Page:LorentzGravitation1916.djvu/44

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which is a measure for the "distance" to the centre. As to x_{1} and x_{2}, we shall put x_{1}=\cos\vartheta, x_{2}=\varphi, after first having introduced polar coordinates \vartheta, \varphi (in such a way that the rectangular coordinates are r\cos\vartheta, r\sin\vartheta\cos\varphi, r\sin\vartheta\sin\varphi). It can be proved that, because of the symmetry about the centre, g_{ab}=0 for a\ne b, while we may put for the quantities g_{aa}

g_{11}=-\frac{u}{1-x_{1}^{2}},\ g_{22}=-u\left(1-x_{1}^{2}\right),\ g_{33}=-v,\ g_{44}=w (80)

where u, v, w are certain functions of r. Ditferentiations of these functions will be represented by accents. We now find that of the complex \mathfrak{t} only the components \mathfrak{t}_{1}^{1}, \mathfrak{t}_{3}^{3} and \mathfrak{t}_{4}^{4} are different from zero. The expressions found for them may be further simplified by properly choosing r. If the distance to the centre is measured by the time the light requires to be propagated from to the point in question, we have w = v. One then finds

\left.\begin{array}{l}
\mathfrak{t}_{1}^{1}=\frac{1}{2\varkappa}\left(-\frac{u'^{2}}{2u}+2u''-\frac{uv'^{2}}{v^{2}}+\frac{uv''}{v}\right),\\
\\
\mathfrak{t}_{3}^{3}=\frac{1}{2\varkappa}\left(-2v+\frac{u'^{2}}{2u}+\frac{uv'}{v}\right),\\
\\
\mathfrak{t}_{4}^{4}=\frac{1}{2\varkappa}\left(-2v-\frac{u'^{2}}{2u}+2u''+\frac{uv''}{v}\right),
\end{array}\right\} (81)


§ 49. We must assume that in the gravitation fields really existing the quantities g_{ab} have values differing very little from those which belong to a field without gravitation. In this latter we should have

u=r^{3},\ v=w=1,

and thus we put now

u=r^{2}(1+\mu),\ v=w=1+\nu

where the quantities \mu and \nu which depend on r are infinitely small, say of the first order, and their derivatives too. Neglecting quantities of the second order we find from (81)

\begin{array}{l}
\mathfrak{t}_{1}^{1}=\frac{1}{2\varkappa}\left(2+2\mu+6r\mu'+2r^{2}\mu''+r^{2}\nu''\right),\\
\\
\mathfrak{t}_{3}^{3}=\frac{1}{\varkappa}\left(\mu-\nu+r\mu'+r\nu'\right),\\
\\
\mathfrak{t}_{4}^{4}=\frac{1}{2\varkappa}\left(2\mu-2\nu+6r\mu'+2r^{2}\mu''+r^{2}\nu''\right),
\end{array}

For our degree of approximation we may suppose that of the quantities T_{ab} only T_{44} differs from 0. If we put