Henri Bergson: Half-relativity – Michelson-Morley experiment (2)

Let us calculate, in fact, according to the usual kinematics, the duration of each of the double paths. In view of simplifying the exposition, we will admit that the direction SA of the luminous ray has been chosen so as to be that of the movement of the earth itself through the ether. We will call v the speed of the Earth, c the speed of light, l the common length of the two lines OA and OB. The speed of light relative to the apparatus, in the path of O to A, will be c-v. It will be c+v on the way back. The time taken by the light to go from O to A and to return will therefore be l/(c – v) + l/(c + v), that is to say 2l/(c² – v²), and the path traveled by this ray in ether 2lc²/((c² – v²) or 2l/(1 – v²/c²).

Now consider the path of the ray which goes from the glass plate O to the mirror B and which comes back. The light moving from O towards B with the speed c, but on the other hand the apparatus moving with the speed v in the direction OA perpendicular to OB, the relative speed of light is here √(c² – v²), and therefore the duration of the total trip is 2l/√(c² – v²). This is what we would still see, without directly considering the composition of the velocities, in the following manner.

(Michelson-Morley experiment)

When the ray returns to the glass plate, this is at 0′, (figure), and it touched the mirror at the moment when it was at B’, the triangle OB’O’ being moreover obviously isosceles. Let us then lower from point B’, on line 00′, the perpendicular B’P. As the path of path OB’O’ took the same time as path OO’, we have OB’O’/c = OO’/v, that’s to say OB’/c = OP/v. As we also have OB’² = l² + OP² , we obtain, by transferring into this last equality the value of OP taken from the first: OB’ = lc/√(c² – v²).

The travel time of the line OB’O’ is therefore indeed 2l/√(c² – v²), and the distance actually traveled in the ether 2lc/√(c² – v²), or 2l/√(1 – v²/c²). This amounts to saying that the movement of the Earth in the ether affects the two paths differently and that, if a rotation imparted to the device brings the arms OA and OB of the device to be switched between them, a displacement of the interference fringes should be observed. However, nothing like that happens.

The experiment, repeated at different times of the year, for different velocities of the Earth with respect to ether, always gave the same result (1). Things happen as if the two double paths were equal, as if the speed of light relative to the Earth were constant, finally as if the Earth were immobile in the ether.

Note

(1) Moreover, it involves conditions of precision such that the difference between the two paths of light, if it existed, could not fail to manifest itself.

Source: Henri Bergson, Durée et simultanéité : à propos de la théorie d’Einstein, Deuxième édition, qugmentée, Paris, 1923. Translation and interpretation Nicolae Sfetcu. © 2023 MultiMedia Publishing