Page:LorentzGravitation1916.djvu/49

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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