Page:LorentzGravitation1915.djvu/15

765

$-\left({\frac {\partial \mathrm {L} }{\partial x}}\right)_{\psi }=-\Sigma (e){\frac {\partial T_{ec}^{g}}{\partial x_{e}}}$

(53)

Remembering what has been said in § 12 about the meaning of $\left({\dfrac {\partial \mathrm {L} }{\partial x_{c}}}\right)_{\psi }$, we may now conclude that the quantities $T_{ab}^{g}$ have the same meaning for the gravitation field as the quantities $T_{ab}$ for the electromagnetic field (stresses, momenta etc.). The index $g$ denotes that $T_{ab}^{g}$ belongs to the gravitation field.

If we add to (53) the equations (44), after having replaced in them $b$ by $e$, we obtain

$K_{c}=-\Sigma (e){\frac {\partial T_{ec}^{t}}{\partial x_{e}}}$

(54)

where

$T_{ec}^{t}=T_{ec}+T_{ec}^{q}.$

The quantities $T_{ec}^{t}$ represent the total stresses etc. existing in the system, and equations (54) show that in the absence of external forces the resulting momentum and the total energy will remain constant.

We could have found directly equations (54) by applying formula (50) to the case of a common virtual displacement $\delta x_{c}$ imparted both to the electromagnetic system and to the gravitation field.

Finally the differential equations of the gravitation field and the formulae derived from them will be quite conform to those given by Einstein, if in $Q$ we substitute for $H$ the function he has chosen.

§ 16. The equations that have been deduced here, though mostly of a different form, correspond to those of Einstein. As to the covariancy, it exists in the case of equations (18), (24), (41), (42) and (44) for any change of coordinates. We can be sure of it because $\mathrm {L} dS$ is an invariant.

On the contrary the formulae (49), (53) and (54) have this property only when we confine ourselves to the systems of coordinates adapted to the gravitation field, which Einstein has recently considered. For these the covariancy of the formulae in question is a consequence of the invariancy of $\delta \int HdS$ which Einstein has proved by an ingenious mode of reasoning.

—————