Time Dilation and the Twin Paradox: A Look at the Relativistic Doppler Effect

[sha1000]

Hello everyone,

Reminder: The key of the explanation of the "twin paradox" resides in the acceleration. We know that the time runs slower for the twin in the spaceship (since it was accelerated).

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Now I would like to share with you a thought experiment.

Let's imagine that we have triplets on Earth. Now we put two of the them into the spaceship and send them away with speed V₁ relative to Earth. After some time one of the spaceship-triplets is send back to Earth in the space-capsule (in the opposite direction with the speed -V₂ relative to Earth).

Let's take |V₁| > |V₂| in the Earth's frame. This means that in the Earth-triplet perspective the time dilation rate of the spaceship is higher than for the returning space-capsule.

Now let's look into this from the perspective of the spaceship. In this frame it was actually the capsule which was accelerated; so time must run slower in the capsule from the spaceships perspective.

So who is right? Time runs slower in the spaceship or in the capsule?

 

[Ibix]

The key of the explanation of the "twin paradox" resides in the acceleration. We know that the time runs slower for the twin in the spaceship (since it was accelerated).

Not really - it's perfectly possible to set up circumstances where the twins undergo the same accelerations yet end up with different ages. The actual resolution is that your elapsed time is a measure of the interval along your worldline ("interval" is to spacetime what distance is to space - so this is closely analogous to the distance traveled along a path through space). Different paths can have different lengths; the traveling twin simply took a shorter path.

Now let's look into this from the perspective of the spaceship. In this frame it was actually the capsule which was accelerated; so time must run slower in the capsule from the spaceships perspective.

The capsule is accelerated from anyone's perspective.

Your example doesn't actually seem to add anything to the twin paradox. There is a stay at home triplet and one triplet who travels out and back. They can compare their ages when they meet again. But for some reason you've added a third triplet who never meets up with either of the other two after leaving them, so can never compare ages. So this is just a twin paradox with a third party who wanders off to do something else part way through.

So who is right? Time runs slower in the spaceship or in the capsule?

Time runs at one second per second everywhere. There is less elapsed time over the course of the experiment for the traveling twin because they took a shorter route through spacetime.

 

[sha1000]

Not really - it's perfectly possible to set up circumstances where the twins undergo the same accelerations yet end up with different ages. The actual resolution is that your elapsed time is a measure of the interval along your worldline (closely analogous to the distance traveled long a path through space). Different paths can have different lengths; the traveling twin simply took a shorter path.
The capsule is accelerated from anyone's perspective.

Your example doesn't actually seem to add anything to the twin paradox. There is a stay at home triplet and one triplet who travels out and back. They can compare their ages when they meet again. But for some reason you've added a third triplet who never meets up with either of the other two after leaving them, so can never compare ages. So this is just a twin paradox with a third party who wanders off to do something else part way through.
Time runs at one second per second everywhere. There is less elapsed time over the course of the experiment for the traveling twin because they took a shorter route through spacetime.

Thank you for your reply.

Is the "out and back travel" and "comparing the ages" is crucial?

I mean can we define the time dilation rates without "coming back and comparing".
(When the spaceship is send away from Earth we know already that the twin in the spaceship ages slower, don't we?).

 

[Ibix]

Is the "out and back travel" and "comparing the ages" is crucial?

Yes. If you don't meet up you can't compare clocks without having to assume a simultaneity convention. Since different frames have different simultaneity conventions, they don't agree on what "at the same time" means in the question "how old is the other guy at the same time I had my 40th birthday party?" unless the other guy is actually in the same place - presumably at the birthday party in this case.

I mean can we define the time dilation rates without "coming back and comparing".

Not without assuming a simultaneity convention.

(When the spaceship is send away from Earth we know already that the twin in the spaceship ages slower, don't we?)

Not according to the twin in the spaceship. He will argue that the stay-at-home twin's clocks are ticking slowly because in this frame the stay-at-home is the one moving.

It's not until they meet up that they unambiguously, and in all frames, share a definition of "now".

 

[sha1000]

Yes. If you don't meet up you can't compare clocks without having to assume a simultaneity convention. Since different frames have different simultaneity conventions, they don't agree on what "at the same time" means in the question "how old is the other guy at the same time I had my 40th birthday party?" unless the other guy is actually in the same place - presumably at the birthday party in this case.
Not without assuming a simultaneity convention.

Not according to the twin in the spaceship. He will argue that the stay-at-home twin's clocks are ticking slowly because in this frame the stay-at-home is the one moving.

It's not until they meet up that they unambiguously, and in all frames, share a definition of "now".

Ok. I think I understand where is the root of my misconception.

I would like to return to this point: "Not according to the twin in the spaceship. He will argue that the stay-at-home twin's clocks are ticking slowly because in this frame the stay-at-home is the one moving.".

But: The spaceship-twin felt the acceleration; so normally he must know that he is the one who is moving (even though after acceleration in his perspective the stay-at-home one is moving)?

 

[Ibix]

The spaceship-twin felt the acceleration; so normally he must know that he is the one who is moving

Only if he assumes he was initially stationary. There is no experiment that can be done in the rocket that will give a different result before and after the acceleration phase, so he has no particular reason to say that he was initially at rest and is now in motion or vice versa.

 

[sha1000]

Only if he assumes he was initially stationary. There is no experiment that can be done in the rocket that will give a different result before and after the acceleration phase, so he has no particular reason to say that he was initially at rest and is now in motion or vice versa.

Now I'm even more confused...

As far as I understand there is no absolute frame in the special relativity. So does it makes sense to talk about things like: "Only if he assumes he was initially stationary"? Stationary compared to what?

All the inertial frames are equal and when we talk about the twin which is accelerated from Earth; normally we don't need to consider the Earth's movement relative to something else or was it initially stationary etc. The spaceship accelerated and have some velocity in Earth's frame so normally the stay-home-twin knows for sure that time is running slower on the spaceship
(the spaceship-twin knows that he was accelerated (since he felt it); so normally he also knows that his clocks are ticking slower compared to the clocks on the Earth).

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Einstein himself based his theory on the fact that the one-way speed of light is equal to c in all inertial frames and by this he abolished the idea of the absolute frame. There are "special relativity test theories" (made by Sexl and others) which consider that there is an absolute frame and that the one-way speed of light is equal to c only in this absolute frame. For moving inertial frames only the two-way speed of light is equal to c.

If we were talking about these "test theories" i would completely agree with you with this statement: "Only if he assumes he was initially stationary". And I would also agree that "out and back travel" and "comparing the ages" are crucial.

I hope you can understand my confusion on this.

 

[Ibix]

As far as I understand there is no absolute frame in the special relativity. So does it makes sense to talk about things like: "Only if he assumes he was initially stationary"? Stationary compared to what?

You said The spaceship-twin felt the acceleration; so normally he must know that he is the one who is moving. I merely pointed out that this sentence assumes that the spaceship twin was not moving before acceleration and was moving after acceleration. But this is not, as you note, an absolute statement. You can perfectly turn it around and say that he was moving before the acceleration and not moving afterwards.

(the spaceship-twin knows that he was accelerated (since he felt it); so normally he also knows that his clocks are ticking slower compared to the clocks on the Earth).

No. The spaceship twin knows that, according to the Earth frame his clocks are ticking slowly. According to his frame, the Earth's clocks are ticking slowly.

That was my point: you are privileging the Earth frame by regarding it as stationary and the traveling twin as moving - exactly what you are (correctly) saying we shouldn't do.

 

[Dale]

I mean can we define the time dilation rates without "coming back and comparing".

You can define them but they are just an arbitrary definition with no physical consequence. Right now, by this arbitrary definition, you are extremely time dilated relative to some particle in the LHC. Do you feel like you are ageing slowly?

@Ibix is trying to get you to look beyond the arbitrary definitions and see the measurable physics.

 

[sweet springs]

Hi.

Velocity observation in

System of Earth
E:0
R:$V_1$
C:$-V_2$,

System of Rocket
E:$-V_1$
R:0
C:$-\frac{V_1+V_2}{1+\frac{V_1 V_2}{c^2}}$, and

System of Capsule
E:$V_2$
R:$\frac{V_1+V_2}{1+\frac{V_1 V_2}{c^2}}$
C:0.

Clock adjust

$t_E=t_R=0$ for event E meets R
$t_R=t_C=S$ for event R meets C

Keeping essence of the case, let us make it simpler. Earth,Capsule and Rocket are on their own inertial system all the time. Rocket does clock adjustment with Earth clock which shows 0 when they cross.Capsule does clock adjustment with Rocket clock which shows S when they cross.

Systems of Earth, Rocket and Capsule are all inertial ones. SR gives time dilation observed in each system accordingly and of course right answer to Twin Paradox by comparing $t_E$ and $t_C$ for event C meets E.

Best.

 

[Dale]

So does it makes sense to talk about things like: "Only if he assumes he was initially stationary"? Stationary compared to what?

More formally, "only if he assumes he was initially stationary" means "your statement is only valid in the inertial reference frame where his initial velocity was zero, but not in some other equally valid inertial frames where his initial velocity was non zero".

 

[sha1000]

You said The spaceship-twin felt the acceleration; so normally he must know that he is the one who is moving. I merely pointed out that this sentence assumes that the spaceship twin was not moving before acceleration and was moving after acceleration. But this is not, as you note, an absolute statement. You can perfectly turn it around and say that he was moving before the acceleration and not moving afterwards.
No. The spaceship twin knows that, according to the Earth frame his clocks are ticking slowly. According to his frame, the Earth's clocks are ticking slowly.

That was my point: you are privileging the Earth frame by regarding it as stationary and the traveling twin as moving - exactly what you are (correctly) saying we shouldn't do.

Thx now I understand. We don't actually know if spaceship is accelerating or decelerating in the absolute sense. But doesn't this mean that there must be a preferred reference frame? I am not talking about aether but some frame to which we compare all the others.

 

[sha1000]

Thank you all for your responses.

 

[Ibix]

Thx now I understand. We don't actually know if spaceship is accelerating or decelerating in the absolute sense. But doesn't this mean that there must be a preferred reference frame? I am not talking about aether but some frame to which we compare all the others.

No. Acceleration is changing your speed as measured in an inertial frame - but depending on your state of motion with respect to a given inertial frame your speed may be increasing or decreasing (or just changing direction).

 

[Vitro]

Thx now I understand. We don't actually know if spaceship is accelerating or decelerating in the absolute sense. But doesn't this mean that there must be a preferred reference frame?

Why do you think it does? Doesn't it actually mean the opposite?

Note here that by saying "accelerating or decelerating" you are making a distinction without a difference. You seem to be using these terms in the sense of accelerating meaning an increase in speed and decelerating meaning a decrease in speed. But speed is relative, an increase in speed in one frame can mean a decrease in another, so there's really just change in speed without having to make a distinction between increasing and decreasing. Therefore there's only acceleration, and really no such thing as deceleration. Furthermore acceleration may not cause a change in speed at all, like you accelerating upwards at 9.81 m/s^2 while sitting in your chair at home.

So acceleration doesn't make you go faster or slower, unless you choose some reference to measure your speed relative to. But then again any such choice of reference is purely arbitrary so it cannot have any physical consequence.

 

[Nugatory]

We don't actually know if spaceship is accelerating or decelerating in the absolute sense. But doesn't this mean that there must be a preferred reference frame?

Only if we try to make an unnecessary (and confusing) distinction between acceleration and deceleration. That distinction does imply a preferred frame; decelerations are those accelerations that reduce speed in some privileged frame and accelerations are those that increase speed in that frame.

But the only physical measurable phenomenon here is proper acceleration: An accelerometer on the accelerated object will read some non-zero value in some direction, and observers in all frames will agree on this. We don't need the preferred frame unless we want to introduce some rule that allows us to call some of these accelerations "decelerations", a distinction that affects no observable phenomena.

 

[Mister T]

The spaceship accelerated and have some velocity in Earth's frame so normally the stay-home-twin knows for sure that time is running slower on the spaceship (the spaceship-twin knows that he was accelerated (since he felt it); so normally he also knows that his clocks are ticking slower compared to the clocks on the Earth).

No, each twin will observe the other's clock to be ticking slower than his own. I suggest you resolve this paradox before you try to resolve the twin paradox. Once you do, you will see the resolution of the twin paradox quite easily.

 

[sha1000]

Well, I needed some time in order to digest your responses and I'm still a little bit confused.

You can define them but they are just an arbitrary definition with no physical consequence.

Now my question is: Does special relativity reject reality? I was always thinking that If spaceship is accelerated from Earth there is already a real time dilation phenomena in the spaceship. But now it seems like it doesn't make sense to talk about the real time dilation until the spaceship is returned to Earth (after the measurement). This somehow is similar to the quantum mechanics philosophy.

 

[sha1000]

After some googling I found the conversation between Einstein and Heisenberg. Is it relevant? From this one can see that Einstein constructed the theory of special relativity only on observable quantities (no reality).

“Heisenberg: "One cannot observe the electron orbits inside the atom. [...] but since it is reasonable to consider only those quantities in a theory that can be measured, it seemed natural to me to introduce them only as entities, as representatives of electron orbits, so to speak."

“Einstein: "But you don't seriously believe that only observable quantities should be considered in a physical theory?"

“"I thought this was the very idea that your Relativity Theory is based on?" Heisenberg asked in surprise. “"Perhaps I used this kind of reasoning," replied Einstein, "but it is nonsense nevertheless. [...] In reality the opposite is true: only the theory decides what can be observed."”

(translated from "Der Teil und das Ganze" by W. Heisenberg)

 

[Ibix]

You'd have to define "reality" to start discussing whether relativity denies it or not, and that would be forbidden by forum rules on philosophical discussions. So I suggest you don't.

The point is that coordinate time is just a convention. You can't measure it with a single clock; you need an array of clocks floating in space to show you time at different places. Then you need some way to synchronise those clocks, and the way you do that will affect the results you get for time dilation. So time dilation depends on your choice of synchronisation convention. Your choice of convention can't have any effect on what's happening - although it may make the mathematical description more complicated.

And this is why time dilation isn't really a physical thing. If a person moving with respect to the array of clocks zeroes their own clock as they pass one of my free-floating clocks then checks their time against the next clock they pass, the answer they get depends on how I synchronised my clocks. And there is no "one true way" to do that. The only way to avoid ambiguity about synchronisation is to turn around and come back and compare to the first clock. In that case, the person who turned round will show a shorter elapsed time - but this isn't time dilation, which is about comparing clocks at rest in one frame to clocks at rest in another. The clock that turned round is not at rest in just one frame.

 

[Dale]

Now my question is: Does special relativity reject reality? I was always thinking that If spaceship is accelerated from Earth there is already a real time dilation phenomena in the spaceship. But now it seems like it doesn't make sense to talk about the real time dilation until the spaceship is returned to Earth (after the measurement)

As soon as you start talking about "reality" you are wandering away from physics and into philosophy (metaphysics, ontology).

The classification that is important here is "frame variant" or "frame invariant". Only frame invariant quantities can have direct physical consequences, since the outcome of any experimental measurement cannot change through a mere change of coordinates.

Time dilation (in the sense described above) is frame variant, so it doesn't have physical consequences.

 

[Mister T]

After some googling I found the conversation between Einstein and Heisenberg. Is it relevant?

I don't see how. That conversation is about interpretations of theories, not about the theories themselves. In other words, it's a debate about philosophical considerations, not physical considerations.

As I said in my previous reply you will never sort out the twin paradox until you sort out the symmetry of time dilation. The fact that the twin paradox illustrates an asymmetry in the aging of the twins does NOT imply an asymmetry in time dilation. In other words, it's not valid to conclude, as you seem to be doing, that when the twins are moving relative to each other the staying twin's clock ticks faster than the traveling twin's. And the reason that that's not a valid conclusion is because the traveling twin will observe the opposite, and there is no way to claim that the staying twin's observation is any more or less valid than the traveling twin's. There is no giant clock in the sky that keeps track of time in such a way that allows one to draw such conclusions.

Your claim that it's the traveling twin who must accelerate to begin his journey is not valid. You can have the traveling twin execute a series of maneuvers in his rocket ship before his journey begins. Those maneuvers get him up to speed, and after they're completed he passes the staying twin at high speed. At that instant they start their clocks and the journey begins.

 

[sha1000]

I thank you all again for your responses.

Now I understand that before, I didn't understand at all the twin paradox. Everything is much more clear for me now.

 

[sha1000]

I would like to ask another question which is out of the main subject.

Is the "out and back travel" is also crucial in GR? As far as I understand now, in SR we can't talk about the "real time dilation" in the spaceship until we compare the clocks (when spaceship returns).

Some time ago I read a thread in this forum about the time dilation near the Black Hole. If we will send the spaceship to the BH it will be destroyed relatively fast (in the absolute sense); but observed from Earth one would never see how the spaceship enters the horizon; from Earths perspective it will slow down until it "freezes completely". Does this mean that in GR we can talk about the "real time dilation independent of observations"; contrary to SR where the real time dilation has no physical sense and we can only reason in terms of what we can actually observe?

Or maybe again, I still understand nothing.

 

[Dale]

I would say that the local comparison is even more crucial in GR, not less. Due to curvature you cannot even define things like relative velocity uniquely at a distance.

 

[jbriggs444]

Does this mean that in GR we can talk about the "real time dilation independent of observations"; contrary to SR where the real time dilation has no physical sense and we can only reason in terms of what we can actually observe?

Let us not involve event horizons here. They muddy the waters. Under general relativity, there are situations where gravitational time dilation can be considered as "real" time dilation.

Under special relativity (and inertial frames), time dilation is always due to relative velocity. Both observers consider their peer's clock to have slowed down.

Under general relativity, there can be time dilation due to gravitational potential. An observer higher looking down at his peer will consider the peer's clock to be running slow. The lower observer looking upward will consider his peer's clock to be running fast. If you wait long enough, an unambiguous difference builds up between the respective clock readings. This could be thought of as real time dilation. It is physically measurable and does not arise due to differing choices of simultaneity conventions.

 

[Ibix]

The problem with time dilation in special relativity is that it only happens between clocks in relative motion. What you actually see is the relativistic Doppler effect, and there is no unique way to divide that into "rate change due to time dilation" and "rate change due to changing distance".

The same is true with bells on in general relativity, except under certain circumstances. For example, if we are both hovering in an unchanging gravitational field, we can keep bouncing radar pulses off each other and confirm that the round-trip time isn't changing (edit: or we could just stand on different floors of a building on a planet). In this case, therefore, there is a sensible way to split Doppler into "rate change due to time dilation" and "rate change due to changing distance", which is that there is none of the latter.

I think it's still arbitrary to some extent, but it's a more physically motivated choice.