Page:LorentzGravitation1916.djvu/55

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should have to ascribe a certain negative value of the energy to a field without gravitation, in such a way (comp. § 57) that the energy in the shell between the spheres described round the origin with radii r and r + dr becomes

-\frac{4\pi c}{\varkappa}dr

The density of the energy in the ordinary sense of the word would be inversely proportional to r^{2}, so that it would become infinite at the centre.

It is hardly necessary to remark that, using rectangular coordinates we find a value zero for the same case of a field without gravitation. The normal values of g_{ab} are then constants and their derivatives vanish.


§ 60. Using rectangular coordinates we shall now indicate the form of \mathfrak{t}_{4}^{'4} for the field of a spherical body, with the approximation specified in § 57. Thus we put

\left.\begin{array}{l}
g_{11}=-(1+\lambda)+\frac{x_{1}^{2}}{r^{2}}(\lambda-\mu),\ etc.\\
\\
g_{12}=\frac{x_{1}x_{2}}{r^{2}}(\lambda-\mu),\ etc.\\
\\
g_{14}=g_{24}=g_{34}=0,\ g_{44}=c^{2}(1+\nu)
\end{array}\right\} (110)

By (109) and (110) we find[1]

  1. Of the laborious calculation it may be remarked here only that it is convenient to write the values (110) in the form

    g_{11}=-1+\alpha+\frac{\partial^{2}\beta}{\partial x_{1}^{2}},\ etc.

    g_{12}=\frac{\partial^{2}\beta}{\partial x_{1}\partial x_{2}},\ etc.

    where \alpha and \beta are infinitesimal functions of r. We then find

    \begin{array}{l}
\mathfrak{t}_{4}^{'4}=\frac{c}{2\varkappa}\left\{ -\frac{1}{2}\sum(a)\left(\frac{\partial\alpha}{\partial x_{a}}\right)^{2}+\sum(a)\frac{\partial\nu}{\partial x_{a}}\frac{\partial\alpha}{\partial x_{a}}+\right.\\
\\
\qquad\left.+\frac{1}{4}\sum(aik)\left[\frac{\partial^{3}\beta}{\partial x_{a}\partial x_{i}^{2}}\frac{\partial^{3}\beta}{\partial x_{a}\partial x_{k}^{2}}-\left(\frac{\partial^{3}\beta}{\partial x_{a}\partial x_{i}\partial x_{k}}\right)^{2}\right]\right\} \\
\\
\qquad\qquad(a,i,k=1,2,3)
\end{array}


    which reduces to (111) if the relations between \alpha,\beta and \gamma,\mu, viz.

    \alpha+\frac{1}{r}\beta'=-\lambda,\ -\frac{1}{r}\beta'+\beta''=\lambda-\mu

    and the equality \alpha'=\nu' involved in (109) are taken into consideration.