Page:Grundgleichungen (Minkowski).djvu/18

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of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.


§ 7. Fundamental Equations for bodies at rest.

After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case \epsilon=1,\ \mu=1,\ \sigma=1, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us — when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every spacetime point as functions of x, y, z, t: — the vector of the electric force \mathfrak{E}, the magnetic induction \mathfrak{M}, the electrical induction \mathfrak{e}, the magnetic force \mathfrak{m}, the electrical space-density \varrho, the electric current \mathfrak{s} (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector \mathfrak{w}, the velocity of matter.

The relations in question can be divided into two classes.

Firstly — those equations, which, — when \mathfrak{w}, the velocity of matter is given as a function of x, y, z, t, — lead us to a knowledge of other magnitude as functions of x, y, z, t — I shall call this first class of equations the fundamental equations

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector \mathfrak{w} as functions of x, y, z, t.

For the case of bodies at rest, i.e. when \mathfrak{w}(x,\ y,\ z,\ t) = 0, the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are ; —

(1) The Differential Equations: — which contain no constant referring to matter: —

(I) & \qquad & curl\ \mathfrak{m}-\frac{\partial e}{\partial t} & =\mathfrak{s},\\
\\(II) &  & div\ \mathfrak{e} & =\varrho,\\
\\(III) &  & curl\ \mathfrak{E}+\frac{\partial\mathfrak{M}}{\partial t} & =0,\\
\\(IV) &  & div\ \mathfrak{M} & =0\end{array};