should not be surprised if such a change does not alter the form of the equations of Lorentz, obviously independent of the choice of axes.
We are thus led to consider a continuous group which we call the Lorentz group and which admit as infinitesimal transformations:
1° the transformation T0 which is permutable with all others;
2° the three transformations T1, T2, T3;
3° the three rotations [T1, T2], [T2, T3], [T3, T1].
Any transformation of this group can always be decomposed into a transformation of the form:
and a linear transformation which does not change the quadratic form
We can still generate our group in another way. Any transformation of the group may be regarded as a transformation of the form:
(1) |
preceded and followed by a suitable rotation.
But for our purposes, we should consider only a part of the transformations of this group; we must assume that l is a function of ε, and it is a question of choosing this function in such a way that this part of the group that I call P still forms a group.
Let's rotate the system 180° around the y-axis, we should find a transformation that will still belong to P. But this amounts to a sign change of x, x', z and z'; we find:
(2) |
So l does not change when we change ε into -ε.
On the other hand, if P is a group, then the inverse substitution of (1)
(3) |
must also belong to P; it will therefore be identical with (2), that is to say that
We must therefore have l = 1.
§ 5. — Langevin waves
Langevin has put the formulas that define the electromagnetic field produced by the motion of a single electron in a particularly elegant form.
Let us remember the equations
(1) |