Let's imagine a spaceship moving in the frame of the Earth with uniform rectilinear motion at speed ** v**, we know that the clock of the astronaut twin slows down compared to all the clocks of the frame of the Earth.

*(t_1 = t / gamma)*In this case it's not ** x_1 = - v * t_1** , the speed of the Earth in the frame of the spaceship is

** - gamma * v**. (The motion of the Earth is in advance of the uniform rectilinear motion

**)**

*x_1 = - v * t_1*In both frames the distance between the twins is the same:

*gamma * v * t_1 = v * t*

The astronaut twin may also mistakenly think that the Earth's clock slows down relative to the clock of the spaceship frame.

The astronaut twin could therefore assume that the elapsed time in the Earth's frame

** t_wrong** is equal to

**.**

*t_1 / gamma*The spaceship would then move at speed

*gamma * gamma * v.*

In both frames the distance between twin is the same:

*v * gamma * gamma * t_wrong = gamma * v * t_1*

But, if we know that the actual speed of the spaceship is ** v**, then

*t = gamma * gamma * t_wrong,*

*t = gamma * gamma * (t_1 / gamma)*

*t = gamma * t_1*

*t_1 = t / gamma.*

Twins can be the same age even though they are in relative motion with respect to each other.

Imagine two astronaut twins, one twin going left and the other heading right of the Earth (at the same speed *v***(v and - v)**relative to the Earth's frame)

Each twin "sees" the other in motion, but neither is younger than the other.

*Why always apply time dilation?*