Page:LorentzGravitation1915.djvu/7

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${\frac {u_{c}w_{a}}{P}}=T_{ac},$

(23)

we find

$K_{c}+\left({\frac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{w}=-\Sigma (a){\frac {\partial T_{ac}}{\partial x_{a}}}.$

(24)

The first three of these equations ($c$ = 1, 2, 3) refer to the momenta; the fourth ($c$ = 4) is the equation of energy. As we know already the meaning of $K_{1},\dots K_{4}$ we can easily see that of the other quantities. The stresses $X_{x},X_{y},X_{z},Y_{x}\dots$ are $T_{11},T_{21},T_{31},T_{12}\dots$; the components of the momentum per unit of volume $-T_{41},-T_{42},-T_{43}$; the components of the flow of energy $T_{14},T_{24},T_{34}$. Further $T_{44}$ is the energy per unit of volume. The quantities

$\left({\frac {\partial \mathrm {L} }{\partial x_{1}}}\right)_{w},\ \left({\frac {\partial \mathrm {L} }{\partial x_{2}}}\right)_{w},\ \left({\frac {\partial \mathrm {L} }{\partial x_{3}}}\right)_{w}$

are the momenta which the gravitation field imparts to the material system per unit of time and unit of volume, while the energy which the system draws from that field is given by $-\left({\dfrac {\partial \mathrm {L} }{\partial x_{4}}}\right)_{w}$.

An electromagnetic system in the gravitation field.

§ 7. We shall now consider charges moving under the influence of external forces in a gravitation field; we shall determine the motion of these charges and the electromagnetic field belonging to them. The density $\varrho$ of the charge will be supposed to be a continuous function of the coordinates; the force per unit of volume will be denoted by $K$ and the velocity of the point of a charge by $v$. Further we shall again introduce the notation (10).

In Einstein's theory the electromagnetic field is determined by two sets, each of four equations, corresponding to well known equations in the theory of electrons. We shall consider one of these sets as the mathematical description of the system to which we have to apply Hamilton's principle; the second set will be found by means of this application.

The first set, the fundamental equations, may be written in the form

$\Sigma (b){\frac {\partial \psi _{ab}}{\partial x_{b}}}=w_{a},$

(25)