Page:AbrahamMinkowski1.djvu/9

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§ 5. Determination of momentum density and energy density.


The various theories of electrodynamics of moving bodies, are differing by the relations assumed between the four vectors arising in the main equations. However, before we pass to the discussion of special theories, we want to pursue the general developments; there, only a quite general presupposition shall be made about the form of these relations: The vectors shall be connected by equations, which namely contain the velocity vector itself, though not any derivatives of it with respect to time or coordinates.

Main equation (IV) gives:

this becomes with respect to the two first main-equations:

From main equation (III) and relation (14) it follows:

(14a)

a condition, which one can also be written in accordance with (4):

(14b)

and which finally, by using main equation (V), passes into:

(15)

This relation serves to determine the densities of energy and momentum in their dependence from the electromagnetic vectors.

They read with respect to (2a):

(15a)

Since the way of time differentiation now employed, is satisfying the calculation rules, it follows when it is set for brevity's sake:

(16)
(17)

As mentioned in the beginning of the paragraph, the relations which connect with shall contain the velocity vector , though not its derivative with respect to time and space. The same is to be demanded from the expressions,