B Understanding twin paradox without math

[PeterDonis]

I am not going to focus on one, or even ten examples when there are dozens in this forum.

I have never seen any post in this forum that meets this description of yours:

One might think that a traveling-twin-centered calculation should be just as valid as the Earth-centered calculation. Yet the two calculations give conflicting answers.

I have seen plenty of incorrect claims by posters that the traveling twin would calculate a conflicting answer, but I have never seen any poster who makes such a claim give an actual calculation to back up such a claim. But in the above quote you are claiming that there is such a calculation. That's why I asked for a reference to one.

 

[FactChecker]

I have seen plenty of incorrect claims by posters that the traveling twin would calculate a conflicting answer, but I have never seen any poster who makes such a claim give an actual calculation to back up such a claim. But in the above quote you are claiming that there is such a calculation. That's why I asked for a reference to one.

The assumption that the situation of the Twins is completely symmetric leads to the Earth-centered calculations being swapped to the traveling-twin and wrongly concluding, from that perspective, that the Earth-bound twin has aged less. There is no need for a new calculation. You must have seen this a hundred times.
(see naive[3][4] application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less )

 

[PeterDonis]

The assumption that the situation of the Twins is completely symmetric leads to the Earth-centered calculations being swapped to the traveling-twin

In a sort of hand-waving way, perhaps. But again, I have never seen anyone who makes this claim actually try to do the math.

There is no need for a new calculation.

Yes, there is: in order to do the math, you need to actually define the traveling twin's rest frame, since it is not the same as the Earth twin's rest frame. If that frame were inertial, as the Earth twin's is, you would be able to find a single inertial worldline that was at rest in the frame for the whole trip. But of course you can't. And any actual attempt to do the math would quickly run up against that insurmountable difficulty. The only reason people are able to make such a claim is that they have not actually tried to do the math.

(see naive[3][4] application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less )

As far as I can tell, all of the references given in that passage of the Wikipedia article start right out by saying explicitly that the viewpoint being described is incorrect, and explain why, along the same lines that I gave above. So anyone who reads those references should already be on notice that this viewpoint is wrong and that any claims made based on it are also wrong.

 

[vanhees71]

This is not correct. An inertial frame and a non-inertial frame are different coordinate charts with different forms for the metric. You can describe all that without mentioning acceleration at all. It is true that there will, in general, be a parameter in the metric in the non-inertial frame that has units of proper acceleration, but that in itself does not mean that parameter has to have the physical meaning of the proper acceleration of something.

What's physical is the spacetime itself, not the coordinate charts used to describe them. In special relativity you have Minkowski space as the spacetime manifold, and this is not changed by choosing another coordinate system, defining a non-inertial frame of reference. In GR the specific spacetime (a pseudo-Riemannian manifold) is determined by the physical situation, but it's also completely independent of the choice of coordinates.

The aging of the twins is also independent of the choice of coordinates, it's given by the proper times of the twins along their world lines (measured from the common start to the common end).

 

[Sagittarius A-Star]

What's physical is the spacetime itself, not the coordinate charts used to describe them.
...
The aging of the twins is also independent of the choice of coordinates, it's given by the proper times of the twins along their world lines (measured from the common start to the common end).

That's true. The twin's world lines are usually mathematically described using coordinates of a reference frame, often the rest-frame of the non-inertial twin.

There is sometimes confusion like "according to special relativity, from twin A's perspective twin B will of aged less but from twin B's perspective twin A will of aged less".

Source (see OP):
https://www.physicsforums.com/threads/symmetrical-twin-paradox.663787/

I think, the error made in such a case is, that the commonly known time-dilation factor $\sqrt{1-v^2/c^2}$ is applied to the rest frames of both twins in the calculations of the elapsed proper time of the respective other twin, although one of the rest-frames is non-inertial. It is not understood, that with reference to a non-inertial frame, an additional "gravitational" time-dilation exists. For example, the time-dilation factor in Kottler–Møller coordinates for a Rindler-frame (with constant acceleration $\alpha$) is
$\sqrt {(1+ \frac{\alpha\,x} {c^2})^2 - \frac{v^2}{c^2}}$.

 

[SiennaTheGr8]

There's a chapter in Taylor/Wheeler's Spacetime Physics that I recall being very helpful for understanding the twin paradox ("Trip to Canopus" I think?). It does emphasize geometry, but it also shows how the relativity of simultaneity comes into play when you analyze the problem in terms of inertial frames. In particular, it discusses what the Earth-twin's wristwatch reads "at the same time as" the turnaround event for both the outbound-frame and the inbound-frame.

 

[vanhees71]

That's true. The twin's world lines are usually mathematically described using coordinates of a reference frame, often the rest-frame of the non-inertial twin.

Sure, how else do you want to describe the worldline? This is no contradiction to the fact that the aging of each twin is a property of the spacetime and not of any choice of coordinates or frames. In SR it's of course most convenient to use a global inertial frame and pseudo-Cartesian coordinates.

There is sometimes confusion like "according to special relativity, from twin A's perspective twin B will of aged less but from twin B's perspective twin A will of aged less".

This cannot be, because the aging of the twins are scalar quantities and thus independent of any choice of coordinates and frames.

Source (see OP):
https://www.physicsforums.com/threads/symmetrical-twin-paradox.663787/

I think, the error made in such a case is, that the commonly known time-dilation factor $\sqrt{1-v^2/c^2}$ is applied to the rest frames of both twins in the calculations of the elapsed proper time of the respective other twin, although one of the rest-frames is non-inertial. It is not understood, that with reference to a non-inertial frame, an additional "gravitational" time-dilation exists. For example, the time-dilation factor in Kottler–Møller coordinates for a Rindler-frame (with constant acceleration $\alpha$) is
$\sqrt {(1+ \frac{\alpha\,x} {c^2})^2 - \frac{v^2}{c^2}}$.

Of course you have to use the metric components according to each choice of coordinates/frames. Otherwise you simply produce errors.

 

[vanhees71]

There's a chapter in Taylor/Wheeler's Spacetime Physics that I recall being very helpful for understanding the twin paradox ("Trip to Canopus" I think?). It does emphasize geometry, but it also shows how the relativity of simultaneity comes into play when you analyze the problem in terms of inertial frames. In particular, it discusses what the Earth-twin's wristwatch reads "at the same time as" the turnaround event for both the outbound-frame and the inbound-frame.

Of course, comparing clock readings of clocks at different places depends on the observers due to the relativity of simultaneity. That's why the clocks have to be synchronized at the beginning of the journey, when the twins start at the same place and then compared after the trip, when both twins again meet at the same place.

 

[FactChecker]

This cannot be, because the aging of the twins are scalar quantities and thus independent of any choice of coordinates and frames.

He says it is a misconception. That is exactly why it is a paradox. The answer to the problem is NOT the same as the explanation of the paradox. There are many ways to get the correct answer to the problem that do nothing to illuminate why the misconception of the paradox is wrong. IMO, most of the "answers" in this forum to the Twin Paradox are like that.

 

[Ibix]

He says it is a misconception. That is exactly why it is a paradox. The answer to the problem is NOT the same as the explanation of the paradox. There are many ways to get the correct answer to the problem that do nothing to illuminate why the misconception of the paradox is wrong. IMO, most of the "answers" in this forum to the Twin Paradox are like that.

You are drawing a distinction between explaining why the right answer is what it is and explaining what's left out of the naive (wrong) analysis. I agree that the paradox arises when one doesn't realise that there's a change in simultaneity when one changes reference frames, and the change in simultaneity exactly accounts for the difference between the inertial twin's elapsed time $T$ and the "paradoxical" $T/\gamma^2$. However, my experience of twin paradox questions at PF is that they don't usually ask how to resolve the paradox, but usually ask some variant on "what mechanism makes the moving clock run slow", to which some variant on "clocks measure interval and interval is path dependent just like distance" is a reasonable answer. The OP's diagrams (with the needed corrections) are another approach. I think that's why a lot of answers don't invoke the simultaneity change explanation you're using.

 

[FactChecker]

However, my experience of twin paradox questions at PF is that they don't usually ask how to resolve the paradox, but usually ask some variant on "what mechanism makes the moving clock run slow",.

Good point! I'll buy that.
It sounds like the more common student response to the Twin Paradox is not really about the twins, but rather that they want to question the very mechanics of time dilation. So their question is more fundamental than I was assuming.

 

[Sagittarius A-Star]

Sure, how else do you want to describe the worldline?

For example, I could describe the worldline of a car and not using coordinates in the following way:

• The car follows a certain street.
• Define start-location and destination on the street.
• Define profile of proper acceleration of the car in the local direction of the street (from start to destination) as function of it's elapsed proper time since start.
• Define as initial condition: Car is at rest at the start-location.

 

[Ibix]

Good point! I'll buy that.
It sounds like the more common student response to the Twin Paradox is not really about the twins, but rather that they want to question the very mechanics of time dilation. So their question is more fundamental than I was assuming.

That's my experience, at least. Ironically, what it's actually supposed to teach is the point you're making, that you can't neglect the relativity of simultaneity (ah good, my new phone has learned the word "simultaneity" in three days). That it doesn't seem to inspire that lesson is, I think, why people are trying "geometry first" approaches.

 

[Sagittarius A-Star]

I agree that the paradox arises when one doesn't realise that there's a change in simultaneity when one changes reference frames, and the change in simultaneity exactly accounts for the difference between the inertial twin's elapsed time $T$ and the "paradoxical" $T/\gamma^2$.

However, my experience of twin paradox questions at PF is that they don't usually ask how to resolve the paradox, but usually ask some variant on "what mechanism makes the moving clock run slow"

Here are other examples in PF for asking how to resolve the paradox:

Each observer determines that all clocks in motion relative to that observer run slower than that observer’s own clock. If the moving observer returns to point A, he will notice the following: the clock at point A is slower than his own clock. but this clock shows that more time has passed since his departure than his own clock. For moving observers this conjucture makes no sense. If a clock is slower than its own clock, it should also show less time. Therefore, the clock in point A should be synchronized according to the rules of relativity ... but how?

But how could Goslo appear to age faster, when time is always dilated?!.

Source:
https://www.physicsforums.com/threads/the-twin-paradox-cant-be-resolved.607210/

I don't know much about the math of SR, but this is what's bothering me: if a moving clock B ticks slower than the stationary one I have (A), then from the viewpoint of B, my clock (A) is ticking slower. So if we turn around and meet each other in the middle, which clock was slower than which?

I am thinking that the traveling twin feeling acceleration should not differentiate between the twins from an SR perspective, since we cannot consider the acceleration asymmetrically (i.e. both twins are identically accelerating or in uniform motion w.r.t. each other at all times, from SR perspective).

Source:
https://www.physicsforums.com/threa...lation-in-twin-paradox-possible-in-sr.646622/

My question is this; what is the actual maths that satisfactorally predicts then the outcome of the situation? I have heard and read the same answer about symetry being broken several times but never been able to find someone who can explain the underlying maths

Source:
https://www.physicsforums.com/threads/symetry-time-dilation-twin-paradox-and-all-that-stuff.146447/

 

[vanhees71]

He says it is a misconception. That is exactly why it is a paradox. The answer to the problem is NOT the same as the explanation of the paradox. There are many ways to get the correct answer to the problem that do nothing to illuminate why the misconception of the paradox is wrong. IMO, most of the "answers" in this forum to the Twin Paradox are like that.

I don't understand, what you mean. The aging of the twins is given by scalar quantities, i.e., the proper times between the departure and arrival "events" along their worldlines. This shows immediately that there is no paradox. I have no clue, which more many ways you have in mind to answer the problem. I've the impression that any attempt to get the simple statement "more pedagogical" rather raises the confusion about a pretty simple "no-brainer".

 

[Dale]

I've the impression that any attempt to get the simple statement "more pedagogical" rather raises the confusion about a pretty simple "no-brainer".

Telling a student that the thing that confuses them is a “pretty simple no brainer” may not be as helpful to the student as you think. Usually they require a teacher to address their actual misunderstanding, not insult them.

 

[FactChecker]

This shows immediately that there is no paradox. I have no clue, which more many ways you have in mind to answer the problem. I've the impression that any attempt to get the simple statement "more pedagogical" rather raises the confusion about a pretty simple "no-brainer".

The paradox is that they believe there are two equally valid calculations that give conflicting answers. It is not enough to say that one calculation is a "no-brainer". It is necessary to convince them that the other calculation is invalid in its own right.

 

[robphy]

"no brainer" likely refers to the person who already accepts the result,
but not so much (as others have said)
for the student who is struggling to understand why it's true
and why alternative attempts against the result aren't true or valid.

 

[SiennaTheGr8]

I think typically somebody struggling with the twin paradox might have a little background in SR from an inertial-frames perspective, and might understand something about time dilation, but presumably doesn't understand the full implications of the Lorentz transformation (read: the relativity of simultaneity) and almost certainly isn't well-acquainted with the invariant-based geometric picture.

So whatever explanation you give them, they're going to try to reconcile it with what they already know (as they should). And if their background knowledge is all about inertial frames and expressing everything in terms of Cartesian coordinates and coordinate time, then the geometric explanation—while "simplest" (and clearly best)—is probably not going to be the easiest for them to digest.

For me, the relativity of simultaneity was the key to understanding the twin paradox.

From a coordinate-based inertial-frames perspective, what distinguishes the rocket-twin's situation from the Earth-twin's? Answer: the Earth-twin remains in one inertial frame, while the rocket-twin switches frames at the turnaround event. [EDITED FOR MORE CAREFUL WORDING] Answer: the Earth-twin's inertial rest frame never changes, while the rocket-twin's does (at the turnaround event).

Why does that matter if both twins are always moving inertially? Don't they both always say that the other's wristwatch is running slow? Answer: yes, but at the turnaround event, the rocket-twin's answer to the question "what time does the Earth-twin's wristwatch display right now?" changes dramatically—the Earth "suddenly skips ahead in time" for the rocket-twin, by decades in the usual setup (though it would of course have to happen "smoothly" in real life, since acceleration is required). By contrast, the Earth-twin's answer to the question "what time does the rocket-twin's wristwatch display right now?" never changes dramatically at any point.

 

[Ibix]

Here are other examples in PF for asking how to resolve the paradox:

Source:
https://www.physicsforums.com/threads/the-twin-paradox-cant-be-resolved.607210/

Source:
https://www.physicsforums.com/threa...lation-in-twin-paradox-possible-in-sr.646622/Source:
https://www.physicsforums.com/threads/symetry-time-dilation-twin-paradox-and-all-that-stuff.146447/

Fair enough, but at a quick glance those threads do cover the issues with simultaneity, or use the Doppler analysis (preferred by @ghwellsjr).

 

[vanhees71]

Telling a student that the thing that confuses them is a “pretty simple no brainer” may not be as helpful to the student as you think. Usually they require a teacher to address their actual misunderstanding, not insult them.

This was not against the students but the teachers. In my experience the explanation that the different aging is expressed in terms of scalars, i.e., the proper times of the twins, and thus independent of the coordinates used to calculate them is well undrstood.

 

[Ibix]

I have no clue, which more many ways you have in mind to answer the problem.

You can explain where the naive analysis that gives the paradoxical $T/\gamma^2$ result for the stay-at-home's elapsed time goes wrong (i.e. they forgot that they changed simultaneity convention, and thus failed to account for some of the stay-at-home's worldline). I agree that there are better and easier ways of getting to the right answer, but it's also good to explain why the wrong method doesn't work.

 

[Dale]

i.e. they forgot that they changed simultaneity convention, and thus failed to account for some of the stay-at-home's worldline

Or you could introduce radar coordinates for the non inertial twin and show that the trip is not symmetrical from his perspective. But I suspect that students that are confused with this “paradox” will probably be totally lost with that.

 

[member 728827]

An (enough) good explanation for me is:

- The Alice (Earth) observer has a lattice of synchronized clocks that fill the whole space. Specifically there's an "Earth's clock" at the star position, that reports back its readings to Earth (at each time, what happened there).

- In Bob's (traveler) perspective, during the outbound path and just when he observes the star reach his position, his clock reads 2.0 and as he knows that clocks run slower in Earth, he predicts Earth's clock to be $2 \cdot \frac{1}{\gamma}=1.6$.

- Leading clocks lag, which means that for two synchronized clocks separated by a distance of D in the same frame, the leading clock (respect the velocity) will lag $\frac{D \cdot v}{c^2}$ respect the rear clock, as observed by a rest frame. So the "Earth's clock" at the star position is ahead its correspondent sibling at Earth by 0.9, and it records 2.5 (1.6+0.9) at the event of the star reaching Bob at his time of 2. When the Earth's observer collect the reported data, all fits nicely.

- The velocity is then suddenly inverted and the inbound path started. As soon as this happens, and Bob is put abruptly into another frame, that "Earth's clock" at the star is now the leading clock of the play, so it lags by 0.9 respect its correspondent sibling clock at Earth. Obviously, the reading of that clock in that "after" instant is the same as "before", but now lags by 0.9 respect the correspondent sibling at Earth. So that Earth's clock must be 3.4. In no time for Alice, Earth clock has gone from 1.6 to 3.4

- Then, it takes another 1.6 Earth-years for Earth to reach Bob's position. And that accounts for the 5 years elapsed in Earth, but only 4 for the traveler twin.

 

[robphy]

Or you could introduce radar coordinates for the non inertial twin and show that the trip is not symmetrical from his perspective. But I suspect that students that are confused with this “paradox” will probably be totally lost with that.

For me, it was radar methods that made things click. This provided a measurement procedure and operational definitions for assigning coordinates. (This was more understandable than first trying to imagine shrunken rulers and slow clocks…. That is, I use the radar methods first to then interpret all of the various traditional but likely confusing ways of thinking about things.)

 

[obtronix]

I think typically somebody struggling with the twin paradox might have a little background in SR from an inertial-frames perspective, and might understand something about time dilation, but presumably doesn't understand the full implications of the Lorentz transformation (read: the relativity of simultaneity) and almost certainly isn't well-acquainted with the invariant-based geometric picture.

So whatever explanation you give them, they're going to try to reconcile it with what they already know (as they should). And if their background knowledge is all about inertial frames and expressing everything in terms of Cartesian coordinates and coordinate time, then the geometric explanation—while "simplest" (and clearly best)—is probably not going to be the easiest for them to digest.

For me, the relativity of simultaneity was the key to understanding the twin paradox.

From a coordinate-based inertial-frames perspective, what distinguishes the rocket-twin's situation from the Earth-twin's? Answer: the Earth-twin remains in one inertial frame, while the rocket-twin switches frames at the turnaround event. [EDITED FOR MORE CAREFUL WORDING] Answer: the Earth-twin's inertial rest frame never changes, while the rocket-twin's does (at the turnaround event).

Why does that matter if both twins are always moving inertially? Don't they both always say that the other's wristwatch is running slow? Answer: yes, but at the turnaround event, the rocket-twin's answer to the question "what time does the Earth-twin's wristwatch display right now?" changes dramatically—the Earth "suddenly skips ahead in time" for the rocket-twin, by decades in the usual setup (though it would of course have to happen "smoothly" in real life, since acceleration is required). By contrast, the Earth-twin's answer to the question "what time does the rocket-twin's wristwatch display right now?" never changes dramatically at any point.

"suddenly skips ahead in time"

is almost never accepted as a reasonable answer by most people, I suppose you need to introduce the block universe and angled slices idk

"Earth-twin's answer to the question "what time does the rocket-twin's wristwatch display right now?" never changes dramatically at any point."

Exactly, this is Brian Greene's explanation of the twin paradox the only measurements you can trust are from an observer in an inertial frame. No time skips. Which is why I drew those diagrams, from three inertial frames.

 

[PeterDonis]

This seems like a good time to reference the Usenet Physics FAQ article on the twin paradox:

https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

"suddenly skips ahead in time"

is almost never accepted as a reasonable answer by most people

See the "time gap" page in the above article.

"Earth-twin's answer to the question "what time does the rocket-twin's wristwatch display right now?" never changes dramatically at any point."

If you use the Earth twin's inertial rest frame to define "now", then of course not. But the best answer is that "now" has no physical meaning at all in relativity.

this is Brian Greene's explanation of the twin paradox the only measurements you can trust are from an observer in an inertial frame.

And this is wrong. It is perfectly possible to construct non-inertial frames that do not have "time skips" and work just as well as inertial frames for analyzing physics. You just need to use the correct equations for the non-inertial frame.

 

[Ibix]

Exactly, this is Brian Greene's explanation of the twin paradox the only measurements you can trust are from an observer in an inertial frame. No time skips. Which is why I drew those diagrams, from three inertial frames

I'd be interested in a reference to where Greene says this. Either there's some context you're missing or Greene is saying wrong things in his popularisations (again). There are plenty of non-inertial frames one can work with where there are no time skips (and Greene knows this - the study of non-inertial frames was a step on the road to the discovery of general relativity). They are just mathematically harder to use than inertial frames.

 

[PeroK]

Exactly, this is Brian Greene's explanation of the twin paradox the only measurements you can trust are from an observer in an inertial frame. No time skips. Which is why I drew those diagrams, from three inertial frames.

It seems to me you are going in the wrong direction here. In GR there are no global inertial frames, so the idea of relying on global inertial frames is no longer possible. If you now try to learn GR, then the whole ground has been pulled out from under your feet.

Also, in GR you can more easily have a version of the paradox where there is no proper acceleration. I.e. by executing half an orbit of a black hole as a turnaround.

There is also the opposite paradox of two clocks in the same circular orbit around the Earth or a star in opposite directions. There is continuous time dilation (same gravitational potential and non-zero relative velocity), yet complete symmetry. Despite the continuous mutually symmetric time dilation, the clocks show the same time when they cross paths twice per orbit.

This is where you need a broader (not narrower) understanding of these concepts. In particular, it is an understanding of spacetime geometry that allows you not only to understand the twin paradox fully, but to move on to GR.

 

[Dale]

In particular, it is an understanding of spacetime geometry that allows you not only to understand the twin paradox fully, but to move on to GR.

This is particularly good advice.