IV.

(Communicated in the meeting of October 28, 1916).

§ 53. The expressions for the stress-energy-components of the gravitation field found in the preceding paper call for some further remarks. If by $\delta _{h}^{e}$ we denote a quantity having the value 1 for $e=h$ and being 0 for $e\neq h$, those expressions can be written in the form (comp. equations (52) and (78))

${\mathfrak {t}}_{h}^{e}={\frac {1}{2\varkappa }}\left\{-\delta _{h}^{e}Q+\sum (ab){\frac {\partial Q}{\partial g_{ab,e}}}g_{ab,h}+\sum (abf){\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fh}-\sum (abf){\frac {\partial Q}{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\right)g_{ab,h}\right\}$ |
(88) |

They contain the first and second derivatives of the quantities $g_{ab}$. Einstein on the contrary has given values for the stress-energy-components which contain the first derivatives only and which therefore are in many respects much more fit for application.

It will now be shown how we can also find formulae without second derivatives, if we start from (88).

§ 54. For this purpose we shall consider the complex ${\mathfrak {u}}$ defined by

${\mathfrak {u}}_{h}^{e}={\frac {1}{2\varkappa }}\left\{\delta _{h}^{e}Q-\sum (abf){\frac {\partial }{\partial x_{h}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)\right\}$ |
(89) |

and we shall seek its divergency.

We have

$(div\ {\mathfrak {u}})_{h}=\sum (e){\frac {\partial {\mathfrak {u}}^{e}}{\partial x_{e}}}={\frac {1}{2\varkappa }}\left\{{\frac {\partial Q}{\partial x_{h}}}-\sum (abfe){\frac {Q^{2}}{\partial x_{e}\partial x_{h}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)\right\}$

or

$(div\ {\mathfrak {u}})_{h}={\frac {1}{2\varkappa }}{\frac {\partial R}{\partial x_{h}}}$ |
(90) |

if we put

$R=Q-\sum (abfe){\frac {\partial Q}{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)$ |
(91) |

Now $Q={\sqrt {-g}}G$ can be divided into two parts, the first of which $Q_{1}$ contains differential coefficients of the quantities $g_{ab}$ of the first order only, while the second $Q_{2}$ is a homogeneous linear function