Regarding the twin paradox, I propose an example of a journey that I will describe in this article. (continuing the treatment already discussed previously in the “TRANSFORMATIONS OF LORENTZ AB OMNI NAEVO VINDICATAE”)
Let’s imagine that one of the two brothers in the spaceship manages to reach the speed v = 0.866*c (about 90% of the speed of light). Someone will think that this is impossible to achieve, but if this value has been chosen it is only because to me it is more congenial in the calculations that will be shown. (and in any case nothing prevents repeating the proposed procedure with a value lower than the speed reached by the spaceship)
When the spaceship proceeds at this speed, since the inverse of the Lorentz factor (which I will indicate with a) is in this case a = 0.5, the time measured with respect to the spaceship’s frame (also called proper time) it is half of the time elapsed with respect to the Earth’s frame; at this speed the astronaut will also evaluate the length of the distance to be traveled as if it were the respective half of that measured by observers of the Earth’s frame.
Considering that the astronaut twin arrives to a distant planet (and then returns to Earth), we know very well that motion can’t occur at a constant speed. On the outward journey, for example, the spaceship accelerates as it moves away from the Earth, then eventually proceeds at a cruising speed (which for us is v = 0.866*c) and finally decelerates reaching the planet. (to then change course, and possibly equipped with engines that allow it not to change its position)
If the speed is not constant, the proper time in case of acceleration (the travel time measured in the frame of the spaceship which I will indicate with t_astr_acc) can be estimated anyway avoiding the resolution of not simple integrals.
By indicating with delta_t the time in which the spaceship accelerates in Earth’s frame, the time t_astr_acc will certainly be a value greater than 0.5*delta_t (this in fact is the extreme case according to which the astronaut twin is always traveling at speed v = 0.866*c) and less than delta_t itself (the second limiting case according to which the phenomenon of time dilation does not occur at zero speed); we consider our function of the increasing speed during acceleration in such a way as to obtain the average, that is, t_astr_acc = 0.75*delta_t. (it is however possible to choose another value between 0.5 and 1, reasonably closer to 1 in order to describe a situation that can actually be realized)
During the deceleration phase (to arrive on the planet), it is convenient to choose the decreasing function of the “symmetrical” speed at the departure speed, so that the spaceship assumes the decreasing speed values as opposed to how they were assumed in the initial phase away from the Earth.
It should also be assumed that the return journey of the astronaut twin is identical (from a temporal point of view) to the outward journey, in this case the speeds on the return journey are opposite to the speeds assumed in the first part of the journey; however, remember that the treatment is the same as the Lorentz factor refers to the square of the speed.
To continue, it is possible to divide the spaceship’s journey into two phases at constant speed and four accelerated phases (two of acceleration and two of deceleration). If each time you accelerate (or decelerate) the time duration is 4 years with respect to the Earth’s frame, only 3 years pass for the astronaut (3 = 0.75*4) and, if for each time you it proceeds at cruising speed v = 0.866*c the time duration is 10 years with respect to the Earth’s frame, while for the astronaut only 5 years pass. (5 = 0.5*10)
In conclusion, when the spaceship returns to Earth, 22 years will have elapsed for the astronaut twin (4*3 + 2*5), while 36 years will have elapsed for the twin left to wait on Earth (4*4 + 2*10); on his return, the traveler twin is therefore “younger” than his brother!
There is no paradox, if in fact we consider the events from the point of view of the traveling twin, in the spaceship’s frame the time taken by the Earth-planet distance in its “apparent” outward and return movement is always 22 years. (and not different as one might mistakenly believe); distances in motion are smaller than as measured by an observer in the spaceship’s frame, just as indicated by the Lorentz Transformations. If then the value of the speed with which one travels increases (without ever exceeding the value of the speed of light c), the effects due to the contraction of the lengths will become more and more pronounced!