Understanding the Twin Paradox: Exploring the Effects of Relativity on Aging

[Fredrik]

This link is helping me somewhat. http://casa.colorado.edu/~ajsh/sr/timebig.html

I understand the clocks and time dilation sections, but when I get to the Lorentz Gamma Factor and the Paradox image I get lost again.

The derivation of gamma is easy if you understand the pythagorean theorem. There are lots of proofs on the Wikipedia page. I like the one in the "algebraic" proof section. You just note that the area of the larger square is equal to the sum of the area of the smaller (light blue) square and the areas of the 4 triangles. When you write that down and simplify the expression, you get the pythagorean theorem.

The fact that the red perspective stays horizontal I get, but why does the blue need to be placed on a diagonal with a skewing of his "cone" area?

The argument can be made more rigorous than I'm making it here, but it still uses the same basic idea. Cerulean's spatial plane has to be tilted by the same amount and in the opposite direction as his time axis, because otherwise a line that's exactly half-way between Vermilion's time axis and his spatial plane wouldn't be exactly half-way between Cerulean's time axis and his spatial plane. (A line that's exactly half-way between the time axis and the spatial plane represents the motion of something that's moving at the speed of light, which is =1 in the units we're using in these diagrams, and is supposed to be the same in all frames. So if we don't tilt the spatial plane, we're either contradicting that assumption, or the assumption that no frame is fundamentally different from any other).

Is this like the Einstein example of the ball being dropped from a train? That it travels on a diagonal to a "stationary" observer at the train station?

Yes, that's sounds like an accurate description of what's going on in the time dilation picture.

 

[stevmg]

Old comic strip character from 50+ years ago - a man hires onto a cargo spcecraft which travels near the speed of light and takes a 50 year round trip, always moving really quickly. He ages a few moths. He returns to Earth and finds his fiancee is now over 70 years old.

Depressed, he goes back to that space cargo ship to work as feels isolated and alone with no one he remembers.

 

[sylas]

Far be it for me to take on Mr. Michio Kaku, the really really smart guy I was referring to. But I have watched the shows he narrated about Time and remember distinctly the leap he made from stating one clock appears to another to be ticking more slowly, to the statement (after a commercial break) that one clock is really ticking more slowly than the one that is implicitly now preferred. The conclusion was unmistakable due to the implications of long distance space travel being accomplished by flying really fast (compared to what? is the question), same as the 'twins paradox'. The popular notion is that one clock, and one twin, experiences time flow more slowly, is pervasive and just plain wrong.
Hope we agree.
Ron

No, the clock really is ticking more slowly when it flies really fast.

This can be ambiguous in some contexts, such as two clocks moving inertially. Each one is running more slowly than the other.

However, there are other cases which are not so symmetrical. Consider a clock moving in circles around another clock which is inertial. The one moving in circles really is moving more slowly, and each clock can agree on this. The distance between the clocks is unchanging, and the signals sent from one to the other come at regular intervals, which can be used to establish how fast each clock is running in the frame of the other. The clock moving in circles runs more slowly.

Similarly, a clock up a mountain runs more quickly, due to gravitational time dilations.

There's no real question about this. There's a good description of a fabulous measurement of the change in time for a clock at different altitudes in Clocks, Kids, and General Relativity on Mt Rainier. See also [post=2532205]msg #163[/post] in thread "Twins paradox and ageing"

Cheers -- sylas

 

[Fredrik]

No, the clock really is ticking more slowly when it flies really fast.

Sylas, when you say "the clock really is ticking more slowly", you really should add "in an inertial coordinate system where the clock has a non-zero velocity", "in the other clock's frame", "in the frame where it was at rest before we changed its velocity", or something like that. When you say "really is" without adding that extra information, you're making it sound like it's ticking more slowly in an absolute sense.

I think W.RonG was just saying that it's a bad idea to leave out those extra words, and I agree with him about that. This is based on his earlier posts in the thread, but I agree with you that this sounds pretty strange:

The popular notion is that one clock, and one twin, experiences time flow more slowly, is pervasive and just plain wrong.

Ron, it almost seems like you want people to misunderstand you.

 

[sylas]

Sylas, when you say "the clock really is ticking more slowly", you really should add "in an inertial coordinate system where the clock has a non-zero velocity", "in the other clock's frame", "in the frame where it was at rest before we changed its velocity", or something like that. When you say "really is" without adding that extra information, you're making it sound like it's ticking more slowly in an absolute sense.

I thought I did give that extra information, in the next line, which says that this is ambiguous; specifically in the case that there are two inertial clocks moving relative to each other. Each one really is ticking more slowly in the frame of the other.

But the circular motion case no longer has that symmetry. Each clock agrees that the one which is moving in circles is the one that is moving more slowly. (I specified that one is inertial, and the other is moving in circles around it).

Each clock perceives the other as moving in circles around it, but only one of the clocks is actually inertial. Let "A" be inertial, and "B" be non-inertial, moving in circles around "A".

In the "frame" (not inertial frame, though) of the non-inertial clock "B", there is a pseudogravitational force directed away from the (apparently) circling clock 'A", and the (apparently) circling clock "A" is running fast. In the frame of the inertial clock "A". there are no pseudogravitational forces, but the circling clock "B" is running slow. Both clocks agree that it is "B" which runs more slowly than "A".

For any other inertial observer, "A" is running slow, and "B" is running even slower than that, on average, with a curiously sinusoidal variation in the dilation factor, unless the other observer is moving perpendicular to the plane of "B"s orbit around "A".

I am probably not helping here; and could phrase the examples more clearly.

However, this example does show that a circularly moving clock "really" does run more slowly than the clock it is circling. Not because there is some absolute frame, but because "B" is slower than "A" in ALL frames, as long as you average out any cyclic variations in the dilation than can occur.

Cheers -- sylas

 

[atyy]

Ron, it almost seems like you want people to misunderstand you.

Yes, what he's saying sounds terribly standard. This sounds like another what's really real discussion. Why not get him to make an experimental prediction? As long as we agree on that, all will be happy, right?

 

[JesseM]

But the circular motion case no longer has that symmetry. Each clock agrees that the one which is moving in circles is the one that is moving more slowly. (I specified that one is inertial, and the other is moving in circles around it).

All inertial frames (and any non-inertial ones that aren't extremely contrived) should agree that over the course of an entire orbit the circling clock elapses less time, but it might be potentially misleading to say everyone agrees the circling clock "is the one that is moving more slowly", since the instantaneous rate of ticking of the circling clock at a particular moment (or its average rate of ticking over a period of time less than one orbit) can still be faster than the inertial clock in some frames.

 

[yuiop]

The conclusion was unmistakable due to the implications of long distance space travel being accomplished by flying really fast (compared to what? is the question), same as the 'twins paradox'. The popular notion is that one clock, and one twin, experiences time flow more slowly, is pervasive and just plain wrong.

Fredrick suggests you are being provocative here and actually know better and if that is the case, all you are doing is confusing the heck out of newcomers like Jackslap who genuinely seek understanding of the subject.

Can you make it clear if you are stating that in the classic 'twins paradox' you are claiming that when the traveling twin returns home and is once again united with his sibling, that they will not have aged by different amounts in a way that is not only measured by their respective clocks but also manifested physically in how they aged biologically?

If that is what you are claiming, then it contradicts your claim that you 'believe in SR", because anyone that understands SR will know the twins age differentially and that experiments with particles such as muons support that prediction.

If on the other hand you are talking about two twins with purely inertial motion relative to each other, with no turning around and no acceleration, then I would agree with you that it is impossible to define which twin is (really) ageing slower, but that wouldn't be the classic 'twins paradox' would it?

 

[sylas]

All inertial frames (and any non-inertial ones that aren't extremely contrived) should agree that over the course of an entire orbit the circling clock elapses less time, but it might be potentially misleading to say everyone agrees the circling clock "is the one that is moving more slowly", since the instantaneous rate of ticking of the circling clock at a particular moment (or its average rate of ticking over a period of time less than one orbit) can still be faster than the inertial clock in some frames.

Quite right... and I mentioned that; speaking of the average time, and indicating that there would be (for an inertial observer) a kind of sinusoidal variation in the dilation factor, unless observed with a frame moving perpendicular to the plane of the orbit.

kev has aksed the crucial question, however, in the preceding post. That should be the first point Ron should clear up, before worrying about my examples.

Cheers -- sylas

 

[Jackslap]

My mind is getting nuked here! I was really understanding what Ron was saying because I thought it was right, but now you are all saying it isn't. And even though I still don't get it fully, I'm coming closer I can feel it.

Now that I know what you guys mean by frames, inertial frames, absolute frames etc...I'm starting to grasp this whole "aging slowly" thing.

Ron had me thinking for a while that SR was some sort of exception to the rule or something, but are you all saying that it isn't?

In the twin paradox, the INERTIAL twin will actually be biologically/physically younger than the other one right? I can accept that even though I don't know it as deeply as you guys. I have come to a deeper understanding of how spacetime works and how the two are related. I've have learned that since speed is simply a measurement of the amount of distance moved in a certain time it has become clearer to me that a person who is traveling close to c is beginning to travel through time more slowly.

I watched a youtube video that said it like this : You have a little graph where the x-axis is time and the y-axis is space. If you are stationary is space, not moving at all, then you are already traveling at the maximum speed through time (x axis). But as soon as you start to move through space, in addition to time, you are covering less time as you cover more space. At the speed of light you are covering essentially only space, and no time.

I can't word it like the video shows it, but it helped to open my eyes a bit. It's hard to accept that one twin will ACTUALLY AGE LESS than the other, biologically included. I still have a tough time with that. Because to him his biological clock is normal, but to the other HIS clock is normal. There can't be two normals only one.

But I guess that is what relativity really is huh? There is no ONE normal, but infinite normals. Am I getting there yet?

 

[Fredrik]

In the twin paradox, the INERTIAL twin will actually be biologically/physically younger than the other one right?

It's actually the other way round. A clock measures the proper time of the curve that represents its motion, and of all the timelike curves from the event where your twin leaves Earth to the event where he comes back, there is no one with longer proper time than yours.

I've have learned that since speed is simply a measurement of the amount of distance moved in a certain time it has become clearer to me that a person who is traveling close to c is beginning to travel through time more slowly.

I watched a youtube video that said it like this : You have a little graph where the x-axis is time and the y-axis is space. If you are stationary is space, not moving at all, then you are already traveling at the maximum speed through time (x axis). But as soon as you start to move through space, in addition to time, you are covering less time as you cover more space. At the speed of light you are covering essentially only space, and no time.

I'm not a big fan of that way of thinking about these things, mostly because I don't find it useful. And I would say that the last sentence is wrong. But don't worry, you're in good company. What you're saying here (including the last sentence) is just what Brian Greene said in "The elegant universe".

Also note that in the "spacetime diagrams" that we keep talking about, time is usually drawn in the "up" direction (i.e, as the "y" axis).

Because to him his biological clock is normal, but to the other HIS clock is normal.
...
But I guess that is what relativity really is huh? There is no ONE normal, but infinite normals. Am I getting there yet?

I'd say that's right, and that you're getting closer.

 

[stevmg]

The atomic clocks in the GPS satellites do move slower than their "clones" on Earth. They have to be juiced up with the inverse of the gamma factor. These orbiting atomic clocks have speed and centripetal force "slowing them down." There is no reason that living creatures would behave differently.

 

[Jackslap]

So the twin in the INERTIAL frame is the twin that is not moving? I must be confused on inertia then. (Visits wiki)

O.K. I looked it up and inertia is resistance to change of motion, so that means that it IS the twin that is not moving in my opinion (hopefully everyones).

So Fredrik, when you say you are not a big fan of "that way of thinking", exactly what do you mean by that? Is there a different/better way to simplify it? The X and Y graph was the best thing I could find on google and youtube. No anger here, just wondering if there's something better. I had a hard time with that one at first too, but after watching it several times and breaking it down statement by statement it became a bit clearer. If I had to say that I'm still slightly confused about something with it, it would be that the graph has 186,000 annotated in both axes, and that the video says as long as you exist in the universe and are not moving, you are traveling at the maximum speed through time.

I guess I get it, but I have a tendency to always second guess myself until something is EMBLAZONED in my brain. For reference, here is a link to the video. The part I'm talking about is at the 2:50 mark. I watched all 4 parts of that series and it helped me with a few major points so all it all I thought it was good.

 

[Dale]

Is there a different/better way to simplify it?

The approach I prefer is the spacetime geometric approach. Do you know how to draw spacetime diagrams? If so, can you understand geometrically how the stay-at-home twin's path forms one side of a triangle and the traveling twin's path forms the other two sides?

 

[Jackslap]

Not a clue about that. But my mind is sufficiently blown for the night. Google is my friend tomorrow.

 

[Fredrik]

So the twin in the INERTIAL frame is the twin that is not moving? I must be confused on inertia then. (Visits wiki)

O.K. I looked it up and inertia is resistance to change of motion, so that means that it IS the twin that is not moving in my opinion (hopefully everyones).

"Inertial" means "not accelerating". In the twin "paradox", the twin that stays on Earth can be described as "inertial". The astronaut twin can't, because he's accelerating when he turns around to go back home. The twin on Earth will be older than the astronaut twin when they meet again.

So Fredrik, when you say you are not a big fan of "that way of thinking", exactly what do you mean by that?

I just don't see how the concept of "speed through time" can help anyone understand anything, but I do see how it can confuse people. If someone tells you that an object X with velocity v=0.8c relative to you is moving slower through time than you are, do you even know what that means?* Is it clear to you if the word "space" refers to what you consider space, or what X considers space?** Did you know that you and X won't agree about which slice of spacetime is space?

*) It means that the invariant square of the the projection of the normalized tangent vector of the curve that represents's X's motion onto the normalized tangent vector of the curve that represents your motion is greater than -1. And that's really just a complicated way of saying that it doesn't have the same velocity (through space) as you.

**) It has to be a reference to what you consider space. Any object has velocity 0 through what it considers space.

The funny thing is, when they proceed to say that a photon moving at c is only moving through space, the word "space" is now referring to what it would consider space if it made any sense to define the photon's point of view by considering speed v and taking the limit v→c. And it doesn't make sense to do that, so that statement is not only misleading in at least two different ways, it's also wrong.

This is why:

The reason why we associate a specific inertial coordinate system with the motion of an inertial observer is that there's a clock synchronization procedure that makes that the natural choice. All the statements about Lorentz contracton, time dilation, etc., are consequences of that choice. The claim that massless particles experience no time comes from applying the usual time dilation formula for speed v and taking the limit v→c, but there's no reason why we should think of the result of that procedure as "a photon's point of view". There is however a good reason not to: The clock synchronization procedure doesn't work for massless particles. See my posts in this thread (at Physics Forums) for more about this.

Is there a different/better way to simplify it? The X and Y graph was the best thing I could find on google and youtube. No anger here, just wondering if there's something better.

Nothing is better than spacetime diagrams. (OK, I can think of a few things, but none of them have anything to do with physics).

the video says as long as you exist in the universe and are not moving, you are traveling at the maximum speed through time.

Don't worry about it. It's a useless idea anyway. I would recommend that you get some practice drawing spacetime diagrams, and pay extra attention to the concept of simultaneity. A spacetime diagram has an x and a t axis, with the t axis drawn in the up direction. The most important thing to understand is that if the world line of another inertial observer has slope v in the diagram, his simultaneity lines have slope 1/v. This is the result of the synchronization procedure that I described briefly in the post I linked to in the quote above.

For reference, here is a link to the video.

You didn't actually post the link.

Not a clue about that.

The quote below links to a spacetime diagram for the twin paradox, with comments about how the twins would describe things at different points on their world lines, in terms of the coordinates of their momentarily co-moving inertial frames.

Check out http://web.comhem.se/~u87325397/Twins.PNG .

I'm calling the twin on Earth "A" and the twin in the rocket "B".
Blue lines: Events that are simultaneous in the rocket's frame when it's moving away from Earth.
Red lines: Events that are simultaneous in the rocket's frame when it's moving back towards Earth.
Cyan (light blue) lines: Events that are simultaneous in Earth's frame.
Dotted lines: World lines of light rays.
Vertical line in the upper half: The world line of the position (in Earth's frame) where the rocket turns around.
Green curves in the lower half: Curves of constant -t^2+x^2. Points on the two world lines that touch the same green curve have experienced the same time since the rocket left Earth.
Green curves in the upper half: Curves of constant -(t-20)^2+(x-16)^2. Points on the two world lines that touch the same green curve have experienced the same time since the rocket turned around.

The black vertical line serves no purpose at all. I should have deleted it a long time ago, but I've been lazy.

 

[stevmg]

R Shakar at Yale University has a series of physics lectures and he goes into that very question. In essence, the guy standing still ages while the guy moving ages more slowly. That's true for clocks and all systems that work on time.

I can't quote him, Google it and you will see. His lectures 12, 13, 14 and 15 cover simple relativity quite well.

 

[W.RonG]

R Shakar at Yale University has a series of physics lectures and he goes into that very question. In essence, the guy standing still ages while the guy moving ages more slowly. That's true for clocks and all systems that work on time.

I can't quote him, Google it and you will see. His lectures 12, 13, 14 and 15 cover simple relativity quite well.

But in Relativity, nobody is standing still. Everybody is 'the guy moving'.
Ron

 

[W.RonG]

Funny that the name Brian Greene came up. I remembered another example of the aging difference fallacy. I was perusing the physics section of a large book store about a year ago, happened on Mr. Greene's book more recent than the Elegant Universe. Not sure the title, but I opened it to the page where he makes the same mistake of confusing the appearance of a clock ticking more slowly to an observer in a different inertial frame, to make the statement that there is a 'moving' clock ticking more slowly than a 'stationary' clock. Oops.

 

[Fredrik]

Ron, I think you're just misunderstanding what other people are saying. You're right that nobody is standing still in an objective sense, but once you have picked a coordinate system, then it's possible to really stand still in that coordinate system. Is it possible that most of the people that you think are wrong are just describing things in terms of a coordinate system?

 

[sylas]

Ron, it will help clarify your position if you could answer kev, in msg #43.

Can you make it clear if you are stating that in the classic 'twins paradox' you are claiming that when the traveling twin returns home and is once again united with his sibling, that they will not have aged by different amounts in a way that is not only measured by their respective clocks but also manifested physically in how they aged biologically?

If that is what you are claiming, then it contradicts your claim that you 'believe in SR", because anyone that understands SR will know the twins age differentially and that experiments with particles such as muons support that prediction.

If on the other hand you are talking about two twins with purely inertial motion relative to each other, with no turning around and no acceleration, then I would agree with you that it is impossible to define which twin is (really) ageing slower, but that wouldn't be the classic 'twins paradox' would it?

Can you respond to this please Ron? It would help clarify things, I think.

Cheers -- sylas

 

[W.RonG]

Recall that the 'Twins Paradox' arose specifically out of Special Relativity as an effect of Time Dilation. It is quite simple; does Time Dilation cause one clock to tick more slowly than another? the consequence of this phenomenom is the paradox, that one clock will count less time than another if they meet up after 'one of them' has traveled for a while. Answer: no.
Thanks,
Ron

 

[sylas]

Recall that the 'Twins Paradox' arose specifically out of Special Relativity as an effect of Time Dilation. It is quite simple; does Time Dilation cause one clock to tick more slowly than another? the consequence of this phenomenom is the paradox, that one clock will count less time than another if they meet up after 'one of them' has traveled for a while. Answer: no.
Thanks,
Ron

That clarifies. And I mean no offense, but this means you don't believe SR... and if you think you do, then you don't understand it either.

In special relativity, the non-inertial clock will count less time, and this is confirmed when they are brought back together. The amount of time difference between the two clocks can be calculated, and it is confirmed by experiment that the calculations do give the amount of time difference experienced between the two clocks.

It was important that you answered this plainly, though, so we could understand your position more clearly. Thanks.

But you are wrong.

Cheers -- sylas

 

[atyy]

Recall that the 'Twins Paradox' arose specifically out of Special Relativity as an effect of Time Dilation. It is quite simple; does Time Dilation cause one clock to tick more slowly than another? the consequence of this phenomenom is the paradox, that one clock will count less time than another if they meet up after 'one of them' has traveled for a while. Answer: no.
Thanks,
Ron

This is wrong. It can be correct, depending on the exact definition of terms, to say that no ideal clock ever ticks more quickly or more slowly depending on its motion. However, the elapsed proper time will be different for an inertial clock and a non-inertial clock that meet up at the start and end of their spacetime trajectories.

 

[W.RonG]

To 'atyy' and 'sylas': you are assigning preference to one inertial frame over another.
Ron

 

[sylas]

To 'atyy' and 'sylas': you are assigning preference to one inertial frame over another.
Ron

Ron, this is not up for debate. It is not a case of two inertial frames.

If you want to understand this better, we can help; but in my experience it is almost impossible to help someone with this if they insist they already understand it but continue to make all the same errors. As you are doing.

The twin that turns around is not inertial. It is in a different frame on the outward journey, and on the inward journey. As I said previously, it is the non-inertial twin which experiences less elapsed time between departure and when they meet up again.

Cheers -- sylas

 

[Fredrik]

Recall that the 'Twins Paradox' arose specifically out of Special Relativity as an effect of Time Dilation. It is quite simple; does Time Dilation cause one clock to tick more slowly than another? the consequence of this phenomenom is the paradox, that one clock will count less time than another if they meet up after 'one of them' has traveled for a while. Answer: no.

Ah, then I was wrong about what you had misunderstood. SR definitely predicts that the twin that goes on the trip is younger when they meet again. This is an immediate consequence of the axiom that says that a clock measures the proper time of the curve that represents its motion.

That's the coordinate independent resolution of the paradox. If you'd like to see it resolved completely in terms of co-moving inertial frames, then see the spacetime diagram that I linked to in #51 (in the last quote box).

 

[JesseM]

To 'atyy' and 'sylas': you are assigning preference to one inertial frame over another.
Ron

No, you can analyze the problem from any inertial frame and all will have the same answer about the age of the inertial twin and the age of the non-inertial twin when they reunite. Let's call the inertial (Earth-bound) twin "Terence" and the traveling twin "Stella", following the Twin Paradox FAQ. First let's look at the numbers in Terence's rest frame. Suppose that in this frame, Stella travels away from Terence inertially at 0.6c for 10 years, at which point she is at a distance of 0.6*10 = 6 light-years from Earth in this frame, then she turns around (i.e. she accelerates, a non-inertial motion which will cause her to experience G-forces that show objectively that she wasn't moving inertially) and heads back towards Terence at 0.6c, finally reuniting with Terence after 20 years have passed since her departure in this frame. Since Terence is at rest in this frame, he has aged 20 years. But since Stella was moving at 0.6c in this frame, the time dilation formula tells us her aging was slowed down by a factor of $$\sqrt{1 - 0.6^2}$$ = 0.8, so she only aged 0.8*10 = 8 years during the outbound leg of her trip, and another 0.8*10 = during the inbound leg, so she has only aged 16 years between leaving Earth and returning.

Now let's analyze the same situation in a different inertial frame--namely, the frame where Stella was at rest during the outbound leg of her trip (she can't also be at rest during the inbound leg in this frame, since this is an inertial frame while Stella accelerated between the two legs of the trip). In this frame, Terence on Earth is initially moving away from Stella at 0.6c while she remains at rest. In Terence's frame, remember that Stella accelerated when she was 6 light-years away from Earth, so we can imagine she turns around when she reaches the far end of a measuring-rod at rest in Terence's frame and 6 light-years long in that frame, with Terence sitting on the near end; in the frame we're dealing with now, the measuring-rod will therefore be moving along with Terence at 0.6c, so it'll be shrunk via length contraction to a length of only 0.8*6 = 4.8 light-years. So, Stella accelerates when the distance between her and Terence is 4.8 light-years in this frame, and since Terence as moving away from her at 0.6c in this frame, they will be 4.8 light-years apart after 4.8/0.6 = 8 years have passed. During these 8 years, it is Terence's aging that is slowed down by a factor of 0.8, so while Stella ages 8 years during this leg, Terence only ages 0.8*8 = 6.4 years. Then Stella accelerates to catch up with Terence, while Terence continues to move inertially at 0.6c. Using the relativistic velocity addition formula, if Stella was moving at 0.6c in Terence's frame and Terence is moving at 0.6c in the same direction in this frame, then in this frame Stella must be moving at (0.6c + 0.6c)/(1 + 0.6*0.6) = 0.88235c during the inbound leg. And since Terence is still moving at 0.6c in the same direction, the distance between Stella and Terence will be closing at a "closing speed" of 0.88235c - 0.6c = 0.28235c. Since the distance was initially 4.8 light years at the moment Stella accelerated, in this frame it will take 4.8/0.28235 = 17 years for Stella to catch up with Terence on Earth. During this time Terence has aged another 0.8*17 = 13.6 years, so if you add that to the 6.4 years he had aged during the outbound leg, this frame predicts he has aged 20 years between Stella leaving and Stella returning, same as in Terence's frame. And since Stella is traveling at 0.88235c her aging is slowed by a factor of $$\sqrt{1 - 0.88235^2}$$ = 0.4706, so during those 17 years in this frame she only ages 0.4706*17 = 8 years during the inbound leg. If you add that to the 8 years she aged during the outbound leg, you find that this frame predicts she has aged 16 years between departing and returning, which again is the same as what was predicted in Terence's frame.

 

[stevmg]

Sports Fans:

Here is that Yale lecture series I spoke of. I think it is lecture 13 or 14. Start with lecture 12 and step forward.

http://www.youtube.com/watch?v=pHfFSQ6pLGU&feature=SeriesPlayList&p=FE3074A4CB751B2B

Yea, everyone is moving BUT start with twin A on the Earth and look at the second twin B in mtion in space relative to A and it is B who age slower than A. all clocks slow down under time dilation including biological clocks.

Listen to this guy - he explains it a lot better than I can and a lot better than I saw on the earlier posts on this blog.

 

[JesseM]

Yea, everyone is moving BUT start with twin A on the Earth and look at the second twin B in mtion in space relative to A and it is B who age slower than A. all clocks slow down under time dilation including biological clocks.

Only in A's rest frame. In B's rest frame, it is A who ages slower than B (assuming both are moving inertially).

 

[stevmg]

JesseM -

You are the person who straightened my brain out before..
Oh well,

Maybe it has something to do with General Relativity...

If you think I knew nothing about Special Relativity, I won't disappoint you with my less than zero knowledge of GR.

Under GR, time slows down by acceleration and gravity. The guy on Earth is attached to something with more mass than a spaceship.. The B guy has to accelerate and turn (centripetal force), so I guess that is the explanation.

I read Einstein's book "Relativity" and if I wasn't lost and confused after SR, I was clueless and pulseless after reading GR.

Loved that bending the light demonstration and the perihelon{?sp) of Mercury.They even did a movie about a group of guys in 1919 who went to Brazil and did some picture taking of an ecclipse and made it with some local Brazilian girls. Kept my interest up.

Too bad that they didn't believe the 1919 pictures and Uncle Al had to wait until 1923 when a more "official version" was taken and Einstein became God on Earth.

 

[stevmg]

The name of that movie is "House of Sand" - great flick!

 

[Dale]

Not a clue about that. But my mind is sufficiently blown for the night. Google is my friend tomorrow.

Hope you had happy Googling Sorry about my terse response earlier, I was in a rush.

Anyway, a spacetime diagram is simply a graphical plot of the path of a particle. By convention, it is drawn so that the time axis is vertical and the space axis is horizontal. Since paper is traditionally 2D usually we just include one spatial axis and politely ignore the other two, but if you had a 3D graphics system you could easily add a second spatial axis. Also by convention, the time axis is scaled by c (i.e. ct instead of just t) so that even though it is "time" the units have dimensions of length (ct -> L/T T = L).

Using these conventions you can think of the path of particles as geometric figures. A particle at rest would be represented by a vertical "worldline". A pulse of light would be represented by a line at 45 degrees. A particle moving inertially at some v<c would be a straight line with some slope in between. An accelerating particle would have a curved worldline. Etc.

In such a diagram a clock measures the length of the worldline, except instead of using the normal Euclidean metric, the metric that a clock measures is called the Minkowski metric $ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2$. Then it is no longer very surprising that one clock measures a different time than another since they are measuring the lengths of different lines.

 

[Jackslap]

HA! I'm such a moron for not posting that link. Can't believe I made that mistake. Here it is:

http://www.youtube.com/watch?v=ksbgDJF9bsM&feature=related

Anyway, now that I've officially ruined relationships on this board I suppose I'll continue this pattern by stating that I was confused by the difference of opinions, and quite frankly still am a little bit.

I am thinking that Ron's input must be taking something into consideration. Something that I'm not yet capable of understanding. I see that some of you have suggested such a thing. I shall try to wade through the wealth of information you all have left me so far today. Specifically the Yale lectures. I'll probably only be able to get through one part tonight though.

You guys are really awesome for helping. This site is such a resource for me lately. Such a convenient way to share knowledge and I'm humbled that some of you would share your learning which probably cost a ton of cash with some dude like me that you don't know and probably will never meet.

 

[JesseM]

JesseM -

You are the person who straightened my brain out before..
Oh well,

Maybe it has something to do with General Relativity...

If you think I knew nothing about Special Relativity, I won't disappoint you with my less than zero knowledge of GR.

Under GR, time slows down by acceleration and gravity. The guy on Earth is attached to something with more mass than a spaceship.. The B guy has to accelerate and turn (centripetal force), so I guess that is the explanation.

Well, usually when people discuss the twin paradox they ignore GR--you can imagine a world where all objects, including the Earth, have negligible mass (or you can imagine that the inertial twin is not an Earth but on a space station deep in interstellar space, with too little mass of its own to cause any measurable gravitational time dilation).