# Why the stay-at-home twin is not considered to be accelerating?

## Summary:

According to relativity, if motion is relative, and if A is moving away at a high but constant speed in relation to stay-home B, B is also moving away from A at the same constant speed; then if A is accelerating away from B, why not B is also considered to be accelerating away from A at the same rate? If movement is relative, why not acceleration is relative? After all, at every moment, when A's speed of moving away from B increases, B's speed of moving away from A should also increase? Or, when

## Main Question or Discussion Point

I am not a physicist. I need your kind help in removing my following doubt about twin paradox.
What I have been able to understand about twin paradox is this-

1. Special relativity deals with non-accelerating (inertial) motion.

2. The travelling twin (A) moves at a high speed in relation to the stay-at-home twin (B), hence A's time dilates and he ages less in comparison to B.

3. The stay-at-home twin (B) also moves at a high speed in relation to the travelling twin (A) (according to relativity), hence B's time too should get dilated and B should also age less in comparison to A. It does not happen so (only A ages less), hence it is a paradox.

4. According to available explanation, only A's time gets dilated and he ages less, because it is A only who accelerates as he leaves home and then as he returns back to home.

If what is I understand is correct, then my problem is this-
According to relativity, if motion is relative, and if A is moving away at a high but constant speed in relation to stay-home B, B is also moving away from A at the same constant speed; then if A is accelerating away from B, why not B is also considered to be accelerating away from A at the same rate? If movement is relative, why not acceleration is relative? After all, at every moment, when A's speed of moving away from B increases, B's speed of moving away from A should also increase? Or, when A's direction of movement in relation to B changes, B's direction of movement from A should also change.

But relativity denies this. It says that only the travelling twin accelerates. Please give me a simple explanation of it (if possible, without mathematics).
Thanks and regards.

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PeroK
Homework Helper
Gold Member
Summary:: According to relativity, if motion is relative,
You missed out two important words:

According to special relativity, inertial motion is relative.

dayalanand roy, Dale and Ibix
Nugatory
Mentor
But relativity denies this. It says that only the travelling twin accelerates. Please give me a simple explanation of it (if possible, without mathematics).
Suppose both twins are carrying accelerometers. The stay-at-at-home accelerometer reads zero throughout, while the traveler’s accelerometer shows non-zero acceleration at the time of the turnaround. That’s how we know that the traveler is accelerating.

You will also want to follow up on what @A.T. says in post #2 above: the distinction between coordinate acceleration and proper acceleration. We have many good threads on this.

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vanhees71, dayalanand roy, Dale and 1 other person
Ibix
Special relativity deals with non-accelerating (inertial) motion.
In addition to the comments by other posters, this is wrong. Special relativity deals with any motion, inertial or otherwise, in flat spacetime.

vanhees71, dayalanand roy and Dale
Dale
Mentor
If movement is relative, why not acceleration is relative?
We can build self-contained accelerometers but not self-contained velocitimeters. The velocity meters that we do have are all based on measuring the velocity with respect to some external reference.

As to “why” there is no particular reason that the universe should be any specific way. It just is this way and so we must build our physical laws to reflect the observed facts of the universe.

cianfa72 and vanhees71
What I have been able to understand about twin paradox is this-
1. Special relativity deals with non-accelerating (inertial) motion.
That is wrong.

Special Relativity deals with things where the observer is not accelerating. ie the observer is in an inertial frame.

Objects the observer measures can be accelerating.

Remember, paradox means apparently wrong until the explanation is given showing it is actually correct.

The apparently wrong "the moving twin ages less" is actually correct. It arises because the moving twin does not remain in an inertial frame for her entire journey so she is not an inertial observer.

Just before she reaches her turn point and while still travelling away from earth, she observes that the time on earth is one value. Note she is an inertial observer at this point.

She then decelerates and accelerates back to speed travelling towards earth. She is now an inertial observer again but she is a different inertial observer than when she was travelling away from earth.

Different inertial observers do not agree on things distant to them being simultaneous.

She now observes the time on earth and finds it is very different - years different - from the time she observed just before she turned. The time has jumped forward dramatically and this jump, together with the time dilation factor due to her speed, accounts for the age difference on her return.

Incidentally, this jump in the "observed time on earth" also explains for something you rarely see mentioned.

A stationary observer sees a moving clock run slow.

So, during all her journey she sees her twin on earth ageing more slowly than her.

Similarly, her twin on earth sees her age more slowly than him.

The only way you can resolve these two facts is by there being a jump in someone's measurement of time.

You are not alone in your confusion. In Professor Jim Al Khahili's 2012 book Paradox, Chapter Six - The Paradox of the Twins deals with the paradox and he gives a completely wrong explanation!

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DifferentialGalois and dayalanand roy
Nugatory
Mentor
That is wrong.

Special Relativity deals with things where the observer is not accelerating. ie the observer is in an inertial frame.

Objects the observer measures can be accelerating.
And I regret to say that this is also wrong. Special relativity handles accelerating observers (and more generally, non-inertial frames in which a hypothetical observer at the origin of the frame is accelerating) just fine. Googling for "Rindler coordinates" will bring up one of the more easiiy understood examples.

These situations are often not covered in introductory courses because the math is appreciably more complicated; unfortunately this leads people to assume that because it's not covered it doesn't work.

You can use special relativity in any situation in which there are no relevant gravitational effects, which is to say any situation in which you can consider spacetime to be flat, free of curvature.

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cianfa72, DifferentialGalois, vanhees71 and 1 other person
Ibix
Special Relativity deals with things where the observer is not accelerating. ie the observer is in an inertial frame.
SR can deal with non-inertial frames just fine.

vanhees71
Special Relativity deals with things where the observer is not accelerating.
The definition of SR changed a few decades ago. Today, SR deals also with accelerated reference frames. The observer may be accelerating. But in those accelerated frames, also pseudo-gravitation time-dilation between clocks located at different pseudo-gravitation potentials must be included in calculations.

GR, according to it's actual definition, is only needed in case of spacetime curvature (tidal gravitation).

DifferentialGalois and dayalanand roy
Dale
Mentor
That is wrong.

Special Relativity deals with things where the observer is not accelerating. ie the observer is in an inertial frame.

Objects the observer measures can be accelerating.
And I regret to say that this is also wrong. Special relativity handles acceerating observers
The kernel of truth that the mistake is based on is that the famous two postulates both refer explicitly to inertial frames.

vanhees71, dayalanand roy and Nugatory
Mister T
Gold Member
1. Special relativity deals with non-accelerating (inertial) motion.
And also with accelerating non-inertial motion.

2. The travelling twin (A) moves at a high speed in relation to the stay-at-home twin (B), hence A's time dilates and he ages less in comparison to B.
Not quite. When we talk about aging of the twins we are talking about them comparing their clocks when they share the same location. Thus the traveling twin must return to the location of the staying twin to compare ages. If all the traveling twin does is move away from the staying twin there is no objective way to compare their ages. Each will observe that the other's clocks are running slow, and the comparison is complicated by the fact that the twins don't share the same location. In addition to the clock that each twin keeps with them, they will each need a second clock synchronized with their near-by clock, and placed at the location of the other twin. Each twin will claim that the other twin has performed the synchronization incorrectly (this is called the relativity of simultaneity) and so they will disagree as to which twin has aged more. They need to share the same location to get around this difficulty.

3. The stay-at-home twin (B) also moves at a high speed in relation to the travelling twin (A) (according to relativity), hence B's time too should get dilated and B should also age less in comparison to A. It does not happen so (only A ages less), hence it is a paradox.
See the above explanation involving the relativity of simultaneity.

4. According to available explanation, only A's time gets dilated and he ages less, because it is A only who accelerates as he leaves home and then as he returns back to home.
It is A who changes direction. B does not. That's why the situation is not symmetrical.

Before you try to understand the twin paradox you must first understand the symmetry of time dilation. When two people are in motion relative to each other, each will claim that the other's clock is running slow. You can't gloss over this apparent contradiction. You must understand how the relativity of simultaneity makes it possible. And it does involve some math, but the math is not complicated, it just involves multiplication and addition.

dayalanand roy and Ibix
See the space-time diagram below.

One twin stays on earth.
One twin leaves earth at high speed (0.4142c), travels for 5 earth years, and then returns.
The stay-at-home twin says that 10 years have elapsed since the traveller left.
The traveller says only 9 years have elapsed since she left.

Two things need to be explained:

Paradox 1. Why is the traveller only 9 years older whereas the stay-at-home twin is 10 years older?

Paradox 2. All the time that the traveller is travelling, the traveller observes stay-at-home's clock to be running slower than traveller's clock. And, all the time that traveller is travelling, stay-at-home observes traveller's clock to run slower than stay-at-home's clock. So each observes the other's clock to be running slower than his own clock so each must be younger than the other!

The resolution is given in the space-time diagram below based on earth = 10 (years) and traveller (purple line), travelling at 0.4142c relative to earth, taking 9 years, 4.5 years on his blue axes to get to the turn round, and 4.5 years on the blue axis to get back. Note that I should really have drawn a yellow grid angled the other way for the traveller's return journey but the angles are the same so the values read off are the same.

An Event is a point on the diagram. The earth observer uses the black grid lines to measure the time and space coordinates of the Event in their frame. The travelling observer uses the sloping blue grid lines to measure the time and space coordinates of that same Event in their frame.

Notice that, as he is travelling away, just before he turns round, traveller measures Event A on earth (earth time t = 4.2) to take place at traveller's time t = 4.5 (dotted purple line of traveller's simultaneity). So traveller measures earth's clock (4.2) to be running slower than traveller's clock (4.5).

Note that earth observes the traveller to turn round at earth time t = 5, but traveller's clock reads only 4.5. So earth observes traveller's clock to run slower than earth's clock.

That is, each observes the other's clock to run slower than their own clock.

When traveller is stationary at the turn round, he says time on earth is now 5 (the dotted purple line would now be horizontal).

When traveller is up to speed going back (but before he has moved any distance) he measures Event B (earth time t = 5.8) to be at traveller's time of 4.5. So traveller jumps (instantaneously) from earth time = 4.2 to earth time = 5.8, and "jumps across" 1.6 years at the turn round point.

Similarly, on the way back, it takes traveller 4.5 years.

So, traveller's journey time = 4.5 + 4.5 = 9 years on traveller's clock. But the 10 years of earth's time is made up of comprises 4.2 earth years out, plus a gap of 1.6 earth years, plus 4.2 earth years back, for a total of 10 earth years.

Note also that the faster the travelling twin moves the bigger the gap between Event A and Event B becomes. A photon travelling at c (the green line on the diagram) sees no time elapse between departing earth and arriving back on earth - the gap between Event A and Event B is now the full ten years.

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vanhees71 and Dale
Ibix
All the time that the traveller is travelling, the traveller observes stay-at-home's clock to be running slower than traveller's clock. And, all the time that traveller is travelling, stay-at-home observes traveller's clock to run slower than stay-at-home's clock. So each observes the other's clock to be running slower than his own clock so each must be younger than the other!
This isn't correct as written. What the twins observe is affected by the Doppler effect, and each will observe that the other's clock runs fast or slow at different times. However, they will also observe that the distance to the other twin, and hence the lightspeed delay in what they observe, is changing. Only when they combine the two facts do they calculate the other's clock to be always running slow.

So traveller jumps (instantaneously) from earth time = 4.2 to earth time = 5.8, and "jumps across" 1.6 years at the turn round point.
Better to say that he switched from using one set of clocks to another, and the clocks are offset by different amounts at different distances. The first set show, near Earth, that "the same time as turnaround" is 1.6 years earlier than the second set. Changing clocks without correcting for the time difference is the mistake, analogous to stepping from one timezone to another and thinking that one step took an hour (or minus an hour!).

There's no "jumping in time" going on, it's just a mistake with the clocks that might make it appear so.

I like your Minkowski diagram, but it's somewhat hard to read. I'd suggest making the grids into paler colours (e.g. 196,196,255 instead of 0,0,255) so that the worldlines are clearer. Also definitely don't mark the angle - the Euclidean angle is meaningless in Minkowski space.

Nugatory
Mentor
The kernel of truth that the mistake is based on is that the famous two postulates both refer explicitly to inertial frames.
Ah - good point. That may also have contributed to the persistence of the "SR doesn't work with acceleration" misunderstanding.

Of course Einstein was perfectly capable of handling coordinate transformations to non-inertial frames, so I doubt that he expected his wording to be taken as a limitation of applicability as oposed to a convenient choice of coordinate system for the subsequent derivation. As plausibly, he didn't anticipate that his audience would ever include people who weren't comfortable with switching coordinate systems as the problem demanded.

dayalanand roy and Dale
Staff Emeritus
2019 Award
The definition of SR changed a few decades ago.
Like eleven.

vanhees71 and Dale
Like eleven.
I don't know the exact date. But Einstein said in a speech, hold in Kyoto on 14 December 1922:
I was dissatisfied with the special theory of relativity, since the theory was restricted to frames of reference moving with constant velocity relative to each other and could not be applied to the general motion of a reference frame. I struggled to remove this restriction and wanted to formulate the problem in the general case.
Source:
https://web.archive.org/web/2015122...winter2012/physics2d/einsteinonrelativity.pdf

dayalanand roy and Dale
This is a translation of Einstein's 1905 paper On the electrodynamics of moving bodies.

As you see he restricted himself to inertial frames throughout. The two principles on which his theory were based are:

1. The laws by which the states of physical systems undergo change are not
affected, whether these changes of state be referred to the one or the other of
two systems of co-ordinates in uniform translatory motion.

2. Any ray of light moves in the “stationary” system of co-ordinates with
the determined velocity c, whether the ray be emitted by a stationary or by a
moving body.

He goes on to say (bearing directly on the twin paradox)

So we see that we cannot attach any absolute signification to the concept of
simultaneity, but that two events which, viewed from a system of co-ordinates,
are simultaneous, can no longer be looked upon as simultaneous events when
envisaged from a system which is in motion relatively to that system.

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dayalanand roy, weirdoguy and PeroK
PeroK
Homework Helper
Gold Member
This is a translation of Einstein's 1905 paper On the electrodynamics of moving bodies.

As you see he restricted himself to inertial frames.
And he didn't have a lot to say about spacetime diagrams or the energy-momentum four-vector either!

As you see he restricted himself to inertial frames throughout.
But according to the actual definition of SR, the following is valid:
It is a common misconception that Special Relativity cannot handle accelerating objects or accelerating reference frames. It is claimed that general relativity is required because special relativity only applies to inertial frames. This is not true. Special relativity treats accelerating frames differently from inertial frames but can still deal with them. Accelerating objects can be dealt with without even calling upon accelerating frames.

This error often comes up in the context of the twin paradox when people claim that it can only be resolved in general relativity because of acceleration. This is not the case.
Source:
https://www.desy.de/pub/www/projects/Physics/Relativity/SR/acceleration.html

dayalanand roy, vanhees71 and PeroK
PeterDonis
Mentor
2019 Award
Einstein's 1905 paper On the electrodynamics of moving bodies.
We have learned a lot in the 115 years since this paper was published. Which is how we know that SR can in fact handle accelerated motion and non-inertial frames just fine, as long as spacetime is flat.

cianfa72 and vanhees71
Ibix
As you see he restricted himself to inertial frames throughout.
So what? That's a paper over a century old, and it's the first paper in an area. The idea that such a paper instantly sets in stone the only way things can be done is just silly.

I think everyone has always agreed that inertial frames in flat spacetime are in the domain of SR, and curved spacetime is outside its domain. I think, as @Sagittarius A-Star points out, that there was some initial discussion as to whether non-inertial frames in flat spacetime were in the domain of SR. But once it became clear that there was no new physics in non-inertial frames in flat spacetime (it's just a more complicated mathematical description of the same thing) but there was new physics in curved spacetime, it became extremely difficult to defend the assertion that non-inertial frames should not be part of SR. Ultimately it's a human decision, but we decided to place it at a conceptual boundary (flat vs non-flat spacetime) that has some physical significance.

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Dale
Mentor
As you see he restricted himself to inertial frames throughout. The two principles on which his theory were based are:
Yes, I already mentioned that in post 11

I am trying to learn about Rindler coordinates and would appreciate if you could point to a good introduction to them.

Are they in some way similar to rotating coordinates? By that I mean do we rewrite things like the Lorenz equations to take account of the acceleration as different rates of acceleration will undoubtedly give different a result?

When we compare inertial and rotating frames we can say "Newton's laws don't apply in rotating frames". This is true. If you stand still in a rotating frame and drop an object it will not fall in a straight line relative to you. It will travel in a curved path. You cannot explain that by Newton's Law F = ma.

However, we can write an equation of motion which describes the motion of an object when moving in a rotating frame using the rotating frame coordinates. This equation is very different from Newton's Laws and invokes new fictitious forces like Coriolis and centrifugal forces.

This isn't correct as written. What the twins observe is affected by the Doppler effect, and each will observe that the other's clock runs fast or slow at different times. However, they will also observe that the distance to the other twin, and hence the lightspeed delay in what they observe, is changing. Only when they combine the two facts do they calculate the other's clock to be always running slow.
I used the word observe advisedly in distinction to see. Observe includes the calculation.

If I see a light 186,000 miles away flash at exactly noon, I observe that it flashed a second before noon.

And Doppler affects only the perceived frequency not the speed.

dayalanand roy, etotheipi and Dale