[geordief (OP)]

I understand that the idea that time dilation is caused by acceleration  is a mistaken one(one that I laboured under for some time)and that it is actually relative  motion that is the cause.

I have also seen it stated that it is possible to produce  two scenarios with the same accelerations but with different resulting time dilations(because of  different relative motions ,presumably)

I did not see(I mean they were only mentioned in passing and not described specifically) any such actual examples .

Can anyone here produce such an example,just for my curiosity?

Ps I have changed the title of the thread so it now reads as a question ,as we are supposed to do.

[Halc]

I have also seen it stated that it is possible to produce  two scenarios with the same accelerations but with different resulting time dilations(because of  different relative motions ,presumably)

There were some specific examples at the bottom of this post:
https://www.thenakedscientists.com/forum/index.php?topic=86033.msg697485#msg697485

The first shows identical accelerations (just at different times), and significantly different differential aging. The second shows very different acceleration (continuous, one a thousand times the other) with identical dilation over indefinite time.

OK, you wanted an example of same magnitude of acceleration, but different dilation. I'm sure I've posted something of that nature, but it's easier to do it again. Remember that time dilation (due to speed at least) is a coordinate effect, not a physical one. Differential aging is a physical effect, meaning the difference isn't frame dependent.

First of all, something flying by Earth at 0.4 c is dilated far less relative to Earth than something flying by at 0.8c. There's no acceleration at all in that example.

Second example: Bob and Alice have identical ships that accelerate at 2g continuously for 7.5 Earth years. They both start on Jan 1, 2000. Bob accelerates a year (his clock) outward, two years to reverse direction, and a year to stop at Earth, and thus ages 4 years.  Alice turns the ship around in a month, so she visits Earth about 15 times instead of just once. Alice ages just under 7.5 years, having aged only 10 days less than had she stayed home.  Bob returns early July, 2007, having aged over 3.5 years less than had he stayed home.

[geordief (OP)]

I have also seen it stated that it is possible to produce  two scenarios with the same accelerations but with different resulting time dilations(because of  different relative motions ,presumably)

There were some specific examples at the bottom of this post:
https://www.thenakedscientists.com/forum/index.php?topic=86033.msg697485#msg697485

The first shows identical accelerations (just at different times), and significantly different differential aging. The second shows very different acceleration (continuous, one a thousand times the other) with identical dilation over indefinite time.

OK, you wanted an example of same magnitude of acceleration, but different dilation. I'm sure I've posted something of that nature, but it's easier to do it again. Remember that time dilation (due to speed at least) is a coordinate effect, not a physical one. Differential aging is a physical effect, meaning the difference isn't frame dependent.

First of all, something flying by Earth at 0.4 c is dilated far less relative to Earth than something flying by at 0.8c. There's no acceleration at all in that example.

Second example: Bob and Alice have identical ships that accelerate at 2g continuously for 7.5 Earth years. They both start on Jan 1, 2000. Bob accelerates a year (his clock) outward, two years to reverse direction, and a year to stop at Earth, and thus ages 4 years.  Alice turns the ship around in a month, so she visits Earth about 15 times instead of just once. Alice ages just under 7.5 years, having aged only 10 days less than had she stayed home.  Bob returns early July, 2007, having aged over 3.5 years less than had he stayed home.

Thanks .
So  Alice's and Bob's acceleration are identical (even if they can be characterized  as  broken up into smaller segments,one than the other) and ,as I requested the time dilations differ.

How does that work out with the relative motions /velocities of either Alice or Bob wrt the Earth?

Is it a complicated (tedious) calculation of all the various relative speeds involved followed by an integration of all the time dilations along the respective paths?

Are the actual distances travelled by Alice and Bob  the same?(I suppose that  is not an important thing to know,or might it be?)

[Eternal Student]

Hi.

   @Halc is fast on the buttons again.  Well done.

The examples I would have gone for would be based on people having different initial velocities   (but the same accelerations).    All of these things,  their initial velocity and their accelerations would need to be determined in some lab frame and not some frame that moves with the people who were accelerating.   In each person's own rest frame they would start to get different notions of what their accelerations were and when.    A fixed inertial lab frame will by-pass this complication and we can consider everyones acceleration to have been the same in that lab frame.

     This will then only answer one version of the question you have asked.   You left enough ambiguity in the question, so the easy interpretation of it provides for an easier answer.

    @Halc 's  reply here is an especially easy example where the acceleration of both people was 0 in the lab frame:

something flying by Earth at 0.4 c is dilated far less relative to Earth than something flying by at 0.8c. There's no acceleration at all in that example.

    If you want these two people to have plenty of opportunities to compare elapsed times and age,  then you do need to send them flying in big circles where they can rendezvous often.   You can assign the faster person a circular path of a bigger radius so that their centripetal acceleration would always be identical to the slower traveller - and I suppose that would be adequate to argue that they had the same (no-zero) acceleration always.

Best Wishes.

[and... @geordief has posted again before I finished.  Sorry if this is irrelevant].

[Halc]

So  Alice's and Bob's acceleration are identical

Their proper acceleration magnitude in my example is identical at all times in the Earth frame. The direction of the acceleration is not identical at all times, so their velocity (and with it, speed) relative to Earth is not always identical, and dilation in SR is all about speed, not about acceleration.

Is it a complicated (tedious) calculation of all the various relative speeds involved followed by an integration of all the time dilations along the respective paths?

For an erratic itenerary, yes, doing it that way always yields the right answer. For simple linear acceleration over a known time, there are shortcuts. I performed no integration to calculate Bob's age after the 7.5 Earth years.

Are the actual distances travelled by Alice and Bob  the same?

Not in that example. At no point is Alice moving faster than Bob relative to Earth, but most of the time Bob is moving faster than Alice. Hence he covers more ground.
Relative to a different frame, and the exact same example, I could have Bob and Alice travel the same distance from start to end. This shows that distance traveled is very frame dependent, and the dilation is not a function of it.

[yor_on]

You could try this one geordief

https://www.physicsoftheuniverse.com/topics_relativity_gravity.html

" Einstein's ground-breaking realization (which he called 'the happiest thought of my life') was that gravity is in reality not a force at all, but is indistinguishable from, and in fact the same thing as, acceleration, an idea he called the "principle of equivalence". "

[alancalverd]

The common error is in trying to derive a relativistic phenomenon starting with a classical concept. Like integration and differentiation, or geographical mapping,  there is a bit missing.

If two objects had a relative velocity from the date of creation of the laws of physics, each would observe a red or blue shift of any signal emanating from the other. Problem is that to assert a shift, they would have to assume that they were both made of the same material  because their spectra would never match.

Time dilation takes identical clocks as axiomatic. Relativity says that you can only demonstrate identity if there is no relative motion. So you cannot demonstrate (rather than assume) dilation unless you invoke acceleration. And if the laws of physics preceded the relative motion of the clocks, v_rel > 0 demands that dv/dt > 0 sometime in the past.

[yor_on]

Think of NIST, and their gravitational time dilation experiments with clocks. Then define gravity as being a geodesic. a geodesic that becomes impossible to differ from any other relative (uniform) motion. Like being weightless inside that satellite orbiting Earth.

makes it a little weirder :)

[yor_on]

The equivalence with an acceleration to a gravity is about the exact same thing. What you feel while being under it, accelerating, or being on earth. A weight.
=

A gravitational acceleration though, normally falls under Special Relativity, whereas gravity and accelerations falls under General Relativity

[alancalverd]

makes it a little weirder

There it is again!
If you start with a newtonian view of physics, relativity looks weird.
If you start with a relativistic view, newtonian physics doesn't look weird, just a good approximation if v <<c.

[yor_on]

Quite so. Still, you can look at gravity as describing geodesic(s), for me it just becomes the other side of the coin. The other side being its equivalence to a 'real' acceleration/deceleration, both giving me a weight. It's quite tricky.