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Henri Bergson: Half-relativity – Lorentz Equations (9) – How light has the same speed for all observers

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We have just looked for how light could have the same speed for the fixed observer and for the moving observer: the deepening of this point revealed to us that a system S’, resulting from the splitting of a system S and moving in a straight line with a speed v, underwent singular modifications. We would formulate them like this:

1° All the lengths of S’ have contracted in the direction of its movement. The new length is the old one in the ratio of √(1 – v2/c2) to unity.

2° The time of the system has expanded. The new second is the old one in the ratio of unit to √(1 – v2/c2)

3° What was simultaneity in system S has generally become succession in system S’. Only those events, contemporaneous in S, which are located in the same plane perpendicular to the direction of movement remain contemporaneous in S’. Any two other events, contemporary in S, are separated in S’ lv/c2 seconds of the system S’, if we designate by l their distance counted on the direction of movement of their system, that is to say the distance between the two planes, perpendicular to this direction, which pass through each of them respectively.

In short, the system S, considered in space and time, is a double of the system S which has contracted, as far as space is concerned, in the direction of its movement; which has dilated, as to time, each of its seconds; and which finally, in time, has successively dislocated all simultaneity between two events whose distance has shrunk in space. But these changes escape the observer who is part of the mobile system. Only the fixed observer notices it.

I suppose then that these two observers, Peter and Paul, could communicate together. Pierre, who knows what to expect, would say to Paul: “The moment you broke away from me, your system flattened, your time swelled, your clocks went out of tune. Here are the correction formulas that will help you get back to the truth. It’s up to you to see what you should do with it.” It is obvious that Paul would respond: “I will do nothing, because, practically and scientifically, everything would become incoherent within my system. Lengths have become shorter, you say? But the same is true of the meter that I carry on them; and as the measure of these lengths, within my system, is their ratio to the meter thus displaced, this measure must remain what it was. Time, you still say, has expanded, and you count more than a second where my clocks mark barely one? But if we suppose that S and S’ are two copies of the planet Earth, the second of S’, like that of S, is by definition a certain determined fraction of the planet’s rotation time; and although they do not have the same duration, they are only a second long. Have simultaneities become successions? Clocks located at points H’1, H’2, H’3 all three indicate the same time even though there are three different times? But, at the different moments where they mark the same time in my system, events happen at the points H’1, H’2, H’3 of my system which, in the system S, were legitimately marked as contemporaneous: I will then agree to call them still contemporaneous, so as not to have to consider in a new way the relationships of these events with each other first, and then with all the others. By this I will preserve all your consequences, all your relationships, all your explanations. By calling succession what I called simultaneity, I would have an incoherent world, or built on a plan absolutely different from yours. Thus all things and all relationships between things will retain their grandeur, will remain within the same frameworks, will fall within the same laws. I can therefore act as if none of my lengths had shrunk, as if my time had not expanded, as if my clocks were in agreement. This is at least what concerns ponderable matter, that which I carry with me in the movement of my system: profound changes have been accomplished in the temporal and spatial relationships that its parts maintain among themselves, but I do not care about them. I don’t notice it and I don’t have to notice it.

Source: Henri Bergson, Durée et simultanéité : à propos de la théorie d’Einstein, Deuxième édition, qugmentée, Paris, 1923. Translation and interpretation Nicolae Sfetcu. © 2024 MultiMedia Publishing

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