# THE TWINS ALEX AND BRAD

Let ** Alex** and

**be twins (both in their thirties, located in the same position and both stationary relative to each other);**

*Brad***decides to leave with his spaceship covering a distance**

*Brad***(to then turn back, always moving at the same speed**

*d***), distance**

*v***is represented by one segment, segment**

*d***. (initially**

*AB***and**

*Alex***are positioned near point**

*Brad***and the distance**

*A***is the measure of the length to travel to get to a distant planet, the same value for both)**

*d**How long is the journey for** Alex?*

** Alex** will have to wait a time equal to

**to see his brother again. (we are simply considering a round trip)**

*2*d/v**The journey, “according to **Brad**”.*

Compared to** Brad**, everything else moves with respect to him in the opposite direction at a speed

**. (for**

*v***also the**

*Brad***segment that was previously stationary is now in motion; it is also remembered that, while for**

*AB***everything else is moving, for**

*Brad***only**

*Alex***is in motion and this is important to reach the conclusion)**

*Brad*As well indicated by ** the Lorentz transformations,** the moving distance

**has a shorter length for**

*AB***and measures**

*Brad*

*d_1. (d_1 < d)*If we consider

*“*

*Brad**stationary”*, at first he “waits” for the second extreme

**of segment**

*B***to meet him at speed**

*AB***(this happens when a time interval equal to**

*v***passes, that is when**

*d_1/v***“arrives at his destination”), then**

*Brad***“waits” for the first extreme**

*Brad***of segment**

*A***to meet him so that the journey ends. (and this still happens when a time equal to**

*AB***passes, that is when**

*d_1/v***returns to**

*Brad***,**

*Alex***is the twin who has always stood stationary waiting for his brother near point**

*Alex***)**

*A*In conclusion, less time (equal to

*2*d_1/v**)*has passed for

**.**

*Brad*

*(d_1 < d, 2*d_1/v < 2*d/v)*To simplify the calculations, let’s imagine that the speed is equal to ** 0.866 c** (about

**of the speed of light, in this case the distance**

*90%***in motion according to**

*AB***measures**

*Brad***;**

*d_1 = 0.5*d***is half of**

*d_1***and also the elapsed time for**

*d***is half the elapsed time compared to**

*Brad***); again to simplify the numerical results we can also choose the value of**

*Alex***so that the elapsed time for Alex**

*d***is equal to**

*(2*d/v)*

*20 years**.*(in this case the travel time according to

**is only**

*Brad***)**

*10 years*When the twin brothers meet again, ** Alex** is fifty and

**is forty. (For**

*Brad***the trip lasted**

*Alex*

*20 years**,*while for

**the trip only lasted**

*Brad***, remember that before the trip both twins were in their thirties)**

*10 years**Conclusions*

It is usually believed that ** Alex** knows

*Special Relativity*and expects

**to be younger when he returns, but strangely instead (in my opinion) it is also believed that**

*Brad***clock should also slow down compared to**

*Alex’s***clock. When I wrote in my previous articles that in**

*Brad’s***frame there is no time difference I meant that there is obviously no time difference between**

*Brad’s***apparent motion and**

*Alex’s***motion in the Earth’s frame where distances are contracted.**

*Brad’s**Each of the twins (not only **Brad**, but also **Alex** if he does not know the Lorentz Transformations) may think (wrongly) that the other brother is the same age.*

And, regardless of what each twin thinks of the other, for the first ** (Alex)** a time equal to

**has passed and for the second**

*2*d/v***a shorter time equal to**

*(Brad)***has passed, the time intervals are different and only**

*2*d_1/v*

*Alex**,*if he knows

**has the right to claim that his brother is younger. (**

*the Lorentz Transformations,***can’t do that)**

*Brad*However, the solution is the one that has been presented again now (and deepened in some of my previous articles in the case of non-constant speed, where it was necessary to resort to differential calculus):

*the twin who has traveled, when he returns to his initial position, is younger than the brother who has remained waiting and the Lorentz Transformations allow us to verify this.*

*Massimiliano Dell’Aguzzo*