-v * t_1 = -d (for any distance d and for t_1 < t) is a contradiction, because it means that the Earth DOES NOT MOVE with uniform rectilinear motion at speed -v.

Come to think of it, the Earth

at time t_1 = d / (v * gamma) reaches the second spaceship (x_1 = -d), the motion of the Earth in the frame of the spaceship is therefore represented by x_1 = - gamma * v * t_1.

The distance of the spaceship in the frame of the Earth is x = v * t, and in the frame of the spaceship the Earth is at distance d = gamma * v * t_1.

(d = v * t)

In the frame of the Earth the elapsed time is

t = gamma * t_1, the preceding contradiction therefore means that, even if the Earth is moving with respect to the spaceship, all the clocks of the Earth's frame DO NOT SLOW DOWN with respect to the clock of the astronaut twin.

THE CONTRADICTION NOW HAS A MEANING.

Condividiamo insieme la conoscenza!