# THE LORENTZ TRANSFORMATIONS AND FRAMES IN MOTION (THE TWIN PARADOX)

Let us consider two frames F (x, t) and F_1 (x_1, t_1), coinciding at the initial time t = t_1 = 0.

I consider the two Lorentz transformations:

a) x_1 = gamma * (x -v * t)

b) x = gamma * (x_1 + v * t_1)

With gamma I…

There are two twins, twin A and twin B. Let’s imagine that the twin A has to reach (in a spaceship) a star at a distance d from the Earth. The star belongs to the frame of the Earth.

# LA SOLUZIONE DEL PARADOSSO DEI GEMELLI

IL GEMELLO ASTRONAUTA E’ PIU’ GIOVANE

Il gemello astronauta si muove a velocità costante v nel sistema di riferimento della Terra. (per dirigersi verso una stella che appartiene al sistema di riferimento terrestre, la distanza Terra-stella è quindi una lunghezza ben definita in tale sistema di riferimento)

Nel sistema di…

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…

-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 CLOCK PARADOX AND THE LORENTZ TRANSFORMATIONS

Solving the CLOCK PARADOX means solving a system of two equations with four unknowns.

I consider the two Lorentz transformations:

a) x_1 = gamma * (x -v * t)

b) x = gamma * (x_1 + v * t_1)

With gamma I obviously indicated the Lorentz factor, and I do…

If x = v * t (the spaceship moves with uniform rectilinear motion in the frame of the Earth), then:

1) in the frame of the spaceship there are gamma copies of the Earth-star distance contracted moving (with uniform rectilinear motion) at speed -v,

the times t = d /…

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…

Solving the twin paradox means solving the system of two equations:

a) x_1 = gamma * (x - v * t)

b) x = gamma * (x_1 + v * t_1)

With gamma I obviously indicated the Lorentz factor, I do not consider the other two Lorentz transformations because they…

# THE TWIN PARADOX: FINAL CONCLUSIONS

Consider two frames F and F_1 in relative motion to each other with speed v, the frames F and F_1 are coincident at the initial time. (t = t_1 = 0 s)

You have to choose:

a) the frame F_1 moves with uniform rectilinear motion in the frame F at… ## Massimiliano Dell’Aguzzo

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