On The Twin Paradox The Symmetry Remains

Question

Please, my question is at the end of this formulated problem.

In the case of the twin paradox, the travel can be made without never accelerating higher than $g$. So the one who remains on the earth will always be subject to $g$, while the traveler can be under $g$ only for some part of the trip for example:

• First he leaves earth, after overcoming $g$, and keeps accelerating at $g$ for 180 days, where he reaches around $0.508 c$;

• Then the acceleration is removed and he travels at $0.508 c$ away from earth for 5 years;

• After this, G is applied in the opposite direction for 180 days so that its speed in relation to earth slows down to 0 m/s;

• When the relative speed to earth is null, the traveler spends 1 day slowly rotating 180 degrees, so that he points back to earth;

• Then he starts accelerating at G for 180 day again;

• After that G is removed and he travels at constant speed of 0.508 C for 5 years towards the earth;

• At the end of this 5 years a contrary G force is applied for 180 days, so that he can slow down and return to earth;

• Now he traveled at constant speed for 10 year, and during this period of inertial trip, his time suffered dilation of 1.16 as supposed by the observer on earth. But he also supposes that the time on earth was extended.

• During this 10 years + 720 days + 1 day he was subject to G for 720 day (Never more than G);

• Meanwhile his brother on earth was subject to G during the whole time;

Who really suffered time dilation and why? What causes time dilatation, speed or acceleration?

Answer 1

You need to use the correct relativistic formulae to calculate these things. Please see The Relativistic Rocket.

It looks like your value of 0.508c after 180 days (ship time) comes from the Newtonian formula $v=at$. The relativistic value is 0.468954c. And 180 days ship time corresponds to almost 187 days, 20 hours, 46 minutes Earth time. If you then cruise at a constant speed of 0.468954c you can use the Lorentz factor $\gamma\approx1.132217$ to calculate the time dilation.

Who really suffered time dilatation and why?

As per usual in the twin paradox, when the traveller gets back to Earth, the Earthbound twin will be older.

What causes time dilatation, speed or acceleration?

In Special Relativity, speed causes time dilation, but with constant speed the situation is symmetrical. If observers A & B have a constant relative velocity then A measures B's clock to be running slow by a factor of $\gamma$, and B measures A's clock to be running slow by a factor of $\gamma$.

To break the symmetry, (at least) one of the observers needs to make one or more changes of reference frame. It's not so much that the acceleration causes time dilation, it's merely the mechanism whereby the reference frame is changed.

Answer 2

Who really suffered time dilation and why? What causes time dilatation, speed or acceleration?

The twin at earth will be older. The reason can be understood by the theorem mentioned in this question.

The twin at earth followed a straight line in spacetime, that is the t-axis between the 2 events: departing and returning of the other twin.

On the other hand, if the path of the traveller twin is plotted in the time x space graph, it consist of several steps between the same initial and end points.

It is only possible to deviate from a straight line with some acceleration. So we could say that it is a necessary condition for the mismatch of clocks at the end point.

But we can not use the equivalence principle in this case, and compare $g$ acting on the twin at earth with $g$ acting inside the rocket on the traveller twin. The EP is valid only locally, and that includes a short $\Delta t$. Ex: both can throw an object in the air and observe the same accelerated path for some seconds until falling on the ground.

But if the earth twin throws on object with 12 km/s it will not return (supposing no air drag). But it will return for the twin in the rocket in less than one hour. So the artificial gravity there is not comparable with the earth's one.