If the astronaut twin launches a rocket at the initial time (t = t_1 = 0 s) at a speed -v (in the direction in which the astronaut twin sees the Earth moving away), then the rocket is younger than the astronaut twin.

The clock of the astronaut twin slows down compared to the Earth's clock, and the rocket's clock slows down compared to the clock of the astronaut twin.

If we denote by (x_R, t_R) every event in the frame of the rocket, the two Lorentz transformations that describe the situation are:

x_1 = gamma * (x_R - v* t_R)

x_R = gamma * (x_1 + v * t_1)

The rocket moves with uniform rectilinear motion in the frame of the spaceship, for the rocket it is

x_1 = - v * t_1.

From x_1 = - v * t_1,

x_R = 0,

- v * t_1 = - gamma * v * t_R,

t_R = t_1 / gamma.

(the clock of the rocket slows down compared to the clock of the astronaut twin)

The rocket and the Earth do not occupy the same position at time t_1 in the frame of the spaceship; the motion of the Earth in the frame of the spaceship is represented by x_1 = -gamma * v * t_1, while the motion of the rocket in the frame of the spaceship is represented by x_1 = -v * t_1

If the astronaut twin launches a second rocket R_2 at the initial time (t = t_1 = 0 s) at a speed -gamma * v (in the direction in which the astronaut twin sees the Earth moving away), then the rocket R_2 and the Earth occupy the same position in the frame of the spaceship. (while the time t_1 flows in the frame of the spaceship)

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