# Page:LorentzGravitation1916.djvu/49

of the second derivatives of those quantities. This latter involves that, if we replace (91) by

$R=Q_{1}+Q_{2}-\sum(abfe)\left(\frac{\partial Q}{\partial g_{ab,fe}}g_{ab,fe}\right)-\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,f}$

the second and the third term annul each other. Thus

 $R=Q_{1}-\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,f}$ (92)

If now we define a complex $\mathfrak{v}$ by the equation

 $\mathfrak{v}_{h}^{e}=-\frac{1}{2\varkappa}\delta_{h}^{e}R$ (93)

we have

 $(div\ \mathfrak{v})_{h}=-\frac{1}{2\varkappa}\frac{\partial R}{\partial x_{h}}$ (94)

If finally we put

$\mathfrak{t'=t+u+v}$

we infer from (90) and (94)

 $div\ \mathfrak{t}'=div\ \mathfrak{t}$ (95)

and from (88), (89), (93) and (92)

 $\begin{array}{c} \mathfrak{t}_{h}^{'h}=\frac{1}{2\varkappa}\left\{ -Q_{1}+\sum(ab)\frac{\partial Q}{\partial g_{ab,h}}g_{ab,h}-\sum(abf)\frac{\partial}{\partial x_{h}}\left(\frac{\partial Q}{\partial g_{ab,fh}}\right)g_{ab,f}-\right.\\ \\ \left.\sum(abf)\frac{\partial}{\partial x_{f}}\left(\frac{\partial Q}{\partial g_{ab,hf}}\right)g_{ab,h}+\sum(abfe)\frac{\partial}{\partial x_{e}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,f}\right\} \end{array}$ (96)

and for $e\ne h$

 $\begin{array}{c} \mathfrak{t}_{h}^{'e}=\frac{1}{2\varkappa}\left\{ \sum(ab)\frac{\partial Q}{\partial g_{ab,e}}g_{ab,h}-\sum(abf)\frac{\partial}{\partial x_{h}}\left(\frac{\partial Q}{\partial g_{ab,fe}}\right)g_{ab,f}-\right.\\ \\ \left.-\sum(abf)\frac{\partial}{\partial x_{f}}\left(\frac{\partial Q}{\partial g_{ab,ef}}\right)g_{ab,h}\right\} \end{array}$ (97)

Formula (95) shows that the quantities $\mathfrak{t}_{h}^{'e}$ can be taken just as well as the expressions (88) for the stress-energy-components and we see from (96) and (97) that these new expressions contain only the first derivatives of the coefficients $g_{ab}$; they are homogeneous quadratic functions of these differential coefficients.

This becomes clear when we remember that $Q_{1}$ is a function of this kind and that only $Q_{1}$ contributes something to the second term of (96) and the first of (97); further that the derivatives of $Q$ occurring in the following terms contain only the quantities $g_{ab}$ and not their derivatives.

§ 55. Einstein's stress-energy-components have a form widely different from that of the above mentioned ones. They are