Page:LorentzGravitation1915.djvu/12

 $\delta\psi_{ab}=-\frac{\partial\psi_{ab}}{\partial x_{c}}\delta x_{c}$|undefined

From (36), (14) and (37) we can infer what values must then be given to the quantities $$q_a$$. We must put $$q_{c}=0$$ and for $$a\ne c$$

$q_{a}=\psi_{ac'}\delta x_{c}.$

For $$\delta\mathrm{L}$$ we must substitute the expression (cf. § 6)

$\left\{ -\frac{\partial\mathrm{L}}{\partial x_{c}}+\left(\frac{\partial\mathrm{L}}{\partial x_{c}}\right)_{\psi}\right\} \delta x_{c},$|undefined

where the index $$\psi$$ attached to the second derivative indicates that only the variability of the coefficients (depending on $$g_{ab}$$) in the quadratic function $$\mathrm{L}$$ must be taken into consideration. The equation for the component $$K_{c}$$ which we finally find from (43) may be written in the form

where

and for $$b\ne c$$

Equations (44) correspond exactly to (24). The quantities $$T$$ have the same meaning as in these latter formulae and the influence of gravitation is determined by $$\left(\dfrac{\partial\mathrm{L}}{\partial x_{c}}\right)_{\psi}$$ in the same way as it was formerly by $$\left(\dfrac{\partial\mathrm{L}}{\partial x_{c}}\right)_{w}$$.

We may remark here that the sum in (45) consists of three and that in (46) (on account of (39)) of two terms.

Referring to (35), we find f.i. from (45)

$T_{11}=\tfrac{1}{2}\left(\psi_{43}\overline{\psi}_{43}+\psi_{42}\overline{\psi}_{42}-\psi_{41}\overline{\psi}_{41}+\psi_{23}\overline{\psi}_{23}-\psi_{31}\overline{\psi}_{31}-\psi_{12}\overline{\psi}_{12}\right),$

while (46) gives

$T_{12}=\psi_{31}\overline{\psi}_{23}-\psi_{41}\overline{\psi}_{42}.$

The differential equations of the gravitation field.

§ 13. The equations which, for a given material or electromagnetic system, determine the gravitation field caused by it can also be derived from a variation principle. has prepared the way