« on: 07/02/2024 00:42:24 »
Thanks .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:
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.
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?)