Page:LorentzGravitation1916.djvu/50

${\mathfrak {t}}_{(E)h}^{e}={\frac {1}{2\varkappa }}\delta _{h}^{e}\sum (abcf)g^{ab}\Gamma _{ac}^{f}\Gamma _{bf}^{c}-{\frac {1}{\varkappa }}\sum (abc)g^{ab}\Gamma _{ac}^{e}\Gamma _{bh}^{c}$

where for the sake of simplicity it has been assumed that ${\sqrt {-g}}=1$. Further we have

$\Gamma _{ab}^{c}=-\left\{{\begin{array}{c}ab\\c\end{array}}\right\}=-\sum (e)g^{ce}\left[{\begin{array}{c}ab\\e\end{array}}\right]$

If now our formulae (96) and (97) are likewise simplified by the assumption ${\sqrt {-g}}=1$ (so that $Q$ becomes equal to $G$), we may expect that ${\mathfrak {t}}'$ will become identical with ${\mathfrak {t}}_{(E)}$. This is really so in the case $g_{ab}=0$ for $a\neq b$; by which it seems very probable that the agreement will exist in general.

In the preceding paper it was shown already that the stress-energy-components ${\mathfrak {t}}_{h}^{e}$ do not form a "tensor", but what was called a "complex". The same may be said of the quantities ${\mathfrak {t}}_{h}^{'e}$ defined by (96) and (97) and of the expressions given by Einstein. If we want a stress-energy-tensor, there are only left the quantities ${\mathfrak {t}}_{0h}^{e}$ defined by (86) and (57), the values of which are always equal and opposite to the corresponding stress-energy-components ${\mathfrak {T}}_{h}^{e}$ for the matter or the electromagnetic field.

It must be noticed that the four equations

$\sum (e){\frac {\partial }{\partial x_{e}}}\left({\mathfrak {T}}_{h}^{e}+{\mathfrak {T}}_{(g)h}^{e}\right)=0$

always express the same relations, whether we choose ${\mathfrak {t}}_{0h}^{e},\ {\mathfrak {t}}_{h}^{e},\ {\mathfrak {t}}_{h}^{'e}$ or ${\mathfrak {t}}_{(E)h}^{e}$ as stress-energy-components ${\mathfrak {T}}_{(g)h}^{e}$ of the gravitation field. If however in a definite case we want to use the equations in order to calculate how the momentum and the energy of the matter and the electromagnetic field change by the gravitational actions, it is best to use ${\mathfrak {t}}_{h}^{'e}$ or ${\mathfrak {t}}_{(E)h}^{e}$, just because these quantities are homogeneous quadratic functions of the derivatives $g_{ab,c}$.

Experience namely teaches us that the gravitation fields occurring in nature may be regarded as feeble, in this sense that the values of the $g_{ab}$'s are little different from those which might be assumed if no gravitation field existed. For these latter values, which will be called the "normal" ones, we may write in orthogonal coordinates

$g_{11}=g_{22}=g_{32}=-1,\ g_{44}=c^{2},\ g_{ab}=0,\quad {\textrm {for}}\quad a\neq b$

(98)

In a first approximation, which most times will be sufficient, the deviations of the values of the $g_{ab}$'s from these normal ones may be taken proportional to the gravitation constant $\varkappa$. This factor also appears in the differential coefficients $g_{ab,c}$; hence, according to the character of the functions ${\mathfrak {t}}_{h}^{'e}$ mentioned above (and on account