# Twin paradox: telling who accelerates (why isn't it arbitrary?)

#### Confusedent

I'm sure this question has been asked thousands of times before, but I can't see how nature determines which twin accelerates (or is subject to a gravitational field) and which one doesn't. People say one twin will feel the acceleration and the other won't, but suppose neither twin had an accelerometer present (internal or otherwise). All they can tell is that the seperation between twins was increasing at a constant rate, and then changes at the moment of the acceleration. The twin in a spaceship sees the Earth moving away and then turn around, while the twin on Earth sees the spaceship moving away and then turn around.

If we use the gravitational time dilation explanation, it would seem to still be completely arbitrary who we treat as being in an inertial frame, and the gravitational field would seem to each observer to simply be acting on the opposite observer with an opposite direction.

I feel I understand everything else about what's going on here with SR, except that this seems completely arbitrary, and even scarier, each one actually should be younger than the other one because each one has seen the other one switch frames (which is truely impossible). If someone can explain this it'll seriously help alleviate my headache, and probably save me days of worrying about it.

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#### matheinste

Hello confusedent.

In the unlikely situation that the travelling twin cannot detect acceleration by any means, if he knows his special relativity, he can conclude that because he is younger than his twin when they meet again, then it is he, the traveller, who has undergone the acceleration.

Matheinste.

#### robphy

Homework Helper
Gold Member
I'm sure this question has been asked thousands of times before, but I can't see how nature determines which twin accelerates (or is subject to a gravitational field) and which one doesn't. People say one twin will feel the acceleration and the other won't, but suppose neither twin had an accelerometer present (internal or otherwise). All they can tell is that the seperation between twins was increasing at a constant rate, and then changes at the moment of the acceleration. The twin in a spaceship sees the Earth moving away and then turn around, while the twin on Earth sees the spaceship moving away and then turn around.
Since we are dealing with special relativity (a flat spacetime), it is probably best to not consider any "gravitational fields" (which is associated with spacetime curvature).

I don't think it's best to think that it is "nature [that] determines which twin accelerates".
Rather, it is that nature (via the Einstein Field Equations, presumably) determines which spacetime paths are inertial and which are not. It is the "traveling twin" (by some mechanism whether he is aware of it or not) that has chosen a non-inertial path between two events.

By the way, a simple accelerometer that can be used in the experiment is simply an object on a frictionless table. If you wish to deny that apparatus to the traveller, I think it should be possible to distinguish the two twins by the results of radar experiments. The spacetime maps that each makes from radar experiments will differ.

#### Mentz114

Gold Member
I'm sure this question has been asked thousands of times before, but I can't see how nature determines which twin accelerates (or is subject to a gravitational field) and which one doesn't.
Nature doesn't determine it - the twins do. This is very hard to understand - what do you mean ? If my twin and I decide to travel by different routes from A to B what's to determine ? I go on the motorbkie and my twin walks. When we meet my clock shows slightly less elapsed time that my twins. What's scary about that ?

#### Dale

Mentor
Hi Confusedent, welcome to PF,

Expanding on robphy's point, here is an arXiv paper I really liked called http://arxiv.org/abs/gr-qc/0104077v2" [Broken]. Note in particular figures 8 and 9 which show the situation in the travelling twin's frame. Also note the metrics on page 7 and 8

The key point to remember is that the acceleration does not cause time dilation, it only serves to indicate an asymmetry of the two reference frames. While an accelerometer reading is one such asymmetry it is not the only asymmetry. Another is the difference in radar experiments, or the differece in Doppler shifts. However, at a fundamental level, if you do not have enough information to calculate the metric along each worldline then you do not have enough information to calculate the relative aging.

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#### Confusedent

Thanks for the help all. I'm still pondering these answers...

-Matheinste: Sorry but that kind of seems like circular logic. It says that nature favors one observer as having "really" accelerated, while treating the other as having been inertial, when it could have just as easily treated them oppositely.

-robphy: I kind of see what you're saying about spacetime paths, but I don't see why one path needs to be considered inertial and the other non-inertial. Wouldn't each observer, from their point of view, see their own spacetime path as being the inertial one? Or am I misunderstanding how spacetime geometry works?

I tried to get past using feelings of acceleration since that just goes back to the idea of making one observers reality seem more real than the others. Plus, if the twins were particles without charge they shouldn't feel anything, each would think itself in an inertial frame.

#### Fredrik

Staff Emeritus
Gold Member
People say one twin will feel the acceleration and the other won't, but suppose neither twin had an accelerometer present (internal or otherwise).
Do you think that SR predicts a different result in that case? That would be like walking off a cliff and expecting not to fall if you don't look down

I feel I understand everything else about what's going on here with SR, except that this seems completely arbitrary, and even scarier, each one actually should be younger than the other one because each one has seen the other one switch frames (which is truely impossible). If someone can explain this it'll seriously help alleviate my headache, and probably save me days of worrying about it.
My standard answer: "Check out #3 and #142 (page 9) in this thread."

#### Confusedent

The key point to remember is that the acceleration does not cause time dilation, it only serves to indicate an asymmetry of the two reference frames. While an accelerometer reading is one such asymmetry it is not the only asymmetry. Another is the difference in radar experiments, or the differece in Doppler shifts.
But it is the acceleration that causes the asymmetry, and it should be indistinguishable who accelerated, so I still don't get why each twin doesn't return being younger than the other (besides that it is impossible).

Is it just that while there is no absolute rest with respect to velocity, there is an absolute rest with respect to acceleration (an absolute reference frame in at which everything else's acceleration is judged against)? Even though we could pick a different frame and treat the absolute frame as undergoing acceleration, nature would somehow know the difference?

I do appreciate your guys help in this, and apologize if I sound like a broken record parroting the same question.

#### Mentz114

Gold Member
Sorry but that kind of seems like circular logic. It says that nature favors one observer as having "really" accelerated, while treating the other as having been inertial, when it could have just as easily treated them oppositely.
Nature doesn't have your problem with symmetry. The situation is not symmetrical.

One observer 'really' did accelerate !

#### Confusedent

Nature doesn't have your problem with symmetry. The situation is not symmetrical.

One observer 'really' did accelerate !
However, at a fundamental level, if you do not have enough information to calculate the metric along each worldline then you do not have enough information to calculate the relative aging.
Does this mean that if we aren't told which one accelerated we can't determine which one is will be younger when they meet? For example, if the situation was stated that the distance between them was increasing at 0.8c, and then after the turn around the distance is decreasing by 0.8c, we would still determine that one will be younger than the other (since there was an acceleration) but we couldn't know which one (since the problem didn't specify)? As the first response said, only when they got back together could they determine that the younger one must have been the one to accelerate.

Assuming that's correct, I think I'm starting to get this now...

Gold Member

#### Fredrik

Staff Emeritus
Gold Member
But it is the acceleration that causes the asymmetry, and it should be indistinguishable who accelerated, so I still don't get why each twin doesn't return being younger than the other (besides that it is impossible).
You may have overlooked the fact that this can't be explained just by logical arguments. SR is a theory of physics. It consists of a mathematical model (Minkowski space) and a set of postulates that identify things we can measure with things in the model. The point is that there must be some way to interpret mathematical statements as predictions about the results of experiments, and it's impossible to derive an interpretation. It must be postulated.

The relevant postulate in this case is that what a clock measures is the proper time of the curve in spacetime that represents its motion. The curves that minimize the proper time (i.e. the time measured) are called timelike geodesics. They represent inertial motion. That can be considered a postulate too. (It may be optional in the sense that we can choose to postulate something else instead and derive this thing from that).

We can imagine "n-tuplets" instead of twins, and let n go to infinity. Have them move on all the different paths from start to finish. The one that's the youngest when they meet again is by definition the one who did inertial motion.

Is it just that while there is no absolute rest with respect to velocity, there is an absolute rest with respect to acceleration (an absolute reference frame in at which everything else's acceleration is judged against)?
Not a "frame", but there's a set of curves (the timelike geodesics) with a special significance. (They represent inertial motion).

Even though we could pick a different frame and treat the absolute frame as undergoing acceleration, nature would somehow know the difference?
You can e.g. choose to use a coordinate system such that the time axis coincides with the accelerating twin's world line. His coordinate acceleration in such a frame is of course 0 since his position is 0 at all times, but that doesn't change the fact that he's going to be younger than his brother when they meet. Nature doesn't care about what coordinate systems we're using.

Also, the acceleration measured by an accelerometer is equal to the coordinate acceleration in a co-moving inertial frame. (It obviously can't be equal to the coordinate acceleration in an arbitrary frame). That's another postulate.

Does this mean that if we aren't told which one accelerated we can't determine which one is will be younger when they meet?
Yes.

#### Dale

Mentor
But it is the acceleration that causes the asymmetry, and it should be indistinguishable who accelerated
This is a contradictory pair of statements. If X causes some physical asymmetry then X must be physically distinguishable, otherwise X would not be capable of breaking the symmetry.

For example, if the situation was stated that the distance between them was increasing at 0.8c, and then after the turn around the distance is decreasing by 0.8c, we would still determine that one will be younger than the other (since there was an acceleration) but we couldn't know which one (since the problem didn't specify)?
Essentially, yes.

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#### Al68

But it is the acceleration that causes the asymmetry, and it should be indistinguishable who accelerated, so I still don't get why each twin doesn't return being younger than the other (besides that it is impossible).

Is it just that while there is no absolute rest with respect to velocity, there is an absolute rest with respect to acceleration (an absolute reference frame in at which everything else's acceleration is judged against)? Even though we could pick a different frame and treat the absolute frame as undergoing acceleration, nature would somehow know the difference?

I do appreciate your guys help in this, and apologize if I sound like a broken record parroting the same question.
Hi Cunfusedent,

The issue of why must we treat the inertial frame as a "preferred" frame is not addressed in the standard resolutions. As you have seen, they presuppose their conclusion (that the inertial frame is "special") and show the math, which is good, but that doesn't answer your question.

One thing I would point out is that while relative velocity is not absolute, and therefore its derivative, coordinate acceleration cannot be absolute, proper acceleration is absolute in the sense that when a ship accelerates due to a force acting on it, its acceleration (and change in velocity) is relative to every other massive body in the universe, not just the other twin. These distant masses, and their relationship to (local) inertia, was discussed by Einstein, Mach, and others.

I would recommend searching the net for Einstein's writings. His way of thinking about these things is very different from anything I've read on the subject by others. And he takes the issue of why inertial frames are special seriously, instead of just accepting the fact and not worrying about why it's true. Einstein's writings may not satisfactorily answer your question, but you will find your point addressed and discussed.

You could even read Einstein's own Twins Paradox resolution, it's the only one I'm aware of that considers it from the point of view of an accelerated reference frame (in which the ship is at rest) during the turnaround. But while interesting, it still doesn't answer your question.

Also, Einstein claimed, in reference to one liquid sphere rotating relative to another, that the cause of the bulging equator on one sphere instead of the other, ie the "cause" of inertia itself, cannot be local, but must be due to distant masses. http://hem.bredband.net/b153434/Works/Einstein.htm [Broken]

Al

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#### Confusedent

This makes things considerably clearer. I guess its hard to accept the idea that you get different answers depending on who accelerated, when it doesn't work this way for velocity (e.g. if they're travelling apart at constant velocity, each sees the other aging slower). I need to study up more on Minkowski space and these geodesics too, the two SR books I've read didn't cover this.

Thanks again for the help everyone.

#### RandallB

This makes things considerably clearer. I guess its hard to accept the idea that you get different answers depending on who accelerated, when it doesn't work this way for velocity (e.g. if they're travelling apart at constant velocity, each sees the other aging slower). I need to study up more on Minkowski space and these geodesics too, the two SR books I've read didn't cover this.
But the issue in SR Twins is not, "relative to every other massive body in the universe".
Nor is it due “if they're travelling apart at constant velocity, each sees the other aging slower”
Because: when they're travelling toward each other at constant velocity, each still sees the other aging slower.

What you need to be clear on is how whoever changes reference frame [and they can always tell if the just set out a trackable buoy to float next to them in “stationay” inertial motion with them]. When you change direction you have to know your away motion (the frame the buoy stays in) is a different frame than your return frame motion; (if you still stationary wrt to the buoy you put out the only way you could see the other twin now coming toward you is they are now in a different frame of motion)

The thing that makes all the difference is what is the rate of time difference is, between the two frames used by the clock that changed frames. Inorder to figure the total time on that clock you must include when it runs MUCH slower in the other frames than the one you pick as a reference. Enough slower to more than make up for the slow time you see on the stay at home twin clock - that net total be slow comared to the stay at home twin clock - to match exactly how that clock POV saw both versions of your clock (outbound & inbound) run slow.

The point is not to mess up and use BOTH frames (outbound & inbound) as part one master reference ie. No “Time Line” of a clock accelerating into a second frame!
You must do all calculations and observations from just one Frame.
Although all frames will disagree about which clocks are going fast or slow – they all agree and give the same times when the two clocks are next to each other (at departure and return). Seeing that is just a matter of doing the SR math completely from each one of the three different frames point of view.

It is clear that all frames must be wrong because none can agree on which one has the correct rate of time. BUT since all can produce and agree on one net solution to such a twin problem – all must be defined as correct in some way when used as a sole reference point “as if preferred”.

The net result was to establish relativistic solutions that were to be applied only wrt a single reference frame – but any one reference frame is OK; which required assuming that the idea of a Newton Absolute Time & Absolute Space be abandoned.
That shows up in GR too, with what is called an indeterminate background – you will learn more about that when you read Smolin (Perimeter Institute).
But before you do that be sure to “do the SR math” yourself enough times from different frames to understand why “Aboslute” was rejected in SR & GR.

#### matheinste

Hello Confusedent

Quote from Al68:-

----You could even read Einstein's own Twins Paradox resolution, it's the only one I'm aware of that considers it from the point of view of an accelerated reference frame (in which the ship is at rest) during the turnaround. But while interesting, it still doesn't answer your question.-----

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As RandallB says so well, it does not matter whose point of view or (single) frame you do the calculation in, they will all, of necessity, give the same END result.

Matheinste.

#### nutgeb

Here's a variation on the twins paradox:

Each twin travels in her own spaceship. The twins synchronize their clocks at the start. One twin starts on the far side of galaxy A, the other starts on the far side of Galaxy C. Each ship accelerates rapidly to .8c past its own starting galaxy, and the paths of the two ships are directly toward each other. By the time each ship actually passes the centerline of its own starting galaxy A or B, it has finished accelerating and thereafter continues on at constant speed for many years. Eventually, both ships simultaneously pass Galaxy B traveling in opposite directions. Galaxy B is located exactly at the midpoint between Galaxies A and C.

As each ship passes its starting Galaxy A or C, each twin communicates with residents of her starting galaxy and synchronizes her ship clock with the galaxy residents' local clock. The galaxy residents of Galaxies A, B and C all have synchronized their own clocks with each other just before then.

When the twins pass very close by each other (say, within 10 meters) at Galaxy B they can communicate with each other and see into the other ship's window. Will they agree that their ship clocks are synchronized with each other then?

During the voyage, the twins approach each other at a relative speed higher than .8c but less than c. They are in separate and distinct inertial frames after passing their starting galaxy.

Special Relativity says that an observer in any inertial frame will measure her own clock ticking slower than a clock in an inertial frame which is in motion relative to the observer's rest frame (at relativistic speed). So is there a paradox when the ships pass by each other and each twin can look in the window of the other ship and observe that the other twin has not aged differently than herself?

During the inertial (constant speed) phase of the trip did both twins age less than the residents of the 3 relatively "stationary" galaxies?

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#### Al68

Hello Confusedent

Quote from Al68:-

----You could even read Einstein's own Twins Paradox resolution, it's the only one I'm aware of that considers it from the point of view of an accelerated reference frame (in which the ship is at rest) during the turnaround. But while interesting, it still doesn't answer your question.-----

------------------------------------------------------------------------------------

As RandallB says so well, it does not matter whose point of view or (single) frame you do the calculation in, they will all, of necessity, give the same END result.

Matheinste.
Yes, Einstein's resolution does give the same result as the standard resolutions. But the end result is not the only thing that matters. The OP was not asking what the end result is, presumably he knew that already.

Al

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#### Al68

But the issue in SR Twins is not, "relative to every other massive body in the universe".
Well, this issue relates to the "cause" of inertia itself that leads to accelerated bodies being distinguishable from inertial ones. Which is relevant if you care about the more profound aspect of the Twins case, instead of just how to get the end result. Surely you wouldn't suggest that the ship's voyage is non-inertial just because its coordinate velocity relative to earth changed.

And the OP was not asking how to just get the right answer. He was asking how we determine which twin to consider inertial. Most of the responses explained how to solve the problem after this determination is made.

Al

#### RandallB

Here's a variation on the twins paradox:

The twins synchronize their clocks at the start. One twin starts on the far side of galaxy A, the other starts on the far side of Galaxy C.
No they cannot. They must be next to each other and directly looking at each others clocks to know all agree that what time is on each clock. This cannot be done if the are apart from each other
Special Relativity says that an observer in any inertial frame will measure her own clock ticking slower than a clock in an inertial frame which is in motion relative to the observer's rest frame.
You stated that backwards, any observer will see all moving clocks as running slow.

However, because that obsever also sees each clock in a moving frame as not being correctly set; such that by reading the times on the clocks as they pass by thet are set out of sync in such a way that although each one is running slow - by only reading the time displayed only when the clocks pass by it will seem to show time there running faster than the observers clock. Just remember, is from reading many diffent clocks passing by as if showing a movie of many snapshots of time.

In SR no paradox at all, just the realization that each frame can think of itself as "correct and prefered"; but no one of them can prove they are abosolutly prefered to any other frame.

#### matheinste

Hello Al68.

Quote:-

--- Surely you wouldn't suggest that the ship's voyage is non-inertial just because its coordinate velocity relative to earth changed. ----

I think the suggestion is that the ship's voyage is non inertial because it underwent acceleration.

There is nothing profound about the twin paradox it is just a consequence of the axioms of SR.

The OP asked how we would know which twin accelerated without him/her measuring or feeling any acceleration. Under these circumstances the only way of knowing is by looking at the end result.

Matheinste

#### nutgeb

OK RandallB, but I think you are dismissing my alternative "paradox" based on technicalities (fair enough), which I shall try to remedy in this updated version:

Twins 1 and 2 begin their journey located together, and synchronize their clocks. Each twin's ship quickly accelerates away from each other to a speed of .8c (relative to their common starting rest frame) and then travel apart a distance of 25 LY each at that constant identical speed. Then they decelerate identically (all pre-arranged), stop, and immediately reaccelerate identically toward each other quickly to a constant speed of .8c at which they continue at until they pass by each other (10 meters apart).

When they pass by each other on the return trip and look in each other's window:

(1) Will the other twin's clock show (approximately, say within 1 second) the same time as theirs? They pass by each other only 10 meters apart, so the light travel time from ship to ship is insignificant.

(2) Will they observe each other to have aged by about the same amount? Or will each twin observe the other twin to have aged about 50 or 100 years less?

#### JesseM

(1) Will the other twin's clock show (approximately, say within 1 second) the same time as theirs?
Yes.
nutgeb said:
(2) Will they observe each other to have aged by about the same amount?
Yes.

#### nutgeb

OK JesseM, then there's a paradox, because in SR each twin is supposed to observe that the clock of the other twin, who is in motion relative to the first twin's rest frame, is ticking slower.

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