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A student asked me a question in class last week about the twin paradox, and I found that although I knew a valid answer, it wasn't an answer that my class had the background to understand, and I wasn't immediately able to come up with one that would work for them.

The standard difficulty in understanding the twin paradox is that it seems like either twin could be considered the one at rest, and therefore there is symmetry between them, and no way to determine which twin should end up younger when they're reunited. The standard resolution of this is that the symmetry is broken by the fact that the traveling twin's motion is noninertial. This all works fine, and it works for students who know what the students in my class know.

But my student raised a question that I'll state in the following way. Let E be the twin on earth and T the traveling twin. If there were symmetry, then the difference in age at reunion, [itex]\delta t=t_E-t_T[/itex], would have to be zero. But the lack of symmetry doesn't immediately prove that δt is nonzero or that it has a particular sign. How, then, do we know that it's nonzero and positive?

If you have the concept of the metric, then the answer is straightforward. But for the level at which I'm teaching this class (algebra-based freshman physics for life science majors), I don't teach them about the metric. They do know about the Lorentz transformation in graphical form.

Without the metric, I guess one natural way to explain the result is to say that the Lorentz transformation involves t and x coordinates, and we need to go back through the steps needed in order to establish t and x coordinates by techniques such as Einstein synchronization. These techniques involve signaling. We don't normally think of signaling as an idea that is an intrinsic part of a discussion of the twin paradox, but maybe it's necessary in this approach. It seems likely to be complicated, although maybe once you got the argument to work it could be simplified.

A nice idea that I got from Rindler's SR book is to explain this by appealing to length contraction. T sees the distance from the Earth to the turnaround point as being length-contracted, but E doesn't. Therefore T correctly expects [itex]t_T[/itex] to be shortened relativistically compared to [itex]t_E[/itex]. This argument has the virtue of being very simple, but it seems to add some complication. I view length contraction is being a statement about the world-sheet of a ruler, which means that it's inherently more complicated and less fundamental than time dilation, which is a statement about the world-line of a pointlike clock.

The standard difficulty in understanding the twin paradox is that it seems like either twin could be considered the one at rest, and therefore there is symmetry between them, and no way to determine which twin should end up younger when they're reunited. The standard resolution of this is that the symmetry is broken by the fact that the traveling twin's motion is noninertial. This all works fine, and it works for students who know what the students in my class know.

But my student raised a question that I'll state in the following way. Let E be the twin on earth and T the traveling twin. If there were symmetry, then the difference in age at reunion, [itex]\delta t=t_E-t_T[/itex], would have to be zero. But the lack of symmetry doesn't immediately prove that δt is nonzero or that it has a particular sign. How, then, do we know that it's nonzero and positive?

If you have the concept of the metric, then the answer is straightforward. But for the level at which I'm teaching this class (algebra-based freshman physics for life science majors), I don't teach them about the metric. They do know about the Lorentz transformation in graphical form.

Without the metric, I guess one natural way to explain the result is to say that the Lorentz transformation involves t and x coordinates, and we need to go back through the steps needed in order to establish t and x coordinates by techniques such as Einstein synchronization. These techniques involve signaling. We don't normally think of signaling as an idea that is an intrinsic part of a discussion of the twin paradox, but maybe it's necessary in this approach. It seems likely to be complicated, although maybe once you got the argument to work it could be simplified.

A nice idea that I got from Rindler's SR book is to explain this by appealing to length contraction. T sees the distance from the Earth to the turnaround point as being length-contracted, but E doesn't. Therefore T correctly expects [itex]t_T[/itex] to be shortened relativistically compared to [itex]t_E[/itex]. This argument has the virtue of being very simple, but it seems to add some complication. I view length contraction is being a statement about the world-sheet of a ruler, which means that it's inherently more complicated and less fundamental than time dilation, which is a statement about the world-line of a pointlike clock.

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