Twin Paradox without acceleration

[confused2]

Comparing xxxxxxx's claim (elsewhere) that time dilation is the result of acceleration..

In the UK we have train stations which are 33km apart in a straight line.. 
Alcester, Bicester, Cirencester and Dorchester *
Alice will stay in Alcester,
Bob will travel at constant velocity v through all the stations.
Colin will travel on a train travelling at velocity -v that passes Bob's at Dorchester station and heads back towards Alice.

Go!
Bob zooms through Alchester and sets his watch to be the same as the station clock.
At Bicester he checks his watch against the station clock and reports back to Alice that the station clock seems to be going faster than his watch.
By the time he reaches Dorchester and compares his watch with the Dorchester station clock he's a fair bit behind station clock time and he tells Alice exactly how much.
As Colin whizzes past in the opposite direction he shows his watch to Colin and Colin sets his watch to match Bob's.

By the time Colin gets back to Alice at Alcester station his wristwatch time is slower by twice what Bob's was in Dorchester.

So..
Bob was already younger than Alice when he reached Dorchester (less time on his watch compared to the station clock).
Colin (arguably) did some sort of acceleration trick but was also (arguably) carrying back to Alice a record of the time elapsed for Bob as he reached Dorchester but was himself a victim of the same effect that wasn't the result of acceleration.

Calculations to follow unless everybody hates me too much.

?

* Americans best just call these A,B,C and D.

[Kornee]

I'm sure that exact scenario has been presented multiple times at PF, but won't try and hunt down specific instances.
One criticism is it lies outside the standard dictum that differential aging is only well defined for a round trip journey between two entities that must begin and end at a common location.
Another is that accelerations have necessarily occurred, even if well outside of the time frame of the scenario's duration.

It is a good head scratcher in the sense it apparently 'defies' the symmetrical nature of SR aging for constant relative velocity observers.

One scenario I used at SciForums to tank Mike Fontenot's now evidently abandoned CADO (Current Age of Distant Object) scheme, involved constant relative motion speed but where there was no change in relative location.
Circular motion of one twin, the other at a fixed location anywhere along the axis of rotation of the circularly traveling twin.
Obviously, centripetal acceleration is involved, but as per clock hypothesis, the magnitude of centripetal acceleration has no effect on the constant rate of differential aging (during the arbitrarily long constant circular speed phase).
Given simply by the relativistic transverse Doppler factor 1/gamma. Or gamma, depending on which twins pov is specified.

[confused2]

One criticism is it lies outside the standard dictum that differential aging is only well defined for a round trip journey between two entities that must begin and end at a common location.
Another is that accelerations have necessarily occurred, even if well outside of the time frame of the scenario's duration.

I see where you're coming from but the 'muon experiment' works for me - unfortunately not possible to send them back.to where they started from.
http://hyperphysics.phy-astr.gsu.edu/hba.../muon.html

[Kornee]

One criticism is it lies outside the standard dictum that differential aging is only well defined for a round trip journey between two entities that must begin and end at a common location.
Another is that accelerations have necessarily occurred, even if well outside of the time frame of the scenario's duration.

I see where you're coming from but the 'muon experiment' works for me - unfortunately not possible to send them back.to where they started from.
http://hyperphysics.phy-astr.gsu.edu/hba.../muon.html

Sure, I have no argument with the reality of SR relative time dilation. As you surely are aware, a full analysis of the muon decay time dilation, that includes the muon's reference frame, must take into account nonsimultaneity to resolve apparent paradox.

[confused2]

Alster,Bister, Sirensester Dorchester

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..to resolve apparent [muon?]paradox.

Paradox? What paradox?

[Kornee]

Alster,Bister, Sirensester Dorchester

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..to resolve apparent [muon?]paradox.

Paradox? What paradox?

I know in advance what is likely your angle. All one needs as per HyperPhysics article, is to take into account relativistic length contraction in the muon's frame, to achieve a consistent picture. No nonsimultaneity required. Yes and no.
The time dilated muon decay phenomenon is actually an inferred one - legitimate given reasonable assumptions but strictly speaking still inferred.
It relies on a model of cosmic ray collisions with air molecules in the upper atmosphere, that then generate the desired relativistic muon showers, which subsequently decay into secondary decay product showers.
A direct measurement of (inherently statistical) muon decay rate would require a detector high up in the atmosphere, that somehow detected and time tagged without otherwise influencing, individually freshly created muons.
In addition to a ground based detector that counted the (reduced) detection rate of somehow individually time tagged muons.
It's in that overall *direct* process that nonsimultaneity comes into play. Can you see why?

[confused2]

If we know the instantaneous velocity of a thing (say 100mph) we infer that something travelling at 100mph will travel 100 miles in an hour unless acted by some force. My confidence level in that inference is high.
Some details of a student lab 'muon experiment' here:
https://www2.ph.ed.ac.uk/~muheim/teachin...9apr04.pdf
While not (I suspect) intended to be flown to 10km altitude it does discuss many of the relevant points.

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I prefer to start with s to get time dilation (cribbed from another post)

In the muon frame elapsed time =t we have
s²=-c²t²
In the Earth frame t' and x' (where x' ~10km) we have
s²=x'²-c²t'²
since s² is the same in both frames
c²t²=x'²-c²t'²
we know x' = vt' from Newton and the definition of velocity
so
-c²t²=(vt')²-c²t'²
c²t²=c²t'²-(vt')²
t= t'√(1-v²/c²)
so t=t_muon is less than t'=t_Earthclock

[Kornee]

A nicely comprehensive article. Obviously the problem would be much easier if somehow travel was through a vacuum.

[confused2]

A direct measurement of (inherently statistical) muon decay rate would require a detector high up in the atmosphere, that somehow detected and time tagged without otherwise influencing, individually freshly created muons.
In addition to a ground based detector that counted the (reduced) detection rate of somehow individually time tagged muons.
It's in that overall *direct* process that nonsimultaneity comes into play. Can you see why?

To check a pipe for leaks you measure the rate water goes in at one end and the rate it comes out at the other - anything missing is a leak - you don't have to wait for the water you're putting in to come out. With the muon experiment you count them in at the top and average for as long as required and count them at the bottom and average - anything missing has decayed or is otherwise lost.

Edit .. nonsimultaneity ? am I supposed to worry about this and if so why?

[Kornee]

A direct measurement of (inherently statistical) muon decay rate would require a detector high up in the atmosphere, that somehow detected and time tagged without otherwise influencing, individually freshly created muons.
In addition to a ground based detector that counted the (reduced) detection rate of somehow individually time tagged muons.
It's in that overall *direct* process that nonsimultaneity comes into play. Can you see why?

To check a pipe for leaks you measure the rate water goes in at one end and the rate it comes out at the other - anything missing is a leak - you don't have to wait for the water you're putting in to come out. With the muon experiment you count them in at the top  and  average for as long as required and count them at the bottom and average - anything missing has decayed or is otherwise lost.

Edit .. nonsimultaneity ? am I supposed to worry about this and if so why?

No you don't have to worry about it. Suitably adjusted for muon deceleration in the atmosphere (as per formula on p12 in the article), sure one determines the 'leakage rate' and time dilation comes out of the calculations given a known rest frame muon decay half-life.
As I hinted at earlier, a hypothetical direct timing of muons requires synchronized clocks (as stop-watches) upper and lower. In the muon frame, to explain the clock readings, nonsimultaneity of clock timings in that frame must be invoked.
But it is just a hypothetical since one can't in practice track individual muons.