# Twin paradox - is acceleration relative? [duplicate]

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The usual answer to the twin paradox is that the twin who undergoes acceleration is the one who finds the other has aged more, because the trajectory of the 'travelling twin' does not amount to a single inertial frame - the travelling twin has accelerated while the stationary twin has not. But isn't acceleration a relative phenomenon? If the universe were empty apart from the two twins, what does it mean to say that one and not the other is accelerating and how could it be decided which was which?

(Inevitably someone will say this is a duplicate - apologies if so but I can't find it)

Edit: thank you for those thoughtful responses - on reflection I was asking two questions and the twin paradox is about velocity not acceleration. I think the interesting question is how we distinguish the velocities or accelerations of two objects, which you need to do in order to calculate the proper time elapsed for each. It can't be enough to say that the accelerating one 'feels it' - that is true only insofar as the force is transmitted from one part of the object to another (such as from the rocketeer's back against the seat to her inner ear, or from the body of the accelerometer on the floor of the rocket to its detector component). Maybe it relates to Mach's conjecture about absolute rotation, ie that you have to make reference to the distribution of matter in the wider universe in order for the distinction of moving vs not moving to have meaning. If so there is a sense in which acceleration is relative rather than absolute. Anyway thank you.

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• The relevance of acceleration to the twin paradox is a separate question from whether or not acceleration is relative. Which of these two questions are you asking? – safesphere May 7 at 20:07
• The "inertial frame" explanation is correct in SR, but "relativity of acceleration" implies a GR explanation. It's a relatively straightforward exercise to show that the gravitational time dilation formula for a mass that generates an equivalent acceleration (for one in a non-inertial frame of course) breaks the symmetry and gives the same result as usual. – user1247 May 7 at 20:10
• Something to note is that acceleration is not required to resolve the twins paradox – Aaron Stevens May 7 at 21:26
• @ user1247 Since the traveling twin can accelerate & decelerate in innumerable different ways, how is the equivalent gravitational acceleration determined? – D. Halsey May 7 at 23:58
• Each twin would claim to be older than the other twin. – Cinaed Simson May 8 at 5:57

## 6 Answers

This is actually an excellent question and it bothered me for an incredibly long time because I thought exactly what you did: acceleration is relative, is it not? One twin will see the other's speed is changing, and vice-vera. So how does the universe know which once is actually accelerating? The answer is that acceleration is detected using a scale. If you stand on a scale on the Earth, you see a number appear: your weight is non-zero. This means that you are accelerating, and certainly you must be, because the Earth is rotating, and because you are on the Earth, you must be moving with it. Thus gravity acts as a centripetal force that pulls you down, and you measure the existence of this force on a scale. Even if the Earth were not rotating, you are still in a non-inertial frame because there is still a force felt and a measurement made on a scale. Consider throwing a ball horizontally on the Earth: it will fall to the ground. If there Earth were an inertial frame, there would be no way for said ball to gain a vertical component of velocity relative to the Earth. If you step on a scale and measure a weight, your motion is said to be non-inertial.

Consider yourself in free fall above the Earth, and ignore wind resistance (to be specific, consider just the tug of gravity). If you stood on a scale moving with you, it would read zero. Thus, as far as we are concerned in relativity, you are NOT accelerating, and there is no force acting on you. This might be a difficult concept to come across because you might think otherwise. Indeed you are gaining speed as you move toward the Earth, but in your own reference frame you feel no force, and thus we say you are not accelerating, and you are in an inertial rest frame. Actually, inertial reference frame are an idealized concept, only a single point can actually be inertial: all extended bodies can only be approximated as inertial. But this is off-topic, so for our purposes we make the rather good approximation that a person in free fall is in an inertial reference frame.

The final answer is then that during the acceleration, the twin in the rocket could stand on a scale and the reading on the scale would reflect the acceleration process, whereas the other twin's scale should theoretically read zero because they should be in an inertial reference frame. It is the twin in the non-inertial reference frame that actually experiences the acceleration, and this is the twin that has a reading on the scale.

What you should take home from this is that forces and acceleration are detected via a scale. Certainly in many physics courses you might take that the acceleration is the time derivative of velocity, and that works; but, we see here that thinking of acceleration in this way leads to some ambiguity and makes the twin paradox seem more daunting than it should be.

• f you stand on a scale on the Earth, you see a number appear: your weight is non-zero. This means that you are accelerating, and certainly you must be, because the Earth is rotating, and because you are on the Earth, you must be moving with it. You would still have a weight on a scale without the Earth rotating – Aaron Stevens May 8 at 3:28
• @AaronStevens That's a good point, I've hopefully updated my post to be more clear. – Kraig May 8 at 12:42

It's easy To spot you're accelerating without any other reference points. For example a twin in an accelerating ship would see he couldn't apply Newton's laws inside the ship since positions of objects within the ship would change their positions at an increasing rate without any forces present. He would realise he had to modify Newton's laws to do physics thus he is in a non inertial ie accelerating frame.

• This logic does not automatically apply in an empty universe as in the OP's question. See: en.wikipedia.org/wiki/Mach%27s_principle – safesphere May 7 at 20:29
• @safesphere: the logic might apply in some empty universes but not others. what matters here, I think, (given the SR tag) is that it applies in minkowski space, which is the universe that SR describes. – WillO May 7 at 20:39
• @WillO I did not say it did not apply. I said it did not automatically apply. An empty universe is tricky. You cannot define it without making additional assumptions. In your case, the assumption is that the spacetime is Minkowski. This assumption would be valid, say, in the Milne universe with no matter, but generally spacetime is Minkowski only locally while globally it is something else (like de Sitter, etc.). – safesphere May 7 at 20:49
• @safesphere : i of course agree with all of that. my only point was that, by the choice of tag, the OP was telling us what universe s/he was asking about. – WillO May 7 at 20:52
• @safesphere : actually, he should have said whatever he meant, which, in the absence of any reason to think otherwise, is what i'm assuming he did. – WillO May 7 at 20:58

Without acceleration the twin paradox cannot be observed, but it is an error to think that the twin paradox was due to acceleration.

The twin paradox is exclusively due to velocity, not acceleration, this is shown by the equation

$$dt = \gamma (v) d\tau$$

that means that the observed time $$dt$$ equals the proper time $$d\tau$$ times the Lorentz factor Gamma (the Lorentz factor is a function of velocity, not of acceleration): The twin remaining on Earth is the observer who will observe the aging of the traveling twin according to the different velocities of the traveling twin during the observation period.

However, it is true that without acceleration the twin paradox cannot be observed, because in order to compare their age the two twins must meet at some moment to compare their clock/ their aging, it is essential that the traveling twin returns to the twin who stayed home, and for this, acceleration and deceleration maneuvers are required.

Another example are two spaceships receding one from the other. While receding, each of them observes (paradoxally) the own time running faster than the time of the other spaceship. Both views are equivalent, and the twin paradox concretizes only in the moment when both of them at some moment are meeting (again after some maneuvers of acceleration/ deceleration of both space ships). But when they compare their clocks, the difference does in no way depend on these maneuvers of acceleration/ deceleration, but exclusively on the history of their relative velocity.

• You can bypass the need of any acceleration by using a third "twin" or observer. So in the end acceleration is not needed at all, even for observing the twin paradox. – thermomagnetic condensed boson May 8 at 10:16

As Kraig already nicely noted, acceleration is absolute, not relative.

I wanted to add one more thing that no one mentioned yet: the resolution of the twin paradox is much more obvious in GR (general relativity) than in special relativity.

EDIT: As pointed out in the comments, this answer doesn't really actually use GR, it just represents the GR "perspective". In GR, if you plot the space-time path of an observer on a space-time diagram, then $$\textrm{Proper Time} = \textrm{Arc Length}.$$ And acceleration is the curvature of the worldline (different thing than spacetime curvature, by the way).

In flat spacetime (like special relativity), the arc-length/proper-time $$\tau$$ is measured in the Minkowski metric $$d\tau^2 = dt^2 - dx^2 .$$

Note the minus sign, so that arc length in spacetime is not the same as arc length on the paper. Because of the minus sign, any side-to-side motion (that is, moving through space), decreases the proper time experienced. A line at a 45 degree angle is therefore "light-like", and has zero arc length. Any observer's worldline is timelike, meaning pointed more up than sideways, so that proper time is strictly positive. In the twin paradox, at least one of the twins must have some acceleration, if the twins are to ever meet up again. The twin that accelerates (that is, the twin whose world line curves) experiences less proper time, because of their side to side motion.

This is the story as told from Twin A's reference frame. It's a nice exercise to work out how this looks in a different inertial frame, where Twin A and Twin B are both moving. But you can't easily translate into Twin B's frame, because by going to a non-inertial frame you would change the metric.

• I agree with everything you've said, but note that none of this requires GR - special relativity is perfectly capable of handling curved worldlines and accelerating observers. – J. Murray May 9 at 4:32
• Absolutely... it doesn't require GR, but I think it's safe to say this is the GR "perspective" on the problem. As far as I've seen, most SR texts focus more on Lorentz transformations and inertial frames, and avoid dealing with non-inertial motion. But I haven't looked really, maybe I just missed the good ones! – Joe Schindler May 9 at 5:00
• Graduate mechanics texts like Goldstein cover quite a reasonable amount of SR dynamics, including the treatment of non-inertial reference frames. That being said, I agree - it's unfortunate that many people (such as myself) are exposed only to a neutered, kinematics-focused view of SR before being plunged into the full framework of GR, which fuels the understandable misconception that SR deals only with straight line motion at constant speed. – J. Murray May 9 at 5:14
• I guess it depends where you draw the SR/GR line. On the one hand, probably the only reasonable choice is "SR is the relativistic physics of flat spacetime", so I would have to agree: this is SR only. On the other hand, if we let SR in flat spacetime go too far, then GR is an almost trivial extension --- and then what happens to the ego of the relativists... we are trivial! ;) – Joe Schindler May 9 at 5:32

The twin in a spaceship that is accelerating by its engines, will feel that acceleration, while the stationary twin will not. The accelerating twin will feel the force his spaceship is exerting on him, and by the equivalence principle it will seem to him that he is in a gravitational field. I don't think the laws of physics will stop working.

Note that this is different from free-fall acceleration where both the vehicle and the person are being accelerated by the same external force and the person will not feel any force from his vehicle.

• But in an empty universe, would you actually feel the acceleration? – Cinaed Simson May 8 at 6:01
• The content of the universe is irrelevant if you are being pushed along by your spaceship's acceleration. You certainly would feel it. – Bill Watts May 8 at 6:14
• No one ever observed empty universe so who knows. In minkowski spacetime however you will. That is due to the fact, that minkowski spacetime is build by definining parralel transport and any accelerated observers 4-velocity vector is not parralel transported. The question wheter parrallel transport has any meaning in really empty universe is for philosophers to ask, but in GR the parrallel transport must be defined for any spacetime – Umaxo May 8 at 19:49
• @BillWatts: matter might matter. – Cinaed Simson May 9 at 5:23

The twin paradox is STR problem, but i want to say something about GR. In GR the geometry of spacetime is defined by parallel transport. That is, when i have some 4-vector and a curve, i can say what it means to move the vector along the curve so that it remains parallel to the original vector in the initial point.

In GR any object in any spacetime concievable has some 4-velocity that determines its motion. The object undergoes acceleration if its 4-velocity is not parrallel transported during the motion. And because any spacetime in GR has paralkel transport defined on it, the acceleration is always distinguished (at least mathematically), no matter how empty our universe is.

Then there is the question of physics. Any physics happening in this spacetime that obeys some form of second newton law - that is if something has initial 4-velocity vector and is left alone, the object will move in such a way, that 4-velocity vector will be parallel transported - you can use this physics to actually measure what parallel transport means. So if there is any physics living in the spacetime that depends on definition of parralel transport, you can measure wheter you are accelerating or not. F.e. in optics light moves on geodesics (i.e. curves that are build by parallel transporting some 4-vector) and you can use this to actually measure what parallel transport means.

So if GR explains the universe, you have to have parallel transport defined, at least mathematically. If we have universe where there is nothing but twins, you still need physics (f.e. light) to exist (for twins to exist) and you can use this physics to measure what it means to parallel transport 4-vectors in this universe. Once you do this, you can check wheter your own 4-velocity is parallel transported along your motion or not, which is equal to finding out wheter you are accelerating or not.