Do Twins Age Differently in Space?

[Ibix]

Ok, sorry. I thought that in the context it would be self-explanatory. It is the two twins in inertial motion in spaceships, each seeing the other's photon clock running slower than his own.

If you want to explain the Twin Paradox in terms of time dilation then you also need to take into account the relativity of simultaneity and the problems that causes at turnover for the traveling twin's accounting. That's a big part of what the OP was missing. It is also missing in your diagram, presumably because that isn't what it set out to show. Hence me wondering why you were posting it.

 

[Tommyboyblitz]

If both are traveling at the same velocity in any direction then time will be slowed down the same for both, so they would see each others clocks staying the same. time has slowed equally for both.

 

[Ibix]

If both are traveling at the same velocity in any direction then time will be slowed down the same for both, so they would see each others clocks staying the same. time has slowed equally for both.

This is incorrect, as has been pointed out several times on this thread. Both twins always see the other twin and the Earth in motion relative to them. Once they correct for the changing light travel time, they will always calculate both the Earth's clocks and the other twin's clock to be running slow.

The underlying error here is the assumption of an absolute rest frame with regard to which the twins are in motion. There is no such frame. The twins may always regard themselves to be at rest.

 

[Mister T]

Ok, sorry. I thought that in the context it would be self-explanatory. It is the two twins in inertial motion in spaceships, each seeing the other's photon clock running slower than his own.

When I look at the original posts I see the OP's variant of the twin paradox as an attempt to eliminate the conflating issue of simultaneity, hoping to come up with a variant that could explain the effect using only time dilation. Of course, that cannot be done.

One way of looking at this issue is to understand that time dilation involves comparing a proper time to a dilated time, and that's the only type of comparison shown in your drawing. The twin paradox, on the other hand, involves a comparison of two proper times. (Each twin ages an amount of proper time, so any difference in their ages is a difference in proper times.)

Why is this important? Because to measure a dilated time you need two events separated along the line of relative motion. And to measure the time that elapses between those spatially separated events you need to know what the two different clocks at those locations read. And those clocks of course have to be synchronized for that process to be in any way meaningful. When measuring the amount of elapsed proper time you need only one clock because you are measuring the time that elapses between two events that occur in the same place.

 

[Mister T]

The underlying error here is the assumption of an absolute rest frame with regard to which the twins are in motion. There is no such frame. The twins may always regard themselves to be at rest.

Which of course implies the erroneous notion of an absolute velocity.

It just seems natural to think that for me to travel at a very high speed I must first accelerate. Thus changing my speed. Deeply seated in there is the misconception that I'm now at rest.

 

[Battlemage!]

That's why I quoted it in my question.
This, and your entire explanation was extremely helpful/interesting (even the cartoon!). Although I still don't understand how, if each (after discounting light travel time) sees the other's clock running slow for the entire journey (as SR would predict) how their (discounted) sums match at the end.

I think this response from Dale, especially the bold part, may be relevant:

No, this is not at all what SR predicts. Not only is it not what they visually observe, there is also no reference frame where that is what they calculate to be true.

What they would observe is the other twins clock being redshifted during the first part, normal in the middle, then blueshifted during the last part. They end with the same elapsed time.

In the Earth's frame they are always equal. They end with the same elapsed time.

In an inertial frame where one twin is initially at rest then the other will start slow, but the first will end slow. They end with the same elapsed time.

In a non inertial frame the math is complicated. They end with the same elapsed time.

No matter what frame you pick, if you actually do the math... They end with the same time.

No, it is nothing like that at all.