Another Time Dilation Question

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TL;DR Summary
Twin Paradox Question
I'm confused about something about the Twin Paradox. To keep it simple, let's ignore acceleration and deceleration.

Twin "A" heads for a point in space 5 light years away and returns at near the speed of light, while twin "B" remains on earth. Approximately 10 years passes for Twin "B" for the 10 light year trip. Due to time dilation, time slows down for the traveler Twin "A" so, when he returns, he has not aged as much as earthbound Twin "B". This is where I get lost. Although time is slower for traveler Twin "A", he is still making a 10 light year trip near the speed of light. So why isn't he also 10 years older when he returns, and the same age as his twin?
 
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  • #2
InquiringMind said:
So why isn't he also 10 years older when he returns, and the same age as his twin?
That’s what makes it a “paradox”, and the resolution of the paradox is that:
1) it’s easy to misanalyze the problem when we use the time dilation formula without understanding its limitations. That’s what happening here.
2) This isn’t a time dilation problem, so it is more easily understood if you don’t start with the time dilation formula. Instead try the sticky FAQ at the top of this subforum: https://www.physicsforums.com/threads/when-discussing-the-twin-paradox-read-this-first.1048697/
3) As with almost all relativity “paradoxes”, the problem is failing to consider the relativity of simultaneity. In this case, space twin and earth twin disagree about what the earth clock reads before and after the turnaround. The “time gap” section of the FAQ addresses this.
 
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  • #3
InquiringMind said:
So why isn't he also 10 years older when he returns, and the same age as his twin?
The first key insight in relativity is understanding that "space" and "time" are combined into spacetime, and there isn't One True Way to think of space and time separately anymore. The second is to realise that a clock is to spacetime as an odometer is to space - it measures the "length" of the path it took through spacetime.

With the latter insight, things like the twin paradox are kind of trivial. The twins didn't take the same path through spacetime from departure to return, so why would you expect the "length" of that path (their own measured times) to be equal? The travelling twin just took a shorter path. That is fundamentally why the twins aren't the same age, although there's a bit of maths needed to justify the claim. That extra maths is probably why popular sources often use the "time runs slow for the travelling twin" explanation. (The whole point of the twin paradox, by the way, is to demonstrate the shortcomings of that approach, so they then have to do a lot of scurrying around to patch up the problems they were supposed to be learning to avoid...)

But it's the first insight that's directly relevant to your question. No, the traveller has not travelled ten light years by his own measurements. He stayed still and the destination came to him at near the speed of light. Then he accelerated briefly and home came to him at near the speed of light. Explaining in detail exactly how he can interpret this is quite complex (understanding interpretation of observations for observers who acelerate actually lead towards general relativity, which is a whole other ball game of maths requirements). But in each inertial phase he would interpret the distance between home and destination as much, much shorter than 5ly - effectively this is the phenomenon called length contraction. Thus he is unsurprised that his own clocks read less than ten years - the destination wasn't far away to begin with.
 
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  • #4
InquiringMind said:
So why isn't he also 10 years older when he returns, and the same age as his twin?
That's because an absolute time does not exist. Each twin has his own time, called "proper time". Not the time-interval between the two meeting-events is invariant, but the spacetime-interval between both events.
 
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  • #5
InquiringMind said:
TL;DR Summary: Twin Paradox Question

I'm confused about something about the Twin Paradox. To keep it simple, let's ignore acceleration and deceleration.

Twin "A" heads for a point in space 5 light years away and returns at near the speed of light, while twin "B" remains on earth. Approximately 10 years passes for Twin "B" for the 10 light year trip. Due to time dilation, time slows down for the traveler Twin "A" so, when he returns, he has not aged as much as earthbound Twin "B". This is where I get lost. Although time is slower for traveler Twin "A", he is still making a 10 light year trip near the speed of light. So why isn't he also 10 years older when he returns, and the same age as his twin?

I'm not sure if this observation will help, but the distance from start to destination being five light years in Earth's frame (the frame of twin B) does not imply that the distance from start to destination is five light years in some other frame. Specifically, the distance can be shorter in a moving frame.

Twin A does not have a single frame, a more exact (and tedious) answer would need better descriptions of what frames you are using. I can make a reasonable guess as to what you _might_ be using, but it'd still be a guess.

There are several ways to work out the problem, in terms of "time dilation" style explanations you need to include time dilation, length contraction, and the relativity of simultaneity. Frequently the third part gets lost :(.

You can also chug through the Lorentz transform equations, if you are familiar with them.

There are other approaches as well, such as concentrating on what twin A "sees" during his journey, which can be interpreted as the doppler shift of some beacon broadcasting from the definition. Pick your favorite approach and stick with it, but as far as your specific approaches goes, only taking into account time dilation won't work because you've ignored the other two effects I've mentioned, length contraction and the Relativity of Simultaneity.
 
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  • #6
InquiringMind said:
TL;DR Summary: Twin Paradox Question

he is still making a 10 light year trip
No. The distance is ten light years in the rest frame of the staying twin but much less than that in the rest frame of the traveling twin.

Distance is not absolute, it's relative. It only makes sense when you state what frame is being used to make the measurement. There is no such thing as the "true" distance.
 
  • #7
I'll chime in with a "me too" answer, in case other didn't do it for you.

  • Twin A has accelerated to near c (measured relative to his point of departure).
  • He observes that he is stationary (by definition) and everything else is rushing toward him, that includes his turnaround target, and everything in between.
  • Because everything in his path is moving relativistically, everything he sees is length compressed.
  • That includes the distance from his point of departure to his turnaround target.
  • From the moment he reaches near c, which might be on Day 1 - he measures his target to be - not 5 light years away but an easy 5 light weeks distant.
  • In other words, he doesn't have to travel 5 light years out and back - the trip is a very short distance for him - 5 light weeks.
  • Thus, when he arrives back home, he has - correctly - aged only 5 weeks.
TL;DR: by moving at a relativistic velocity, Twin A's actual measured journey is length compressed from 5 light years to 5 light weeks. And he correctly only ages 5 weeks during each 5 light week leg of his trip.

Diagrammatically:

1737577888916.png
 
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  • #8
InquiringMind said:
Due to time dilation, time slows down for the traveler
@InquiringMind, just to be sure you understand ... NO ... time does NOT slow down for the traveler. This is a very common misconception but you are confusing time dilation with differential aging.

Both twins see their clocks ticking at the same one second per second, it's just that, as has already been pointed out, they take different paths through spacetime and so age at the same rate but by differing amounts.
 
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