Page:LorentzGravitation1916.djvu/59

and thus to derive the result (112) from (113), we should have to determine the quantity $T$ (comp. 120)), accurately to the order $\varkappa$. The surface integrals in (115) too would have to be considered more closely. We shall not however dwell upon this.

§ 62. From the expression for ${\mathfrak {t}}_{4}^{'4}$ given in (113) and the value

$E=E_{1}+E_{2}$

derived from it, it can be inferred that, though ${\mathfrak {t}}'$ is no tensor, we yet may change a good deal in the system of coordinates in which the phenomena are described, without altering the value of the total energy. Let us suppose e.g. that $x_{4}$ is left unchanged but that, instead of the rectangular coordinates $x_{1},x_{2},x_{3}$ hitherto used, other quantities $x_{1}',x_{2}',x_{3}'$ are introduced, which are some continuous function of $x_{1},x_{2},x_{3}$, with the restriction that $x'_{1}=x_{1},x'_{2}=x_{2},x'_{3}=x_{3}$ outside a certain closed surface surrounding the attracting matter at a sufficient distance. If we use these new coordinates, we shall have to introduce other quantities $g'_{ab}$ instead of $g_{ab}$ however outside the closed surface the quantities $x_{1},x_{2},x_{3}$ and their derivatives do not change, the value of $E_{1}$ will approach the same limit as when we used the coordinates $x_{1},x_{2},x_{3}$, if the surface $\sigma$ for which it is calculated expands indefinitely. The value which we find for $E_{1}$ after the transformation of coordinates will also be the same as before. Indeed, if $d\tau$ is an element of volume expressed in $x_{1},x_{2},x_{3}$-units and $d\tau '$ the same element expressed in $x_{1}',x_{2}',x_{3}'$-units, while $Q'$ represents the new value of $Q$, we have

$Qd\tau =Q'd\tau '$

It is clear that the total energy will also remain unchanged if $x_{1}',x_{2}',x_{3}'$ differ from $x_{1},x_{2},x_{3}$ at all points, provided only that these differences decrease so rapidly with increasing distance from the attracting body, that they have no influence on the limit of the expression (115).

The result which we have now found admits of another interpretation. In the mode of description which we first followed (using $x_{1},x_{2},x_{3}$), $\varrho$^[1]) and $g_{ab}$ are certain functions of $x_{1},x_{2},x_{3}$; in the new one $\varrho '$, $g'_{ab}$ are certain other functions of $x_{1}',x_{2}',x_{3}'$. If now, without leaving the system of coordinates $x_{1},x_{2},x_{3}$, we ascribe to the density and to the gravitation potentials values which depend on $x_{1},x_{2},x_{3}$, in the same way as $\varrho '$, $g'_{ab}$ depended on $x_{1}',x_{2}',x_{3}'$ just now, we shall obtain a new system (consisting of the attracting body and the gravitation field) which is different from the original system

1. By $\varrho$ we mean here what was denoted by ${\bar {\varrho }}$ in § 56.