Well here is the problem I have already presented with some little changes to explain it very well:
Here is a "perfect" experiment thought to show the inconsistence of Relativity Theory.
Is a new version of the well known twins paradox.
Just to not consider the movement of Hearth we will think in a mother-ship that goes to the most "fixed place" you can imagine. May be some point at a fixed position relative to the known "fixed stars" of the Universe.
The mother-ship goes there brakes and stop remaining there. After that, two small space-ships with twins accelerate in opposite directions, travel some time and brake in the same manner making a perfect symmetric travel to stop at some far distance.
After that, they turn their space-ships in the opposite direction and at some time (may be synchronized by the mother-ship that is at equal distance from them) they accelerate and travel in a second symmetrical flight deviating a negligible little (to not collide) just to pass very near of them and the mother-ship at the same instant but they don't brake!
The intention is to capture the movement as they are flying at some considerable velocity to detect some relativistic effects.
We must consider that the state of both twins can be directly observed by them and by the people in the mother-ship! For example photographs can be taken at the instant of "crossing" and be sent to everybody, even the twins, even to us, to analyze the phenomenon!
Now the situation is:
Both are travelling at some velocity v but in opposite directions just in front of the mother-ship.
For simplicity we will consider time zero this instant they are in front of the mother-ship.
Now we will apply Lorentz Transform to the twins to see how they are aging. Note that age is an intrinsic property of living individuals.
We are going to consider the results in different frames and compare it. Any phenomenon of Nature is independent of the referential we chouse to observe and describe it so the results should be consistent (no contradictions should exist).
First we choose a referential in the mother-ship pointing in the same direction as the velocity of one of the twins. We must replace x=+vt and x=-vt for each twin in the equations of time.
We assume k = (1-v2/c2)exp-1/2
Then for one twin we will have (x=+vt):
t = k(1-v2/c2)t' = t'/k
and for the other (x=-vt):
t = k(1+v2/c2)t'
We can see that for each twin time t' is different what means they age differently. The first one is smaller what means the first twin will get younger than the other.
But the direction of the referential was arbitrary choused with the velocity of one of the twin! If we select the other twin the equations are inverted and that twin will get now older than the other!
This means opposite contradictory results.
Now we will consider the problem as seen by the twins themselves. they see each other travelling at a velocity w (classicaly is 2v but with the relativistic addition of velocities is something different) chousing the directions of the referentials as the directions of the relative velocity.
For them we must consider k = (1-w2/c2)exp-1/2
Then for both twins we will have the same:
t = t'/k
This means that for each one the other twin is getting younger than himself.
This means also opposite contradictory results.
We must also note that the rate of aging is different as seen in the mothership than seen by the twins.
We must pay attention that they have made a perfect symetrical travel, they accelerated the same amount, they traveled at the same velocity for the same time so there's no privileged direction in the experiment to decide for one of the cases.
Then three inconsistences were found because of contradictory results.
NOTE:
We can also note with some surprise that the results for the mother-ship referential is not the same aging as we would expect in this totally symmetric problem.