What Causes Time Dilation in the Twin Paradox?

[granpa]

The "SR simultaneity rule" is not a law of physics, it's just true by convention.

it may not be a law of physics but if you use something different then you will have to rewrite the laws of physics for that frame.

after all the whole point of relativity is that the laws of physics are the same in every frame. that is where the Einstein simultaneity rule comes from.

 

[Al68]

it may not be a law of physics but if you use something different then you will have to rewrite the laws of physics for that frame.

after all the whole point of relativity is that the laws of physics are the same in every frame. that is where the Einstein simultaneity rule comes from.

Of course, but we're never required to even consider what Earth's clock reads simultaneous with any event in the ship's frame. I didn't say the rule was wrong, just that we don't need to use it.

Al

 

[Al68]

why not consider 3 grids of synchronized clocks. one for Earth frame. one for outbound rocket frame. and one for inbound rocket frame.

I like this one. No need to worry about what a distant clock in a different frame reads, nothing "weird" happens at the turnaround, and the ship can just look out the window and see what time it is in Earth's frame (local to the ship) at any time. And that clock is the time that the Earth clock would read "now" if the ship decides to return to Earth's inertial frame (stop) near that clock.

Al

 

[Fredrik]

Of course, but we're never required to even consider what Earth's clock reads simultaneous with any event in the ship's frame.

We don't have to think about simultaneity to prove that there's no paradox, or to find the correct final ages of the twins, but we have to do it if we want to explain what's wrong with the calculation that says A is younger.

why not consider 3 grids of synchronized clocks. one for Earth frame. one for outbound rocket frame. and one for inbound rocket frame.

That's what my spacetime diagram does. The jump from 7.2 years to 32.8 years is the correction that's needed for the error that's introduced by simply switching from the first of B's inertial frames to the second.

No need to worry about what a distant clock in a different frame reads, nothing "weird" happens at the turnaround, and the ship can just look out the window and see what time it is in Earth's frame (local to the ship) at any time. And that clock is the time that the Earth clock would read "now" if the ship decides to return to Earth's inertial frame (stop) near that clock.

You only need one of those "grids" (inertial frames) for that, so I assumed that he had something else in mind when he started talking about three of them.

 

[Al68]

Although the sudden jump simultaneity convention works in this particular version of the twin paradox, it's kinda weird. George Jones posted a paper with a nice simultaneity convention that's a lot smoother for B (surprisingly simple too): https://www.physicsforums.com/showthread.php?p=1893032&highlight=accelerated#post1893032

It's only weird because looking at the coordinate distance of the ship from Earth (or equivalently, clock time on earth) in the ship's frame before, during, and after the turnaround is, IMO, just a weird way to look at it. It's like the only reason to look at it this way is to have an exercise in the simultaneity rule for its own sake.

I'll have to look at that other thread.

Al

 

[granpa]

It's only weird because looking at the coordinate distance of the ship from Earth (or equivalently, clock time on earth) in the ship's frame before, during, and after the turnaround is, IMO, just a weird way to look at it. It's like the only reason to look at it this way is to have an exercise in the simultaneity rule for its own sake.

I'll have to look at that other thread.

Al

as has already been pointed out several times, we don't need simultaneity to figure out how much less the twin will age. we need it to explain why the twin that ages less observes the other twin aging more slowly on the outbound and inbound parts of the trip. that is the whole point of the paradox.

 

[Al68]

We don't have to think about simultaneity to prove that there's no paradox, or to find the correct final ages of the twins, but we have to do it if we want to explain what's wrong with the calculation that says A is younger.

Sure, if someone makes the erroneous calculation.

The jump from 7.2 years to 32.8 years is the correction that's needed for the error that's introduced by simply switching from the first of B's inertial frames to the second.

Sure, if we care about Earth's clock reading simultaneous with the turnaround in the ship's frame.

You only need one of those "grids" (inertial frames) for that, so I assumed that he had something else in mind when he started talking about three of them.

Well, the other two grids could be used to look at the ship's frame(s) from Earth's pov.

Fredrik, do you believe that the "jump" in the coordinate position of the Earth (and Earth's clock, since it's directly dependent) in the ship's frame during the acceleration is more than just a result of changing the pov/reference frame?

Al

 

[Al68]

as has already been pointed out several times, we don't need simultaneity to figure out how much less the twin will age. we need it to explain why the twin that ages less observes the other twin aging more slowly on the outbound and inbound parts of the trip. that is the whole point of the paradox.

Only if the ship's twin looks at it in this "weird" way. If he ignores what time on Earth is simultaneous with local time, he could just look out his window at a clock in the grid. That won't tell him what time it is on Earth "now", but that's OK if he's not asking. It will tell him how much time will have elapsed on Earth if he chooses to return to Earth's frame (stop) near any of the clocks. That should make it clear to him that, since each clock shows shows more elapsed time than his own (by the gamma factor), that his twin on Earth will be older whenever and wherever he returns to Earth's frame.

Al

 

[granpa]

Only if the ship's twin looks at it in this "weird" way. If he ignores what time on Earth is simultaneous with local time, he could just look out his window at a clock in the grid. That won't tell him what time it is on Earth "now", but that's OK if he's not asking. It will tell him how much time will have elapsed on Earth if he chooses to return to Earth's frame (stop) near any of the clocks. That should make it clear to him that, since each clock shows shows more elapsed time than his own (by the gamma factor), that his twin on Earth will be older whenever and wherever he returns to Earth's frame.

Al

it wouldn't seem weird to the people on the rocket. thear time would seem perfectly natural to them.

if the Earth twin accelerates and catches up with the moving twin and they spent the rest of their lives traveling at relativistic speed then Earth time would weird. its all relative.

 

[neopolitan]

It's only weird because looking at the coordinate distance of the ship from Earth (or equivalently, clock time on earth) in the ship's frame before, during, and after the turnaround is, IMO, just a weird way to look at it. It's like the only reason to look at it this way is to have an exercise in the simultaneity rule for its own sake.

I'll have to look at that other thread.

Al

Woot, that is a far more complex version of what I was thinking. Although I never thought about in terms of that prism-like effect in Figures 5 and 6, I do think I understand how it works.

Thanks for that link, Al.

cheers,

neopolitan

 

[Al68]

Woot, that is a far more complex version of what I was thinking. Although I never thought about in terms of that prism-like effect in Figures 5 and 6, I do think I understand how it works.

Thanks for that link, Al.

cheers,

neopolitan

Well, atyy provided the link, I only quoted his post.

Al

 

[Fredrik]

Fredrik, do you believe that the "jump" in the coordinate position of the Earth (and Earth's clock, since it's directly dependent) in the ship's frame during the acceleration is more than just a result of changing the pov/reference frame?

No.

(What else would it be? I can't even think of a wrong answer to that question.)

 

[Al68]

No.

(What else would it be? I can't even think of a wrong answer to that question.)

OK, something in another post made me wonder if I was missing something. My bad.

Al

 

[Dale]

We don't have to think about simultaneity to prove that there's no paradox, or to find the correct final ages of the twins, but we have to do it if we want to explain what's wrong with the calculation that says A is younger.

Q: What's wrong with the calculation that says A is younger?

A: It treats a non-inertial frame as an inertial frame.

No simultaneity needed.

 

[Fredrik]

Q: What's wrong with the calculation that says A is younger?

A: It treats a non-inertial frame as an inertial frame.

No, it doesnt. It doesn't even try to associate a coordinate system with B's world line. All it does is to combine the result of two calculations performed in two different coordinate systems. It must seem quite plausible to someone less experienced with relativity calculations that that should work, since the two frames agree that A's aging rate is 60% of B's. You clearly don't have to believe that B's path is a geodesic to think that the incorrect calculation looks correct.

Usually when someone makes the mistake of switching coordinate systems in the middle of a calculation, the reason why that doesn't work is that the two coordinate systems disagree about the specific thing you're calculating. How long did your plane trip take? Calculate it as local arrival time minus local departure time and you get the wrong result if the destination is in another time zone. The answer is wrong because the coordinate systems (time zones) disagree about the time. But in the twin paradox, the coordinate systems don't disagree about the aging rate. The only relevant thing they disagree about is simultaneity.

 

[granpa]

Q: What's wrong with the calculation that says A is younger?

A: It treats a non-inertial frame as an inertial frame.

No simultaneity needed.

but it is a fact that the moving twin perceives the stationary twin to be aging more slowly both on the outbound and the inbound parts of the trip. how do you explain this to a student?

 

[JesseM]

but it is a fact that the moving twin perceives the stationary twin to be aging more slowly both on the outbound and the inbound parts of the trip. how do you explain this to a student?

But he doesn't really "perceive" this in any direct sense (that certainly isn't what he sees), it's just that during each part of the trip, if he uses an inertial coordinate system where he is at rest during that phase, then in that coordinate system the other twin will be aging more slowly than himself. I would just explain to the student that you can't combine the elapsed ages for the inertial twin in the two coordinate systems for each leg of the trip, because the definition of simultaneity used in the first coordinate system is different from the definition in the second, so the inertial twin's age at the moment of the turnaround in the first one is very different from the inertial twin's age at the moment of the turnaround in the second one.

 

[granpa]

But he doesn't really "perceive" this in any direct sense (that certainly isn't what he sees),

it is exactly what he 'sees' as long as he takes light travel time into account. where are you getting the idea that it isnt. this is simple relativity.

 

[Fredrik]

I would just explain to the student that you can't combine the elapsed ages for the inertial twin in the two coordinate systems for each leg of the trip, because the definition of simultaneity used in the first coordinate system is different from the definition in the second,

What we're really discussing here is if the second half of the quoted text above needs to be included at all. DaleSpam is of the opinion that all we need to say is that there's no inertial frame in which B is stationary during the whole trip. "The End". My opinion is that this doesn't really explain why it's wrong to just use the time dilation formula on the two straight parts of B's world line separately. I think the only thing that can explain that is what you just said.

 

[Dale]

No, it doesnt. It doesn't even try to associate a coordinate system with B's world line.

Yes it does, specifically it tries to associate a coordinate system where B is at rest the whole time (B's world line is straight and vertical at all points). That is a non-inertial coordinate system.

All it does is to combine the result of two calculations performed in two different coordinate systems.

If you want to combine results performed in two different coordinate systems you must always properly transform your results from one into the other. Since you don't explicitly perform a coordinate transform you are implicitly working in a single coordinate system and that coordinate system is non-inertial.

 

[granpa]

so you accept that the moving twin sees the stationary twin aging more slowly during the outbound part of his trip and also during the inbound part of his trip. and you accept that it is the fact that he accelerates during the turn around that causes him to actually age less. so what exactly are you arguing?

 

[Dale]

but it is a fact that the moving twin perceives the stationary twin to be aging more slowly both on the outbound and the inbound parts of the trip. how do you explain this to a student?

Again, this is the same point I have been making all along. While the twins are not together they cannot make local comparisons of their clocks, so any "relative aging" claims are actually statements about the time coordinate in a given reference frame. Any reference frame where the stationary twin is aging more slowly on both legs is a non-inertial reference frame.

so you accept that the moving twin sees the stationary twin aging more slowly during the outbound part of his trip and also during the inbound part of his trip.

Only in a non-inertial reference frame.

and you accept that it is the fact that he accelerates during the turn around that causes him to actually age less.

The fact that he accelerates during the turn around is what indicates that his rest frame is non-inertial.

so what exactly are you arguing?

I am arguing that the spacetime geometric approach is completely sufficient for resolving the paradox because the metric is different in non-inertial reference frames and you cannot get a twin paradox in flat spacetime without using a non-inertial reference frame.

IMO, it is more important to teach a student to identify and avoid non-inertial reference frames than to teach them about confusing and arbitrary simultaneity conventions that can arise in non-inertial reference frames. Most students struggle with the relativity of simultaneity more than any other concept even in inertial reference frames. The extra confusion of simultaneity conventions in non-inertial frames is not necessary to the resolution and therefore should be avoided.

 

[granpa]

Again, this is the same point I have been making all along. While the twins are not together they cannot make local comparisons of their clocks, so any "relative aging" claims are actually statements about the time coordinate in a given reference frame.

they are statements about what each will actually see. while they are not accelerating

Any reference frame where the stationary twin is aging more slowly on both legs is a non-inertial reference frame.

that is exactly what we are saying. why do you say that as though it disagrees with what we are saying.
non-inertial=accelerating. acceleration=change in simultaneity.

 

[granpa]

if we assume that the laws of physics are the same in every frame then we must conclude that simultaneity is lost and that it follows the Einstein convention. so do you not believe that the laws of physics are the same in every frame?

 

[JesseM]

What we're really discussing here is if the second half of the quoted text above needs to be included at all. DaleSpam is of the opinion that all we need to say is that there's no inertial frame in which B is stationary during the whole trip. "The End". My opinion is that this doesn't really explain why it's wrong to just use the time dilation formula on the two straight parts of B's world line separately. I think the only thing that can explain that is what you just said.

If this is an accurate summary of DaleSpam's view I think I'd agree with you and disagree with DaleSpam--after all, if we want to integrate along a curve to find its length from point A to C in Euclidean space, we're free to pick some point B along the curve between A and B, and use one cartesian coordinate system to do an integral which gives us the length from A to B, and a different cartesian coordinate system to do an integral which gives us the length from B to C, and then add these two lengths. Here we are not using a single non-cartesian coordinate system, but rather adding results from two different cartesian coordinate systems, which is perfectly valid in this situation. If different inertial frames in SR agreed on simultaneity you could do something similar in the twin paradox, using one inertial frame to find the time elapsed on the inertial twin's clock between the moment the other twin departed and the moment the other twin turned around, and using a different inertial frame to find the time elapsed on the inertial twin's clock between the moment the other twin turned around and the moment they reunited. So knowing about the relativity of simultaneity is key to understanding why you can't just add partial elapsed times from two different frames in this way to get the total elapsed time.

 

[JesseM]

it is exactly what he 'sees' as long as he takes light travel time into account. where are you getting the idea that it isnt. this is simple relativity.

Light travel time depends on knowing the distance that the signal was when it was emitted, and this depends on your choice of coordinate system as well. I object to your use of the word "perceive", which makes it sound like it's just a straightforward observation that doesn't depend on choosing a particular coordinate system in which to define measurements. For example, if the twin that turns around uses an inertial coordinate system where he is at rest during the outbound phase, and then continues to use that same coordinate system during the inbound phase (rather than switching to a new coordinate system where he is at rest during the inbound phase), then he will find that the inertial twin's clock is ticking faster than his own during the inbound phase, not slower. This will be true even if the only thing he uses the coordinate system for is to calculate the distance of the inertial twin from himself when the light from each clock tick was emitted, in order to figure out how long ago each tick "really" happened by subtracting the light travel time.

 

[granpa]

the phrase 'what he sees' implies that he is using his current frame in which he is at rest.

 

[JesseM]

the phrase 'what he sees' implies that he is using his current frame in which he is at rest.

Most physicists seem to use the word "observes" for what is happens in a given observer's rest frame, while "sees" refers to actual visual appearances.

 

[granpa]

lol.

 

[JesseM]

Well, whatever you think of the terminology, my original point stands: "perceives" makes it sound too physical, when in fact your statement depends on the non-inertial observer first picking one coordinate system during the outbound leg and another during the inbound leg, any time you have a non-inertial observer the choice of what coordinate system(s) represent his "perceptions" is pretty arbitrary. I could equally well invent a coordinate system where the non-inertial observer is at rest and in which the inertial twin's clock alternates between ticking faster and slower, and then say based on this that the non-inertial twin "percieves" the inertial twin's alternates between fast and slow ticking throughout the journey--this statement would be no more or less physical than your own.

 

[neopolitan]

...(therefore - ed.) you accept that it is the fact that he accelerates during the turn around that causes him to actually age less ...

This, along with the discontinuity, is a sticking point.

Having confirmed that Fredrik does not believe there is a real discontinuity, I thought this reduced ageing due to acceleration was dead too.

Is it possible to garner informed opinion on this: which is true?

A. Acceleration has no effect on relative ageing, per se, only the resultant trajectory through spacetime is relevant.

B. Acceleration does affect relative ageing directly but for some reason only at the middle of a two way journey which ends up with the traveller back at the origin. (Note this is acceleration, not resisted force such as gravity. I am trying to stick to SR as much as possible.)

C. Acceleration does affect relative ageing directly, during all accelerations during the journey. (Note this is acceleration, not resisted force such as gravity. I am trying to stick to SR as much as possible.)

In my understanding, the answer is A. I also interpret DaleSpam's response as A.

cheers,

neopolitan

 

[granpa]

This, along with the discontinuity, is a sticking point.

Having confirmed that Fredrik does not believe there is a real discontinuity, I thought this reduced ageing due to acceleration was dead too.

Is it possible to garner informed opinion on this: which is true?

A. Acceleration has no effect on relative ageing, per se, only the resultant trajectory through spacetime is relevant.

B. Acceleration does affect relative ageing directly but for some reason only at the middle of a two way journey which ends up with the traveller back at the origin. (Note this is acceleration, not resisted force such as gravity. I am trying to stick to SR as much as possible.)

C. Acceleration does affect relative ageing directly, during all accelerations during the journey. (Note this is acceleration, not resisted force such as gravity. I am trying to stick to SR as much as possible.)

In my understanding, the answer is A. I also interpret DaleSpam's response as A.

cheers,

neopolitan

now please don't take my comments out of context. I was speaking to one particular person with whom I was looking for semething we could agree on. and I was speaking in the broadest possible terms. acceleration doesn't effect aging directly. time dilation affects aging.

but we all agree that it is acceleration that is the key to understainding the twin paradox.

 

[neopolitan]

now please don't take my comments out of context. I was speaking to one particular person with whom I was looking for semething we could agree on. and I was speaking in the broadest possible terms. acceleration doesn't effect aging directly. time dilation affects aging.

but we all agree that it is acceleration that is the key to understainding the twin paradox.

I wasn't ascribing a particular point of view, granpa, I was trying to clarify exactly what your point of view is, and whether there are informed people who actually think that acceleration does affect the ageing.

I agree that, in so much as the twin which undergoes acceleration and returns to the unaccelerated twin and that means that one twin experiences two inertial frames while the other experiences one, then, yes, acceleration is key. But it is not the acceleration, per se, but the consequence of the acceleration.

And acceleration causes no time dilation directly. Different inertial frames cause time dilation.

I assume that you do not think that acceleration causes time dilation, but I can't put my hand on my heart and say that I know that you don't. (I could I guess, but I would not be being genuine :) )

cheers,

neopolitan

 

[Dale]

Any reference frame where the stationary twin is aging more slowly on both legs is a non-inertial reference frame.

that is exactly what we are saying. why do you say that as though it disagrees with what we are saying.

I haven't been following what you have been saying, but this disagrees with what Frederik has been saying. He has been claiming that you can obtain the twin paradox without reference to a non-inertial frame. Therefore I have been pointing out how every time he thinks he is avoiding a non-inertial frame he is in fact not avoiding it.

If this is an accurate summary of DaleSpam's view I think I'd agree with you and disagree with DaleSpam--after all, if we want to integrate along a curve to find its length from point A to C in Euclidean space, we're free to pick some point B along the curve between A and B, and use one cartesian coordinate system to do an integral which gives us the length from A to B, and a different cartesian coordinate system to do an integral which gives us the length from B to C, and then add these two lengths. Here we are not using a single non-cartesian coordinate system, but rather adding results from two different cartesian coordinate systems, which is perfectly valid in this situation.

But that is exactly what you are not doing. If you are to correctly perform such an integration you must do a change of variables. If you do not do a change of variables then you are implicitly using a single non-cartesian coordinate system.

\int_a^c f(x) \, dx=\int_a^b f(x) \, dx+\int_b^c f(x)<br /> \, dx=\int_a^b f(x) \, dx+\int_{b&#039;}^{c&#039;} f\left(x&#039;\right) \, dx&#039;\neq<br /> \int_a^b f(x) \, dx+\int_b^c f\left(x&#039;\right) \, dx&#039;
where the primed variables are the same points but in a different cartesian coordinate system.

Since in the analysis of the twins paradox you do not properly transform between the two inertial coordinate systems you are not doing two separate integrals in cartesian/inertial coordinates, but rather one integral in a non-cartesian/non-inertial coordinate system.

You cannot possibly obtain the twin paradox without using a non-inertial coordinate system and treating it as an inertial coordinate system. Therefore all that needs to be taught is how to identify and avoid non-inertial coordinate systems.

 

[JesseM]

If this is an accurate summary of DaleSpam's view I think I'd agree with you and disagree with DaleSpam--after all, if we want to integrate along a curve to find its length from point A to C in Euclidean space, we're free to pick some point B along the curve between A and B, and use one cartesian coordinate system to do an integral which gives us the length from A to B, and a different cartesian coordinate system to do an integral which gives us the length from B to C, and then add these two lengths. Here we are not using a single non-cartesian coordinate system, but rather adding results from two different cartesian coordinate systems, which is perfectly valid in this situation.

But that is exactly what you are not doing. If you are to correctly perform such an integration you must do a change of variables. If you do not do a change of variables then you are implicitly using a single non-cartesian coordinate system.

Of course I'm doing a change of variables--for the first integral from A to B you'd use the position coordinates of the first coordinate system in order to get the coordinate-independent length of the curve from A to B, and for the second integral from B to C you'd use the position coordinates of the second coordinate system in order to get the coordinate-independent length from B to C, then you'd add up the two coordinate-independent lengths to get the total length from A to C. I don't understand why you think this change of variables conflicts with what I said above in the quoted section, though.

Since in the analysis of the twins paradox you do not properly transform between the two inertial coordinate systems you are not doing two separate integrals in cartesian/inertial coordinates, but rather one integral in a non-cartesian/non-inertial coordinate system.

I don't understand what you mean by "transform between the two inertial coordinate systems" in this context. If different coordinate systems didn't disagree about simultaneity, you could certainly use one coordinate system to find the coordinate-independent proper time elapsed on the inertial twin's worldline between the moment the other twin departed and the moment the other twin turned around, and then use a different coordinate system to find the coordinate-independent proper time elapsed on the inertial twin's worldline between the moment the other twin turned around and the moment they reunited. Since proper times are coordinate-independent, you could then just add these to find the total time elapsed on the inertial twin's clock between the moment they separated and the moment they reunited, no need for any transformation of these times. Of course, since the two coordinate systems don't define simultaneity the same way this procedure is wrong because it leaves a section of the inertial twin's worldline unaccounted for, but that difference in simultaneity between the two frames is really the only reason to reject this procedure.

 

[phyti]

The twin paradox appeared after considering the effects of SR. To avoid
accelerations, the travel times can be extended so as to make the reversal
neglible. The gamma expression for time dilation specifically states that speed
relative to c is the determining factor. When you eliminate the reversal, you trade
off the gradual catching up of signals from the Earth twin for a discontinuous
'jump' in Earth time, and must insert this (reset the Earth clock).
It can be shown geometrically that the twin on the two leg trip, ages slower on at
least one of the legs, and less in total than the twin on the one leg trip.

 

[Dale]

Of course I'm doing a change of variables--for the first integral from A to B you'd use the position coordinates of the first coordinate system in order to get the coordinate-independent length of the curve from A to B, and for the second integral from B to C you'd use the position coordinates of the second coordinate system in order to get the coordinate-independent length from B to C, then you'd add up the two coordinate-independent lengths to get the total length from A to C.

If you do such a change of variables then the twin paradox does not occur. The twin paradox only occurs if you fail to transform coordinates at the turnaround (thereby implicitly using a non-inertial reference frame).

Of course, since the two coordinate systems don't define simultaneity the same way this procedure is wrong because it leaves a section of the inertial twin's worldline unaccounted for, but that difference in simultaneity between the two frames is really the only reason to reject this procedure.

That and the fact that it is mathematically incorrect. Just because there might be some pairs of coordinate systems where the transformation of one or more coordinates is identity doesn't mean that you can skip the transform.

 

[JesseM]

If you do such a change of variables then the twin paradox does not occur. The twin paradox only occurs if you fail to transform coordinates at the turnaround (thereby implicitly using a non-inertial reference frame).

What specifically do you mean by "transform coordinates at the turnaround" in this example? Let's stick to the example of calculating the length of a curve in ordinary 2D Euclidean space, where I say that using two different coordinate systems to calculate two parts of the length is perfectly valid. Do you agree that length along a curve is coordinate-invariant, so if I calculate the length from point A to point B in one cartesian coordinate system, and I calculate the length from point B to C in another, I can just add them together to get the length from point A to C? Do you think I have to "transform coordinates at point B" here, and am therefore "implicitly using a non-cartesian coordinate system"?

That and the fact that it is mathematically incorrect. Just because there might be some pairs of coordinate systems where the transformation of one or more coordinates is identity doesn't mean that you can skip the transform.

I don't understand what you mean by "the transformation of one or more coordinates is identity"--what specific coordinates do you think I am transforming in this example?

 

[Dale]

Do you think I have to "transform coordinates at point B" here, and am therefore "implicitly using a non-cartesian coordinate system"?

Most definitely you must transform the coordinates. Let's say that in some Cartesian coordinate system A = (ax,ay) and the coordinates of that same point in some other Cartesian coordinate system is A' = (ax',ay'). Similarly for B and C. Because the norm is invariant we have |A-B| + |B-C| = |A-B| + |B'-C'| != |A-B| + |B-C'| != |A-B| + |(bx,by')-C'|.

Any coordinate system in which either of the last two equations would be true would necessarily be a non-Cartesian coordinate system and would therefore use a different form of the norm that the standard Cartesian norm.

 

[JesseM]

Most definitely you must transform the coordinates. Let's say that in some Cartesian coordinate system A = (ax,ay) and the coordinates of that same point in some other Cartesian coordinate system is A' = (ax',ay'). Similarly for B and C. Because the norm is invariant we have |A-B| + |B-C| = |A-B| + |B'-C'| != |A-B| + |B-C'| != |A-B| + |(bx,by')-C'|.

What does notation like |A-B| represent in this context? Is it \sqrt{(ax - bx)^2 + (ay - by)^2 }? And when you say "the norm is invariant", the norm of what? I thought we were talking about curves, not vectors. But let's pick a very simple example of a curve--a straight line segment with endpoints A and C, and a point along the line between them labeled B. Say we have two cartesian coordinate systems, in the first the coordinates of A are (ax,ay), the coordinates of B are (bx,by), the coordinates of C are (cx,cy), and in the second coordinate system the coordinates are (ax',ay'), etc. So can't we say that, using the pythagorean theorem in the first coordinate system, the distance from A to B is \sqrt{(bx - ax)^2 + (by - ay)^2 }, and using the pythagorean theorem in the second coordinate system, the distance from B to C is \sqrt{(cx&#039; - bx&#039;)^2 + (cy&#039; - by&#039;)^2 }? And once we have actually evaluated those expressions, since each length is coordinate-invariant, can't we just add the two to get the total length from A to C? I don't see where in this a coordinate transformation has been done.

 

[Dale]

But let's pick a very simple example of a curve--a straight line segment with endpoints A and C, and a point along the line between them labeled B.

Sorry I wasn't clear. That is exactly what I was thinking. We should probably also be explicit that we are considering only rotations and translations.

So can't we say that, using the pythagorean theorem in the first coordinate system, the distance from A to B is \sqrt{(bx - ax)^2 + (by - ay)^2 }, and using the pythagorean theorem in the second coordinate system, the distance from B to C is \sqrt{(cx&#039; - bx&#039;)^2 + (cy&#039; - by&#039;)^2 }? And once we have actually evaluated those expressions, since each length is coordinate-invariant, can't we just add the two to get the total length from A to C? I don't see where in this a coordinate transformation has been done.

The coordinates (bx',by') and (cx',cy') are the result of the coordinate transformation of B and C into the primed frame.

 

[JesseM]

The coordinates (bx',by') and (cx',cy') are the result of the coordinate transformation of B and C into the primed frame.

Sure, but any time you use multiple coordinate systems to analyze the same physical scenario, naturally you must know the coordinate transformation that relates them. Are you arguing that anytime we juggle different cartesian coordinate systems (or different inertial coordinate systems in SR) when making calculations about coordinate-invariant quantities, we are implicitly using a single non-cartesian/non-inertial coordinate system? Suppose instead of breaking the path from A to C into two segments, we instead calculated the entire length from A to C in the first coordinate system, then again calculated the entire length from A to C in the second coordinate system. Aren't we just doing calculations in two separate cartesian coordinate systems here, rather than implicitly using a single non-cartesian coordinate system?

 

[Dale]

Are you arguing that anytime we juggle different cartesian coordinate systems (or different inertial coordinate systems in SR) when making calculations about coordinate-invariant quantities, we are implicitly using a single non-cartesian/non-inertial coordinate system?

No that is not my argument. I'm sorry JesseM, but I don't know how much clearer I can be than posts 189 and 184, and it really frustrates me that you are arguing against such an obvious point as the fact that the twin paradox does not occur unless you try to treat the traveling twin's non-inertial frame as an inertial one. Geometrically it is stunningly obvious that any frame that straightens the travelers worldline is non-inertial.

Suppose instead of breaking the path from A to C into two segments, we instead calculated the entire length from A to C in the first coordinate system, then again calculated the entire length from A to C in the second coordinate system. Aren't we just doing calculations in two separate cartesian coordinate systems here, rather than implicitly using a single non-cartesian coordinate system?

Yes, and when you do that the twin paradox does not arise since all inertial frames agree on the length of each path.

 

[JesseM]

No that is not my argument. I'm sorry JesseM, but I don't know how much clearer I can be than posts 189 and 184

If you would address my questions it might clear it up--I really don't see how the notation you gave in 189 and 184 is relevant to what I'm doing, which is computing the length of two segments of a curve in two separate coordinate systems and then adding the lengths of these segments to find the total length.

and it really frustrates me that you are arguing against such an obvious point as the fact that the twin paradox does not occur unless you try to treat the traveling twin's non-inertial frame as an inertial one.

The most "naive" version of the twin paradox arises because one imagines that the non-inertial twin has a single frame which is used throughout the problem, but I have gotten into arguments with people who put forward a more "sophisticated" version, where they acknowledge that the non-inertial twin has different inertial rest frames at different times and suggest there's no reason why you can't just use different frames for each segment of the journey. My point is that if it weren't for simultaneity issues, this more sophisticated argument would be totally valid, just as it's totally valid to use two different cartesian coordinate systems to calculate the length of different segments of a curve, and then add up the length of each segment to get the total length.

Suppose instead of breaking the path from A to C into two segments, we instead calculated the entire length from A to C in the first coordinate system, then again calculated the entire length from A to C in the second coordinate system. Aren't we just doing calculations in two separate cartesian coordinate systems here, rather than implicitly using a single non-cartesian coordinate system?

Yes, and when you do that the twin paradox does not arise since all inertial frames agree on the length of each path.

Only if the segments you are using actually join with one another correctly at the ends. The point is that if you erroneously say "Ok, I'll use frame #1 to calculate the proper time on the inertial twin's worldline between the moment the twins separate and the moment the non-inertial twin turns around, and I'll use frame #2 to calculate the proper time on the inertial twin's worldline between the moment the non-inertial twin turns around and the moment they reunite", then these two segments do not in fact cover the entire worldline of the inertial twin, because the point on the inertial twin's worldline that's simultaneous with the turnaround in frame #1 is not the same as the point on the inertial twin's worldline that's simultaneous with the turnaround in frame #2. If different frames agreed on simultaneity, the procedure I describe above would be fine, but they don't so it's not.

 

[Al68]

And acceleration causes no time dilation directly. Different inertial frames cause time dilation.

Hi neoploitan,

Sounds like a matter of semantics to me, since the change of reference frames is caused by the acceleration. And gravitational time dilation is even derived from SR, considering accelerated reference frames.

Might I suggest looking at a little known and very different (and not very popular) Twins paradox resolution: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

This is Einstein's resolution, and actually considers the ship to be stationary during the turnaround, and Earth's clock to run faster than the ship's in the ship's accelerated frame, resulting in the Earth twin aging more overall.

Al

 

[neopolitan]

Hi neoploitan,

Sounds like a matter of semantics to me, since the change of reference frames is caused by the acceleration. And gravitational time dilation is even derived from SR, considering accelerated reference frames.

Might I suggest looking at a little known and very different (and not very popular) Twins paradox resolution: http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

This is Einstein's resolution, and actually considers the ship to be stationary during the turnaround, and Earth's clock to run faster than the ship's in the ship's accelerated frame, resulting in the Earth twin aging more overall.

Al

Hi Al,

I don't mean it to sound like semantics. What I am trying to get at is an idea whether anyone actually believes that something funny happens during the turnaround due to the acceleration.

It seems to me that Fredrik and Jesse are saying that during the turnaround there is a change of trajectory which results in a simultaneity view change, sort of like "B" looking at "A"'s 7.2 year clock before turn around and looking past 25.6 years worth of clocks during the change of trajectory to end up looking at "A"'s 32.8 year clock. But this is a simultaneity perspective change, not any sudden ageing of "A".

It also seems to me that some people want to say that during the turnaround, either "A" ages suddenly, or "B" somehow jumps forward or moves quickly forward from a moment simultaneous with "A" having aged 7.2 years to a moment simultaneous with "A" having aged 32.8 years.

I don't think that is semantics.

If everyone pretty much agrees with the first then there is no problem, if a significant number of people agree with the second, I think there is a problem. It sounds terrible to say it, but I wonder if Einstein was on the ball with that text referenced.

I thought that it is "resisting" gravity, as we do by virtue of standing on the surface of a planet, that is instrumental in creating the time dilation effects. A body which is allowed to fall with gravity takes the shortest path through spacetime and will age more slowly, won't it? A body which is allowed to fall takes on a velocity in the same way as rocket ship alters trajectory when accelerated. For that reason, I would have thought that acceleration would not cause any "speeding ahead".

My understanding is also that acceleration has been shown, experimentally, not to cause any time dilation effect.

I would appreciate informed comment, and a polite request for clarification if anyone doesn't understand what the hell I am talking about.

cheers,

neopolitan

 

[Jonathan Scott]

The thing that happens suddenly is no more mysterious than the fact that if you turn your head, things which were ahead of you are SUDDENLY beside you and so on. It isn't the turning of your head which moves things; it is merely that once you have turned it, you see things differently.

Changing velocity instantly changes your point of view. Our normal view of time, space and simultaneity is a simplified illusion which works only because we do not often encounter relativistic speeds, so we have to use mathematics to model how it would really look. The set of events considered to be simultaneous with one's own clock depends on velocity in a very similar way to the way that "beside me" depends on the direction in which you are facing.

 

[granpa]

The thing that happens suddenly is no more mysterious than the fact that if you turn your head, things which were ahead of you are SUDDENLY beside you and so on. It isn't the turning of your head which moves things; it is merely that once you have turned it, you see things differently.

Changing velocity instantly changes your point of view. Our normal view of time, space and simultaneity is a simplified illusion which works only because we do not often encounter relativistic speeds, so we have to use mathematics to model how it would really look. The set of events considered to be simultaneous with one's own clock depends on velocity in a very similar way to the way that "beside me" depends on the direction in which you are facing.

first post to make good sense in a long time.

 

[Al68]

It sounds terrible to say it, but I wonder if Einstein was on the ball with that text referenced.

Hi neoploitan,

Well, I've asked about it on this forum and was very surprised that no one seems to have an opinion, or be familiar with it. The result is the same as the standard version, which makes sense if gravitational time dilation is just SR time dilation in an accelerated frame.

A body which is allowed to fall takes on a velocity in the same way as rocket ship alters trajectory when accelerated.

I think it would be more accurate to say that a body which is allowed to fall takes on a velocity relative to Earth's surface in the same way as Earth takes on a velocity relative to an accelerated rocket. If I throw a ball up in the air, it will turnaround due to my (proper) acceleration relative to the ball, not the other way around.

My understanding is also that acceleration has been shown, experimentally, not to cause any time dilation effect.

Some would object to the word "cause" semantically, but I would say that gravitational time dilation is proven, and it's "caused" by acceleration (relative to inertial frames). And time dilation is a result of relative velocity which is "caused" by acceleration.

I would appreciate informed comment, and a polite request for clarification if anyone doesn't understand what the hell I am talking about.

Well, I would agree that the standard resolution is not a satisfactory explanation. It really only explains how the SR math works. And the jump ahead in Earth's clock is a result of looking at a single event from two frames. Notice that right before the ship reaches earth, in the ship's return frame, the time the ship first left Earth is 66.66 yrs ago, and only 40 yrs has elapsed on earth. In the ship's return frame, the Earth clock reading zero is simultaneous with the ship's clock reading -42.66 yrs, if I did the math right. It's just a convention used to assign times to events in different frames, "simultaneity issues" don't "cause" anything to happen to anyone or their clocks.

Unless one is of the school of thought that the laws of physics are caused by mathematics. (A school with way too many members, IMO.)

Al

 

[Al68]

I thought that it is "resisting" gravity, as we do by virtue of standing on the surface of a planet, that is instrumental in creating the time dilation effects.

There's also the example of differential clock rates between clocks at the "top" and "bottom" of a long rocket. The result is the same for 1G acceleration whether the rocket is firing its thrusters in deep space or sitting on a launchpad on earth. The rocket is accelerating the same relative to an inertial frame either way.

Al