Page:LorentzGravitation1916.djvu/46

It is also remarkable that in real eases the first term in (83) can be much larger than the following ones. If we consider e. g. a point $P$ outside the attracting sphere, we can prove that the ratio of the first term to the third is of the same order as the ratio of the square of the velocity of light to the square of the velocity with which a material point can describe a circular orbit passing through $P$.

The following must also be noticed. In the system of polar coordinates used above there will exist in the field without gravitation the stress ${\mathfrak {t}}_{1}^{1}={\tfrac {1}{\varkappa }}$. If a stress of this magnitude were produced by means of actions which give rise to a stress-energy-tensor, the passage to rectangular coordinates would give us a stress which becomes infinite at the point $O$. In those coordinates we should namely have

${\mathfrak {t}}_{1}^{'1}={\frac {\sin ^{2}\vartheta }{r^{2}}}\cdot {\frac {1}{\varkappa }}$

§ 52. Evidently it would be more satisfactory if we could ascribe a stress-energy-tensor to the gravitation field. Now this can really be done. Indeed, the quantities ${\mathfrak {s}}_{0h}^{e}$ determined by (57) form a tensor and according to (58), (79) may be replaced by

$K_{h}=-div_{h}{\mathfrak {T}}-div_{h}{\mathfrak {t}}_{0}$

(85)

if ${\mathfrak {t}}_{0}$ is defined by a relation similar to (78), viz.

${\mathfrak {t}}_{0h}^{e}={\frac {1}{2\varkappa }}{\mathfrak {s}}_{0h}^{e}$

(86)

Equation (85) shows that, just as well as ${\mathfrak {t}}_{h}^{c}$, we may consider the quantities ${\mathfrak {t}}_{0h}^{e}$ as the stresses etc. in the gravitation field. This way of interpretation is very simple. With a view to (41) we can namely derive from the equations for the gravitation field (65)

$G=\varkappa T$

and

$T_{ab}=-{\frac {1}{\varkappa }}\left(G_{ab}-{\frac {1}{2}}g_{ab}G\right)$

Further we find from (66)

${\mathfrak {T}}_{h}^{e}={\frac {1}{2\varkappa }}G\sum (a){\mathfrak {g}}^{ae}g_{ah}-{\frac {1}{\varkappa }}\sum (a){\mathfrak {g}}^{ae}G_{ah}$

and from (57) and (86)

${\mathfrak {t}}_{0h}^{e}=-{\mathfrak {T}}_{h}^{e}$

(87)

At every point of the field-figure the components of the stress-energy-tensor of the gravitation field would therefore be equal to