3 clocks thought experiment - Absolute vs. relative aspects?

[Friggle]

TL;DR Summary

The classical "travelling twin" thought experiment, but with three clocks. Then expanded to two travelling clocks. Wondering about effects of relative motion vs. (absolute) acceleration.

Here are two similar, quite simple thought experiments, followed by assumptions on final clock readings. In the end, my most important question to them.

Exp1:
- we have two space ships, ss1 and ss2, both have clocks on board, named cl1 and cl2
- we have a third clock, cl3, somewhere located in space, initially at rest and at the very same location with both cl1 and cl2.
- all clocks are initially synchronized while they are at rest and at the same point in space.
- now, both sh1 and sh2 accelerate, so that they fly away in exactly opposite directions from cl3.
- after some significant acceleration, sh1 and sh2 may continue to fly away from each other and from cl3 at constant speed.
- sh1 and sh2 then slow down at the same rate, turn around, accelerate back again and slow down again, so that they both arrive back and at rest with cl3 at the same time.
- now, again at rest and at the same point in space, all three clocks are compared.
--> What do the three clocks cl1, cl2 and cl3 show, in relation to each other?
--> My asumption: cl1_1 = cl2_1 < cl3_1
(The indices "_1" refer to "Exp1")

Exp2:
- Similar to Exp1 but now cl1 and cl2 are mounted on one and the same spaceship (we only have one space ship here). I.e., cl1 and cl2 always accelerate and move together.
--> This is the classic "travelling twin" thought experiment, just with two clocks in the space ship
--> In the end of the experiment: What do the three clocks cl1, cl2 and cl3 show, in relation to each other?
--> Assumption: cl1_2' = cl2_2 < cl3_2
(The indices "_2" refer to "Exp2")

--> It seems like if the relative motion between cl1 and cl2 does not matter in terms of their clock speed as long as the absolute values of their accelerations are always the same.
--> It seems like only acceleration and the times travelled in accelerated states are of matter
--> Acceleration is an absolute quantity.
--> Where is SRT in these thought experiments and what about them is "relative" as opposed to what is "absolute"?

Thanks for all answers in advance!

 

[PeroK]

--> It seems like if the relative motion between cl1 and cl2 does not matter in terms of their clock speed as long as the absolute values of their accelerations are always the same.

The final clock reading for any clock is the length of the spacetime path (or worldline) taken by the clock from first to final reading. Certainly, if the acceleration profiles (as measured in some inertial reference frame) are the same, then the final clock readings are the same.

The role of acceleration, however, is only to change the speed of an object. There is no additional time dilation caused by the acceleration itself. This is the clock hypothesis.

--> It seems like only acceleration and the times travelled in accelerated states are of matter
--> Acceleration is an absolute quantity.

Proper acceleration is an invariant quantity, yes.

--> Where is SRT in these thought experiments and what about them is "relative" as opposed to what is "absolute"?

Thanks for all answers in advance!

If we have a clock moving in some inertial reference frame, then the time shown on the clock is determined by its speed profile, as measured in that reference frame: $v(t)$, where $t$ is the coordinate time in that reference frame. It's usual to denote the (proper) time as measured by the clock as $\tau$, and we have:$$\tau = \int_{t_0}^{t_1} \sqrt{ 1 - \frac{v(t)^2}{c^2}} \ dt$$In that sense, therefore, it's all about speed and the acceleration itself is not a direct factor.

 

[Dale]

TL;DR Summary: The classical "travelling twin" thought experiment, but with three clocks. Then expanded to two travelling clocks. Wondering about effects of relative motion vs. (absolute) acceleration.

Where is SRT in these thought experiments and what about them is "relative" as opposed to what is "absolute"?

SR is in the fact that the times of the rocket clocks are not equal to the time on the non-rocket clock.

All of SR is contained in this one equation: $$ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$$ And the times on the clocks follow directly from this core equation.

 

[Friggle]

Thanks to both of you!

The role of acceleration, however, is only to change the speed of an object. There is no additional time dilation caused by the acceleration itself. This is the clock hypothesis.

Doesn't GR say that in proximity to mass (i.e. in a gravity field, i.e. curved spacetime), time goes slower? And since gravity is the same as acceleration I thought that there should be some share of the observed time dilation between cl1 and cl3 or cl2 and cl3 caused by the acceleration they experienced during the travel.

 

[jbriggs444]

Thanks to both of you!

Doesn't GR say that in proximity to mass (i.e. in a gravity field, i.e. curved spacetime), time goes slower? And since gravity is the same as acceleration I thought that there should be some share of the observed time dilation between cl1 and cl3 or cl2 and cl3 caused by the acceleration they experienced during the travel.

No. GR is consistent with SR in this respect. Gravitational time dilation scales with gravitational potential, not with the local acceleration of gravity.

Clocks do not tick slower because they are under a greater local acceleration of gravity (greater local spacetime curvature) but because they are at a lower gravitational potential (on the far side of a large region where spacetime curvature exists).

This in turn has to do with the $t-\frac{vx}{c^2}$ term in the Lorentz transform for time. That $vx$ in there is crucial. When comparing two nearby clocks to one another (small $x$), little time dilation results, even though $v$ may be changing rapidly (high acceleration).

But when $x$ is large (way far away at the bottom of the gravitational well) then a little change in $v$ (small acceleration) can result in a significant time dilation.

 

[PeroK]

Thanks to both of you!

Doesn't GR say that in proximity to mass (i.e. in a gravity field, i.e. curved spacetime), time goes slower?

There is gravitational time dilation. But, that's GR and not SR.

And since gravity is the same as acceleration

It isn't. The equivalance principle equates (locally) an accelerating reference frame with a gravitational field. In other words, if someone is in an accelerating rocket, then they cannot distinguish by local experiments (over a short period of time) between their acceleration and a gravitational field.

But, if you are an observer outside the elevator, then there is no additional element of time dilation due to the acceleration: only due to the speed of the rocket. The equivalence principle does not apply in that case. An accelerating elevator and a gravitational field are not the same thing.

, I thought that there should be some share of the observed time dilation between cl1 and cl3 or cl2 and cl3 caused by the acceleration they experienced during the travel.

"Acceleration does not cause time dilation" is a good thing to remember.

 

[Friggle]

And back to the not-occuring time dilation between cl1 and cl2 in Exp 1 above: Is there some well comprehensible statement (ideally in natural language without maths) why the velocity between cl1 and cl2 does not matter? Time dilation between them is always zero, right? No matter how fast they move with respect to each other. As long as the absolute values of their acceleration is always the same. Correct?

 

[Dale]

I thought that there should be some share of the observed time dilation between cl1 and cl3 or cl2 and cl3 caused by the acceleration they experienced during the travel.

Both GR and SR are fundamentally geometric theories (pseudo Riemannian, and Lorentzian or Minkowski respectively). So you can get good insights by thinking geometrically.

The length of a Lorentzian spacetime line is the time a clock traveling that path measures. Acceleration is a bend in the line. Different frames are just rotated coordinate systems.

Now, in Euclidean geometry, the triangle inequality says that if you have a triangle ABC then the length AC is always less than the length AB plus the length BC. The direct path AC is a straight path whereas the other path bends at B.

What share of the difference in length is caused by the bend at B?

 

[Dale]

Is there some well comprehensible statement (ideally in natural language without maths) why the velocity between cl1 and cl2 does not matter?

Not as far as I know*. But is the math really so difficult that you need to avoid it? It just comes from the Pythagorean theorem. We start with $$A^2+B^2=C^2$$ where $A$, $B$, and $C$ are the sides of a right triangle. Then we decide that instead of just calculating the sides of a triangle we would like to calculate the length, $s$, of any shape in the $x,y$ plane. So we break the overall shape into lots of little segments and apply the Pythagorean theorem to each to get $$ds^2=dx^2+dy^2$$ where the $d$ indicates taking an infinitesimal segment. Then it is a small step to add a third dimension $$ds^2=dx^2+dy^2+dz^2$$ which gives us the standard formula for computing the length of any path in $x,y,z$ space. Notice how that formula is just the Pythagorean theorem that you learned in middle school, just written for little triangles.

Now, we get Relativity just by taking this idea of geometry from space to spacetime. So we are going to add time as a geometric dimension. It is a weird idea, but the universe works this way, and the modification is easy. We just take the previous formula and include time. Since time is different from space, we include it with the opposite sign: $$ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$$ And that is it. All Relativity comes from that one formula, and it is not such a scary formula. It is just the Pythagorean theorem for $ct,x,y,z$ spacetime.

*While writing the above, I did come up with one easy justification. It works for your specific scenario, not in general. In your specific scenario you set it up so that cl1 and cl2 were symmetrical. You can simply rotate by 180 degrees to change to a new scenario where they swap. Since the laws of physics are the same in every direction, then by that same symmetry they must come out the same.

 

[Sagittarius A-Star]

And back to the not-occuring time dilation between cl1 and cl2 in Exp 1 above: Is there some well comprehensible statement (ideally in natural language without maths) why the velocity between cl1 and cl2 does not matter? Time dilation between them is always zero, right? No matter how fast they move with respect to each other. As long as the absolute values of their acceleration is always the same. Correct?

No. The overall aging difference of cl1 and cl2 is zero when they meet. But time dilation between them is not always zero. For example the time dilation of cl1 in the restframe of cl2 varies and depends on the speed of cl1 in the restframe of cl2 and also on the location of cl1 in the restframe of cl2 (pseudo-gravitational time-dilation), because the restframe of cl2 is not inertial.

 

[PeroK]

And back to the not-occuring time dilation between cl1 and cl2 in Exp 1 above: Is there some well comprehensible statement (ideally in natural language without maths) why the velocity between cl1 and cl2 does not matter? Time dilation between them is always zero, right? No matter how fast they move with respect to each other. As long as the absolute values of their acceleration is always the same. Correct?

There is time dilation. But, there isn't differential ageing.

One answer is that time dilation, despite its prominence in introductory SR, is not really a "thing". It's just a coordinate effect. The invariant thing is the length of the spacetime path.

If we take the example of two objects moving inertially towards each other. There is symmetric, velocity-based time dilation. But, by the first postulate of SR, all inertial reference frames are equally valid. So, neither can be ageing relative to the other.

In general, accelerating reference frames are a distraction in SR. If you stick to inertial reference frames, then all is clear. If you insist on analysing things in an accelerating reference frame, then things do get trickier. Note that the basics of SR (including the formula for time dilation) are derived using an inertial reference frame. Hence, that formula simply is not applicable in an accelerating reference frame.

That would be my answer: if you want to complicate things by using an accelerating reference frame, then you cannot use the basic formulas you derived assuming an inertial reference frame.

 

[Janus]

And back to the not-occuring time dilation between cl1 and cl2 in Exp 1 above: Is there some well comprehensible statement (ideally in natural language without maths) why the velocity between cl1 and cl2 does not matter? Time dilation between them is always zero, right? No matter how fast they move with respect to each other. As long as the absolute values of their acceleration is always the same. Correct?

You are conflating "time dilation" with "The difference in accumulated time when they are brought back together." During different points of the experiment, cl1 and cl2 will tick at different rates in Exp 1 when measured from cl1 or cl2. It is only from cl3 that they tick at the same rate at all times.
For example, let's say that each ship measures that it is moving at 0.6c relative to cl3 after the initial acceleration. This means, that due to the way velocities add up in Relativity, cl1 will measure its relative velocity to cl2 as being ~0.882c and cl2 ticking ~47% as fast. (cl2 will measure the same for cl1: that cl1 ticks slow)
This happens again when both clocks are on their way back to cl3 after their turn around.
Both cl1 and clock cl2 measure cl3 as running 80% as fast during these periods.
It is when cl1 and cl2 do their turn around that they would say that the other clocks tick fast. How fast is determined by not just the magnitude of the acceleration, but also the distance between them and the other clock. So cl1 during its turn around, would determine not only that clock both cl2 and cl3 run fast, but that of the the two, cl2 is running faster than cl3.
In addition, according to cl1, cl2 has not yet reached its turn around point when cl1 begins its own turn around. (Even though according to cl3, both cl1 and cl2 turn around at the same time.)
Relativity is not time dilation alone, it is also length contraction and Relativity of simultaneity. And it takes all 3 three to paint a picture of what occurs according to any given observer.
And to really grasp Relativity generally requires one to abandon previous notions on the nature of "time" and "space". For example, the idea that if cl1 measures cl2 as running slow, cl2 must measure cl1 as running fast.)

 

[Friggle]

OK, so I summarize my current understanding:
1.) When two objects move relative to each other and "look" at each other, then the way or magnitute they recognize or measure things like distance, speed or time with respect to the other object solely depends on their relative speed. The history, how this relative speed was established, doesn't matter. This is what SR is about.
2.) Once these two objects come to rest at the same point in spacetime (i.e. at the same Event), by applying appropriate acceleration forces to either or both of them, then their individual acceleration history (as measured by their own clocks - not sure here?) is what determines their accumulated relative ageing since they had been at rest at the same Event the last time in the past.
--> Are these correct statements? Thanks again!

Edit: I'm talking about "proper acceleration" in the above.

 

[PeroK]

OK, so I summarize my current understanding:
1.) When two objects move relative to each other and "look" at each other, then the way or magnitute they recognize or measure things like distance, speed or time with respect to the other object solely depends on their relative speed. The history, how this relative speed was established, doesn't matter. This is what SR is about.

SR is more fundamentally about reference frames than individual particles. For example, when you are using SR in particle physics, you may consider the rest frame of one particle or the centre of momentum frame. What SR is saying is that you can do physics in any (inertial) reference frame.

Once you have established the baseline for spacetime, it's the relativistic energy and momentum of the particles that is critical. Things like an "acceleration history" for each particle is not a thing. It's the relative energy-momentum of the collision that is important.

Particles can be created in collisions, so there is no acceleration history at all in the theory. If a particle is moving inertially, it has an inertial rest frame and a relative velocity in any other frame.

I already said that acceleration is a distraction. You're going off on a fruitless tangent in this respect, IMO.

2.) Once these two objects come to rest at the same point in spacetime (i.e. at the same Event), by applying appropriate acceleration forces to either or both of them, then their individual acceleration history (as measured by their own clocks - not sure here?) is what determines their accumulated relative ageing since they had been at rest at the same Event the last time in the past.
--> Are these correct statements? Thanks again!

Again, acceleration is not a direct factor. It's the length of the spacetime paths.

An analogy would be to try to understand why you got a speeding ticket by studying the acceleration profile of your car since you bought it! It's much simpler to look at your speed and you get a ticket if your speed goes above a certain limit. You could recast the speeding laws in terms of acceleration profiles, but it would be neither useful nor enlightening.

Finally, the twin paradox, for example, can be done without acceleration and with entirely inertial motion (in flat spacetime). This proves that acceleration is not a direct factor.

 

[Sagittarius A-Star]

OK, so I summarize my current understanding:
1.) When two objects move relative to each other and "look" at each other, then the way or magnitute they recognize or measure things like distance, speed or time with respect to the other object solely depends on their relative speed. The history, how this relative speed was established, doesn't matter. This is what SR is about.

In 1.) you seem to speak about time-dilation (frame-dependent), in 2.) about different aging (not frame-dependent).

For example the time dilation of cl1 in the restframe of cl2 depends solely on the speed of cl1 in the restframe of cl2 only, if this restframe is inertial. Else it depends also on the location of cl1 in the restframe of cl2, see posting #10.

2.) Once these two objects come to rest at the same point in spacetime (i.e. at the same Event), by applying appropriate acceleration forces to either or both of them, then their individual acceleration history (as measured by their own clocks - not sure here?) is what determines their accumulated relative ageing since they had been at rest at the same Event the last time in the past.
--> Are these correct statements? Thanks again!

Edit: I'm talking about "proper acceleration" in the above.

That's almost correct. Elapsed proper time of an object between two events is the length of the spacetime paths along it's worldline. The worldline is determined by the initial conditions and the proper acceleration profile.

the same Event the last time in the past

... is not correct. In the past was a different event.

 

[PeroK]

Here's an example. If you have rockets moving in uniform circular motion about a central space station, then in this case the time dilation is not symmetric and it leads directly to differential ageing. The rockets are accelerating centripetally. The calculation for the time-dilation depends only on the rocket's orbital speed and not on the acceleration. Note that for uniform circular motion, you can have a large orbital speed and low magnitude of centripetal acceleration and vice versa - as it depends on the radius.

In any case, for any motion in an inertial reference frame, we have $\frac{d\tau}{dt} = v(t)$. In other words, the time dilation depends explicitly and solely on the speed of the object.

 

[Friggle]

... is not correct. In the past was a different event.

Certainly agree, that one was nonsense ;-)

 

[Friggle]

Here's an example. If you have rockets moving in uniform circular motion about a central space station, then in this case the time dilation is not symmetric and it leads directly to differential ageing. The rockets are accelerating centripetally. The calculation for the time-dilation depends only on the rocket's orbital speed and not on the acceleration. Note that for uniform circular motion, you can have a large orbital speed and low magnitude of centripetal acceleration and vice versa - as it depends on the radius.

In any case, for any motion in an inertial reference frame, we have $\frac{d\tau}{dt} = v(t)$. In other words, the time dilation depends explicitly and solely on the speed of the object.

I get your point here. But the allelerated mass in terms of energy or momentum is still clearly the rocket. And this acceleration to its tangential speed has been brought in in its past. Isn't the centripedal acceleration youbtalk about only coordinate acceleration as opposed to proper acceleration?

 

[Nugatory]

Doesn't GR say that in proximity to mass (i.e. in a gravity field, i.e. curved spacetime), time goes slower?

That "time goes slower" phrase is common in popular and non-serious attempts to explain relativity, mainly because people writing popular non-serious descriptions don't want to spoil the entertainment by using non-popular serious stuff like math, and "time goes slower" is an easy math-free way of talking about time dilation.
But it is hugely misleading, and it is one of the things that we must unlearn before we can start learning relativity. Time always flows, tautologically, at the rate of one second per second. How could it be anything else? It follows from Einstein's "time is what a clock measures" - when I'm looking at my wristwatch I know that one second has passed when it changes from 12:07:05 to 12:07:06.

All of the "time goes slower" claims are actually comparing the number of seconds along different paths through spacetime or between different points in spacetime, and not surprisingly, coming up with different results. Velocity-based time dilation, the twin paradox in its various incarnations, and gravitational time dilation do this in different ways, but all have in common that once we understand which intervals through spacetime we're talking about, we won't be surprised to find that they're different.

 

[Friggle]

An analogy would be to try to understand why you got a speeding ticket by studying the acceleration profile of your car since you bought it! It's much simpler to look at your speed and you get a ticket if your speed goes above a certain limit. You could recast the speeding laws in terms of acceleration profiles, but it would be neither useful nor enlightening.

Admittedly, this is a nice analogy but I feel it doesn't quite fit the purpose: I'm not asking for the relative speed of the objects in the experiment (which would be responsible for the speed ticket). I'm interested in their accumulated ageing, and it seems that the acceleration profile is actually key here: Take the OT post, Exp 1: When I would look at only the pair of clocks cl1 and cl2 from a distance, I could observe their relative speed (not even sure about this, I must admit) but this wouldn't tell me anything about which one has experienced the larger ageing after they unite again at rest in an event. I can only calculate the accumulated ageing of the two individual clocks by knowing their proper acceleration profile since the event where they had been synched the last time. I can accept that it's the paths through spacetime. But these paths actually depend on proper acceleration, too, I suppose, and not only on relative velocity between the objects.

Examples:
- Only cl1 accelerates. Then cl2 > cl1 when reunited. The relative velocity profile between the two lets me calculate the accumulated ageing difference. Since I know that only cl1 was accelerated.
- Only cl2 accelerates. Then cl1 > cl2 when reunited. The relative velocity profile between the two lets me calculate the accumulated ageing difference. Since I know that only cl2 was accelerated.
- Both cl1 and cl2 accelerate, symmetrically. Then cl1 = cl2 when reunited. The bare relative velocity profile information doesn't help me there. I need to know about the symmetric acceleration profiles.
- Both cl1 and cl2 accelerate, asymmetrically. Then cl1 <> cl2 when reunited. Which one is larger and to what extent can only be determined by knowing the individual proper acceleration profiles. The bare relative velocity profile information doesn't help me there
--> Correct?

 

[PeroK]

Admittedly, this is a nice analogy but I feel it doesn't quite fit the purpose: I'm not asking for the relative speed of the objects in the experiment (which would be responsible for the speed ticket). I'm interested in their accumulated ageing, and it seems that the acceleration profile is actually key here: Take the OT post, Exp 1: When I would look at only the pair of clocks cl1 and cl2 from a distance, I could observe their relative speed (not even sure about this, I must admit) but this wouldn't tell me anything about which one has experienced the larger ageing after they unite again at rest in an event. I can only calculate the accumulated ageing of the two individual clocks by knowing their proper acceleration profile since the event where they had been synched the last time. I can accept that it's the paths through spacetime. But these paths actually depend on proper acceleration, too, I suppose, and not only on relative velocity between the objects.

Examples:
- Only cl1 accelerates. Then cl2 > cl1 when reunited. The relative velocity profile between the two lets me calculate the accumulated ageing difference. Since I know that only cl1 was accelerated.
- Only cl2 accelerates. Then cl1 > cl2 when reunited. The relative velocity profile between the two lets me calculate the accumulated ageing difference. Since I know that only cl2 was accelerated.
- Both cl1 and cl2 accelerate, symmetrically. Then cl1 = cl2 when reunited. The bare relative velocity profile information doesn't help me there. I need to know about the symmetric acceleration profiles.
- Both cl1 and cl2 accelerate, asymmetrically. Then cl1 <> cl2 when reunited. Which one is larger and to what extent can only be determined by knowing the individual proper acceleration profiles. The bare relative velocity profile information doesn't help me there
--> Correct?

Not really. Age is the proper time experienced by the particle. That can be calculated directly and generally from the length of the spacetime path. Or, from the velocity function in any inertial reference frame. Of course, acceleration is related to velocity, so you can always derive the velocity from the acceleration. But, as I said, there are plenty of experiments where there is no acceleration, per se. E.g.counting muons created in the upper atmosphere

The important point is that there is a simple expression for proper time involving spacetime length or velocity, but none for acceleration.

By focusing on acceleration, as the key concept in relativity, you are heading down a blind alley.

 

[Sagittarius A-Star]

The bare relative velocity profile information doesn't help me there
--> Correct?

You need to define a reference frame to do the calculation. If you know the initial conditions and the velocity profile of each relevant object relative to this reference frame, then this helps. The missing acceleration information (proper- vs. coordinate-acceleration) comes then from the acceleration profile of the selected reference frame, which is zero in case the reference frame is inertial.

Also, there is no such thing as a "relative velocity profile" without defining a reference-frame.

 

[PeroK]

Correct. But you need to define a reference frame to do the calculation. If you know the initial conditions and the velocity profile of each relevant object relative to this reference frame, then this helps. The missing acceleration information comes then from the acceleration profile of the selected reference frame, which is zero in case the reference frame is inertial.

The critical question is whether SR text books and problems are filled with complicated calculations involving proper acceleration profiles? The answer is no.

In the twin experiment, the end result depends almost entirely on the inertial phases, and the specifics of the acceleration phases have only a small effect on the overall ageing. It's definitely not the reverse where a detailed study of the precise acceleration phases is required.

 

[Sagittarius A-Star]

The critical question is whether SR text books and problems are filled with complicated calculations involving proper acceleration profiles? The answer is no.

In the twin experiment, the end result depends almost entirely on the inertial phases, and the specifics of the acceleration phases have only a small effect on the overall ageing. It's definitely not the reverse where a detailed study of the precise acceleration phases is required.

Yes. I removed in posting #22 the "correct".

 

[Sagittarius A-Star]

I get your point here. But the allelerated mass in terms of energy or momentum is still clearly the rocket. And this acceleration to its tangential speed has been brought in in its past. Isn't the centripedal acceleration youbtalk about only coordinate acceleration as opposed to proper acceleration?

The centripetal acceleration is a proper acceleration. The scenario, @PeroK described in #16, was experimentally carried-out in a particle storage ring with muons by Bailey et al. (1977).

Clock hypothesis - lack of effect of acceleration

The clock hypothesis states that the extent of acceleration does not influence the value of time dilation. In most of the former experiments mentioned above, the decaying particles were in an inertial frame, i.e. unaccelerated. However, in Bailey et al. (1977) the particles were subject to a transverse acceleration of up to ∼1018 g. Since the result was the same, it was shown that acceleration has no impact on time dilation.[28] In addition, Roos et al. (1980) measured the decay of Sigma baryons, which were subject to a longitudinal acceleration between 0.5 and 5.0 × 1015 g. Again, no deviation from ordinary time dilation was measured.[30]

Source:
https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation

 

[Friggle]

In the twin experiment, the end result depends almost entirely on the inertial phases, and the specifics of the acceleration phases have only a small effect on the overall ageing. It's definitely not the reverse where a detailed study of the precise acceleration phases is required.

I agree, and I didn't intend to express any doubts to this. What I meant is the whole acceleration profile instead. This certainly also includes all times where acceleration is zero but time is spent in the accelerated state. And clearly, the bulk of time dilation comes from these zero-acceleration phases.

 

[Friggle]

You need to define a reference frame to do the calculation. If you know the initial conditions and the velocity profile of each relevant object relative to this reference frame, then this helps. The missing acceleration information (proper- vs. coordinate-acceleration) comes then from the acceleration profile of the selected reference frame, which is zero in case the reference frame is inertial.

I feel that this one really helps. Need to let this sink in for a while, thanks :-)

Seems that if I know my own acceleration (the acceleration of my reference frame which I can measure locally with an accelerometer) and then observe the velocities of other objects within my reference frame I can from there calculate the spacetime paths of the observed objects which then gives me their ageing. Correct?

 

[Dale]

Seems that if I know my own acceleration (the acceleration of my reference frame which I can measure locally with an accelerometer) and then observe the velocities of other objects within my reference frame I can from there calculate the spacetime paths of the observed objects which then gives me their ageing. Correct?

Yes. That is correct. Here is the math, using the radar convention for determining the frame of the accelerating object.

https://arxiv.org/abs/physics/0412024

 

[jbriggs444]

I feel that this one really helps. Need to let this sink in for a while, thanks :-)

Seems that if I know my own acceleration (the acceleration of my reference frame which I can measure locally with an accelerometer) and then observe the velocities of other objects within my reference frame I can from there calculate the spacetime paths of the observed objects which then gives me their ageing. Correct?

There are complications, but yes.

"the acceleration of my reference frame" is not well defined. Not all parts of your reference frame move at the same speed at the same time. And "same time" is problematic as well.

Yes, one can transform from the sequence of coordinates for events on any trajectory as reported against your frame of reference to the sequence of coordinates for events on the same trajectory as reported against some inertial frame of reference. Then one could integrate $(\Delta S)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - \Delta ct)^2$ for that inertial frame to get the elapsed proper time along the trajectory.

One could not integrate $(\Delta S)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 - \Delta ct)^2$ directly in your accelerated frame because the "metric" when expressed in coordinate form for your chosen accelerating frame would be a far nastier expression. Not just the square root of a sum of squares.

The term for an accelerating frame defined in the obvious way by the world line of an object is "fermi normal coordinates". [@Dale adopted a different set of coordinates above]. Fermi coordinates are valid for a world tube surrounding the object's world line. They cannot usually be extended to cover all of the spacetime. Outside the tube, you can get double mappings, two coordinates for the same event.

 

[Sagittarius A-Star]

Seems that if I know my own acceleration (the acceleration of my reference frame which I can measure locally with an accelerometer) and then observe the velocities of other objects within my reference frame I can from there calculate the spacetime paths of the observed objects which then gives me their ageing. Correct?

Yes, with some limitations, which @jbriggs444 listed. If you describe a uniformly accelerated frame for example with Kottler–Møller coordinates, then you are limited to scenarios, in which the other clock is not behind the Rindler horizon. In this accelerated frame, the time-dilation formula can be derived from the Minkowski metric expressed in terms of the coordinates of the accelerated frame.

A moving clock with coordinate-velocity $v$ at a location with coordinate $x$ has the following time dilation factor:

$d\tau = dt\sqrt {(1+ \frac{\alpha\,x} {c^2})^2 - \frac{v^2}{c^2}}$

Here, $\alpha$ is the proper acceleration of the observer at rest at $x=0$.
Only in the special case of $\alpha=0$ this simplifies to the time dilation factor, which is used in inertial reference frames. Alternatively, in the special case of $v=0$, this simplifies to the usual gravitational time dilation factor.

Also, the relative velocity between the two clocks is in general not reciprocal, because the coordinate-velocity of the "moving" clock is affected by the pseudo-gravitational time-dilation in the accelerated frame.

 

[A.T.]

Admittedly, this is a nice analogy but I feel it doesn't quite fit the purpose: I'm not asking for the relative speed of the objects in the experiment (which would be responsible for the speed ticket). I'm interested in their accumulated ageing, and it seems that the acceleration profile is actually key here:

Try this analogy:

The rate at which an odometer reading increases is not directly affected by the steering, just by speed. But the steering on the way between two points affects the total distance accumulated by the odometer, and knowing the complete steering profile allows you to compute it.

odometer distance : proper time interval
steering : proper acceleration

 

[A.T.]

And since gravity is the same as acceleration ...

Make sure not to confuse the acceleration of a frame of reference with the acceleration of the observed clocks. This is very common in the "gravity as acceleration" context.

 

[Friggle]

First, thanks to all of you, that has already helped a lot, overall.
I again stumbled on the above from PeroK:

Here's an example. If you have rockets moving in uniform circular motion about a central space station, then in this case the time dilation is not symmetric and it leads directly to differential ageing. The rockets are accelerating centripetally. The calculation for the time-dilation depends only on the rocket's orbital speed and not on the acceleration.

I now fully agree that it's proper acceleration, since somebody in the rocket clearly could measure it using just a local accelerometer. So, that's just fine.

What I'm struggling with is that there is asymmetric time dilation at all in this setup (a clock in the rocket vs. a clock in the base station). Reason for my issues with this is that there is (from my perspective, you will surely correct me if I'm wrong) no relative motion between the base station and the rocket. I mean: The distance between them never varies, right? So how can there be a velocity between them? May be that is naive thinking, please advise...
Nevertheless, this might be a new approch for me to grasp the idea of the root cause for both time dilation and ageing differently: May be it is not the relative motion between an observer and an observed object which drives time dilation and accumulated relative ageing but rather the movement path of the observed object through the reference spacetime frame of the observer. Because, in the rocket example, clearly the rocket moves in relation to the reference frame of the observer (it performs circular movements within that frame), Although it does not move in relation to the observer. Is this correct or am I totally wrong here?

 

[Sagittarius A-Star]

The distance between them never varies, right? So how can there be a velocity between them?

In his popular book "Relativity: The Special and General Theory" from 1917, Einstein explained this scenario with an inertial reference-frame (velocity-based time-dilation) and with a rotating reference-frame (pseudo-gravitational time-dilation).

Source (see Appendix III, sub-chapter "(c) Displacement of Spectral Lines Towards the Red":
https://en.wikisource.org/wiki/Rela...isplacement_of_Spectral_Lines_Towards_the_Red

 

[jbriggs444]

Reason for my issues with this is that there is (from my perspective, you will surely correct me if I'm wrong) no relative motion between the base station and the rocket. I mean: The distance between them never varies, right? So how can there be a velocity between them? May be that is naive thinking, please advise...

Inertial frames do not rotate. In any inertial frame where the central observer is at rest, the rocket will have a non-zero velocity. We can use a rotating frame and get an answer of zero. But then we are using a non-inertial frame and our result for time dilation will be complicated by this.

There is no inertial frame where the orbitting rocket is at rest. But again, we can use a variety of non-inertial frames where the answer is zero. But then we are using a non-inertial frame and our result for time dilation will be complicated by this.