Ambiguity of time dilation/twin paradox

[ShreyasR]

The first thing we learn in relativity is that what we measure is relative motion, i.e., with respect to a particular frame of reference. The values of position, velocity, acceleration differs with the frame of reference chosen.
When we go on, we come across time dilation. The theory mentions that "a moving clock(1) tics slower than a clock(2) at rest". The terms "moving" and "rest" actually creates a lot of ambiguity.
Even if it is mentioned that this is explained with respect to the frame of reference in which the clock(2) is fixed at the origin of a coordinate system taken as the reference, and clock(1) is moving, clock(1) ticks slower than clock(2).

Using this, the "Twin paradox" is explained.

"Consider a pair of twins "A" and "B"(age 20). Let "B" blast off in a spaceship on Jan 1st 2013, moving at about 2.8x10^8 m/s, close to the speed of light, and return to Earth on Jan 1st 2023. The observations of "A" are:
"B" is in the spaceship.
"B" and spaceship are moving away at 2.8x10^8 m/s.
"B"'s clock ticks slower.
After "B" returns to the earth, "A" would be physically 30 years old, and "B" would physically be about 21 years. In other words, To "A", it seems that "B" is 9 years younger. In this situation, obviously, it should appear to "B" that "A" is 9 years older than "B".

If the same theory is explained with a frame of reference in which the spaceship of "B" is taken as the origin (B is at rest), the observations are:
"A" is on the earth.
"A" and Earth are moving away at 2.8x10^8 m/s.
"A"'s clock is ticking slower,
And hence, after "B" returns to the earth, "B" would be physically 30 years old, and "A" would physically be about 21 years. In other words, To "B", it must seem that "A" is 9 years younger. In this situation, obviously, it should appear to "A" that "B" is 9 years older than "B".

The theoretical results obtained from the first case is exactly opposite to that of the second case.

Now my question is, as the theory of relativity itself is quite ambiguous, have I not understood it properly, or is there something wrong with this twin paradox?

If there is something wrong with my explanation, please correct it line by line.

 

[tensor33]

Here's a good explanation for why that is not the case,
http://www.phys.vt.edu/~jhs/faq/twins.html

 

[Ibix]

If the same theory is explained with a frame of reference in which the spaceship of "B" is taken as the origin (B is at rest)

That is your mistake. There is no inertial reference frame in which B is at rest for the whole trip. There is one where he's at rest outbound, but he's moving on the return leg. There is one where he's at rest returning, but he's moving on the way out. A is moving in both.

 

[1977ub]

That is your mistake. There is no inertial reference frame in which B is at rest for the whole trip. There is one where he's at rest outbound, but he's moving on the return leg. There is one where he's at rest returning, but he's moving on the way out. A is moving in both.

Experimentally, B will find after he turns around, if he is using radar to generate his view of the world, that suddenly it is much later on the Earth than before he turned around.

"In special relativity there is no concept of absolute present. The present from the point of view of a given observer is defined as the set of events that are simultaneous for that observer. The notion of simultaneity depends on the frame of reference, so switching between frames requires an adjustment in the definition of the present... In a sense, during the U-turn the plane of simultaneity... very quickly sweeps over a large segment of the world line of the Earth-based twin. The traveling twin reckons that there has been a jump discontinuity in the age of the Earth-based twin." - http://en.wikipedia.org/wiki/Twin_paradox

 

[Dale]

Experimentally, B will find after he turns around, if he is using radar to generate his view of the world, that suddenly it is much later on the Earth than before he turned around.

Not quite. This paper describes what he will see if he is using radar. There is no "jump" in time using radar.

http://arxiv.org/abs/gr-qc/0104077

 

[1977ub]

Not quite. This paper describes what he will see if he is using radar. There is no "jump" in time using radar.http://arxiv.org/abs/gr-qc/0104077

For any given IRF, I was under the impression that radar yields the same results as Einstein method. I'll read on.

[2:54] There still appears to be a jump for B's view of A, IRF-wise. There is a discontinuity between the before-turn IRF and the after-turn IRF. This method appears to solve "over-determination" where B ends up with multiple coordinates for some events, but that isn't our problem, right? A certain time range *disappears* on Earth when for B's IRF when he turns suddenly around. That isn't still true here? Then it sounds like radar yields something different than Einstein...

 

[ShreyasR]

That is your mistake. There is no inertial reference frame in which B is at rest for the whole trip. There is one where he's at rest outbound, but he's moving on the return leg. There is one where he's at rest returning, but he's moving on the way out. A is moving in both.

How is that a mistake...? can you be a bit more clear please? Ok now if i consider B is observing the earth, holding his hand in front of him, and marking the tip of his index finger as the origin, Wont he note that the Earth is moving away at 2.8x10^8 m/s? if yes, according to the equation of time dilation, he must notice that every event on the Earth is taking place slower than the events on his spaceship, isn't it? in this case B is aging faster than A... Please clarify this point?!

 

[1977ub]

How is that a mistake...? can you be a bit more clear please? Ok now if i consider B is observing the earth, holding his hand in front of him, and marking the tip of his index finger as the origin, Wont he note that the Earth is moving away at 2.8x10^8 m/s? if yes, according to the equation of time dilation, he must notice that every event on the Earth is taking place slower than the events on his spaceship, isn't it? in this case B is aging faster than A... Please clarify this point?!

To "observe" events on Earth is to receive light signals from Earth and then decide *when* they were sent. Any time you turn your ship in a different direction or change your speed that is an "acceleration" and your whole perspective regarding when events took place and which events are simultaneous shifts. When you then perform calculations about [edit] how much time has passed on Earth, you will end up with different answers than you did before your acceleration.

 

[ShreyasR]

To "observe" events on Earth is to receive light signals from Earth and then decide *when* they were sent. Any time you turn your ship in a different direction or change your speed that is an "acceleration" and your whole perspective regarding when events took place and which events are simultaneous shifts. When you then perform calculations about [edit] how much time has passed on Earth, you will end up with different answers than you did before your acceleration.

Yes i agree to your point. But when when you divide the "main" event, that is the trip of "B", into many intervals/parts... That is,
1) Leaving the Earth by spaceship with an initial acceleration to reach a speed of 2.8x10^8 m/s (wrt earth): time taken = t1,
2) Once the speed is reached, the spaceship moves with a constant speed of 2.8x10^8 m/s for a time interval t2.
3) after that the spaceship decelerates to rest in time t3,
4) The spaceship now accelerates back to 2.8x10^8 m/s in the reverse direction in time t3,
5) the spaceship moves with the constant speed of 2.8x10^8 m/s for time t2,
6) and decelerates in time t1 to land back on earth.
(all time measured by B)

total time spent = 2(t1+t2+t3)

Can you give me a mathematical explanation about the time in each interval measured by A and the total time.
Please explain it like in a tabular column, like in the Left side column, every interval,what A sees, what time he measures, and on the right hand side of the column, what B sees and measures... (According to the concept of time dilation, time measured by B should not be equal to the time measured by A) But i feel if the problem is mathematically illustrated in the form of a tabular column, total time measured by both will be the same... Please try it and reply?!

 

[ShreyasR]

Also, u can include a 3rd person C, who moves such that A and B are equidistant from him, So he'll measure the speeds and accelerations of A and B to be exactly the same wrt himself (but in opposite directions), throughout the whole trip. This should mean that The calculations of C should result in A and B being the same age after the trip isn't it?

 

[1977ub]

Also, u can include a 3rd person C, who moves such that A and B are equidistant from him, So he'll measure the speeds and accelerations of A and B to be exactly the same wrt himself (but in opposite directions), throughout the whole trip. This should mean that The calculations of C should result in A and B being the same age after the trip isn't it?

Person C might be able to position himself so that he can calculate accelerations in his coordinate system that are the same for A & B. However the gorilla in the room is the "source of inertia" which is simply not understood at the current time. Real world measurements will find that A will experience no accelerations and so all of his measurements make sense in an inertial frame. B *will* experience them and so this will throw off the measurements of simultaneity which he might attempt to make. No assignment of coordinates by C to A will cause A to *experience* acceleration, and it is this - the change between different inertial frames - which leads to real differences in elapsed time once B comes back to A. http://www.calphysics.org/haisch/science.html

 

[1977ub]

Please try it and reply?!

Let's see.

It's easier if you consider the acceleration period to be negligibly rapid and short in duration. And why not.

B leaves point X.

1) acceleration = negligible.

2) once B moving, the landscape which is at rest to X appears contracted to him, and so he feels he arrives at point Y much more quickly than he anticipated. meanwhile it appears to take the expected amount of time for for B to reach Y as measured by observer A at point X, who does however find B's ship to be contracted.

3) deceleration = negligible.

4) negligible as #1

5) as #2, landscape is contracted, trip takes less time than expected.

6) negligible.

So I reckon B's time to be 2*t2 shrunken by the lorenz factor.

So what accounts for the difference? B "Changed inertial frames" aka "accelerated," unlike A.

There are multiple descriptions elsewhere in this site and also here.
http://en.wikipedia.org/wiki/Twin_paradox

I honestly think people become so distracted that each observer has their own IRF and is entitled to declare the other person's clock as moving slowly that they become distracted by this effect and lose sight of the fact that it is something related to the change in inertial frame which gives rise to the objectively agreeable difference in clock rates.

 

[ShreyasR]

But then isn't Earth itself supposed to be a non inertial frame when it comes to observing B? I mean even the Earth is rotating and revolving around the sun so we must consider the spaceship of B also revolving around the sun, and moving radially outwards away from the line joining the Earth and sun. So all 3 frames A,B,C are revolving around the sun, and hence my argument is that All three frames are non inertial...

and this quote: "B *will* experience them and so this will throw off the measurements of simultaneity which he might attempt to make. No assignment of coordinates by C to A will cause A to *experience* acceleration" contradicts the statement "ALL MOTION IS RELATIVE" doesn't it?

 

[1977ub]

But then isn't Earth itself supposed to be a non inertial frame when it comes to observing B? I mean even the Earth is rotating and revolving around the sun so we must consider the spaceship of B also revolving around the sun, and moving radially outwards away from the line joining the Earth and sun. So all 3 frames A,B,C are revolving around the sun, and hence my argument is that All three frames are non inertial...

and this quote: "B *will* experience them and so this will throw off the measurements of simultaneity which he might attempt to make. No assignment of coordinates by C to A will cause A to *experience* acceleration" contradicts the statement "ALL MOTION IS RELATIVE" doesn't it?

Generally for these SR discussions we find the Earth's non-inertial aspects to be negligible. Think of someone sitting on the rock in the middle of space and calling it "Earth". Just as you can't draw a perfect circle, no observer is in a perfectly inertial frame. To isolate the effects of SR, you draw up scenarios where other effects can be disregarded as very small.

All motion is relative, but not all acceleration is. That's what you might not realize until you get a little more up to your elbows in this stuff. Any why? Nobody really knows. It takes empirical observation to determine who is accelerating and who is not.

 

[ShreyasR]

OH! The last point has caught my attention... The thing is, the second time i read the basics of general relativity, i read that experimental results have shown that the length of a moving plane is found to be shorter than the plane when it was at rest. this proves that the theory of relativity is valid. Personally, i partially don't believe in this theory. I immediately felt that the difference/error found in the measurement of the length of the plane must be because of the reason that light has got finite speed, and we start and stop a clock when the light reflected from the body under observation to our eyes, creates the error in the time measured, as light has to travel a longer distance before reaching our eyes when the plane is moving away... Who knows? I never made any calculations by imagining a hypothetical experiment, and stressing on the light reflected by the body under observation and so on...

 

[Nugatory]

I immediately felt that the difference/error found in the measurement of the length of the plane must be because of the reason that light has got finite speed, and we start and stop a clock when the light reflected from the body under observation to our eyes, creates the error in the time measured, as light has to travel a longer distance before reaching our eyes when the plane is moving away...

Nope. Time dilation and length contraction are what's left over AFTER we've corrected for the light-travel time.

 

[ShreyasR]

Oh! Okay... Thats awesome. In the books that i have read, they haven't mentioned about the correction in light travel time. This is the reason why i was driven nuts! :P

 

[Ibix]

Can you give me a mathematical explanation about the time in each interval measured by A and the total time.

Why not try it yourself? Say there is a star a distance d away from Earth, as measured in the Earth's frame. A ship travels to it at speed v, turns round and comes back. How long does the trip take according to a stay-at-home observer, and according to an observer on the rocket?

You'll need the Lorentz transforms, which relate time, t, and distance-from-Earth, x, as measured by someone on Earth to the time, t', and distance-from-Earth, x', as measured by someone doing speed v. These are:
<br /> \begin{eqnarray}<br /> x&#039;&amp;=&amp;\gamma(x-vt)\\<br /> t&#039;&amp;=&amp;\gamma(t-vx/c^2)<br /> \end{eqnarray}<br />
where
<br /> \gamma=\frac{1}{\sqrt{1-v^2/c^2}}<br />
Post what you can do if you get stuck, and we'll help.

 

[ShreyasR]

@Ibix, I have understood the calculations given in the book, using the equations mentioned... I was only confused as I was only aware that all motion is relative but I was not aware that all acceleration is NOT relative... Now the equations make lots of sense... Thank you all! :)

 

[1977ub]

...i read that experimental results have shown that the length of a moving plane is found to be shorter than the plane when it was at rest.

Only true for someone who is not moving with the plane, who is measuring the length of a moving plane. It is intimately related to the fact that for someone who is not moving relative to the plane, they will find the clocks at different ends of the plane to show different times while someone on the plane would find them to be in synch.

If you are on the plane, you will not find these results.

 

[Dale]

For any given IRF, I was under the impression that radar yields the same results as Einstein method. I'll read on.

For an inertial observer radar yields the same result as Einstein's method for the frame where the inertial observer is at rest at the origin. But B is not inertial. See figure 9 of the paper I referenced for a diagram of B's frame using radar coordinates.

[2:54] There still appears to be a jump for B's view of A, IRF-wise. There is a discontinuity between the before-turn IRF and the after-turn IRF

Could you point out what you are referring to here? There is no discontinuity in figure 9, so I am unsure what you mean.

A certain time range *disappears* on Earth when for B's IRF when he turns suddenly around. That isn't still true here? Then it sounds like radar yields something different than Einstein...

Correct, on both counts (for a non inertial observer).

 

[Dale]

if i consider B is observing the earth, holding his hand in front of him, and marking the tip of his index finger as the origin, Wont he note that the Earth is moving away at 2.8x10^8 m/s? if yes, according to the equation of time dilation, ...

Yes, he will notice that the Earth is moving away, but the equation of time dilation is specifically for an inertial frame, so it doesn't apply to any frame where B is at rest the whole time.

 

[1977ub]

For an inertial observer radar yields the same result as Einstein's method for the frame where the inertial observer is at rest at the origin. But B is not inertial. See figure 9 of the paper I referenced for a diagram of B's frame using radar coordinates.

B is initially outward, in IRF1. Then at the turnaround point, B is suddenly traveling exactly coincident with C, who was always in IRF2 earthward and then they both return to Earth together in IRF2. So we are suddenly shifting from IRF1 outward to IRF2 inward. In IRF1, certain Earth times were in the future. In IRF2, many of them are suddenly in the past. If we define IRF1 and IRF2 using the Einstein method, which should be the same as the radar method for these IRFs, there must indeed be a discontinuity. [ this method appears to be different than plain old radar, since they are defining coordinates differently in the different regions? ]

 

[1977ub]

[corrected url]
http://arxiv.org/abs/physics/0411008


"Title: A note on Dolby and Gull on radar time and the twin 'paradox'

Abstract: Recently a suggestion has been made that standard textbook representations of hypersurfaces of simultaneity for the traveling twin in the twin 'paradox' are incorrect. This suggestion is false: the standard textbooks are in agreement with a proper understanding of the relativity of simultaneity."

 

[Dale]

B is initially outward, in IRF1. Then at the turnaround point, B is suddenly traveling exactly coincident with C, who was always in IRF2 earthward and then they both return to Earth together in IRF2. So we are suddenly shifting from IRF1 outward to IRF2 inward. In IRF1, certain Earth times were in the future. In IRF2, many of them are suddenly in the past. If we define IRF1 and IRF2 using the Einstein method, which should be the same as the radar method for these IRFs, there must indeed be a discontinuity. [ this method appears to be different than plain old radar, since they are defining coordinates differently in the different regions? ]

Suppose that A, B, and C are each using radar to generate their coordinate systems. Then A's radar coordinates will be the same as Einstein's method for the frame where A is at rest at the origin. Also, C's radar coordinates will be the same as Einstein's method for the frame where C is at rest at the origin. Now, B's coordinates never match with A's coordinates, but if you look in figure 5 in region F you can see that there is a region where B's coordinates do match C's coordinates.

The text below the figure describes it in some detail. Basically, in any region where the radar user is inertial between sending and receiving the radar pulse, radar coordinates give the same result as Einstein's method. However, where there is some acceleration between sending and receiving then you do get a difference.

 

[1977ub]

Suppose that A, B, and C are each using radar to generate their coordinate systems. Then A's radar coordinates will be the same as Einstein's method for the frame where A is at rest at the origin. Also, C's radar coordinates will be the same as Einstein's method for the frame where C is at rest at the origin.

C is not at rest wrt the origin. C is traveling along with B's return trip.

Outbound, B agrees with A. inbound, B, agrees with C. B changes IRF at the far terminus. Dolby and Gull are creating something which is more elaborate the Einstein's method or indeed more than plain old radar.

 

[Alain2.7183]

So we are suddenly shifting from IRF1 outward to IRF2 inward. In IRF1, certain Earth times were in the future. In IRF2, many of them are suddenly in the past.

You got it backwards. The traveler says that the home twin's age suddenly increases during the turnaround.

 

[Dale]

C is not at rest wrt the origin. C is traveling along with B's return trip.

There does exist an inertial frame where C is at rest wrt the origin. For brevity, that is called "C's frame". C's frame is different from A's frame. C is at rest in C's frame and A is travelling. A is at rest in A's frame and C is travelling.

Outbound, B agrees with A.

B never agrees with A. B is always moving wrt A.

inbound, B, agrees with C. B changes IRF at the far terminus.

Yes, and yes.

Dolby and Gull are creating something which is more elaborate the Einstein's method or indeed more than plain old radar.

Dolby and Gull's distance measurement is EXACTLY the same as "plain old radar".

"Plain old radar" usually doesn't bother with the time coordinate. I.e. a radar operator doesn't really care that the radar distance was from 50 μs ago.

 

[1977ub]

There does exist an inertial frame where C is at rest wrt the origin. For brevity, that is called "C's frame". C's frame is different from A's frame. C is at rest in C's frame and A is travelling. A is at rest in A's frame and C is travelling.

I regarded "origin" as being where A is. The "non traveling" twin.

B never agrees with A. B is always moving wrt A.

Oops. Right. I meant that B is identical to an outward IRF Then switches to inward IRF.

Dolby and Gull's distance measurement is EXACTLY the same as "plain old radar".

"Plain old radar" usually doesn't bother with the time coordinate. I.e. a radar operator doesn't really care that the radar distance was from 50 μs ago.

Since the whole endeavor here is to examine simultaneity, it's the time part we care about. Presumably there isn't too much argument about where B thinks the Earth is, but rather what time by his own clock a particular "simultaneous" event on Earth takes place. That reckoning shifts abruptly once he joins C. "O₂" in Eagle's piece.

 

[1977ub]

You got it backwards. The traveler says that the home twin's age suddenly increases during the turnaround.

Oops. Thanks.

 

[Dale]

Presumably there isn't too much argument about where B thinks the Earth is, but rather what time by his own clock a particular "simultaneous" event on Earth takes place. That reckoning shifts abruptly once he joins C. "O₂" in Eagle's piece.

Right, but Eagle's piece isn't using radar time. If you use radar time then there is no abrupt shift. Which brings us back to posts 4 and 5.

 

[1977ub]

Right, but Eagle's piece isn't using radar time. If you use radar time then there is no abrupt shift. Which brings us back to posts 4 and 5.

Eagle's piece defends the textbook version of the abrupt turnaround the Gully piece attempts to "remedy" with their more complex function.

Perhaps I don't understand what you mean by "radar time." For B on his outgoing IRF leg, we have the Einstein approach to determining what time it is "now" for him on Earth. Is the "radar time" method different? Does it given different planes/lines/surfaces of simultaneity than normally given in SR ?

 

[Dale]

Perhaps I don't understand what you mean by "radar time."

Radar coordinates are pretty standard in the SR literature. The Dolby and Gull definition is the standard one. See the first section on page 3 for a detailed explanation. Given that you referenced them in post 4 I am surprised that you are not fully aware of what they mean.

For B on his outgoing IRF leg, we have the Einstein approach to determining what time it is "now" for him on Earth. Is the "radar time" method different? Does it given different planes/lines/surfaces of simultaneity than normally given in SR ?

This is getting repetitive. Please see my previous answers to this question in posts 21 and 25 and figure 5 in Dolby and Gull together with the discussion below it.

 

[1977ub]

Ok so outgoing B's radar coordinates are the same as if calculated according to Einstein. Incoming C's radar coordinates are the same as if calculated according to Einstein. However, B, upon turnaround, has his own unique radar coordinates which are different from C's who is traveling alongside.

 

[Dale]

Ok so outgoing B's radar coordinates are the same as if calculated according to Einstein. Incoming C's radar coordinates are the same as if calculated according to Einstein. However, B, upon turnaround, has his own unique radar coordinates which are different from C's who is traveling alongside.

Basically, in any region where the radar user is inertial between sending and receiving the radar pulse, radar coordinates give the same result as Einstein's method. However, where there is some acceleration between sending and receiving then you do get a difference.

 

[1977ub]

Thank you for your patience. I was aware of the general idea but not all the ways that it works out. Since B's motion was composed of 2 discreet stretches of inertial motion, I couldn't think past treating each as an inertial case which needed to be treated with radar coordinates - i.e. ending up the same as Einstein coordinates. I understood radar to be interesting in the case of *periods* of acceleration. In this example, there is no *period* of acceleration, so I didn't get my head around the implications of using one "session" of radar coordinates for the whole trip. Also, It really hadn't occurred to me that two observers could end up on the same trajectory and yet have different radar coordinates with different information regarding, say, the current time on Earth.

 

[Dale]

It really hadn't occurred to me that two observers could end up on the same trajectory and yet have different radar coordinates with different information regarding, say, the current time on Earth.

Yes, that is a surprising result. It is essentially due to the fact that the point that is identified as "now" on a distant worldline in my radar coordinates depends not only on my motion now, but on my motion between when I sent the radar pulse and when I receive the echo.

So after B meets up with C, for a while, B continues to receive echoes from pulses sent out before the acceleration. These are the ones where B and C disagree. Once that first radar pulse after the turnaround reaches A, from then on B and C agree about the current time at A.

 

[ShreyasR]

Hey guys! Well, I'm from India. I am an engineering student who is studying relativity as a side hobby or whatever you want to call it. And from all the replies that followed my last reply, you people are discussing about the physics of A,B and C. Reading them slowly, I am confused whether the 3 people A,B and C the ones I mentioned? Or are they some other A,B, and C?
I mean I had mentioned the problem A on earth, B in a spaceship going for a linear trip (wrt earth) and coming back, and C traveling such that he measures A and B to be equidistant from him...?
Since DaleSpam mentioned something about B meeting up with C, which according to me will happen only after they return to earth... I am confused :-( :-(

 

[1977ub]

Hey guys! Well, I'm from India. I am an engineering student who is studying relativity as a side hobby or whatever you want to call it. And from all the replies that followed my last reply, you people are discussing about the physics of A,B and C. Reading them slowly, I am confused whether the 3 people A,B and C the ones I mentioned? Or are they some other A,B, and C?
I mean I had mentioned the problem A on earth, B in a spaceship going for a linear trip (wrt earth) and coming back, and C traveling such that he measures A and B to be equidistant from him...?
Since DaleSpam mentioned something about B meeting up with C, which according to me will happen only after they return to earth... I am confused :-( :-(

Different C. Sorry for any confusion.

 

[ShreyasR]

Different C. Sorry for any confusion.

Oh then its ok... Thanks.

 

[bobc2]

ShreyasR, If you follow in the traditions of the great mathematician Ramanujan and one of our physics heroes Natyendra Nath Bose (notwithstanding your engineering rather than mathematics and physics endeavors) you will be a marked man here. A special welcome to you coming all of the way from India via Bose photon paths (it is only fitting that you arrive on Bose photons).

 

[ghwellsjr]

ShreyasR, I'm going to draw some spacetime diagrams to illustrate a scenario similar to the one you proposed. In your scenario, you had A remaining stationary while B traveled away and back at 2.8x10^8 m/s which is about 0.933c. You stated that B would come back to find that A had aged by 10 years and B would age about 1 year. In fact, at that speed, B would age 3.59 years. So I want to change B's speed to 0.96c because that will make the drawing easier to make and understand since the Relativistic Doppler factor is exactly 7.

Here is the first diagram for the Inertial Reference Frame (IRF) in which A remains at rest:

Please note that B ages by exactly 1.4 years at the point of his turnaround and then ages another 1.4 years for a total age gain of 2.8 years compared to A's 10 years.

Now you also suggested a different IRF in which a 3rd person C remains at rest while A and B start off traveling away from him at the same speed in opposite directions. That speed would be 0.75c and here is a diagram showing that:

Now you will quickly see what was brought to your attention in earlier posts that C does not remain equidistant from A and B.

Finally, I want to show you how A can use radar to measure B's trajectory. I added in three radar signals in green, red and orange to the first diagram:

Remember, A cannot see B as we see him on the diagram. A needs to wait for the image of B at each different location to propagate to him. So when A sends out a radar signal, he has to wait the entire time before the return signal comes back to him.

His first radar signal shown in green is sent at 0.1 years into the mission and he receives the return signal at his time of 4.9 years. Note how the signal propagates along diagonal lines of exactly 45 degrees. The calculation that A does is simple. He first figures out how much time progressed between sending and receiving the radar signals. That's just the difference between the emit and return times or 4.8 years. He divides this time in half to get 2.4 years which tells him that A was 2.4 light-years away, but when?. For that, he takes the average of the two times, which is the midpoint of the interval, to get 2.5 years. This tells him that when his clock was at 2.5 years, B was 2.4 light-years away from him. You can confirm this on the diagram.

The red radar signal was sent at 0.2 years and received at 9.8 years so he concludes that B was located 9.6/2 or 4.8 light-years away at his time of 5 years.

The orange radar signal was sent at 5.1 years and received at 9.9 years so he concludes that B was located 4.8/2 or 2.4 light-years away at his time of 7.5 years.

Obviously, I hand-picked these three particular measurements because they were easy to make and illustrate on the diagram. Twin A would likely be sending out a new radar signal at a repetitive interval such as every 0.1 years but this would merely fill in more points along B's path.

You can also use the second spacetime diagram to calculate the same radar measurements. I would suggest that you copy the diagrams and paste them into Paint documents so you can add your own radar signal paths along 45-degree trajectories.

After you get proficient at determining the location of B as a function of A's time in either IRF diagram, you can use the same process to calculate A's location as a function of B's time. This will be a little more challenging but it will be worth it for you to do it. You asked us in post #9 to tabulate this kind of information but now that I have shown you how to do it, I'm asking you to do it yourself. If you need help or have any questions, just ask.

 

[1977ub]

Also, u can include a 3rd person C, who moves such that A and B are equidistant from him, So he'll measure the speeds and accelerations of A and B to be exactly the same wrt himself (but in opposite directions), throughout the whole trip. This should mean that The calculations of C should result in A and B being the same age after the trip isn't it?

Now you also suggested a different IRF in which a 3rd person C remains at rest while A and B start off traveling away from him at the same speed in opposite directions.

I took OP's invention of C to be a figure whose distance from A & B is kept the same though they are the same original A & B, i.e. A remains at rest. I took it to be the usual Twin Paradox confusion over the primacy of coordinate systems without regard for inertial / non-inertial.

 

[ShreyasR]

Thanks a lot ghwellsjr for making time to give such a nice explanation! I have understood how to plot space-time graphs now... So if i consider any A on Earth and his twin bro B traveling in a spaceship away and back at a speed v, the angle of B's trajectory with space axis will be tan^-1(c/v), and i have to mark the dilated time of B accordingly along his trajectory.

But then i don't understand where to start when i try to find A's position as a function of B's time. Now i have to consider the length contraction factor too. And also i have to plot the graph showing the total time spent by B is 2.8 years. But B's frame of reference is non inertial. So how do i go about it?

If i consider the same spacetime plot you sent me, and B sending a radar signal simultaneously as the green radar signal is reflected back by the spaceship (0.6(B)years), it will travel along the same path as the green radar signal, reflected back and reaches the spaceship in about 2.06(B) years. though i have this data i am not able to determine anything! :(

 

[ghwellsjr]

I took OP's invention of C to be a figure whose distance from A & B is kept the same though they are the same original A & B, i.e. A remains at rest. I took it to be the usual Twin Paradox confusion over the primacy of coordinate systems without regard for inertial / non-inertial.

You just quoted the OP saying that C measures the speed of A and B to be the same but in opposite directions, so how can you now say that A remains at rest? Do you realize that I transformed the first diagram into the second diagram and then added C at rest?

I don't understand your statement "the primacy of coordinate systems without regard for inertial / non-inertial". What are you talking about?

 

[ShreyasR]

I took OP's invention of C to be a figure whose distance from A & B is kept the same though they are the same original A & B, i.e. A remains at rest. I took it to be the usual Twin Paradox confusion over the primacy of coordinate systems without regard for inertial / non-inertial.

What do you mean by OP's invention?
Yes that's what i meant... A is on earth... B and C leave the Earth in different spaceships in same direction such that C measures A and B to be equidistant from him (after considering the time dilation and length contraction of C). When B turns back, C also turns back at the same time. Please help me with my problem in my previous reply so that i can try solving this one on my own.

 

[1977ub]

You just quoted the OP saying that C measures the speed of A and B to be the same but in opposite directions, so how can you now say that A remains at rest?

In the original scenario, A remains inertial - and still does once we add C to the mix. Does A remain inertial in your diagram? OP states "So he'll measure the speeds and accelerations of A and B to be exactly the same wrt himself (but in opposite directions)" - this can be done if we use the original scenario of A inertial, B goes out, turns around, and comes back. If C is simply creating coordinate system of distance and time, neglecting inertia, he can indeed merrily declare the movements of A (who remains inertial) and B (who does not) to have identical coordinate movements and accelerations, though only B *experiences* acceleration throughout the trip.

 

[1977ub]

What do you mean by OP's invention?

Only that C is *added* by you to your original scenario.

 

[ShreyasR]

Oh so I am OP? is it an abbreviation? Yeah i know that the second diagram of george is not according to what i mentioned. I meant it this way...
*________________*________________*
A(on earth) C(in spaceship) B(in spaceship)
B and C are moving in --------> direction. A and B are moving away in opposite directions as observed by C. So C measures A's speed as say 'v' (same as B's speed in opposite direction)

 

[ghwellsjr]

Thanks a lot ghwellsjr for making time to give such a nice explanation! I have understood how to plot space-time graphs now... So if i consider any A on Earth and his twin bro B traveling in a spaceship away and back at a speed v, the angle of B's trajectory with space axis will be tan^-1(c/v), and i have to mark the dilated time of B accordingly along his trajectory.

But then i don't understand where to start when i try to find A's position as a function of B's time. Now i have to consider the length contraction factor too. And also i have to plot the graph showing the total time spent by B is 2.8 years. But B's frame of reference is non inertial. So how do i go about it?

You don't have to do anything special to consider length contraction, that's what the radar method does for you automatically. I did A's measurements for B on a diagram which already showed lengths correctly for B. That's why I wanted you to repeat the same process on the second diagram where A is not stationary and where the distances to B will calculate differently than the diagram shows but they will come out the same as what were calculated using the first diagram. If you first tabulate the data for each measurement you will see that. After you get the tabular data, you can reconstruct the first spacetime diagram from measurements obtained from the second spacetime diagram.

If i consider the same spacetime plot you sent me, and B sending a radar signal simultaneously as the green radar signal is reflected back by the spaceship (0.6(B)years), it will travel along the same path as the green radar signal, reflected back and reaches the spaceship in about 2.06(B) years. though i have this data i am not able to determine anything! :(

The first green radar signal is reflected back at 0.7(B) years, not 0.6. So you take the difference between 2.06 and 0.7 which is 1.36 and you divide that by 2 to get 0.68 light-years as the distance. Then you take the average of 2.06 and 0.7 which is 1.38 years as A's time for when B is 0.68 light years away. Just repeat the process for several other radar signals. Do some earlier and some later in time. I know you can do it, it's easy. Post your results.