Relativity - Oh dear, here we go again!

[Fredrik]

I don't have any objections. You seem to have the correct idea about all of those things.

Just a few minor details...

"they share the same frame": The motion of an object that isn't being accelerated only defines a time axis, and (in any theory where space has more than one dimension) there's an infinite number of frames that share the same time axis.

Speed is the magnitude of velocity, and velocity is a vector. So it would have been more appropriate to use the word "velocity" instead of "speed" in the last part.

None of these things have anything to do with relativity. They would have been true even if the speed of light had been infinite.

 

[ynot]

I don't have any objections. You seem to have the correct idea about all of those things.

Just a few minor details...

"they share the same frame": The motion of an object that isn't being accelerated only defines a time axis, and (in any theory where space has more than one dimension) there's an infinite number of frames that share the same time axis.

Speed is the magnitude of velocity, and velocity is a vector. So it would have been more appropriate to use the word "velocity" instead of "speed" in the last part.

None of these things have anything to do with relativity. They would have been true even if the speed of light had been infinite.

Sure, but I’m just considering the basics of what motion is from a universal perspective, to check that the foundation I’m coming from is valid. Relativity and math don’t come in to it at that level. Hope you can understand what I mean.

 

[sphenisc]

This is not the first time and probably not the last time that I've been involved in a discussion about the twin "paradox". So I decided to draw a space-time diagram that shows a lot of details, including the ages of the twins at several different events on their respective world lines.

I have explained a lot of this already, in the posts I listed in #238. In this diagram, 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 away from 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 left the position (in the Earth's frame) where it turned around.

[qimg]http://web.comhem.se/~u87325397/Twins.PNG[/qimg]

Any questions?

Great, thanks for that.

So from that I get, for the first part of the journey.

Age of travelling twin in earthbound's frame is sqrt(t²(1-v²))

I assume 20 and 16 are t and x at the midpoint, before I try and work out the second part.

Cheers

 

[Fredrik]

So from that I get, for the first part of the journey.

Age of travelling twin in earthbound's frame is sqrt(t²(1-v²))

I assume 20 and 16 are t and x at the midpoint, before I try and work out the second part.

That's correct. The equation -t²+x²=-t'²+x'² holds everywhere in this diagram. (The primed coordinates are the ones used by the twin in the rocket as it's moving away from Earth). On the first half of the rocket's world line, we have x'=0, so we get that t' is...exactly what you said. With this particular v (=0.8), this is just 0.6*t.

And yes, the rocket turns around at t = 20 years, x = 16 light-years.

I should probably mention that if we had not been using units such that c=1, that equation would have taken the form -c²t²+x²=-c²t'²+x'² instead.

 

[BillyJoe]

Fredrik,

That's a very good diagram...but totally irrelevant to the question of whether time dilation is caused by acceleration.

In fact, your diagram almost implies that time dilation IS caused by acceleration. From B's point of view, A instantaneously ages 25.6 years when B accelerates.

Although Ziggurat's scenario, which had B revolving around a distant star and passing A with each revolution, also involved acceleration (towards the distant star), it at least clearly showed that the time dilation was independant of acceleration.

But then again, perhaps I'm just annoyed that my own scanario, which, I believe, clearly shows that time dilation is totally independant of acceleration (because no accleration is actually occuring at any time, has been more or less completely ignored.




(At least Paulhoff can have a pleasant little chuckle. )

 

[Fredrik]

That's a very good diagram...but totally irrelevant to the question of whether time dilation is caused by acceleration.

I wasn't trying to answer that question. I just wanted to explain why the "twin paradox" isn't a paradox.

In fact, your diagram almost implies that time dilation IS caused by acceleration. From B's point of view, A instantaneously ages 25.6 years when B accelerates.

But that isn't time dilation. It's another effect. I will call it a "simultaneity shift" here.

From A's point of view, this is what happens: Because of time dilation, B ages at only 60% of A's aging rate, so when the rocket gets back to Earth after 40 years, B has only aged 24 years. Nothing at all happens when the rocket turns around, except that for a brief moment B's aging rate catches up with A's.

From B's point of view, this is what happens: Because of time dilation, A ages at only 60% of B's aging rate, so the moment before the rocket turns around after 12 years, A has only aged 7.2 years. When the rocket turns around, the simultaneity shift makes A's age jump ahead by 25.6 years, so the moment after the rocket has turned around, A is 7.2+25.6-12=20.8 years older than B. Because of time dilation, A continues to age at only 60% of B's aging rate for the rest of the trip, so when the rocket gets back to Earth after 12 more years, A has only aged another 7.2 years, so the age difference is only 20.8+7.2-12=16 years.

Acceleration doesn't cause time dilation in this scenario. Instead it causes the simultaneity shift that prevents the paradox that would arise if time dilation had been the only relativistic effect in this problem. This would be "woo" if we had to assume that there's a simultaneity shift that resolves the paradox, but we can prove that there is using the same methods that proved time dilation, and without making any new assumptions.

But then again, perhaps I'm just annoyed that my own scanario, which, I believe, clearly shows that time dilation is totally independant of acceleration (because no accleration is actually occuring at any time, has been more or less completely ignored.

Maybe because the drawing was in ASCII code.

I've seen the scenario you describe before. It's the twin paradox without acceleration. I see now that you made it a bit more complicated than it needs to be by using the frame o. You could have drawn it from from B's point of view, like this:

You also made the slope of line C such that it would indicate a superluminal speed in a space-time diagram with c=1, and you made the time coordinate increase in the down direction, which is also unconventional.

If you take the velocity of A to be 0.8 and the velocity of C to be -0.8, you get the same numbers as in my diagram.

And, finally, you don't need to consider a version of the twin paradox to see that time dilation is caused by speed rather than acceleration. That can be seen directly from the invariance of the quantity -t²+x².

 

[ynot]

Fredrik,

That's a very good diagram...but totally irrelevant to the question of whether time dilation is caused by acceleration.

I think any question whether time dilation is caused by acceleration has been answered. At least I’m happy to accept that according to Relativity it’s not. The point I’m making regarding acceleration is this - When an object is accelerating it experiences inertial forces. The faster the acceleration, the stronger the inertial force. We can know objects are accelerating differently due to the different inertial forces being experienced. We can therefore attribute different speeds (velocities?) to different objects when they are accelerating (but we can’t say that acceleration is speeding up or slowing down). We can’t do this however when objects are not accelerating, or more precisely, when they are accelerating so slowly that inertial forces aren’t detected. When two objects move apart and come back together again, one or both have to go through periods of acceleration. If we ignore the periods of acceleration, and just consider the periods that the objects move apart and come back together again, we can’t attribute any particular speed to either object, only to the rate at which they move apart and come together. In other words, we can’t say that one is moving and the other is stationary, or that one is moving at a different speed than the other. I can’t see therefore that any time dilation that occurs to one object, doesn’t occur in the same measure and effect to the other, or that time dilation even occurs.

 

[Ziggurat]

When two objects move apart and come back together again, one or both have to go through periods of acceleration. If we ignore the periods of acceleration, and just consider the periods that the objects move apart and come back together again, we can’t attribute any particular speed to either object,

We cannot attribute any particular speed to either object unless we choose a reference frame. The choice of reference frame is arbitrary, but everything that comes after that is not.

only to the rate at which they move apart and come together.

Actually, this is reference-frame dependent as well (though it is not in Gallilean relativity - ie, Newtonian mechanics).

In other words, we can’t say that one is moving and the other is stationary, or that one is moving at a different speed than the other.

Not in an absolute sense, but with respect to any particular reference frame, yes, we can.

I can’t see therefore that any time dilation that occurs to one object, doesn’t occur in the same measure and effect to the other, or that time dilation even occurs.

Time dilation, like velocity, is only measurable with respect to a particular reference frame. So in a sense, yes, you cannot assign a single value of time dilation to any particular object. But there's nothing wrong with choosing a particular reference frame, even if the choice is completely arbitrary, from which to examine time dilation. And once you've picked your reference frame, time dilation for a given trajectory is uniquely determined, and it is quite real.

 

[ynot]

We cannot attribute any particular speed to either object unless we choose a reference frame. The choice of reference frame is arbitrary, but everything that comes after that is not.



Actually, this is reference-frame dependent as well (though it is not in Gallilean relativity - ie, Newtonian mechanics).



Not in an absolute sense, but with respect to any particular reference frame, yes, we can.



Time dilation, like velocity, is only measurable with respect to a particular reference frame. So in a sense, yes, you cannot assign a single value of time dilation to any particular object. But there's nothing wrong with choosing a particular reference frame, even if the choice is completely arbitrary, from which to examine time dilation. And once you've picked your reference frame, time dilation for a given trajectory is uniquely determined, and it is quite real.

The twins (can’t get away from the brats) start in the “same frame“, then “different frames“, then the “same frame” again. I can’t see that the properties of the “different frames” are actually different as observed by each twin (equal and opposite). For simplicity, let’s say that the twins are just travelling together. They each view that they are stationary and the other is travelling toward them. The velocity/speed and distance of the “approaching twin” would be exactly the same for both. They make their final age comparisons in the “same frame” and any time dilation would apply exactly the same to both.

 

[Fredrik]

I don't understand what you're saying. Did you just put both twins on the rocket? Or are you talking about what happens when the twin paradox scenario is viewed from some other frame, say the frame of another rocket?

Hm, maybe you just need to re-read the last part of Ziggurat's most recent reply to you.

 

[Ziggurat]

The twins (can’t get away from the brats) start in the “same frame“, then “different frames“, then the “same frame” again.

That's not a required part of the problem. It is enough that they start in the same location and end in the same location. Their initial and final velocities are actually irrelevant. Furthermore, they're not just in different frames. One twin is in a single frame for the entire journey, and the other twin is in at least two frames.

I can’t see that the properties of the “different frames” are actually different as observed by each twin (equal and opposite).

They are equal and opposite at the start, but they don't remain so because one twin changes reference frames and another does not.

For simplicity, let’s say that the twins are just travelling together. They each view that they are stationary and the other is travelling toward them. The velocity/speed and distance of the “approaching twin” would be exactly the same for both. They make their final age comparisons in the “same frame” and any time dilation would apply exactly the same to both.

They would indeed conclude that. But if they don't start in the same place, then you haven't addressed how their clocks start, and if they're not in the same place, that part of the problem ends up reference-frame dependent as well.

 

[ynot]

That's not a required part of the problem. It is enough that they start in the same location and end in the same location. Their initial and final velocities are actually irrelevant. Furthermore, they're not just in different frames. One twin is in a single frame for the entire journey, and the other twin is in at least two frames.



They are equal and opposite at the start, but they don't remain so because one twin changes reference frames and another does not.



They would indeed conclude that. But if they don't start in the same place, then you haven't addressed how their clocks start, and if they're not in the same place, that part of the problem ends up reference-frame dependent as well.

Their clocks were started from a flash of light emitted from a point exactly mid-distance between them. Or perhaps they are both travelling in opposite directions around a common circular path. They synchronise their watches when they pass. Or did an omnipresent God say to them both “Start your clocks now“.

 

[ynot]

I don't understand what you're saying. Did you just put both twins on the rocket? Or are you talking about what happens when the twin paradox scenario is viewed from some other frame, say the frame of another rocket?

Hm, maybe you just need to re-read the last part of Ziggurat's most recent reply to you.

I’m saying that, forgetting any acceleration effects, or let’s say the twins both accelerate way from and return to a common point equally, they should both experience identical time dilation. I know this is not the conditions of the “twins paradox” and I wish I hadn’t bought them up again.

 

[Fredrik]

If they e.g. both leave Earth in rockets moving at 0.8c in opposite directions, and both turn their rockets around when they have experienced 12 years, they will of course both have aged 24 years when they get back to Earth, even though 40 years will have passed on Earth.

Even in the twin paradox scenario, time dilation affects them both the same way: From A's point of view B is aging at 60% of A's aging rate. From B's point of view A is aging at 60% of B's aging rate. The reason this isn't a paradox is that the moment before B turns around, he's in a frame where A has aged 7.2 years, and the moment after he's turned around, he's in a frame where B has aged 32.8 years.

 

[ynot]

If they e.g. both leave Earth in rockets moving at 0.8c in opposite directions, and both turn their rockets around when they have experienced 12 years, they will of course both have aged 24 years when they get back to Earth, even though 40 years will have passed on Earth.

I take it that this is only from the Earth’s frame, not each of the twin’s frames? But when the twins are back on Earth, they’re all in the same frame.

Even in the twin paradox scenario, time dilation affects them both the same way: From A's point of view B is aging at 60% of A's aging rate. From B's point of view A is aging at 60% of B's aging rate. The reason this isn't a paradox is that the moment before B turns around, he's in a frame where A has aged 7.2 years, and the moment after he's turned around, he's in a frame where B has aged 32.8 years.

And doesn’t it also follow that the moment before A turns around, he's in a frame where B has aged 7.2 years, and the moment after he's turned around, he's in a frame where A has aged 32.8 years?

When the Earth and the twins are all back together again (in the same frame), the Earth observes that the twins have both been time dilated the same amount, but each twin observes that the other has been time dilated a different amount. Each twin has three ages. Each twin doesn’t think itself has been time dilated, but the Earth thinks they have been time dilated one amount, and the other twin thinks a different amount.

 

[Fredrik]

I take it that this is only from the Earth’s frame, not each of the twin’s frames? But when the twins are back on Earth, they’re all in the same frame.

I said "when they have experienced 12 years" to indicate that the 12 years are in their respective frames. I just meant that A turns around when his own clock shows that 12 years have passed, and that B turns around when his own clock shows that 12 years have passed.

And doesn’t it also follow that the moment before A turns around, he's in a frame where B has aged 7.2 years, and the moment after he's turned around, he's in a frame where A has aged 32.8 years?

No, because I was talking about the standard twin paradox here, where one of the twins stay on Earth. And in that scenario, nothing special happens in A's frame when B turns around.

In the alternative scenario, where they both travel at 0.8c in opposite directions, something like what you described must of course happen. The situation is perfectly symmetrical, so the age of B in A's frame makes a big jump ahead, and the age of A in B's frame makes the same jump ahead. The numbers wouldn't be 7.2 and 32.8 in this scenario, because their relative speed isn't 0.8c, it's 0.96c. (The relativistic formula for addition of velocities isn't u+v, it's (u+v)/(1+uv/c²)).

When the Earth and the twins are all back together again (in the same frame), the Earth observes that the twins have both been time dilated the same amount,

Yes. (I assume that we're talking about the scenario where both twins leave Earth in opposite directions).

but each twin observes that the other has been time dilated a different amount.

Yes. (I assume that you mean a different amount than what an observer on Earth would say that they have). Both twins would say that the other aged at only 21% of their own aging rate, but jumped ahead by a huge amount at a certain moment. This jump ahead exactly cancels the time dilation.

Each twin has three ages. Each twin doesn’t think itself has been time dilated, but the Earth thinks they have been time dilated one amount, and the other twin thinks a different amount.

No. If they were 20 years old when they started and their buddy Rumpelstiltskin was 20 years old too, the twins are now 44 years old, and Rumpelstiltskin is 60 years old. No one is any other age, in any frame. (There are never any disagreements between different frames about what happens at a certan event. There are only disagreements about what happens at a certain moment or at a certain position).

 

[BillyJoe]

I wasn't trying to answer that question. I just wanted to explain why the "twin paradox" isn't a paradox.

Fair enough. I thought we were trying to help ynot understand that time dilation is real and doesn't depend on acceleration.

But that isn't time dilation. It's another effect. I will call it a "simultaneity shift" here.

From A's point of view, this is what happens: Because of time dilation, B ages at only 60% of A's aging rate, so when the rocket gets back to Earth after 40 years, B has only aged 24 years. Nothing at all happens when the rocket turns around, except that for a brief moment B's aging rate catches up with A's.

From B's point of view, this is what happens: Because of time dilation, A ages at only 60% of B's aging rate, so the moment before the rocket turns around after 12 years, A has only aged 7.2 years. When the rocket turns around, the simultaneity shift makes A's age jump ahead by 25.6 years, so the moment after the rocket has turned around, A is 7.2+25.6-12=20.8 years older than B. Because of time dilation, A continues to age at only 60% of B's aging rate for the rest of the trip, so when the rocket gets back to Earth after 12 more years, A has only aged another 7.2 years, so the age difference is only 20.8+7.2-12=16 years.

Acceleration doesn't cause time dilation in this scenario. Instead it causes the simultaneity shift that prevents the paradox that would arise if time dilation had been the only relativistic effect in this problem. This would be "woo" if we had to assume that there's a simultaneity shift that resolves the paradox, but we can prove that there is using the same methods that proved time dilation, and without making any new assumptions.

Yeah, I said "your diagram almost implies that time dilation IS caused by acceleration. From B's point of view, A instantaneously ages 25.6 years when B accelerates."
I should perhaps have said "your diagram makes it look like...."
I understand your explanation, but I think it is a bit complicated for someone trying to understand some basics to read:

"When the rocket turns around, the simultaneity shift makes A's age jump ahead by 25.6 years..."

Of course this could never happen because the acceleration that causes the "simultaneity shift" is .8c per instant for one instant!!! In other words, the abrupt ageing of A by 25.6 years is impossible because the acceleration of B that would be needed to cause this "simultaneity shift" is impossible. So, I think we are unnecessarily complicating things here.

I've seen the scenario you describe before. It's the twin paradox without acceleration. I see now that you made it a bit more complicated than it needs to be by using the frame o. You could have drawn it from from B's point of view, like this:

You have? But I just made it up!
And I used o because ynot seemed a bit hung up on one of the twins being stationary on Earth, so I decided to have everyone moving relative to some independent frame. As I said before, sometimes you can't win!

You also made the slope of line C such that it would indicate a superluminal speed in a space-time diagram with c=1, and you made the time coordinate increase in the down direction, which is also unconventional.

Hey? I don't even have any units or speeds on my diagram! And no time coordinate at all!!!

And, finally, you don't need to consider a version of the twin paradox to see that time dilation is caused by speed rather than acceleration. That can be seen directly from the invariance of the quantity -t²+x².

Try telling ynot that!

 

[BillyJoe]

I think any question whether time dilation is caused by acceleration has been answered.

Great!

If we ignore the periods of acceleration, and just consider the periods that the objects move apart and come back together again, we can’t attribute any particular speed to either object, only to the rate at which they move apart and come together. In other words, we can’t say that one is moving and the other is stationary, or that one is moving at a different speed than the other. I can’t see therefore that any time dilation that occurs to one object, doesn’t occur in the same measure and effect to the other, or that time dilation even occurs.

Hence my idea to not use acceleration at all in my example!!!
Good Grief!!!

 

[Paulhoff]

When the Earth and the twins are all back together again (in the same frame), the Earth observes that the twins have both been time dilated the same amount, but each twin observes that the other has been time dilated a different amount. Each twin has three ages. Each twin doesn’t think itself has been time dilated, but the Earth thinks they have been time dilated one amount, and the other twin thinks a different amount.

No, now you know that only one twin and that is the twin that left Earth, moved, no matter how you may misread this, only one twin left at a high speed and came back at a high speed. It was only the twin that left that seen the Sun's, the Moon's, the star's and everything else's time change. The twin that stayed didn't see any change to any of them, it was only to the twin that left that did, and both of them didn’t experience the same conditions.

Paul

 

[Fredrik]

You have? But I just made it up!
...
Hey? I don't even have any units or speeds on my diagram! And no time coordinate at all!!!

OK, I just assumed that it was a space-time diagram. In that case, I haven't seen that exact scenario before. I think it's better to use one spatial dimension instead of two because that makes it much easier to draw a space-time diagram. The scenario I've seen before is basically your scenario projected onto the x axis.

I understand your explanation, but I think it is a bit complicated for someone trying to understand some basics to read:

"When the rocket turns around, the simultaneity shift makes A's age jump ahead by 25.6 years..."

Of course this could never happen because the acceleration that causes the "simultaneity shift" is .8c per instant for one instant!!! In other words, the abrupt ageing of A by 25.6 years is impossible because the acceleration of B that would be needed to cause this "simultaneity shift" is impossible. So, I think we are unnecessarily complicating things here.

It wasn't my choice to start discussing the twin paradox into this thread. It's certainly not necessary to discuss problems that are that difficult to learn the basics of SR.

However, the resolution of the paradox can't be made much simpler than I've made it. The problem would have been more complicated if I had chosen to make the acceleration finite.