# THE INFINITESIMAL TRANSFORMATIONS OF LORENTZ

# (in the case of a moving body)

Let us consider two frames ** S** and

**in relative motion to each other, which coincide when the times**

*S_1***and**

*t***are equal to**

*t_1***(i.e. the instants of time in which the frames are coincident constitute the same initial time instant for both) and with the**

*0***axis and**

*x***axis always superimposed.**

*x_1*In particular, reference is always made to the well-known case in which the origin ** O_1** of the frame

**moves away from the origin**

*S_1***of the frame**

*O***. (but with a velocity**

*S***which, even if it has only a horizontal component along the abscissa axis**

*v***, is not always constant)**

*x*Now let’s imagine that in the origin of the frame ** S_1** a body is positioned which, starting from the origin of the frame

**(at the zero initial time**

*S***and at zero initial speed), arrives at a point of the**

*t = 0***axis of positive abscissa**

*x***with respect to the frame**

*d***.**

*S*For the frame ** S**, the frame

**at time**

*S_1***is in motion at speed**

*t***while, for the frame**

*v(t)***, the frame**

*S_1***S**at time

**is in motion at speed**

*t_1***.**

*v_1(t_1)*We thus obtain that

**(in modulo and for**

*v(t) = v_1(t_1)***which corresponds to the travel time measured in**

*t_1***S_1**when for the frame

**the measured travel time is**

*S***) and the two Lorentz factors**

*t***and**

*γ(t)***coincide.**

*γ(t_1)*From the Lorentz transformation ** x_1 = γ*(x -v*t)**, for

**as the body is positioned in the origin of the frame**

*x_1=0***, we obtain**

*S_1***, from which**

*x -v*t=0***. Turning to the differentials we obtain the relation**

*x=v*t***.**

*dx = v(t)*dt*From the other Lorentz transformation

**, again for**

*x = γ*(x_1+v*t_1)***, we verify that**

*x_1=0***and passing to the differentials we obtain the following relation**

*x = γ*v*t_1***.**

*dx = γ(t_1)*v _1(t_1)*dt_1*The last equation is correct, in fact in the meantime ** dx_1 = v_1(t_1)*dt_1**; for the frame

**the infinitesimal distance**

*S_1***is in motion at speed**

*dx***and, in the frame**

*v_1(t_1)***, it is less than**

*S_1***and measures**

*dx***. From**

*dx/γ (t_1)***and from**

*dx_1=v_1(t_1)*dt_1***we get**

*dx_1 = dx/γ(t_1)***.**

*dx = γ(t_1)*v_1 (t_1)*dt_1*From

**we then pass to**

*dx = γ (t_1)*v_1(t_1)*dt_1*** v(t)*dt = γ(t_1)*v_1(t_1)*dt_1**; finally it is possible to obtain the known result

**, and this is true since**

*dt_1 = γ^(-1)(t)*dt***and**

*v(t) = v_1 (t_1)***.**

*γ (t)=γ (t_1)*Since ** dt_1 < dt**, proceeding to integrate we also obtain

**. (the body in motion is “younger” than all the other stationary bodies in the frame**

*t_1 < t***)**

*S*** dx_1** is less than

**, also**

*dx***and this is what causes the symmetry to break.**

*x_1(t_1) < x(t)**The infinitesimal time **dt** is the time of the body to travel the infinitesimal distance **dx** as measured in the frame **S**, the infinitesimal time **dt_1** instead represents the time for the distance **dx_1** (**dx** in motion) “overtakes” the origin of the frame **S_1**. In the frame **S_1**, **dt_1** therefore corresponds to the infinitesimal travel time of the body which, even if stationary in its frame, waits for the infinitesimal distance **dx_1** to meet it with the opposite speed to its own.*

For accuracy, if the abscissa ** d** is positive, in the two Lorentz transformations examined the two speeds

**and**

*v(t)***are also to be considered as positive values, even the exact expression of the differential**

*v_1(t_1)***is**

*dx_1*** dx_1 = -v_1(t_1)*dt_1.** In the demonstration shown

**an infinitesimal distance (and therefore positive) has been considered, also pay attention to the signs of the various quantities in case the body goes back.**

*dx_1**It should also be considered that if the body crosses areas where gravitational fields are particularly intense, it will be even “younger” than all the bodies that remain stationary in the frame** S**, but ideally it is always possible to consider a point-like body of mass negligible in motion (regardless of the gravitational forces acting on massive bodies).*

As regards the distance between the two origins of the frames ** S** and

**measured at the instant of time**

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

*t_1***, it, indicated with**

*S_1***, at the instant of time**

*d_1m (t_1)***is equal to**

*t_1***or**

*x(t)*γ^(-1)(t_1)*

*d_1m (t_1) = x (t)*γ^(-1) (t).*The last relation is important as it also means that, in the case where the speed of the body is equal to zero, ** d_1m(t_1)= x(t).** When the body is stationary in the frame

**all distances have the same measure for all observers of the frame itself. The now stationary body will therefore seem to have traveled a distance equal to**

*S***, although this does not correspond to the actual distance traveled from the origin of the frame**

*d***with respect to the origin of the frame**

*S***.**

*S_1*In summary, the useful relationships are:

*dx = v(t)*dt*

*dt_1 = γ^(-1)(t)*dt*

*dx = γ(t_1)*v_1 (t_1)*dt_1*

*dx_1 = dx/γ(t_1)= dx/γ(t) = v_1(t_1)*dt_1*

*d_1m(t_1) = x(t)*γ^(-1) (t) (the only one in which there are no differentials).*

This discussion also applies in the event that the body goes back along a further distance ** d** to return to the origin of the frame

**with zero final speed. (situation that represents the so-called**

*S***)**

*“twin paradox”*Although the relation ** dx = v(t)*dt** is trivial at first sight, it is important because it states that there is a body in motion within the frame

**. (for example the motion of a body in the Earth’s frame or better in the frame of the fixed stars).**

*S*

*If a body is moving within a frame S and travels a distance d, in the body’s frame S_1 it is not only the origin O that moves.(but it is the whole frame S that “travels” a distance less than d)**Remember that the frame S in which the movement takes place is not a privileged frame, it is the frame in which the body at the end of the journey is “younger” than the other bodies that have always remained stationary in the frame S.*

*And, as has been shown in this article, the Lorentz transformations are sufficient to prove this simply and elegantly.*

*Massimiliano Dell’Aguzzo*