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 $e\neq h$

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

(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