II.c.2.a) Pythagoras’ theorem and Einstein’s Relativity
The Theory of Special Relativity is associated with great mathematical complexity, but I believe the complexity is conceptual rather than mathematical, as it derives from the application of Pythagoras’ theorem.
One must not forget that we express concepts by means of words, and our brain has some meanings extremely ingrained, as they are very basic in a vital sense. Words like space and time are preconceptions recorded deep in our brain.
Of course, at the same time, not only references to space are relative, but space itself is too.
To top it off, it was necessary to add an explanation with the idea of the relativity of perception of time by living beings, including twins; or alternatively, the subjective relativity of time. Even love appeared in the middle of it, I suppose that it was more convincing. Who would dare deny that…?
Because this perception or subjective reality does indeed exist, they ended up accepting a scientific model, which states that if two objects are moving away from each other at the speed of light, the speed at which they are separating will still be the speed of light, as in the experiment of the antipodal photons.
An additional element is that everything is relative; and when it is convenient, because something does not quite fit, one can say “Well, actually the matter is a lot more complex, but we were simplifying it implicitly, for… you.”
Anyway, if required they simply go back to tensors in the formulae of General Relativity and… lights out!
Why they do not explain that relativity of time means an asymptotic conversion of the speed of light so that it cannot exceed c and that it simply deduces from Pythagoras’ theorem? Alternatively, even simpler, that the reason for the folding and unfolding of time is inverse of cosine of the corresponding sides of the rectangular triangle. I am referring to the mathematical reason or proportion, not logical reason, unless the former implies the latter.
In case one consider relativistic physics complex, let us do an exercise and try to imagine how it must have been discovered in its time, and what Pythagoras’ theorem involves of (assuming that they knew of the postal envelope and a bit of mechanics).
The trick entails thinking of the envelope as open and closed simultaneously, as if we were dealing with a quantum envelope. We look at the geometric curvature of the flap when it folds over inside the big square B (side=b), unfolds, expands or comes out and forms the small square A (side=a)
One can easily observe that the area of B is double the area of A. Then, as the area of B is b² and that of A is a², we find that [b² = a² + a²], and when we take the square root of this, we obtain Pythagoras’ Theorem.
Area of A = a² = b² / 2
When teaching children how to find the area of a square knowing its diagonal, they usually say you must calculate the length of the side using Pythagoras’ theorem and then, square it, instead of telling you that it is equal to the diagonal squared divided by two.
A concrete application of Pythagoras’ Theorem to Special Relativity is in the figure of the thought experiment Unreal or contradictory hypotheses.
This thought experiment shows the rectangular triangle that comes from different perspectives of two observers, plus the implicit idea that light conserves the inertia of the spacecraft, but only a hypothetical observer perceives this…
The figure is similar to the aforementioned assumption when talking about the element of the relativity of time; from where it was deduced that the temporal difference could be easily calculated:
t = t0 /(1 - v²/c²)½
Effectively, a small calculation based on the modern Pythagoras’ theorem, where the legs and the hypotenuse are distances travelled by light and by the object in relative motion, seen from different imaginary observers and conveniently mixed, gives us the result previously shown.
In order to help assimilate the above, I show the following equations, which allow to get an immediate idea of where this Pythagorean time is going. In addition, bearing in mind the difficulty in its recognition, and because it has terrified the little neurons of half the world, it could be a phantasmagorical time.
The idea is to normalize the hypotenuse of the triangle as c or the speed of light. If the velocity u and c are identical –because they are those of light– whilst v is that of the spacecraft, we will find that the dilation of time must be proportional to the inverse of the cosine of the angle α.
Analytical deduction Pythagoras’ Theorem u² = c² - v² Normalization c² u² / c² = 1 - v²/c² Square root and we get Cos α = u / c = (1 - v²/c²)½ Find c c = u * (1 - v²/c²)-½ Substitute auxiliary Lorentz
c = u * γ
If light incorporates the inertia for an observer, we would have to conclude that he would think that we had the typical case of inertial systems with additive velocities, unless it was an observer of the zigzag but unaware of its meaning.
As we will see further on, he would not be far off from the vision of reality that consciously is proposed!
Subsequently, the Lorentz transformations are perfect in order to ward off the ghost of Pythagoras. However, note the similarities of the two forms that the cos a takes with the two auxiliary constants of the equations.
You could say that Pythagoras’ theorem is a particular application of the specific case of Thales’ theorem when a straight angle exists.
It is also widely known that the fundamental theorem in trigonometry, sine squared plus cosine squared equal to one, is an elemental implication of Pythagoras’ theorem. Both the quantifications of sine and cosine are by definition by considering the hypotenuse equal to unity. In other words, they would mean the number of hypotenuses in the adjacent leg or opposite leg to the angle in question.
At least travelling back in time is not possible. Thank goodness, because it would be a supreme act of boldness. What they do not explain very well is how –after a lapse of relative time– one goes back to normal time. I suspect we might have to tug on the tensions of General Relativity!
The cooktop that could appear with time games is an apotheosis. Ovens that are simultaneous for intelligent observers but that are asynchronous for the other observers, stretching distances, geometric effects stimulating the imagination, etc.