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 S and S_1 measured at the instant of time t_1 by an observer in the frame S_1, it has the following value:
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 O of S with respect to the frame S_1, d_1m(t_1) is a different value that also depends on v_1(t_1). The distance between the two origins O and O_1 is instead represented at time t by x(t) in the frame 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 S; in the frame S_1 it is not a unique measure, in fact it depends on the speed value v_1(t_1). In the frame S_1, the measurement of the entire distance to be traveled depends on the speed v_1(t_1) and is equal to: 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 d -x (t) remains to travel for the observers in the frame S, while for the observers in the frame S_1 there remains a distance to travel which, if measured instantly of time t_1, that is d_1(t_1) -d_1m(t_1), which we can also indicate with 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” x_1(t_1) certainly have their importance, but everything depends on future speeds. (there are “no particular surprises” only if you proceed at a constant speed)
The estimate of the measure of proper times
The time t_1 is always less than the time t (a speed is always associated with the body) and, if in certain instants of time the speed is high, in this cases the differentials dt_1 are considerably smaller than their respective dt (since dt_1 = γ^(-1) (t)*dt); the difference between dt and dt_1 is instead irrelevant when the body has just departed from the origin O (with zero initial speed) and when it is about to arrive at its destination at the abscissa d point (decreasing its speed until it stops). Obviously the difference between dt and dt_1 is minimal if the body moves at low speed during its journey and it is dt = dt_1 if the body stops during the journey for possible stops.
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, γ^(-1) (t) is the inverse of the Lorentz factor and, if the velocity function v(t) does not have “a simple trend”, the function γ^(-1) (t) will also be difficult to integrate.
However, it is not difficult to verify that, if the function v(t) in the time interval (t_a, t_b) in S is an increasing (or decreasing) function, the measure of time t_1_b -t_1_a in S_1 can be estimated, in this case we can help with the mathematical average of the values γ^(-1) (t_a) and γ^(-1) (t_b), as explained in more detail in the article “THE TWIN PARADOX” (TRAVEL EXAMPLE).
Indicating with m this mean, there exists an increasing or decreasing function v(t) for which it occurs that (t_1_b -t_1_a) = m * (t_b -t_a).
Dear readers, thank you again for reading my remarks on the “phenomenon of time dilation” and for being interested in Special Relativity. (unique theory, elegant and so “simply beautiful”)