Why Is the Green Observer Always the Bent One in the Twin Paradox?

[Ibix]

Do you understand that if I give you a blank piece of paper then you have the freedom to choose which lines will be bent and which won't?

I can draw anything at all on a piece of paper. But if I'm hoping to depict a twin paradox scenario, the line denoting the traveller needs to have a corner. As Orodruin says, it doesn't matter why it has a corner because the interesting result depends only on the fact that the line is bent, not why it bends.

You can subsequently deform the page to hide the curve of the line if you wish, but you also distort the worldlines of clocks and rulers used to measure things, and the overall result is unchanged.

 

[Ibix]

unless subject to a potential

Unless subject to a force. And the reason we don't draw (for example) the path of the rocket exhaust is that we don't care why the ship turned round - only that it did.

 

[SamRoss]

Again, it is completely irrelevant why the worldline is curved.

You're right - it is irrelevant. The "why" is completely arbitrary but there still must be a why. One observer will have a why and the other won't. To put the confusion with the twin paradox in yet another way, I can accept that the reason one line is curved is arbitrary but I cannot accept it is nonexistent. Omitting the reason for the curve in Minkowski space makes it seem like the author is saying the reason for the curve is nonexistent.

 

[Orodruin]

Omitting the reason for the curve in Minkowski space makes it seem like the author is saying the reason for the curve is nonexistent.

No it is not. It means that it is irrelevant for the treatment. The only relevant thing is the length of the worldline and the shape of the worldline is physical.

 

[SamRoss]

Unless subject to a force. And the reason we don't draw (for example) the path of the rocket exhaust is that we don't care why the ship turned round - only that it did.

I am using force and potential interchangeably as the negative derivative of the potential is the force. Maybe this is only true in Newtonian physics? In any case, I'll accept your correction. Moving on, yes - thank you! - the path of the rocket exhaust is not drawn but it is still the explanation for the deviation of the green guy's path from a straight line. Whether it is drawn or not, it needs to be acknowledged. As I've said to others, it is this omission that is the source of the confusion (at least my confusion).

 

[Orodruin]

I am using force and potential interchangeably as the negative derivative of the potential is the force. Maybe this is only true in Newtonian physics? In any case, I'll accept your correction. Moving on, yes - thank you! - the path of the rocket exhaust is not drawn but it is still the explanation for the deviation of the green guy's path from a straight line. Whether it is drawn or not, it needs to be acknowledged. As I've said to others, it is this omission that is the source of the confusion (at least my confusion).

Again, it is irrelevant. In GR you can setup a similar scenario without any acceleration whatsoever. The important thing is the length of the worldline.

 

[SamRoss]

No it is not.

I assume you mean "No it is not nonexistent" in which case I think we are finally on the same page. I agree that treatment of the problem will continue as described without any further mention of the source of the bend. However, the fact that the source of the bend was not mentioned is why I have found the twin paradox so confusing, despite following the rest of the treatment.

 

[Orodruin]

I assume you mean "No it is not nonexistent" in which case I think we are finally on the same page. I agree that treatment of the problem will continue as described without any further mention of the source of the bend. However, the fact that the source of the bend was not mentioned is why I have found the twin paradox so confusing, despite following the rest of the treatment.

No, I mean omitting the reason is not the same as saying that it is not there. The bend is physical, that is all you need to know. You do not need to know the reason why the line you draw is bent, maybe you drew it like that on purpose, maybe someone bumped into you while you were drawing. When the line is on the paper this is completely irrelevant for the length of the curve. All that is relevant is the curve itself.

 

[Ibix]

negative derivative of the potential is the force

This is only true where potential is defined - it is not the case for all force fields and definitely not for all forces, even in Newtonian physics.

You seem to me to be stuck on a detail, essentially. Yes there is always a physical mechanism that turns the traveller around. It might be a rocket exhaust, or it might be the force of your shoes against the floor (walking to the coffee machine and back makes you a traveling twin, if only you had clocks precise enough to measure the differences), or almost anything.

But this is only the reason for there being a corner in your worldline. All the interesting things happen because the worldline is curved. Why it is curved doesn't matter.

You cannot simply draw the traveller's worldline as straight, because doing so is using different rules to relate what you are drawing to actual physical measurements of location and time. A Minkowski diagram shows the x-position of an object at each time t, using a chosen frame's definition of x and t. An object that turns around therefore results in a curved line. You can't represent a curved line as straight in this scheme. You would need to come up with a new rule for recording x and t values so that the line is straight, and work out the implications of your new rule for all other lines. This can be done, but it's messy.

 

[SamRoss]

No, I mean omitting the reason is not the same as saying that it is not there. The bend is physical, that is all you need to know. You do not need to know the reason why the line you draw is bent, maybe you drew it like that on purpose, maybe someone bumped into you while you were drawing. When the line is on the paper this is completely irrelevant for the length of the curve. All that is relevant is the curve itself.

Again, I think we're in agreement here. Perhaps we're just emphasizing different things. Better yet, maybe when you and I hear or say "twin paradox", we are referring to two different parts of the problem. I take it that when you hear someone say "twin paradox", you think they mean, "How is it that the length of this line is unambiguously longer than the length of this one?" That's not what I mean when I say "twin paradox". I mean, "Why was this line drawn this way while this other one was drawn this way?" I need to hear some physical argument for why the lines have been drawn differently. So let me ask this... Yes or no, is there any physical justification for drawing a curved line in Minkowski space that has not been subject to a force? Also, is there any justification for drawing a straight line in Minkowski space that has been sujbect to a force?

 

[SamRoss]

All the interesting things happen because the worldline is curved. Why it is curved doesn't matter.

It matters to me because the lack of that explanation in many of the proofs I have seen made it seem like there was no physical reason for the curve and that is what led to my confusion. To me, the "interesting" thing is the reason for the curve, not the analysis that comes after it.

 

[stevendaryl]

I understand that perfectly well. Do you understand that if I give you a blank piece of paper then you have the freedom to choose which lines will be bent and which won't?

No, you don't have that freedom. A blank piece of paper has an associated "metric", which determines the distance between nearby points. This determines the length for any curve on the paper. For Euclidean geometry (which applies to a piece of paper), a straight line between two points on the paper is the curve with the shortest length connecting those points.

 

[Nugatory]

It matters to me because the lack of that explanation in many of the proofs I have seen made it seem like there was no physical reason for the curve and that is what led to my confusion. To me, the "interesting" thing is the reason for the curve, not the analysis that comes after it.

On the other hand, we can produce twin paradox situations in which there is no curve in either worldline (in the presence of gravity there can be more than one straight line between two points) and ones in which both worldlines are curved, yet the more curved (depending on how one judges “how curved” a world line is) one is that of the more aged traveler.

At its heart, the twin paradox is just saying that two different paths between the same two points may have different lengths. It so happens that in flat spacetime there is only one straight path between two points, so if we have two different paths at least one of them will not be straight.

 

[Dale]

Mathematically, the acceleration and velocity of an object-centered coordinate system can only be defined relative to another object.

I am not sure that I understand the terminology you are using here. Usually when I use the word "relative" I mean "not invariant". But proper acceleration is invariant, so it is not "relative" in this sense. Also, proper acceleration most definitely does not require reference to another object. Inside a closed box with no external input you can still unambiguously determine the proper acceleration of the box without reference to another object outside the box. So I don't see how the acceleration is "relative".

 

[Ibix]

To me, the "interesting" thing is the reason for the curve, not the analysis that comes after i

"There has to be a force for the ship to turn around" is true in Newtonian physics too. The differential aging is different.

And, as PeroK pointed out, it's possible to do a twin paradox with no acceleration at all - you have one observer who stays at home. Another observer moves inertially past the stay at home and they zero their clocks when they pass. Some time later this second observer meets a third who is traveling in the opposite direction. As they pass, the third observer sets his clock to whatever the second observer's clock reads. When this third observer reaches the stay-at-home they compare clock readings. The result will be the same as the regular twin paradox. So the acceleration can't be important, except inasmuch as (if you do this in flat spacetime with actual twins) at least one of the twins must accelerate for them to meet up again.

 

[Dale]

It matters to me because the lack of that explanation in many of the proofs I have seen made it seem like there was no physical reason for the curve and that is what led to my confusion. To me, the "interesting" thing is the reason for the curve, not the analysis that comes after it.

I think that is a valid point. There is always a reason for the bend. Possible reasons include:
1) some real force, like a rocket engine
2) stitching together two inertial observers that are not mutually at rest
3) curved spacetime

The bend is not where the extra time is "stored" or "created", but it does distinguish the two paths. With that distinction in mind you can then unambiguously examine the length of the respective paths to determine the time difference.

 

[Orodruin]

Yes or no, is there any physical justification for drawing a curved line in Minkowski space that has not been subject to a force? Also, is there any justification for drawing a straight line in Minkowski space that has been sujbect to a force?

No. This is the definition of force. If the curve bends, then by definition there is a force so the logic goes the exact opposite way. The question is like asking if there is any justification for drawing a red figure using a red pen. If you drew it with a blue pen, it would not be red by definition.

3) curved spacetime

I disagree. This in itself does not introduce a bend in the curve. However, in some limit it may be seen as changing the direction of the cuve.

 

[FactChecker]

I am not sure that I understand the terminology you are using here. Usually when I use the word "relative" I mean "not invariant". But proper acceleration is invariant, so it is not "relative" in this sense. Also, proper acceleration most definitely does not require reference to another object. Inside a closed box with no external input you can still unambiguously determine the proper acceleration of the box without reference to another object outside the box. So I don't see how the acceleration is "relative".

Sorry. I probably should have said that the second derivative of position is relative to some other position. If the term "acceleration" is reserved for the felt force, then that is different. But that relies on a physical effect that is external to the intrinsic mathematics of a twin-centered reference frame.

 

[Dale]

I disagree. This in itself does not introduce a bend in the curve. However, in some limit it may be seen as changing the direction of the cuve.

Yes, this is fine. Things get even more complicated to describe in English when we use curved spaces. Better I should stick with the math on that comment.

 

[SamRoss]

No, you don't have that freedom. A blank piece of paper has an associated "metric", which determines the distance between nearby points. This determines the length for any curve on the paper. For Euclidean geometry (which applies to a piece of paper), a straight line between two points on the paper is the curve with the shortest length connecting those points.

As I said later in that quote - I don't remember my exact wording - something else must be specified in order for that freedom to disappear. Your own reply implies this. The freedom will go away only if I specify the lengths of the lines I want you to draw. If I don't tell you how long I want the lines to be then you can do whatever you want. In the twin paradox, unless you say which twin was under the influence of the force then you won't be able to justify picking one specific twin as the one with the shorter world line.

 

[SamRoss]

No.

Good. That's the answer I was hoping to hear. :)

 

[Orodruin]

Good. That's the answer I was hoping to hear. :)

I'm sorry, but I feel that you are simply ignoring what I said after "No", which is the important thing. Your statement is true because it is a tautology.

 

[PeterDonis]

The freedom will go away only if I specify the lengths of the lines I want you to draw. If I don't tell you how long I want the lines to be then you can do whatever you want. In the twin paradox, unless you say which twin was under the influence of the force then you won't be able to justify picking one specific twin as the one with the shorter world line.

The lengths of the lines and the force are two different things, and only specifying the former takes all of the freedom away. There is no general rule that says "the one under the influence of the force has the shorter worldline". Both twins could be subjected to a force. And there isn't even a general rule that says "the one who is subjected to more force has the shorter worldline". The only general rule is that specifying the lengths of the lines is what takes the freedom away.

 

[SamRoss]

I'm sorry, but I feel that you are simply ignoring what I said after "No", which is the important thing. Your statement is true because it is a tautology.

So your reasoning is that a curved line is the definition of a force, not the result of a force. Therefore, I am being redundant by saying that we must draw a curved line because of the force. Maybe we do have a disagreement here. As I see it, the geometry is only a representation of what's happening in the real world. The force has physical meaning; the curved line is the representation of the trajectory of the object under the influence of that force. The trajectory is not arbitrary - it does not "come first" - but rather follows from physical laws. We could look at the position and velocity of all objects related to the experiment, come up with a Lagrangian, and then use the principle of least action to calculate what the trajectory must be. We are not being redundant if we say that the trajectory follows from the physical constraints in the problem and is not the definition of those constraints. Of course, it would be possible to define the word "force" simply with respect to the curvature of a line, but then we would be doing pure math and not physics. We would lose the connection to the real world.

 

[SamRoss]

The only general rule is that specifying the lengths of the lines is what takes the freedom away.

Are we doing pure math or physics? How can we specify the lengths of the lines without looking at what's actually going on in the situation? How can we justify making one line longer or shorter than the other unless there is some physical distinction between the two observers in the real world?

 

[PeterDonis]

How can we justify making one line longer or shorter than the other unless there is some physical distinction between the two observers in the real world?

The different lengths of the lines correspond to the different elapsed times on the observers' clocks. That's a physical distinction. And it's the only physical distinction that will always be there in any "twin paradox" situation and will always correspond to the thing that defines the kind of situation you are interested in (different elapsed times for the twins between the same two points in spacetime). Sure, there will be other physical distinctions between the twins, but none of them correspond to a general rule that let's you predict which one will have more elapsed time.

 

[SamRoss]

Sure, there will be other physical distinctions between the twins, but none of them correspond to a general rule that let's you predict which one will have more elapsed time.

So if one twin stays on Earth while the other flies off in a rocket and comes back, I can't use those physical distinctions as a general rule to decide which one will be older when they meet? Sometimes it will be the Earth twin and sometimes it will be the rocket twin?

 

[Dale]

There is no general rule that says "the one under the influence of the force has the shorter worldline".

But there is a specific rule for flat spacetime that says that the one that is not under the influence of a force has the longer worldline.

 

[PeterDonis]

there is a specific rule for flat spacetime that says that the one that is not under the influence of a force has the longer worldline

In the special case where there is one who is not under the influence of a force, yes. But that's a special case.

 

[Orodruin]

So your reasoning is that a curved line is the definition of a force

The definition of a curved line is one that has non-zero acceleration. The concept of force is introduced (for a fixed mass object) as mass multiplied by acceleration. This is what a force is, nothing more and nothing less. It turns out that modelling forces in certain ways provides a useful description of some natural phenomena but this is a connection to those phenomena that is utterly irrelevant to what is being regarded here. What is relevant is kinematics, not dynamics.

As I see it, the geometry is only a representation of what's happening in the real world.

This is true for anything yoy describe in physics, including forces.

The force has physical meaning; the curved line is the representation of the trajectory of the object under the influence of that force

The force does not have any more physical meaning than the acceleration. Both are mathematical descriptions of observable phenomena. One is relevant to kinematics, the other not. Both are irrelevant to proper time - only geometry is relevant to proper time. Then you can connect some of that geometry to statements about acceleration in Minkowski space, but the force is still irrelevant to the description. Force only becomes relevant when you ask the question how an object came to follow a particular curved world-line. It is irrelevant to the proper time.

The force has physical meaning; the curved line is the representation of the trajectory of the object under the influence of that force. The trajectory is not arbitrary - it does not "come first" - but rather follows from physical laws.

Obviously, but you are still wrong. You do not need to know anything about the force to compute proper time, all you need is the world-line. If you observe an object following a particular world-line it does not matter in the slightest how it came to follow that world-line. The world-line is every bit as physical as the force.

Of course, it would be possible to define the word "force" simply with respect to the curvature of a line, but then we would be doing pure math and not physics. We would lose the connection to the real world.

No, you are simply wrong here. It is how we define force. Then we make a mathematical model of how different forces appear and there is your connection to dynamics. However, again, this is not needed for the physics under consideration. Compare with the situation in classical mechanics where you want to know how long it takes an object to move a distance with constant acceleration. If you are given the acceleration (kinematics), dynamics are irrelevant. Obviously, if you want to know why the object moved that way you need to do dynamics, but this is totally irrelevant to computing the time. If you do not agree with this you essentially have a beef with the entire middle school physics curriculum.

We would lose the connection to the real world.

Again, wrong, we are not trying to explain why an object follows a particular world-line. We are computing the proper time along it. This is a physical observable measured by a clock. You are essentially saying clocks are not connected to the real world.