Relative Motion vs. 'Relative' Acceleration: Twin 1 & 2

[Martyn Arthur]

TL;DR Summary

Given the relativity concept between twin i and twin 2 what is the differnce between relative motion and 'relative' acceleration?

Given the relativity concept between twin 1 and twin 2 what is the difference between relative motion and 'relative' acceleration? Are not either twin experiencing the differential between them as a consequence of acceleration.

When twin 1 instigates motion relative to twin 2, a moving carriage v a ‘stationery’ carriage for twin 2 relativity seems to apply.

So we are physically causing a change in velocity via acceleration but it is still relative motion.

So why is there a distinction there with acceleration?
Thanks Martyn

 

[Dale]

Summary:: Given the relativity concept between twin i and twin 2 what is the differnce between relative motion and 'relative' acceleration?

So why is there a distinction there with acceleration?

Because acceleration is NOT relative. You can attach accelerometers to each twin and unambiguously determine if a given twin accelerated, by how much they accelerated, and how long they accelerated, all without any reference to or observation of the other twin and irrespective of any coordinate system.

Note, this refers to proper acceleration which is physical acceleration. Coordinate acceleration is non-physical and is relative to the chosen reference frame.

 

[robphy]

Both relative motion (in general) and relative acceleration miss the key distinction in the twin paradox: it’s really about inertial vs non-inertial (an absolute sense of acceleration).

 

[Orodruin]

Both relative motion (in general) and relative acceleration miss the key distinction in the twin paradox: it’s really about inertial vs non-inertial (an absolute sense of acceleration).

I’d go so far as to say it is not about acceleration at all, but about geometry. (Although, yes, in an affine space lines will not meet more than at most once unless at least one of them bends at some point.)

 

[robphy]

I’d go so far as to say it is not about acceleration at all, but about geometry. (Although, yes, in an affine space lines will not meet more than at most once unless at least one of them bends at some point.)

Indeed, there is no twin-paradox clock-effect in a Galilean/Newtonian spacetime,
whether or not the worldlines are those of accelerated astronauts.

 

[Ibix]

Given the relativity concept between twin 1 and twin 2 what is the difference between relative motion and 'relative' acceleration?

The experimental physicist's answer is that there is no inside-a-closed-box experiment you can do to determine your velocity, but there is to determine your acceleration: put a known mass on a scale.

The theoretical physicist's answer is that velocity corresponds to the angle your worldline makes with another one (formally this is called rapidity). "Angle" is meaningless without two worldlines to measure the angle between. Acceleration, on the other hand, corresponds to the worldline curving, and that is definable without reference to any other worldline.

So velocity is relative and acceleration is not.

 

[FactChecker]

The term "acceleration" must be used more carefully in physics than in general mathematics. In physics, "acceleration" is not the derivative of the relative velocity, it is the result of a force, as in F = mA. The two are very different. Clearly, the derivative of relative velocity is just as relative as the relative velocity was. But that is not true of acceleration.

 

[Nugatory]

Are not either twin experiencing the differential between them as a consequence of acceleration.

The acceleration is something of a red herring in the twin paradox (one hint that it's not just about the acceleration is that the longer the journey the greater the age difference even if the acceleration to turn the traveling twin around is the same).

The two twins age differently because they've followed different paths through spacetime and there are fewer seconds along the traveler's path through spacetime than along Earth twin's path. We can't have a twin paradox (at least in gravity-free flat spacetime) without acceleration but that's not because acceleration causes the twin paradox, it's because there's no way of setting the twins on different paths through spacetime without accelerating at least one of them.

 

[hutchphd]

I’d go so far as to say it is not about acceleration at all, but about geometry. (Although, yes, in an affine space lines will not meet more than at most once unless at least one of them bends at some point.)

Oh please let's not go there...

 

[robphy]

Oh please let's not go there...

The geometric-essence of the twin-paradox/clock-effect already happens in Euclidean space.
The "paradox" arose because we use a Galilean geometry on a position-vs-time graph.

 

[Orodruin]

Oh please let's not go there...

Oh, I disagree. Understanding the geometrical nature of spacetime is fundamental to understanding special relativity. Understanding differential ageing (and hence resolving the twin paradox) is not much more than understanding the triangle inequality.

The entire reason paradoxes like the twin paradox arise is that prople get stuck in a coordinate mindset (and the resulting relativity of simultaneity), blocking out the essentials.

 

[Mister T]

Are not either twin experiencing the differential between them as a consequence of acceleration.

No. In the usual twin paradox one twin accelerates and the other doesn't. This difference breaks the symmetry and resolves the paradox. (That is, it explains why the twins' experiences are not the same.)

But it does not explain the difference in aging (that gives rise to the paradox). The difference in aging occurs because the twins take different paths through spacetime. In addition to explaining the difference in aging, the different paths also displays the twins' different experiences and resolves the paradox, too.

The bottom line is that when you go to calculate the difference in aging you cannot do it by knowing only the acceleration. In fact, by making the twin trip long enough and fast enough you can make the effects of the acceleration on the difference in ages negligible.

 

[FactChecker]

No. In the usual twin paradox one twin accelerates and the other doesn't. This difference breaks the symmetry and resolves the paradox. (That is, it explains why the twins' experiences are not the same.)

To amplify a little on that, the traveling twin can independently tell that he is not following a geodesic in an inertial reference frame. He has to change frames between the leaving and the returning leg of his trip. So his situation is not symmetric with the Earth twin. In any simple scenario of his turn-around (acceleration over time or instantaneous), the correct calculations can be done that match the results that are directly obtained from the Earth twin's reference frame. So there is no real paradox.

 

[PeterDonis]

the traveling twin can independently tell that he is not following a geodesic in an inertial reference frame

The part I have crossed out in the above is superfluous; whether or not a particular curve is a geodesic is an invariant, independent of any choice of frame.

He has to change frames

He has to experience proper acceleration at the turnaround (in the usual formulation of the scenario). Describing this as "changing frames" is misleading; any object is "in" all reference frames, not just one. Which inertial frame the traveling twin is at rest in does change at the turnaround; but the best way to describe that is to state it explicitly, as I just did.

 

[FactChecker]

The part I have crossed out in the above is superfluous; whether or not a particular curve is a geodesic is an invariant, independent of any choice of frame.

Perhaps the correct way to say what I wanted to say is that he can not consider himself to be at a stationary point in anyone inertial reference frame throughout the entire trip.

He has to experience proper acceleration at the turnaround (in the usual formulation of the scenario). Describing this as "changing frames" is misleading; any object is "in" all reference frames, not just one. Which inertial frame the traveling twin is at rest in does change at the turnaround; but the best way to describe that is to state it explicitly, as I just did.

So here I should say that if he wants to consider himself at a stationary point in an inertial reference frame, then he must change frames between the trip out and the trip back.

 

[PeterDonis]

Perhaps the correct way to say what I wanted to say is that he can not consider himself to be at a stationary point in anyone inertial reference frame throughout the entire trip.

This is correct.

here I should say that if he wants to consider himself at a stationary point in an inertial reference frame, then he must change frames between the trip out and the trip back.

Yes.

 

[Orodruin]

Describing this as "changing frames" is misleading; any object is "in" all reference frames, not just one.

Just to add that this is a very common error and the source of much confusion. The only similar statement that I can make proper sense of is that an object’s instantaneous rest frame changes, which some instructors may phrase as ”changing its rest frame”, which then gets garbled into ”changing its frame”. This kind of underlines the importance of stressing proper formulation both in teaching and in examination.