THE TRANSFORMATIONS OF LORENTZ “AB OMNI NAEVO VINDICATAE”

The arguments presented here relate to the so-called “TWIN PARADOX”, and allow its resolution within the Theory of Special Relativity, only by properly interpreting the Lorentz Transformations. Readers may wonder who the publication of this article is dedicated to, obviously the Lorentz Transformations and their History, starting from when they were conceived until now, because they still continue to represent the essence of “flat” space-time today! And why, nicely, “ab omni naevo vindicatae”?

Whenever we discuss the twin paradox (also called paradox of watches), we inevitably refer to an inertial reference frame (privilegied frame) and to a second not inertial frame(not privilegied frame), and one almost always ends up recalling the Theory of General Relativity or the relativistic Doppler effect; I prefer “to avenge” the Lorentz Transformations to remove this “neo ” that they unfairly carry with them.

Let’s now deal with the well-known paradox: there are two twins on Earth, one of the two decides to leave (towards a distant planet for example and at high speed with a spaceship, to ensure that the phenomenon of time dilation is more evident) and on his return he will be younger than his brother left on Earth.

But if we consider the astronaut twin stationary, why can’t it be assumed that the opposite happens, that is, that the twin left on Earth is younger than the traveling brother? Lorentz transformations are symmetrical, it seems that this contradictory situation is possible.

Keep in mind that, according to the spaceship’s frame, the whole Universe is in motion with respect to it at an opposite speed and, since the moving distances contract (they are in fact smaller as indicated by the Lorentz transformations ), the contracted Earth-planet distance is covered and not the actual one according to the Earth’s reference frame.

(IMPORTANT, IN THIS CASE IT IS NOT ONLY THE EARTH APPARENTLY MOVING, BUT THE WHOLE UNIVERSE)

When, on the other hand, we believe (rightly) that it is the astronaut twin who moves, it is a different situation, because there it is only the spaceship that contracts with respect to the twin remaining on Earth. (the Earth-planet distance is instead fixed and does not change)

(IMPORTANT, IN THIS CASE ONLY THE SPACESHIP IS MOVING)

It is also essential to emphasize that the travel time to reach the planet, if measured with respect to the spaceship’s reference frame, is equal to the travel time of “the apparent motion of the Earth” (again if measured with respect to the spaceship’s frame); the two motions are symmetrical to each other and for the astronaut twin “the apparent motion of the Earth” therefore does not occur with a travel time less than its own!

Although the astronaut is stationary in his own frame of reference, we can still imagine him in motion in the new Earth frame (where distances are contracted), this is what I mean when I refer to two symmetrical motions.

The two motions have in common the distance traveled (the contracted Earth-planet distance) and the same speed.
The times are the same, it can’t be otherwise!

(this is what Lorentz transformations tell us!)

However, it would be different if the astronaut remained stationary floating in Space and if the Earth (for some strange reason) only began to move (and then go back). Then yes that the twin left on Earth would be “younger” than his brother on his return! If only the Earth were to move (whatever the length of the distance traveled), the Universe would not be seen contracted from the spaceship as seen previously, but only our planet!

In reality, when a body is in motion, it does not always proceed at a constant speed; for example, a spaceship first accelerates when it moves away from the Earth and naturally decelerates in the vicinity of the planet where it arrives before reversing its course (to later re-accelerate) and decelerates again when it returns to Earth, in this case integrals must be considered to determine the proper travel times (which I learned from reading Sir Roger Penrose), but in this publication I have preferred to avoid mathematical details, anticipating however that in the next article I will publish it will be clarified how such proper times can be estimated.

Sir Roger Penrose himself has rightly stated that a body can accelerate both in Special Relativity and in General Relativity, the difference is only in the kind of metric that is used to calculate proper times!

Then when the speed of the spaceship decreases until it disappears, the distances traveled (that traveled by the spaceship with respect to its frame and that traveled by the Earth always with respect to the frame of the spaceship) expand (again due to the Lorentz Transformations) and it will appear that these two distances (coincident) are both equal to double the Earth-planet distance, just as if measured with respect to the Earth’s frame. On the other hand, when the astronaut twin ends the journey, he is also in the frame of the Earth and each length will have the same measurement for both twins.

It is also obvious that, if the astronaut twin crosses areas where gravitational fields are particularly intense, he will be much “younger” than his brother on his return, but I repeat that the “twin paradox” can be solved within the Special Relativity since, ideally, we can always consider two point-like bodies of negligible mass in relative motion. (regardless of the gravitational forces acting on massive bodies)

THE TWIN PARADOX IS NOT A “TRUE PARADOX” IF THE OBSERVER WHO HAS TO DECIDE WHICH TWIN IS YOUNGER BELONGS TO THE EARTH’S FRAME (AS IT HAS ALWAYS BEEN CONFIRMED BY ALL THE EXPERIMENTS CARRIED OUT). WHY SHOULD THE TWIN REMAINING ON EARTH BE YOUNGER? LET’S CHANGE THE WAY OF THINKING!

With this, I say that the problem is solvable in the frame of the Earth and we do not worry about the measurements made in other frames!

The Lorentz transformations, never contradicted as the postulate relating to the constancy of the speed of light has so far always been verified, are self-sufficient in a “flat” space-time.

From what I know and surfing the Internet, I have never read considerations related to the twin paradox similar to mine (and I apologize in advance if the approach I proposed has already been indicated by others). Finally, I thank all those who read the article and, if comments follow, I will make myself available for any clarifications, also illustrating in more detail in future publications the interesting mathematical details that I have left out here. (from the correct application of the Lorentz transformations, to the further considerations on the apparent motion of the Earth)

Massimiliano Dell’Aguzzo