The twin paradox, but with a twist.
The twin that accelerates is the one that is younger on return. Statement one
It doesnt matter which direction the accelerating twin goes. Ie if she leaves from the equator and goes due north from the earth and returns, it doesnt make any difference if she had gone due south and returned. Statement two
But what if there are triplets? one (a) goes due north, one (b) due south and one (c) stays on the earth. ( a and b's acceleration and velocity are identical except the one is opposite the other in sign (+/-) and we can by convention decide to make the + acceleration away from the earth in a due north direction)
so when a and b are back they are exactly the same age, but younger than c. If statement one and two are correct, this must also be correct - Statement 3
Even though the velocity difference between a and b was greater at all times than the velocty difference between a and c was or between b and c was.
Therefore it is not the velocity or even relative velocity per se, but the acceleration which causes the clocks to slow down such that when a certain RELATIVE velocity is reached, the clock travelling at the faster velocity compared to the ORIGINAL velocity is going slower than the ORIGINAL clock which has not accelerated. statement 4
Note. By different relative velocities, the two clocks as observed by different moving observes will not tick at the same rate.
Now the question is, between a and b , whose clock is going faster or slower? the answer is that if a and b leave from the same point at the same time, travel exactly the same distance and get back to the same orignial point at the same time and are the same age when they get back, both their clocks must have ticked at the same OVERALL AVERAGED dilated rate compared to the original clock, but not necessarily always syncronously. If A would observe Bs clock during the travelling and B would observe A's clock, then there are two components to the time dilation either observe. One is the time dilation due to relative velocity, and the other is the time dilation (or contraction) due to the relative acceleration ( deceleration) (which is not the same as the absolute acceleration) These two different dilations (and contraction) effects exactly cancel out, so that A and B arrive back the same age. statement 5
Now as c stayed on earth, and was subject the whole time to an acceleration too ( ie gravity) her clock is slightly dilated too.....( as there is no difference to clocks as to what causes the acceleration ie either gravity or rocket engine)
This then brings the following conclusion.....
A clock that has accelerated ticks more slowly than a clock which has not accelerated so there is time dilation at a higher velocity.
So during the acceleration, the clock will start getting slower and slower.
Now as a gravitational acceleration has the same effect on clocks as a rocket powered acceleration, and as a clock in a rocket powered acceleration will get slower and slower the longer the acceleration occurs, then a clock that exists say for 100 million years on the earth and is subject to gravity for the entire period and therefore an acceleration for 100 million years will tick more slowly now in 2015 than it did 100 million years ago.