# Q: How does the Twin Paradox work?

The original question was: I have a question about the twin paradox.  Is it true that faster aging of the twin who stayed at home happens only when the other twin’s spaceship is accelerating/deceleration (btw, does it matter whether he is accelerating or decelerating?)?  Consequently, do they age at the same rate when the spaceship moves inertially?

Physicist: The very short answer is: geometry works different in spacetime than it does in just space.

The twin paradox is a result from special relativity that states that if one person, Alice, remains “stationary” and another person, Bob, takes any kind of round trip, then the stationary Alice will experience more time.  The twin paradox isn’t a paradox at all, it’s just strange and off-putting (like twins).
In relativity (that is to say: “in reality”) there’s no difference between being stationary and having smooth (non-accelerating) movement.  On the surface of it, the only difference between Alice and Bob is that, in order to return home, Bob has to accelerate (turn around) at some point.  So is acceleration the secret to the twin paradox?  Nope.

In all of the pictures that follow the “time direction” is up, and one of the (three) space directions is left/right.

Out and back: In both situations Bob (blue) experiences the same acceleration, while Alice (aqua) sits around on Earth. The only difference is that the second situation involves twice the distance, and twice the difference in experienced time. Acceleration is not what's important.

The trick is: spacetime doesn’t obey the “triangle inequality”. As a result, the bendier a path is, the shorter it is (that shouldn’t make any sense, so please read on).

The triangle inequality says that (in space) the sum of any two sides of a triangle is greater than or equal to the third side. In spacetime the inequality can be reversed. One side is often longer than the other two: a round-about route is shorter than the direct route. In this case, A+C<B.

The equation for distance that we’re used to is: $D^2=\Delta x^2+\Delta y^2+\Delta z^2$ (this is just the Pythagorean theorem).  But you find that when you start involving time and movement, this isn’t a particularly good measure of the distance between two points. Specifically, it’s different for different observers because of length contraction.
It so happens that the effects of length contraction and time dilation cancel each other perfectly, so that we can use a new (better) measure for spacetime distance, called the “Interval” or “spacetime interval” or “Lorentz interval”:

$L^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$

(as often as not the sign on the right hand side is reversed, not to worry)

The advantage to the Interval is that, no matter what, the Interval between any two points in spacetime (two locations and times) are always the same, despite relativistic weirdness.  Here’s another bonus!  The Interval of a path is the same as the amount of time experienced on that path!

No one every really feels like their own position is changing, so: $L^2=c^2\Delta t^2-0^2-0^2-0^2\Rightarrow L=c\Delta t$

Now all that’s left is to draw a picture and do a little calculating.  Here’s an example situation from Alice’s perspective, and then Bob’s (initial) perspective.  The difference between Alice and Bob’s velocity is 0.6C (60% of light speed).

The same situation from two perspectives, but since L is invarient you get the same values. Alice experiences 10 units of time while Bob experiences 8 units (since 5^2-3^2=16=4^2).

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### 29 Responses to Q: How does the Twin Paradox work?

1. Rich says:

I still don’t quite follow why the situation isn’t symmetrical.

Why can we say that Alice is sitting still (moving only through time) and Bob is taking a round trip, but we can’t say that Bob is sitting still and Alice is moving? Is it because Bob changes velocity? But it’s not the acceleration that’s important, it’s just that he’s had two different velocities?

2. The Physicist says:

Regardless of your perspective, Alice is always moving in a straight line, and Bob is always moving on a crooked path. As a result of the “messed up triangle inequality” all crooked paths are shorter than straight paths (through spacetime). That’s the essential difference.
Another (clearer) way to say that is Bob has two velocities.

3. Rani Sunarso says:

Bob has two velocities, that if we say Alice is sitting still on earth, while Bob is moving away.

But from Bob’s perspective, he is sitting still (on the rocket), while Alice is moving away from him with a speed of 0.6c.

4. Anthony Rose says:

Rani, I think that if you accelerate, you can’t treat your frame of reference as inertial. We cannot treat Bob as still and Alice on Earth as accelerating and decelerating, because there is a force (the rocket jet) affecting Bob’s frame of reference. I stand to be corrected but i think that at constant velocities you can apply either frame of reference, but not when one body experiences any kind of acceleration.

5. The Physicist says:

@Anthony Rose
That sounds exactly right! I tried to cover that idea with the last picture in the post, but it’s not terribly clear.

6. Gene Kim says:

If Bob leaves Earth and travels to some point in space, he will have aged slower than Alice who stayed on Earth, but if Bob stays at that point and Alice leaves Earth and joins Bob, will they both have aged the same amount of time?

7. The Physicist says:

Yup. Weirdly, it’s impossible to objectively say which has aged more or less until they’re brought back together.

8. Chrobry says:

Are you aware of any other credible explanations for the Twin Paradox?

I don’t believe the Twin Paradox is solved by determining which twin accelerated (non-inertial frame) or which twin changed inertial reference frames. I can provide a thought experiment where the twin on Earth (who neither accelerates or changes inertial frames) ages slower than the twin in a spaceship (who does accelerate and/or change inertial frames). No tricks.

These two explanations appear to be ubiquitous. I am only looking for “mainstream worthy” credible explanations, not quack explanations such as “special relativity is not real” or some variety of Mach’s principle.

9. John Frey says:

My understanding of the twin paradox is that the uniform accelerating motion of the traveiling twin plays a big role in why the principles of special relativity and this motion cause time for the traveling twin to pass slower than before this motion but, beause this accelerating motion it not the motion of the stay-at-home twin, it causes no slowing of his time.

My questions:

1) Is the above understtanding correct?

2) Does the relativity principle of special relativity function through this uniform motion because the motion keeps slipping from one uniform moving frame to another and thus, this motion cfan be considered uniform as demanded by this principle?

3) Would the relativity prinicple of special relativity function through this accelerating motion if it were rough, erratic, and swerving from side to side?

John

10. Walter Roberts says:

Einstein claims that the laws of physics apply equally to all observers in any inertial field (at rest or moving at a constant speed). He also claims that there is no preferential inertial field. That is Alice can say she is stationary and Bob is moving and Bob can say he is stationary and Alice is moving. In this situation they are both correct and you can not arbitrarily pick one view and say this is the correct view.
From an example above:

“Why can we say that Alice is sitting still (moving only through time) and Bob is taking a round trip, but we can’t say that Bob is sitting still and Alice is moving? ”

Sure you can, Einstein stated it. Either Alice or Bob can claim they are sitting still and the other is moving. Otherwise you have a preferential inertial field and Einstein says you can’t have that.

I think the issue here is Reality vs Relativity. Everyone knows that the rocket ship is what takes off and travels around and returns. But according to Einstein either observer can claim they are sitting still so as weird as it sounds Bob in the rocket ship can claim he is sitting still and the earth is moving away traveling at close to the speed of light and then returns. Although this sounds ridiculous, it is within the postulates of special relativity.

I believe too many people only take one observers view and then claim things like time dilation, length contraction, time running slow etc. They fail to also consider the view from the other observer who also claims that he is sitting still and it is the other that is moving.

The bottom line is that both Alice and Bob claim they are sitting still and the other is moving. When they reunite at the end of the trip they will both sway the other has aged less. So basically they are both exactly the same as before the trip started.

The theory of relativity only means something if taken and observed from one observers view. You have to use both views.

11. John Frey says:

Concering the twins, Alice and Bob, upon being reuinted aftter the motion between them, you wrote: “The bottom line is that both Alice and Bob claim they are sitting still and the other is moving. When they reunite at the end of the trip they will both say the other has aged less. So basically they are both exactly the same as before the trip started..” However, acccording to Einstein in chapter 4 of ‘On the Electrodynamics of Moving Bodies’, when the traveling twin (clock A) returns to the stay-at-home twin (clock B), the twins no longer have the same age (the clocks are no longer synchronied) but the traveling twin is younger than the stay-at-home twin (the traveling clock lags behind the stay-at-home clock). It seems to me that you are disagreeing with Einstein. John

12. Walter Roberts says:

Actually I am not disagreeing with Einstein. From a preferential inertial field view, if “Twin A traveling and Twin B stays at home at home” then Einsteins prediction would hold.

However Einstein also claims that there is no preferential inertial field. That means either A or B can claim to be stationary and the other traveling.

traveling twin (clock A) returns to the stay-at-home twin (clock B)
traveling twin (clock B) returns to the stay-at-home twin (clock A)

Both scenarios are equally valid. Which one would choose? Either one you choose is a preferential inertial field view.

Imagine both twins waking up in outer space. There are no stars, planets or anything else around. What each twin sees is the other passing by. Which one is stationary and which one is moving? How would you pick one and then try to prove that you are correct?

What people don’t seem to understand is that both views exist at the same time, whatever happens to one happens to the other as well. The only way Einsteins theory works is if you pick a preferential view: “A is stationary and B is traveling” or “B is stationary and A is traveling”. But Einstein also said you can’t choose just one because that is preferential.

If you look at all the examples of time dilation, twin paradox etc. they all are from one view only, the stationary observer. This makes the theory work. I’d like to see someone explain the the theory from both observers view and then see their results.

13. John Frey says:

You wrote: “If you look at all the examples of time dilation, twin paradox etc. they all are from one view only, the stationary observer. This makes the theory work. I’d like to see someone explain the the theory from both observers view and then see their results.” However, acccording to Einstein in chapter 4 of ‘On the Electrodynamics of Moving Bodies’, when the traveling twin (clock A) returns to the stay-at-home twin (clock B), the travelling clock lags behind the stationary clock relative to the observers with the stationary clock and those those who travelled with the moving clock – or in terms of twins – the travelling twin was younger than the stationary twin relative to the stationary twin and observers who remained with the twin and relative to the travelling twin and those who travelled with him. In other words, when the travelling clock and the stationary clock are placed side by side after the motion between them, the traveling clock lags behind the stationary clock before all viewers of the two clocks. Some these viewers cannot see clock A lag behind clock B and some see clock B lag behind clock A but all see the same thing: in the words of Eiinstein: “Clock A (the travelling clock) lags behind clock B.

twins no longer have the same age (the clocks are no longer synchronied) but the traveling twin is younger than the stay-at-home twin (the traveling clock lags behind the stay-at-home clock). It seems to me that you are disagreeing with Einstein. John

14. Walter Roberts says:

Once again you are looking at the problem from a preferred view, the twin at home. And yes you are right. Look at the problem from the traveling twin position also.

Traveling Twin: I am stationary in the rocket. It is the earth that moves away at close to the speed of light. When the earth returns to me from traveling I compare my clock with the twin at home (on the earth). The twin on earth would experience the time dilation and hence age slower.

Trying to view this problem from this position is hard to do. Of course the earth could never do this, but Einsteins theory permits us to say it does because there is no preferred reference frame. What you are doing is looking at only the results from one of the twins reference frame. Doesn’t the other twin have the right to claim that it was him that was stationary instead of being the traveling one?

Forget the rocket and earth example for a moment. Imagine both twins being in outer space looking at each other. Pick one of the twins, A or B, and do the mind experiment, ie. twin travels away from the other and returns later. You get the results you mentioned, twins no longer have the same age (the clocks are no longer synchronized) but the traveling twin is younger than the stay-at-home twin (the traveling clock lags behind the stay-at-home clock).

Now redo the same experiment using the other twin as the traveling twin. When this twin returns home you again have the same results as above. Compare the results from both twins trips. How can you say which one is older or which one experienced time dilation. How can one twin be any different from the other? They both traveled the exact same path, distance, time, speed etc. So which one is older?

The confusion for the relativity theory is that Einstein claims, in his first postulate:

- The laws of physics work equally well for any inertial reference frame
- There is no preferred inertial reference frame.

Einstein wrote the theory. I just find it very interesting that people don’t understand what he is actually saying and the implications it causes.

A simple explanation of the first postulate could be: The laws of physics work equally well in any inertial reference frame and you can’t arbitrarily pick the inertial reference field you like and disregard the other.

So think about your comment above. Are you arbitrarily picking the twin at home?

15. The Physicist says:

@Walter
Sorry about the long response time.
The fact you bring up, that there is no preferred frame, is true. The fact that from each of their perspectives the other experiences less time is also true (although understanding how that works is a little tricky).
Addressing why one twin experiences more or less time than the other, despite neither of their frames being “preferred”, is what the post above is about. The difference ultimately comes down to the shape of the path they take through space-time. The twin with the more circuitous, longer path always experiences less time. That distance is defined using this notion for special relativity, and using this notion more generally (when you take into account gravity).
The twin on Earth is assumed to take a “straight line” from their mutual starting time and place to their mutual ending time and place. The “traveling” twin is assumed to take a curved or bent path. The straight path between the two meeting points experiences less time (counterintuitively).
While relativity does assume an equivalence of velocities (no preferred frame), it does not assume an equivalence of accelerations (when someone steps on the gas or the brakes, you’ll always know). The bending of the traveling twin’s path cannot be ignored, because (unlike velocity) it’s a real and objectively measurable quantity. As well, the time difference of the two paths can be objectively measured (and has been!).
Hope that helps!

16. John Frey says:

You wrote: “The bending of the traveling twin’s path cannot be ignored.” However, acccording to Einstein in chapter 4 of ‘On the Electrodynamics of Moving Bodies,’ the traveling twin A (clock A) does not start his travels at the side of the stay-at-home twin, B, but apart from him. Then, according to Einstein, twin A (clock A synchronized with clock B) travels from his distant position to the side of twin B and that indicates that his journey was on a straight, non-bending path. When twin A reaches B, his accelerating, uniform, decelerating motion causes him to be younger than the stay at home twin relative to all observers.

17. John Frey says:

According to Einstein, we have twins at rest with each other with the same age. Twin A goes into motion relative to twin B and B remains stationary. When twin A is at the side of twin B after this motion, he is younger than B relative to all observers. How can this be the case if, as you claim, the motion between the twins is the motion of A relative to B and this motion causes A to age slower than B, and if this motion is the motion of B relative to A and this motion causes B to age slower than A relative to all observers?

18. The Physicist says:

The issue here is that B’s reference frame is used to define the starting time for both twins. From B’s perspective A’s clock will read a lower value when they meet because A was moving.
Twin A will agree that when they meet his clock reads a lower value (this comparison, at the same place and time, is objective). However, A explains this as being caused by B’s clock starting with a higher value. They can’t start at the same time because, without simultaneity, they can’t agree on what the “same time” is.
I thought about using an argument like this, but tilted “now planes” and whatnot seemed like it would be a lot more difficult to convey or understand than the “longer path = shorter time” argument.

19. Walter Roberts says:

Wow, this is getting interesting.

You say “The difference ultimately comes down to the shape of the path they take through space-time. ” and “The twin on Earth is assumed to take a “straight line” from their mutual starting time and place to their mutual ending time and place. The “traveling” twin is assumed to take a curved or bent path. The straight path between the two meeting points experiences less time (counterintuitively).”

How can you assume this? By your logic then you could assume anything you want to make the theory work?

Use my example of the two twins in space, side by side looking at each other, both of their clocks are synchronized. Nothing else is around. They close their eyes and when they open their eyes they both see the other twin as traveling. In reality only one twin is moving the other is stationary, but they can’t tell which is which because the motion looks the same from each perspective. After they return from identical trips both twins claim the other is younger and had the slower clock. Is not the net result then zero? Each twin claimed they were stationary and the other was traveling, and their routes were exactly the same in speed, length, turns etc.

I try to look at this problem from the simplest view possible to try and understand it.

From my simple example of the two twins in orbit same path etc. can you show me why that won’t work?

20. John Frey says:

You wrote: “However, A explains this as being caused by B’s clock starting with a higher value.” You seem to be saying that when the journey of twin A begins while twin B remains stationary, twin B is already older than A and, hence, when A returns, twin B is still older that A.

21. Walter Roberts says:

The Twin Paradox is one of the most well known and debated paradoxes associated with Relativity theory. Opponents challenge Relativity theory on the
grounds that the Twin Paradox reveals an underlying flaw in the theory. Such opponents feel that the existence of a paradox, in and of itself, is sufficient to disqualify the theory. Supporters explain the paradox by introducing the concept of acceleration into the theory, thus limiting the interpretation to the twin that was undergoing the force of acceleration. (Frederick J. Hutchison)

This is my point exactly. When ever you claim things like different legs of travel, acceleration, different path, different starting clocks you change the original parameters to suit your own agenda. In my example of the two twins in space, they both see and experience the exact same path. Upon return they are exactly both the same age etc. even though they both observed the other twin as being in a slower time frame.

Einsteins theory only works from a preferential view. And then the result is basically meaningless. So what if the twin traveling through space came back to earth and was 10 years younger than his twin. Do you actually think that he would live the rest of his life on earth being 10 years younger than his twin, and that life would continue as before he left? That would be the result Einstein would like you to believe. But don’t you think having a brother all of a sudden 10 years younger than you would be sort of weird and maybe challenge some laws of science?

Think of it, there could be an opportunity here to make a fortune here. Give people a ride on the “Youth Space Ship” and after a 1 year trip they can be 10 years younger.

I’m really amazed at that scientists haven’t figured this out yet.

22. Phaedrus says:

Walter, I agree that physicists have some splainen to do! Although I don’t agree that the thought experiment you are giving will prove why Einstein’s theory of relativity is invalid. If You have 2 people floating in space, if one moves away and the other doesn’t you could determine which one is staying in their inertial frame via an accelerometer they could be holding, ok? One stays in their inertial frame, the other moves away (accelerates), moves away at near the speed of light, causing their time to slow down (according the Special Theory of Relativity) with respect to the other twin which is staying in their inertial frame. And the moving away twin will have to make a total deceleration at the end of his/her trip and then reaccelerate in the opposite direction to get back to their twin (or some variation of this via a curved path). So that’s the excuse the believers in this aspect of the SToR will tell you. But I want to present to you a thought experiment I came up with which is much harder to counter, I think. Here it is (I wrote this more than a year ago):

Ok, let’s say we are standing on a huge inertial frame, a flat 2-dimensional space (just like people living on this planet used to think of the ground they were standing on). Off in the distance ahead of us are 2 train tracks, parallel to each other and separated from each other by a few meters. They are both going off to the east and the west from us, the viewer (using east & west directions assumes we are looking north). And our line-of-sight to them is perpendicular to the straight-lines of these tracks, which go off to the right and left for as far as we want. Let’s say if we go way up above this scene, the main points of interest are at the vertices of a huge equilateral triangle, but with the base of the triangle at the top. There is a train on one end of the tracks to the left, sitting there ready to go towards the right. There is a train on the right side of the tracks at that vertex of the triangle, pointed to the left ready to go. There are clocks at all three vertices of this triangle, as well as people: A clock at the train on the left (call this train A), a clock at the train on the right (call this train B). And we the viewer at the 3rd vertex of the triangle also have a clock. All these clocks are connected to this vast inertial frame we a standing on, and they are all still with respect to the inertial frame and with respect with each other. So they are all synchronized (am I allowed to do that, to synchronize clocks at various points of an inertial frame?). They all show the same time and they are all ticking at the same rate.

Now, people at train A get on their train and start going at a predetermined time (say 10 o’clock AM). Likewise, people at train B get on their train (train B) and start their train engine moving towards train A, also at 10 o’clock. They both are going to increase their speed from a dead stop at exactly the same rate. They will go faster and faster. They will both move a faster and faster rate, which will start to approach the speed of light. But they will both be moving with respect to their inertial frame at the same rate, only they will be going in opposite directions and on separate tracks. Finally they have their speed up to something like 1/3 of the speed of light (with respect to the inertial frame this is taking place on) and that will be their cruising speed.

Now, according to the special theory of relativity, people in train A will see the clock in train B as going at a slower pace then their own clock, because train B is going at a high speed in relation to train A. Like wise, the people in train B will see the clock in train A as ticking at a slower pace than their own clock. For example, when the clock in train A gets to show 11 AM, they might see the clock in train B as showing 10:50 AM. So they are both barreling along towards each other. Finally they are both approaching the midpoint between where they started from (this will be directly between the left vertex and right vertex of this equilateral triangle, and it will be directly “above” the bottom vertex where us the viewers are watching this from). They put on their brakes at the same time and slow down. Slower and slower and finally come to a stop right next to each other. The people in train A will have been watching the clock in train B tick along slower than their own clock, so once train A has stopped at let’s say 12 o’clock their time, they expect the clock on train B to be showing something like 11:30 AM. Likewise, the people on train B have been watching the clock on train A, seeing it ticking at a slower rate than their own clock, so when they stop and get off train B at a point when their own clock says 12 o’clock, they expect the clock in train A to show something like 11:30 AM. Now from the vantage point of us the viewers of these 2 trains from the lower vertex of this huge equilateral triangle, we who are sitting on this huge inertial frame that everything is taking place on, we would have seen both the clock on train A and the clock on train B as ticking at the exact same rate, although they would be ticking slightly slower than our own clock, since they are both moving with respect to us and even some significant fraction of the speed of light. But now everyone is stopped and standing on this one huge inertial frame of reference again. Now, how will the clocks on train A and on train B compare? To us the viewers of these trains, they should both show the same time. To train A occupants, train B’s clock should be retarded in relation to their own. To the train B occupants, train A’s clock should be retarded in relation to their own. But only one reality will manifest itself once the occupants of train A and train B meet at the middle, correct? What will they see when they view each other’s clocks?

The above thought experiment, to me, shows that Einstein’s Special Theory of Relativity has a definite problem, at least in regards to this time dilation business. There is a paradox, and I don’t see how it can be resolved. If a theory leads to an unresolvable paradox, then how can that theory stand? Of course, I shouldn’t say it definitely IS unresolvable. Perhaps I just haven’t thought of the solution. But what I love about physicists is the way they talk about this stuff like it is a fact, like its all settled. They can’t say “we believe this is the way it is”, they’ve got to say “this is the way it is” as if they are gods or something.

23. walter roberts says:

Hi Phaedrus, thank you for the very simple and interesting explanation.

I think you finally understood my original question, although you expanded it with the third observer, good job.

My whole point was that in any thought experiment only one side of it is shown (ie, The train passes the observer standing on the platform, who sees the clock on the train running slower, therefore time is running slower on the train.)

No one ever seems to look at the same thought experiment from the observer on the trains view, who would of course see exactly the same thing.

I would guess the reason for this is people can’t visualize a “stationary train and a moving train station/platform experiment”. So they add acceleration, turns and so on to the “stationary side” to show it was different from the real moving object (the train).

Therefore from a non preferential view thought experiment, when they both get back together the net result will be zero. No one traveled faster or slower than the other. Like you expressed in your answer above.

Also if the theory was true look at some unexpected consequences that could occur.

Say in the Twin Paradox, the traveling twin returns home and is 50 years younger than his twin brother on earth. Does he live the rest of his life on earth with a 50 year age difference to his brother? Think of the problems that this raises in how life works on earth, , biology etc.

24. Xerenarcy says:

the twin paradox is a nonparadox if you only involve special relativity, which deals with velocities.

true, both twins would argue that, while in constant motion, the other’s clock is what is running slow and not theirs. the difference is, unfortunately as stated, the path taken.

twin A, when B accelerates away, does not notice anything different about the rest of the universe.
twin B, when it accelerates, notices the entire universe accelerating with twin A.

that is an asymmetrical experience, resulting B aging independently of the rest of the universe as seen by A, and A as seen by B, ages identically to wherever A happens to be (that is, stationary). simple.

25. Andy Nguyen says:

Fist I want to say this is very exciting discussion. Brought new lights to my mind.

Just a thought, but no one seems to take SPACE into account here. If you spin in emty space (take the rest of matter in the universe away), would you feel a force pulling your hand outward eventhough you are spinning relative to nothing? You would feel a force pulling your hand outward, because you are spinning relative to space.

Can’t we say that SPACE is STATIONARY.
- If we say A is moving, A is moving relative to Space.
- If we say B is stationary, B is stationary relative to Space.
In this situation the 3rd observer is Space itself, and Space knows who is moving and who is not moving.

26. Ed says:

I think we are all missing something very important, time may not even exist.

I am not going to address The Twins paradox because without knowing if time exist there is no point at all.

Think about in more simple terms, the way physicist decided to quantify something (Time) they have no real proof of it existence at all, is prone to errors.

All it was proven in all the time dilation experiments is that, interval measuring on machines created by humans are susceptible to quantification errors and or interference.