moving twin that can be considered to be his present.

This is irrelevant, because motion is frame-dependent, but proper acceleration is not. So you can't look at motion to determine what the lines of simultaneity do; you have to look at proper acceleration, as above.

 

[pervect]

I do, but what's bothering me is in fact the difference between simultaneity in two perspectives.

Why does this matter to you? I could go on about simultaneity being conventional, etc etc, but I think everyone has already done that, and you still have some sort of mental reservation. But I don't think you know what it is. So it's rather hard to address the issue :-). So take it as a given that simultaneity is not the same for the two perspectives, exactly why is this an issue?