RocketGirl receives a radio message from Earth

[smaricic]

TL;DR

RocketGirl receives a radio message from her twin, EarthGuy, telling her what his clock shows. It conflicts with what she "knows" (due to time dilation).

This is based on Professor Brian Greene's wonderful 11-hour course on Special Relativity which I watched recently on YouTube.

RocketGirl is travelling to Proxima Centauri, which is 4.25 light years from Earth. She is moving at a constant velocity of 80 percent of the Speed of Light and will approach that star after 5.31 years (according to a clock on Earth).

Since she has reason to believe that she is at rest while the Earth is moving away from her, she "knows" that her twin's clock back on Earth is moving more slowly than her own (due to time dilation).

But after 1.06 years on Earth, her twin, EarthGuy, sends her a radio message: “My Earth-Clock now reads 1.06 years (one year and 22 days).” That message takes 4.25 years to get to Proxima Centauri (at the speed of light), and arrives there around the same time that Rocket-Girl meets that star.

She does some math – she adds 1.06 years (on the Earth-Clock) to 4.25 years (the radio signal’s travel time at the speed of light) and gets 5.31 years. She figures that is how much time has passed on Earth since she and Earth-Guy synchronized their clocks at the start of her trip.

But now she is confused. She "knows" that Earth-Clock was not reading 1.06 years when that radio message was sent. It had to be reading considerably less time due to time dilation. Problem: the radio message from Earth-Guy tells her that the Earth-Clock was not moving slowly. To me, that is a real paradox, not a seeming paradox.

Another part of this paradox: the rocket ship's clock, upon arrival at Proxima Centauri, will claim that only 3.19 years have gone by (due to time dilation). Since Proxima Centauri is 4.25 light years from Earth, it occurs to her that either she or the star travelled faster than the speed of light. That's a no-no.

So, how do we solve this paradox?

 

[Nugatory]

Problem: the radio message from Earth-Guy tells her that the Earth-Clock was not moving slowly. To me, that is a real paradox, not a seeming paradox.

You have overlooked the relativity of simultaneity here. What did the ship clock read at the same time using the frame in which the ship is at rest that the radio signal was emitted? What did the earth clock read at the same time using the frame in which the earth is at rest that the radio signal was received?

You shouldn’t be using the length contraction and time dilation formulas at all here, they only work in very limited situations and this isn’t one. Instead, identify the two events of interest, write down their coordinates using one frame, then use the Lorentz transformations to calculate the coordinates of these events in the other frame. Here the two events are signal sent at x=0,t=1.06 using the earth frame; and signal received at x=4.25, t=5.31 using the earth frame. The Lorentz transformations will give you the coordinates of these events using the frame in which the ship is at rest.

As an aside, this will get much easier if you restate the problem using round numbers: say that the destination is one light year away using the frame in which earth is at rest, measure speeds in light years per year, and now c=1, the speed of the ship relative to earth is .8, and the light travel time using the earth frame is one year. SSame problem, same principles to understand but less arithmetic to get in the way.

Another part of this paradox: the rocket ship's clock, upon arrival at Proxima Centauri, will claim that only 3.19 years have gone by (due to time dilation). Since Proxima Centauri is 4.25 light years from Earth, it occurs to her that either she or the star travelled faster than the speed of light. That's a no-no.

if you do the exercise above you’ll already have found the answer: using the frame in which the ship is at rest proxima centauri is not 4,25 light years away when the ship passes the earth. Thanks to length contraction it is closer.

 

[PeterDonis]

She figures that is how much time has passed on Earth since she and Earth-Guy synchronized their clocks at the start of her trip.

More precisely, she figures that that is the elapsed time that an Earth clock will show at an event on Earth that is simultaneous, in the Earth-Proxima Centauri rest frame, with the event of her arrival at Proxima Centauri. So if Proxima Centauri's clocks are synchronized with Earth's clocks in that frame, they will read 5.31 years when RocketGirl arrives.

But that doesn't mean quite what you appear to think it means. See further comments below.

She "knows" that Earth-Clock was not reading 1.06 years when that radio message was sent.

No, she knows no such thing. Indeed, she knows exactly the opposite, that the Earth clock was reading 1.06 years when the message was sent. How? Because the message itself says so. The message could even include a picture of the Earth clock showing that reading.

It had to be reading considerably less time due to time dilation.

No, that's not how time dilation works. Time dilation can't change the actual reading on a clock; that's an invariant.

You are ignoring length contraction and relativity of simultaneity. When you take those into account, you will find that, first, in RocketGirl's frame the message covers a shorter distance than you are thinking, and second, in RocketGirl's frame the time at which the message is sent is later than you are thinking.

the rocket ship's clock, upon arrival at Proxima Centauri, will claim that only 3.19 years have gone by (due to time dilation).

More precisely, if the rocket ship's clock is set to zero when it leaves Earth, it will read 3.19 years when the ship arrives at Proxima Centauri.

But again, that doesn't mean quite what you appear to think it means. See below.

Since Proxima Centauri is 4.25 light years from Earth, it occurs to her that either she or the star travelled faster than the speed of light.

No, it occurs to her that, in her rest frame, the distance between Earth and Proxima Centauri is shorter than 4.25 light years, due to length contraction. In fact it will be contracted to 2.55 light years, which will take 3.19 years to cover at 0.8 light speed.

Whenever analyzing any relativity scenario like this, it never works to just look at one thing in isolation. Time dilation, length contraction, and relativity of simultaneity all play interlocked roles, and unless you take all three into account, you'll get wrong answers.

how do we solve this paradox?

There is no paradox, just you not doing the correct calculations.

I strongly suggest that you read the Insights article referenced here:

Insights Thread 'When Discussing the Twin Paradox: Read This First'

Jan 2, 2023

This article is intended for anyone who wants to start a thread here at PhysicsForums on the twin paradox. There are already many, many threads here on this topic, and they tend to cover the same ground over and over again, so it seems useful to put a description of that ground into an article that everyone can read before starting yet another thread. If you are in that category, and what is here answers your question, great! But even if it doesn’t, hopefully, this article will help you to frame whatever questions you still have after reading it, in a way that will help you to get...

• PeterDonis
• Paradox Twin paradox
• Replies: 2
• Forum: Special and General Relativity

• 
• 

And also the Usenet Physics FAQ article on the twin paradox that it references.

 

[Nugatory]

And a comment about the thread title: this is not a twin paradox variation, the age difference seen in the twin paradox has nothing to do with the time dilation between observers moving relative to one another. The crucial element in the twin paradox is the travelers are at the same place when the age comparison is done, so relativity of simultaneity is not involved.

 

[PeterDonis]

this is not a twin paradox variation

And that being so, I have removed that part of the thread title.

 

[Matterwave]

But after 1.06 years on Earth, her twin, EarthGuy, sends her a radio message: “My Earth-Clock now reads 1.06 years (one year and 22 days).”

But now she is confused. She "knows" that Earth-Clock was not reading 1.06 years when that radio message was sent.

I think if you read these two snippets back to back you will realize the error in logic.

 

[smaricic]

You have overlooked the relativity of simultaneity here. What did the ship clock read at the same time using the frame in which the ship is at rest that the radio signal was emitted? What did the earth clock read at the same time using the frame in which the earth is at rest that the radio signal was received?

You shouldn’t be using the length contraction and time dilation formulas at all here, they only work in very limited situations and this isn’t one. Instead, identify the two events of interest, write down their coordinates using one frame, then use the Lorentz transformations to calculate the coordinates of these events in the other frame. Here the two events are signal sent at x=0,t=1.06 using the earth frame; and signal received at x=4.25, t=5.31 using the earth frame. The Lorentz transformations will give you the coordinates of these events using the frame in which the ship is at rest.

As an aside, this will get much easier if you restate the problem using round numbers: say that the destination is one light year away using the frame in which earth is at rest, measure speeds in light years per year, and now c=1, the speed of the ship relative to earth is .8, and the light travel time using the earth frame is one year. SSame problem, same principles to understand but less arithmetic to get in the way.
if you do the exercise above you’ll already have found the answer: using the frame in which the ship is at rest proxima centauri is not 4,25 light years away when the ship passes the earth. Thanks to length contraction it is closer.

Thank you for your answer. It is obvious to me that you know much more about physics than I do. I passed high school physics years ago, and I am no kind of scientist. I am what a physicist named Clemenza once called "a pain in the ___ innocent bystander."

Still, I am pretty good with basic logic and arithmetic. By the way, I got the numbers I used from Brian Greene's lecture on Special Relativity -- specifically, this part of it:

https://worldscienceu.com/lessons/12-1-time-dilation-examples/

Let me start with the second part of your answer first: RocketGirl (Professor Greene calls her Gracie) is a smart cookie. She KNOWS that Proxima Centauri is 4.25 light years away. When she arrives at Proxima Centauri, is she supposed to forget what she knows? She looks at her clock and it says that 3.19 years have passed. She must wonder: how did I get here so quickly -- faster than the speed of light?

In addition, to me there is an unreal aspect to this whole Frame of Reference thing. RocketGirl MUST know that she is not at rest. After all, she is the one in the rocket ship! It's not as if someone strapped giant rockets on the Earth and on Proxima Centauri and moved them while she was just sitting there.

Moving on to the first part of your answer: Lorentz transformations are a little beyond me, but here is my guess as to the readings on both clocks when the radio message was emitted and received:

What did the earth clock read at the same time using the frame in which the earth is at rest that the radio signal was received? 5.31 years. Why? 1.06 on the clock + 4.25 travel time = 5.31.



What did the ship clock read at the same time using the frame in which the ship is at rest that the radio signal was emitted?

My guess: .638 years. Here is how I figured that: EarthGuy calculates that it will take 5.31 years for RocketGirl to reach Proxima Centauri. 1.06 years is a little less than 20 percent of that time. RocketGirl's clock, when she makes a rendezvous with Proxima Centauri, reads 3.19 years. Twenty percent of 3.19 years is approximately .638 years.

How did I do?

I'm not sure how that explains the paradox she faces: RocketGirl thought the EarthClock was moving slowly, but the radio message says it was not.

One more thing puzzles me. You said, in the first part of your answer: "You shouldn’t be using the length contraction and time dilation formulas at all here, they only work in very limited situations and this isn’t one." But in the second part of your answer, you wrote: "using the frame in which the ship is at rest proxima centauri is not 4,25 light years away when the ship passes the earth. Thanks to length contraction it is closer." As far as time dilation, I didn't make that up -- that's from Professor Greene's lecture.

 

[smaricic]

I think if you read these two snippets back to back you will realize the error in logic.

Perhaps I wasn't as clear as I should have been. Since she believes that she is at rest, and the Earth is moving away from her, and since she believes in time dilation, she "knows" that the Earth clock must have read a smaller amount.

This is what I wrote in the opening summary: RocketGirl receives a radio message from her twin, EarthGuy, telling her what his clock shows. It conflicts with what she "knows" (due to time dilation).

So, again, she must say to herself: "How the heck did that happen? I thought the EarthClock was moving more slowly."

 

[Nugatory]

RocketGirl MUST know that she is not at rest.

This may be the root of your misunderstanding -until you get over it relativity will continue to confuse you.

It is equally valid to analyze the problem using the frame in which rocket is at rest and earth is moving, or the frame in which the sun is at rest so both are moving (the earth is orbiting the sun, right?), or the frame in which the center of the galaxy is at rest so sun, ship, and earth are all moving, or the frame in which a cosmic ray particle moving past the earth at .99c is at rest while the earth is moving at .99c, or ....
The key point is that all speeds are relative to something else, so the statement "X is moving at speed v" is meaningless. A meaningful statement will be of the form "Y is at rest and X is moving at speed v relative to Y", and it is equally valid to say that X is at rest and Y is moving at speed v relative to X.
(This, by the way is not Einstein and Special Relativity: It is Galilean relativity and as the name suggests it was discovered by Galileo some five centuries ago)

RocketGirl (Professor Greene calls her Gracie) is a smart cookie. She KNOWS that Proxima Centauri is 4.25 light years away.

She know it is 4.25 light years away using the frame in which the earth is at rest. But it is not that distance using the frame in which she is at rest, and if she's going to be using her clock to measure the how long it take Proxima centauri to reach her, she has to use her frame.

 

[Nugatory]

This is what I wrote in the opening summary: RocketGirl receives a radio message from her twin, EarthGuy, telling her what his clock shows. It conflicts with what she "knows" (due to time dilation).

She will only arrive at this incorrect result if she forgets to allow for the relativity of simultaneity, as you are in your calculation. There really is no substitute for doing the general calculation based on the Lorentz transformations as I described in post #2 above, and until you at least attempt it you will be wasting your time trying to understand this apparent paradox.
You may also find drawing a Minkowski spacetime diagram helpful - we have many good examples in other threads in this forum, and one of the members who have software for that purpose may volunteer to post one into this thread.

(Be aware that the length contraction and time dilation formulas are derived from and are special cases of the Lorentz transformations. It would be a good exercise to derive these to formulas from the more general and powerful Lorentz transformations).

 

[robphy]

Maybe this is jumping ahead... but I think the quicker you get to this topic (below),
the better off you'll be.

Have you tried to draw a position-vs-time graph (also called a spacetime diagram)?
Points on this diagram are called events, and they are assigned coordinates for time t and position x
according to the inertial "frame" at rest.
(By convention, time runs upwards, instead of horizontally-to-the-right in PHY 101.)
[In that Briane Greene course, this is in Ch 26:
https://worldscienceu.com/lessons/26-1-spacetime-diagrams/ ]

With practice, one learns to correctly translate
"information given in the problem" into geometric features in the diagram.
For example,

• "an inertial particle at rest in this inertial frame" is drawn as a vertical line (the line is called the "worldline")
• "a distant inertial particle (at x=d) at rest in this inertial frame" is drawn as a vertical line though the event (t=0,x=d)
• "an inertial particle with velocity 0.8c in this inertial frame" is a drawn as sloped line with slope 4/5 with respect to the [vertical] time-axis.
• "when particle-1's clock reads T (an event on particle-1's worldline), it sends a message"
  is drawn as a set of rays (along the (+/-)45-degree lines) from that event.
  (These rays trace out the future-lightcone of that event.)
• One of those rays is received by particle-2 at an event at the intersection of the ray with the worldline of particle-2.
• and so forth.

 

[PeterDonis]

RocketGirl (Professor Greene calls her Gracie) is a smart cookie. She KNOWS that Proxima Centauri is 4.25 light years away.

No, she does not know that.

What she knows is that Earth and Proxima Centauri are 4.25 light years away in the frame in which Earth and Proxima Centauri are both at rest.

But distance is frame dependent. In the frame in which RocketGirl is at rest during her trip, Earth and Proxima Centauri are not 4.25 light years apart. They are only 2.55 light years apart. And both of them are moving in this frame (in the minus $x'$ direction at $v = 0.8$), so you can't even use the 2.55 light years number everywhere, only at the instants in this frame where RocketGirl is either at Earth or Proxima Centauri.

to me there is an unreal aspect to this whole Frame of Reference thing.

This is not a position that we can discuss here. Relativity works; its predictions, including the ones about length contraction, time dilation, and relativity of simultaneity, have been confirmed by thousands of experiments. That's the exact opposite of "unreal".

It is true that "frames of reference" are tools of analysis, and as such are somewhat abstract. But that doesn't mean you can just wave your hands and ignore them if you don't like what they seem to be telling you.

RocketGirl MUST know that she is not at rest.

No, she must not, because "at rest" is relative.

She does know that she is not at rest relative to Earth and Proxima Centauri. But it is equally true to say that Earth and Proxima Centauri are not at rest relative to her. Both viewpoints are valid; neither one is "unreal". And that means there is no absolute sense in which anything is "at rest".

Lorentz transformations are a little beyond me, but here is my guess

You should not be guessing. This is physics, not a quiz show.

This is going to sound blunt, but if you can't be bothered to learn the tools of the physics trade, you shouldn't even try to discuss physics. Lorentz transformations are nothing more than simple equations that you plug numbers into. Someone with the background you claim to have should have no problem with them. And trying to understand any scenario in SR without them is like trying to do arithmetic using Roman numerals. It's not impossible (and it still requires you to make quite a few conceptual leaps that it doesn't seem like you've made), but why would you pound yourself over the head like that?

Also, bear in mind that none of us get paid for posting here. Basically, you're claiming a "paradox" but you can't even use the basic tools that are standard in the field. That means, to be blunt again, that you're wasting not just your own time, but ours. We're here to help people learn what SR says, but that doesn't mean we're going to do all the work.

How did I do?

Your guess is wrong. Please either do the math properly or stop wasting our time.

 

[PeterDonis]

I got the numbers I used from Brian Greene's lecture on Special Relativity

Unfortunately, Brian Greene is notorious for only giving you part of the story. His pop science books and TV shows are even worse. I would not recommend him as a pedagogical source. Taylor & Wheeler's Spacetime Physics is much better.

 

[PeterDonis]

What did the ship clock read at the same time using the frame in which the ship is at rest that the radio signal was emitted?

My guess: .638 years.

Here, I'll give you the correct answer to this question: 1.77 years. I got it by a simple Lorentz transformation, with $v = 0.8$ (since we're using units in which $c = 1$, one light-year per year), of the coordinates of the event of signal emission in the Earth-Proxima Centauri rest frame, which are $x = 0$ (since the emission takes place on Earth and we're treating that as the spatial origin of that frame) and $t = 1.06$ (since that's what Earth clocks read when the signal is emitted). As I said before, Lorentz transformations are simple equations that you plug numbers into; with your claimed background, you should have no problem handling them. So, can you do it?

 

[PeterDonis]

Since she believes that she is at rest, and the Earth is moving away from her, and since she believes in time dilation, she "knows" that the Earth clock must have read a smaller amount.

This is actually correct--if you correctly understand what "a smaller amount" means here. If you look at the numbers I gave in post #14, you will see that the Earth clock time elapsed between RocketGirl leaving Earth (which occurs at the common spacetime origin, coordinates $(0, 0)$, of the two frames) and the radio signal being emitted at Earth is indeed smaller than the RocketGirl time elapsed, by the correct $\gamma$ factor for $v = 0.8$. So you were interpreting "a smaller amount" backwards--you thought it had to mean "smaller than 1.06", but that's not what it means in this context. The 1.06 is the "smaller amount"--RocketGirl's clock time here is the larger amount.

 

[PeterDonis]

You shouldn’t be using the length contraction and time dilation formulas at all here, they only work in very limited situations and this isn’t one.

Actually, as my post #15 indicates, you can use the time dilation formula to get the correct answer for that particular question, because the starting event is the spacetime origin of both frames. But you have to use it the right way around, so to speak. The time dilation analysis the OP was trying to do was in RocketGirl's rest frame, which means RocketGirl's clock time in that frame is larger by the time dilation factor than the Earth clock time, and the latter is what we already know to be $1.06$.

 

[smaricic]

Here, I'll give you the correct answer to this question: 1.77 years. I got it by a simple Lorentz transformation, with $v = 0.8$ (since we're using units in which $c = 1$, one light-year per year), of the coordinates of the event of signal emission in the Earth-Proxima Centauri rest frame, which are $x = 0$ (since the emission takes place on Earth and we're treating that as the spatial origin of that frame) and $t = 1.06$ (since that's what Earth clocks read when the signal is emitted). As I said before, Lorentz transformations are simple equations that you plug numbers into; with your claimed background, you should have no problem handling them. So, can you do it?

Thanks for your answer. I'll try to understand it.

 

[smaricic]

This may be the root of your misunderstanding -until you get over it relativity will continue to confuse you.

It is equally valid to analyze the problem using the frame in which rocket is at rest and earth is moving, or the frame in which the sun is at rest so both are moving (the earth is orbiting the sun, right?), or the frame in which the center of the galaxy is at rest so sun, ship, and earth are all moving, or the frame in which a cosmic ray particle moving past the earth at .99c is at rest while the earth is moving at .99c, or ....
The key point is that all speeds are relative to something else, so the statement "X is moving at speed v" is meaningless. A meaningful statement will be of the form "Y is at rest and X is moving at speed v relative to Y", and it is equally valid to say that X is at rest and Y is moving at speed v relative to X.
(This, by the way is not Einstein and Special Relativity: It is Galilean relativity and as the name suggests it was discovered by Galileo some five centuries ago)
She know it is 4.25 light years away using the frame in which the earth is at rest. But it is not that distance using the frame in which she is at rest, and if she's going to be using her clock to measure the how long it take Proxima centauri to reach her, she has to use her frame.

Thank you for your answer. But aren't some frames of reference more valid than others? Isn't the Copernican view more valid than the view that says the Earth is the center of the Universe? If the Earth is the center, and all the stars rotate around us every day, aren't many of those stars going faster than the speed of light?
Again, I can't get it out of my feeble mind that RocketGirl is the one who took off in a rocket. She must know that she is the one who is moving at high velocity -- not the Earth, not Proxima Centauri.
Don't waste any more time with me.

 

[PeterDonis]

aren't some frames of reference more valid than others?

No.

Isn't the Copernican view more valid than the view that says the Earth is the center of the Universe?

No.

What is the case is that using a frame with the Sun at the center (or to be more precise, with the barycenter of the Solar System at the center) makes the math much simpler for many purposes, than using a frame with the Earth at the center. But that doesn't make a Sun-centered frame, or a barycentric frame, more "valid". It just makes it more convenient.

And even that doesn't go for all purposes. When NASA sent the Artemis mission to the Moon and back recently, they used an Earth-centered frame to track and control the spacecraft's trajectory. Why? Because for that purpose, it's simpler.

If the Earth is the center, and all the stars rotate around us every day, aren't many of those stars going faster than the speed of light?

In coordinate speed, yes. So what? Coordinate speeds are just numbers.

The actual physics is that objects must move within the local light cones in spacetime. That is the frame-independent way of stating it. Which just means that no object can move faster than a light ray right next to it that is going in the same direction. And that is true for the Earth, Sun, planets, stars, and what have you regardless of what frame you choose. Light itself goes "faster than the speed of light" if it's far enough away in a rotating frame centered on the Earth. That doesn't mean the frame is wrong. It just means that coordinate speeds are numbers and don't have any physical meaning in themselves.

 

[PeterDonis]

Isn't the Copernican view more valid than the view that says the Earth is the center of the Universe?

If by "the Copernican view", you mean the view that the Sun is the center of the universe, what makes the Sun special? The Sun itself is orbiting the center of our galaxy. Our galaxy is moving within its Local Group of galaxies. That Local Group is moving in a larger cluster of galaxies, and clusters of galaxies are all moving in the overall universe. There's nothing anywhere in that that is picked out as a "center". It's all relative motion. That's why relativity, with its insistence that all frames of reference are equally valid as far as doing physics, has won as our best current theory: because it best reflects the reality I've just described.

You might want to rethink your intuitions in the light of the above.

 

[PeterDonis]

Thanks for your answer. I'll try to understand it.

Here are some more things to consider as food for thought:

In RocketGirl's rest frame, the Earth is moving in the minus $x'$ direction at $v = 0.8$. So how far away will Earth be from RocketGirl when RocketGirl's clock reads 1.77 years?

The radio signal emitted at that event travels back towards RocketGirl at the speed of light, i.e., $v = 1$ in the units we're using. That means its travel time in years is the same as the distance in light-years. So you can add 1.77 years to the distance that's the answer to the previous question above, to get the time by RocketGirl's clock when the radio signal is expected to arrive. What does that give you?

 

[Matterwave]

Perhaps I wasn't as clear as I should have been. Since she believes that she is at rest, and the Earth is moving away from her, and since she believes in time dilation, she "knows" that the Earth clock must have read a smaller amount.

"He knows, she knows" is an easy way to spin your head around in circles and not get anywhere.

Relativity is non-intuitive but it can't break basic logic.

If the person on Earth waits 1.06 Earth years, sends a message that says "I sent this message 1.06 years after we synced clocks" then that's just a fact. The content of the message gives the proper time elapsed on Earth between syncing clocks and sending the message.

It might be easier to think in terms of proper time. Since it's an invariant, it will be the same in every frame.

 

[sophiecentaur]

So, how do we solve this paradox?

There is no paradox, just you not doing the correct calculations.

We used to have Magic and Miracles to explain / describe situations which were difficult to explain. Now, the word 'paradox' is used instead. I think this is why 'Hard Maths' is not popular; you're not allowed to use such terms in Maths. Instead, people blame Mathematicians for being too nerdy

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

@smaricic your latest post referencing Brian Greene's lecture on SR has been deleted, since its content makes clear that we all are wasting our time trying to help you. I'm going to give a summary of the key points below, even if it doesn't get through to you, for the benefit of others who might be reading this thread and truly want to understand.

(1) Nothing Brian Greene says in the video you referenced [1] either supports your claims or contradicts anything the rest of us have said in this thread.

(2) The scenario in that video has the "traveling twin" returning to Earth. As has already been pointed out, that is a different scenario from the one you are analyzing. Your scenario is basically the first half of that one, with Proxima Centauri as the "turnaround point" for the traveling twin, which is RocketGirl in your scenario.

(3) The question you gave the wrong answer to in post #7, and I gave the correct answer to in post #21, can be posed in Brian Greene's scenario as well: just look at a light signal that is emitted from Earth in such a way that it reaches the traveling twin just at the turnaround point, and ask what time in the traveling twin's outbound frame the signal is emitted. With the numbers adjusted appropriately, the answer is still the one I gave, not the one you gave.

(4) Which clock is "running faster" or "running slower" is frame dependent. In the frame in which RocketGirl is at rest on her way to Proxima Centauri, her clock is running faster than Earth's clock. And that frame is the only frame you need to use to answer the question you asked. But that in no way implies that RocketGirl's clock is "running faster" for the entire scenario. Even if you stick to that one frame (RocketGirl's outbound frame), if RocketGirl turns around at Proxima Centauri (which she has to to fulfill Brian Greene's scenario), she is no longer at rest in the outbound frame--indeed, she is moving faster than Earth in that frame, so her clock runs slower in that frame--by just enough to account for the final result.

(5) Drawing a horizontal line on a spacetime diagram reflects the simultaneity convention of the Earth frame. But simultaneity is relative. In a diagram drawn in the Earth frame (which the one in Brian Greene's lecture that you reference is--note that a similar diagram appears in the Usenet Physics FAQ article on the twin paradox which you were advised to read), lines of simultaneity in RocketGirl's outbound rest frame are not horizontal. They slope up and to the right. If you draw such a line correctly from the event on Earth's worldline where the signal is emitted, you will find that it hits RocketGirl's worldline at the clock reading I gave.

(6) To answer the real question you are probably thinking of, which is: if Earth clocks run slower than RocketGirl's clocks on the outbound leg and on the inbound leg, how can Earth clocks show more elapsed time at the end? I suggest that you read the Usenet Physics FAQ article on the Twin Paradox. It discusses exactly this point.

(7) A final note: @smaricic you are trying to use intuitive reasoning, but intuitive reasoning only works if your intuitions are right. Your intuitions for this scenario, and for relativity in general as far as I can see, are wrong. You can either retrain your intuitions, by learning the correct math and the physical interpretation of it, or keep getting wrong answers. Your call.

This thread will remain closed.

[1] https://worldscienceu.com/lessons/38-1-twin-paradox-the-twins-communicate/