My paraphrased explanation of the Twin Paradox

[ynojunin]

TL;DR Summary

My paraphrased explanation of the Twin Paradox; please poke holes in it and tell me how wrong I am! :D

So I've been trying really hard to understand the theory of relativity at its most basic level lately, and recently I dove down the rabbit hole of the Twin Paradox. This has led me through a series of youtube videos, each one claiming to present a different solution, explain why the other videos presenting other solutions are wrong, and finally ended me up on a video that claimed to poke holes in the logic used by all the other videos, leaving me very confused.

At first I felt like I really wasn't understanding any of it, but I think I was finally able to formulate an answer that makes sense. To help me with this, I made my own, simplified, version of the Twin Paradox below. If anyone would be willing to take the time to read 1) My simplified/slightly modified Twin Paradox, 2) My paraphrasing of the video I saw that claims the paradox is still unresolved, and 3) An explanation, in my own words, of what I understand to be the real resolution of the Twin Paradox, and then let me know if what I have said makes sense or if I'm still missing the whole point, I would be extremely grateful!

(Note: I shorten "inertial frame of reference" to "inertial frame" throughout this whole thing; hopefully that's not too inaccurate or confusing!)

1) My "Simplified" Twin Paradox

Consider a universe that behaves exactly like our own: all the laws of physics are the same, the theory of relativity applies, etc.

This universe contains nothing whatsoever except two spaceships, each of which has one observer and one clock in it. The observers are named Bob and Rob. At the start of our thought experiment, the two spaceships are sitting next to each other, motionless relative to each other. (Gravity in this thought experiment is considered negligible, and thus ignored.)

An event occurs in which the two spaceships move apart from each other, then come back together.

From observer Bob's perspective, it appears that Rob's spaceship accelerates rapidly away from him, quickly reaching a speed very close to the speed of light, then, when Rob's spaceship is several light years away, it suddenly appears to accelerate rapidly in the other direction, and begin to travel back towards Bob, again quickly approaching the speed of light, then finally when it is close to Bob, Rob's spaceship appears to decelerate extremely rapidly, and comes to rest next to Bob's spaceship, leaving the two spaceships motionless relative to each other, in the exact same position as they were at the start of our thought experiment (again all of this from Bob's perspective.)

What does Rob see from his perspective? Well, just copy and paste the above, switch Bob and Rob's names, and the event will look exactly the same to Rob, only mirrored.

Finally, ask the question; for whom was time experienced more quickly relative to the other? Did Bob experience a longer time and Rob a shorter one, or vice versa?

2) The Claim That the Paradox is Unresolved

Now, the consensus of all the videos I've seen on the Twin Paradox seems to be that the Twin who remains still will age more quickly than the one who moves.

HOWEVER, the whole point of the relativity of motion, is just that; motion is relative. In the above example, Rob observes Bob move away from him, then back to him, himself remaining still. But observer Bob sees himself as remaining still, while Rob moves away, then back to him. According to relativity, neither Bob nor Rob is more correct or incorrect in their perception; both are correct given themself as the observer.

So how do you know which one experienced time more quickly? Taking planets and other things that we perceive as stationary out of the equation, Bob and Rob have no way of knowing which one of them actually moved, and which one didn't, therefor, there cannot be any logical way of claiming that one would have experienced time any differently than the other.

3) My Paraphrasing of the Real Explanation (or what I hope is...)

It seems to me that much of the confusion surrounding what is the correct solution to the paradox stems from conflating acceleration with changing one's inertial frame. It appears to me that acceleration simply meaning changing one's speed relative to an observer. If this is the correct definition of acceleration, than a person in free fall will be accelerating relative to an observer on the ground. However, they will NOT be changing their inertial frame; to an observer on the ground they will be in motion, but in their own space-time a person in free fall is not moving or accelerating at all; they are simply remaining in the same space as that space curves towards the mass that is bending space (I'm sure there's some inaccuracies in how I described that, but that's the best I can put it.)

Thus we can see that acceleration and changing one's inertial frame are not the same thing. You can accelerate without changing your inertial frame, like the person who is in free fall does. You can also change your inertial frame without accelerating relative to a given observer, like two people who are in cars that go from 0-60 in the same direction at the same time: relative to each other they haven't accelerated or even moved, but they've both definitely changed their inertial frame. Acceleration is something relative to an observer; it can change based on which observer you're looking from. Changing one's inertial frame is something objective, and is not at all connected to acceleration, other than the fact that in the physical world we often see objects that are experiencing a force that changes their inertial frame also accelerate relative to us at the same time, so we tend to confuse the two.

In the final video I watched, I think the guy was basically stuck on the question: "In a universe devoid of all other objects (ie, empty space,) how do you objectively tell if an observer is changing their inertial frame or not?" Since he conflated changing one's inertial frame with changing one's acceleration, he then argued that since acceleration is relative, and both twins can claim that they experienced no acceleration (from their own perspective as an observer) that it followed they can both claim to have not changed their inertial frame. But this is where he's wrong; acceleration, that is, changing one's speed, is relative to an observer. But changing one's inertial frame is an objective thing, and not relative to any observer. Anything that is changing its own inertial frame will appear to be changing its inertial frame to ALL observers.

So if changing one's inertial frame doesn't mean accelerating, what does it mean? My layperson's explanation of changing one's inertial frame would go like this: "Any time an observer is experiencing a force that their body's mass resists, they are changing their inertial frame." That's probably not a very precise way of putting it, but what I'm basically trying to say is that if there is a force that cause an observer to change the way it would travel through space-time if that force hadn't existed, than the inertial frame of that observer has changed.

(Side note: In a way, I think it would be accurate to say that those who say the Twin Paradox is resolved because one twin experienced acceleration and one did not are partially right (or at least they have the spirit of the correct answer, and are just putting it wrong.) What they really ought to say is that since the "travelling twin" experienced a force that changed his inertial frame (which is the very thing that made him accelerate/decelerate from the perspective of the "earth twin") THAT is what makes the travelling twin experience time differently than the earth twin, and what makes their situation asymmetrical.)

So the final answer to my above modified "Twin Paradox" thought experiment is this: In order for Bob and Rob to have moved relative to each other, one (or both) of them had to change their inertial frame. If Bob changed his inertial frame, resulting in the relative motion, while Rob remained in the same inertial frame, then Bob will experience a shorter time during the relative motion, while Rob will experience a longer time. But if Rob changed his inertial frame while Bob did not, then when they come back together it will be Rob who experienced less time passing. During the motion, the one who is experiencing the change in inertial frame will experience evidence of this because the mass of his body will resist the change, meaning that he experience something that feels like gravity (even though, of course, it has nothing to do with gravity) pulling him towards the side of his spaceship opposite of the direction he is moving in. The one who is not changing his frame of reference will not experience this. Of course, a third possibility exists in all this, which is that the relative motion that Bob and Rob perceived was caused by both of them changing their inertial frame. For instance, they could have both had a change in their inertial frame that was equal but opposite, and thus, to an observer who started out at the same location as them but didn't change their inertial frame, Bob and Rob would have appeared to move in opposite directions at the same speed, and for the same time and distance, and then started moving back towards each other and come back to rest at the same starting point as this third observer. If this is what happened, then Bob and Rob would both have experienced the same length of time at the end of the motion. In short, you can't answer the question "which of them, if any, experienced more time than the other?" without first knowing who changed their inertial frame in order to cause the relative motion in the first place.

(Other side note: There's also a really funny aspect to this thought experiment, which is this: Pretend that Bob and Rob are creatures who cannot feel the effects of the forces that change their inertial frame, and they and their clocks are both strapped down so they won't slide around as the inertial frame changes. Also pretend that the windows to their spaceships are one-way mirrors, so they can both look out and see the other spaceship moving off into the distance (they have really good eyesight and can see lightyears away.) But they can't see inside the other person's spaceship, and thus have no clue how the other person is experiencing time relative to themselves. If this is the case, then during the time that they are moving apart then back together, they will have no way of knowing which one is changing their inertial frame, and which one isn't, thus, at whatever point in time the two spaceships come back together, one of them will have just experienced a longer span of time than the other one, but neither one will know if they are the one who experienced the shorter or longer time, unless they are able to dock their ships together so they can go in and look at each other's clocks.)

So, if you read through the whole thing, thank you so very much! Please try to poke any holes in it that you can; I'm really trying to understand the basic concepts at play hear, and I'd love to be educated in how I can improve my understanding.Some of the videos I watched, in case anyone is wondering:

 

[PeroK]

I think you're missing the concept of proper acceleration, which is detectable experimentally and invariant across all inertial reference frames. As it is in Newtonian physics. That's why Newton's second law is $F = ma$.

In SR proper acceleration entails changing your inertial reference frame.

 

[ynojunin]

I think you're missing the concept of proper acceleration, which is detectable experimentally and invariant across all inertial reference frames. As it is in Newtonian physics. That's why Newton's second law is $F = ma$.

In SR proper acceleration entails changing your inertial reference frame.

Ah, thank you! Yes, it sounds like I am missing that. I don't remember that term coming up in any of the videos I watched, and many of them kept talking about acceleration and changing one's inertial frame as if they were the same thing, which got really confusing, and left me thinking "that can't be right."

So "proper acceleration" is the term I'm looking for to describe changing one's inertial frame of reference (basically) as opposed to "acceleration" which just means changing one's speed relative to an observer?

 

[Ibix]

If this is the case, then during the time that they are moving apart then back together, they will have no way of knowing which one is changing their inertial frame, and which one isn't, thus, at whatever point in time the two spaceships come back together, one of them will have just experienced a longer span of time than the other one, but neither one will know if they are the one who experienced the shorter or longer time, unless they are able to dock their ships together so they can go in and look at each other's clocks.)

This isn't right. If they can see the other ship they can examine its Doppler shift, and the history of the Doppler shift of light coming from the other is different in a twin paradox scenario. The traveller sees redshifted light for half the time and blueshifted light the other half, while the stay-at-home sees redshifted light for most of the journey ($(c+v)/2c$ of it, if my mental arthmetic is correct) and blueshifted light only for a short period at the end ($(c-v)/2c$ of it).

As a general rule, trying to learn physics from YouTube videos is a mistake. Taylor and Wheeler's Spacetime Physics is available for free download, and Morin's Relativity for the Enthusiastic Beginner is cheap.

 

[PeroK]

Ah, thank you! Yes, it sounds like I am missing that. I don't remember that term coming up in any of the videos I watched, and many of them kept talking about acceleration and changing one's inertial frame as if they were the same thing, which got really confusing, and left me thinking "that can't be right."

So "proper acceleration" is the term I'm looking for to describe changing one's inertial frame of reference (basically) as opposed to "acceleration" which just means changing one's speed relative to an observer?

Yes, in the usual twin paradox, one twin undergoes proper acceleration and the other twin remains inertial. The inertial twin undergoes coordinate acceleration, as measured by the accelerating twin. Confusing that coordinate acceleration with "real" proper acceleration is the basis of the paradox, as usually presented.

 

[Ibix]

So "proper acceleration" is the term I'm looking for to describe changing one's inertial frame of reference (basically) as opposed to "acceleration" which just means changing one's speed relative to an observer?

Proper acceleration is acceleration you can measure inside a closed box (proper here comes from the same root as property - something that is one's own, with no reference to anything else). If you are measuring your rate of change of velocity with respect to some notionally fixed point, that would be coordinate acceleration.

 

[Orodruin]

Ultimately, the twin paradox is the spacetime equivalent of the hypothenuse being shorter than the sum of the lengths of the other sides in a triangle in Euclidean space. The ”paradox” arises from misapplying the time dilation concept while not taking relativity of simultaneity properly into account. Nobody who actually understands the theory (and is not blindly trying to apply the time dilation formula) is going to get the result that the staying twin is older. You can try to make a popular version of ”explaining” the twin paradox using different arguments, but ultimately it boils down to spacetime geometry.

You have my full take on it here: https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

 

[PeterDonis]

@ynojunin please read the Usenet Physics FAQ article on the twin paradox:

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

IMO this article should be required reading for anyone before they even try to "solve" the paradox.

 

[Nugatory]

I dove down the rabbit hole of the Twin Paradox....

You mean that you were led into a rabbit hole by a bunch of YouTube videos....
It is pretty much impossible to learn relativity from these - few are any good and there's no way to tell the good ones from the bad if you don't already understand the subject. Instead, follow @Ibix's advice in #4 and work through a decent textbook. I am especially partial to Taylor and Wheeler, one of his suggestions.

Another good online source is the Twin Paradox FAQ: https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html. Pay particular attention to the second and third sections: Doppler Shift Analysis and Spacetime Diagram Analysis.

And what's actually going on with the Twin Paradox is astoundingly simple, has nothing to do with time dilation and is best explained without mentioning frames, changing frames, or time dilation between frames:
Both twins move through spacetime (not just space!) from point A, the separation event, to point B, the reunion event. But they follow different paths through spacetime, a fact that should be obvious because only one twin's path passes through the turnaround event. These different paths have different lengths, again obvious because one path is a straight line and the other is not. And.... the length of a path through spacetime is measured by a clock that follows that path, so different lengths mean different elapsed times for their respective journeys from A to B. The only trick is that the geometry of spacetime is non-Euclidean, so a straight line is not the shortest but instead the longest path between two points. So stay-at-home is the one on the longer path so more time passes for them.

 

[jbriggs444]

The "best" explanation of the twin paradox may vary from one person to the next.

For me, the explanation that first "clicked" was the one that uses coordinate systems but focuses in on the hyperplane of simultaneity sweeping past the stay-at-home twin as one tries to smoothly graft the traveller's outgoing coordinate system to his returning coordinate system.

This allowed me to directly attack my broken intuition about the relativity of simultaneity. Seeing clearly the analogy with a "stay at home" twin on an interstate highway, a travelling twin taking an off-ramp and then an on-ramp with simultaneity tracked by what each driver can see out the side window facing the other. Seeing that making the slight left turn (for a U.S. interstate with a diamond intersection) from off-ramp to on-ramp changed the notion of "simultaneity" and thereby accounted for a discrepancy in elapsed distance.

Then with that intuition safely updated, I could re-examine the analogy, slap myself on the forehead and clearly see the simplicity of the geometric explanation.

For me it is not about finding an explanation that is manifestly correct. It is about identifying the error(s) in the incorrect explanation(s).

 

[ynojunin]

This isn't right. If they can see the other ship they can examine its Doppler shift, and the history of the Doppler shift of light coming from the other is different in a twin paradox scenario. The traveller sees redshifted light for half the time and blueshifted light the other half, while the stay-at-home sees redshifted light for most of the journey ($(c+v)/2c$ of it, if my mental arthmetic is correct) and blueshifted light only for a short period at the end ($(c-v)/2c$ of it).

As a general rule, trying to learn physics from YouTube videos is a mistake. Taylor and Wheeler's Spacetime Physics is available for free download, and Morin's Relativity for the Enthusiastic Beginner is cheap.

Thanks for the explanation, and for the book recommendation. I've been wanting to read a book on the subject for a while, but didn't know where to start, so if there's one available for free that's even better!

 

[member 728827]

I did the following graphic, and perhaps could be useful to understand what happens.

In the graphic, Bob starts accelerating always at 0.5g until it reaches speed 0.7c, and there're several scenarios where he deaccelerates at 0.5g, 1g, 3g and 21g. Dashed lines are the lines of simultaneity at the proper times in which the deacceleration begins.

 

[PeterDonis]

Dashed lines are the lines of simultaneity

According to what frame?

 

[DAH]

I like to use inertial frames when solving the Twin Paradox, and I think a crucial part is to assign coordinates to three events. Event 0 departure, event 1 turnaround and event 2 reunion. The stay home twin is only present at two of those events, whereas the travelling twin is present at all events. Using the correct transformations and spacetime separations, if the stay home twin has a proper time between events 0 and 2 of $T$, then the sum of the proper times for the travelling twin $\tau_{0,1}+\tau_{1,2}$ will always be $\frac{T}{\gamma}$.

 

[member 728827]

According to what frame?

(After consulting my lawyer,) according to the inertial frame of Bob that travels at a constant speed of 0.7c relative to Alice, just an infinitesimal time before Bob starts deaccelerating.

 

[PeterDonis]

if the stay home twin has a proper time between events 0 and 2 of $T$, then the sum of the proper times for the travelling twin $\tau_{0,1}+\tau_{1,2}$ will always be $\frac{T}{\gamma}$.

This will only be true if the turnaround is instantaneous, which of course it can't be. It is a reasonable approximation if the turnaround is short enough that it takes negligible proper time for the traveling twin as compared to the outbound and inbound travel times. This will generally involve proper accelerations that are much larger than humans can endure. (For example, in the scenario described in the Usenet Physics FAQ article linked to above, the proper acceleration of the traveling twin at turnaround, if I've done my math right, is about 3000 g.)

 

[gmax137]

changing one's inertial frame

Here's a question for the respondents, not so much for the OP: Is this standard terminology? I can't figure out what it means. In fact, I can't figure out what "one's inertial frame" means.

 

[PeterDonis]

Is this standard terminology?

No. It's just a common sloppiness.

I can't figure out what it means. In fact, I can't figure out what "one's inertial frame" means.

Most of the time what is actually meant is "changing which inertial frame one is at rest in".

 

[gmax137]

Sorry, I'm still confused by this. Take the traveling twin, during turnaround he experiences proper acceleration - so is there an inertial frame where he is at rest? Maybe I just need to think about it some more.

 

[PeterDonis]

Take the traveling twin, during turnaround he experiences proper acceleration - so is there an inertial frame where he is at rest?

Yes, but it changes every instant. The more technical term would be "momentarily comoving inertial frame".

 

[gmax137]

Yes, but it changes every instant.

OK, thanks, that's what I was beginning to envision.

 

[Orodruin]

momentarily comoving inertial frame

… ”instantaneous rest frame” being another commonly used nomenclature.

 

[Ibix]

OK, thanks, that's what I was beginning to envision.

Note that, while these momentarily comoving inertial frames are perfectly valid and useful concepts, they have many of the same issues that arise if you try to analyse an instantaneous turnround twin paradox using the two inertial rest frames. They work fine on the worldline of interest and "nearby", but things start to go wrong at larger distances and you need to use them carefully.

 

[PeterDonis]

while these momentarily comoving inertial frames are perfectly valid and useful concepts, they have many of the same issues that arise if you try to analyse an instantaneous turnround twin paradox using the two inertial rest frames. They work fine on the worldline of interest and "nearby", but things start to go wrong at larger distances and you need to use them carefully.

One needs to be careful here. A single inertial frame in flat spacetime is global and there are no issues with using it to analyze any experiment whatever. The fact that a particular object happens to be at rest in the frame only momentarily doesn't change that.

The issue you describe comes if one tries to naively construct a single non-inertial "frame" by just "combining" pieces of the momentarily comoving inertial frames at different points along a worldline.

 

[Ibix]

The issue you describe comes if one tries to naively construct a single non-inertial "frame" by just "combining" pieces of the momentarily comoving inertial frames at different points along a worldline.

Yes - that's what I was meaning by "using two inertial frames" in the instantaneous twin paradox - stitching the traveller's outbound rest frame "during" the outbound leg to the inbound rest frame "during" the inbound leg causes trouble. There are smilar problems with stitching together results from momentarily comoving inertial frames - perhaps unsurprisingly since this is just a special case of momentarily comoving frames where the co-movement is more than momentary.