# INSIGHTS RELATING TO THE ARTICLE “THE INFINITESIMAL TRANSFORMATIONS OF LORENTZ” (in the case of a moving body)

*Distances in motion*

In the article it was shown that, if we denote by ** d_1m(t_1)** the distance between the two origins of the frames

**and**

*S***measured at the instant of time**

*S_1***by an observer in the frame**

*t_1***, it has the following value:**

*S_1*

*d_1m(t_1) = x(t)*γ^(-1) (t_1) or d_1m(t_1) = x (t)*γ^(-1) (t).*Regardless of the distance ** x_1(t_1)** “apparently traveled” from the origin

**of**

*O***with respect to the frame**

*S***,**

*S_1***is a different value that also depends on**

*d_1m(t_1)***.**

*v_1(t_1)***The distance between the two origins**

**and**

*O***is instead represented at time**

*O_1***by**

*t***in the frame**

*x(t)***.**

*S*** (d_1m(t_1) = x(t) **if

*v (t) = v_1(t_1) = 0)*As for the entire distance to be covered, it is always ** d** in the frame

**; in the frame**

*S***it is not a unique measure, in fact it depends on the speed value**

*S_1***In the frame**

*v_1(t_1).***, the measurement of the entire distance to be traveled depends on the speed**

*S_1***and is equal to:**

*v_1(t_1)*

*d_1(t_1) = d * γ^(-1) (t_1).**It is important to consider that in the frame*

*S_1**at the instant of time*

*t_1**it is not only the infinitesimal distance*

*dx**that moves with speed*

*v_1(t_1)**but the entire distance*

*d**.*

At the instant of time ** t**, the distance

**remains to travel for the observers in the frame**

*d -x (t)***, while for the observers in the frame**

*S***there remains a distance to travel which, if measured instantly of time**

*S_1***, that is**

*t_1***, which we can also indicate with**

*d_1(t_1) -d_1m(t_1)*

*d*γ^(-1) (t_1) -x (t)*γ^(-1) (t_1).*The measure of the distance still to be traveled ** d_1(t_1) -d_1m(t_1)** and the measure of the distance “apparently traveled”

**certainly have their importance, but everything depends on future speeds. (there are “no particular surprises” only if you proceed at a constant speed)**

*x_1(t_1)**The estimate of the measure of proper times*

The time ** t_1** is always less than the time

**(a speed is always associated with the body) and, if in certain instants of time the speed is high, in this cases the differentials**

*t***are considerably smaller than their respective**

*dt_1***(since**

*dt***); the difference between**

*dt_1 = γ^(-1) (t)*dt***and**

*dt***is instead irrelevant when the body has just departed from the origin**

*dt_1***(with zero initial speed) and when it is about to arrive at its destination at the abscissa**

*O***point (decreasing its speed until it stops). Obviously the difference between**

*d***and**

*dt***is minimal if the body moves at low speed during its journey and it is**

*dt_1***if the body stops during the journey for possible stops.**

*dt = dt_1*Furthermore, as regards the measurement of proper times and avoiding the resolution of not simple integrals, they can sometimes be estimated.

We know that ** dt_1 = γ^(-1) (t)*dt**,

**is the inverse of the Lorentz factor and, if the velocity function**

*γ^(-1) (t)***does not have “a simple trend”, the function**

*v(t)***will also be difficult to integrate.**

*γ^(-1) (t)*However, it is not difficult to verify that, if the function

**in the time interval**

*v(t)***in**

*(t_a, t_b)***is an increasing (or decreasing) function, the measure of time**

*S***in**

*t_1_b -t_1_a***can be estimated, in this case we can help with the mathematical average of the values **

*S_1***and**

*γ^(-1) (t_a)***, as explained in more detail in the article “THE TWIN PARADOX” (TRAVEL EXAMPLE).**

*γ^(-1) (t_b)*Indicating with ** m** this mean, there exists an increasing or decreasing function

**for which it occurs that**

*v(t)*

*(t_1_b -t_1_a) = m * (t_b -t_a).**Conclusions*

Dear readers, ** thank you** again for reading my remarks on the

*“phenomenon of time dilation”*and for being interested in

**. (unique theory, elegant and so**

*Special Relativity**“simply beautiful”*)

*Massimiliano Dell’Aguzzo*