Page:LorentzGravitation1916.djvu/20

To each of these quantities corresponds a definite direction, viz. that in which we have to proceed in order to make the considered quantity change in positive sense while the other three remain constant. If we denote these directions by ${\displaystyle 1^{*},2^{*},3^{*},4^{*}}$ and in the same way the directions of the coordinates ${\displaystyle x_{1},x_{2},x_{3}x_{4}}$ by 1, 2, 3, 4, it is evident that ${\displaystyle 1^{*}}$ is conjugate with 2, 3 and 4, ${\displaystyle 2^{*}}$ with 3, 1 and 4, and so on; inversely 1 with ${\displaystyle 2^{*},3^{*},4^{*}}$; 2 with ${\displaystyle 3^{*},1^{*},4^{*}}$, and so on. From what has been said above about the algebraic signs of ${\displaystyle g_{11},g_{22},g_{33},g_{44}}$ it follows further that, if directions opposite to 1, ${\displaystyle 1^{*}}$ etc. are denoted by — 1, ${\displaystyle -1^{*}}$ etc., the directions — 1 and ${\displaystyle 1^{*}}$ will point to the same side of an extension ${\displaystyle x_{1}=\mathrm {const} .}$. The same may be said of the directions —2 and${\displaystyle 2^{*}}$ or —3 and ${\displaystyle 3^{*}}$ with respect to extensions ${\displaystyle x_{2}=\mathrm {const} .}$, or ${\displaystyle x_{3}=\mathrm {const} .}$, while with respect to an extension ${\displaystyle x_{4}=\mathrm {const} .}$, the directions 4 and ${\displaystyle 4^{*}}$ point to the same side.
Finally, we shall fix (§11) as far as is necessary, which direction corresponds to three others. For that purpose we shall imagine the directions of coordinates ${\displaystyle 1,\dots 4}$ to pass into mutually conjugate directions, which will also be called ${\displaystyle $1,\dots 4}$$, by gradual changes, in such a way that never three of them come to lie in one plane. We shall agree that after this change —4 corresponds to 1, 2, 3.
Let ${\displaystyle a,b,c,d}$ be the numbers 1, 2, 3, 4 in an order obtained from the natural one by an even number of permutations. Then the rule of § 11 teaches us that the direction ${\displaystyle -d}$ corresponds to ${\displaystyle a,b,c}$. It is clear that this would be the ease with ${\displaystyle d}$, if ${\displaystyle a,b,c,d}$ were obtained from 1, 2, 3, 4 by an odd number of permutations. If further it is kept in mind that, always in the new case, the directions ${\displaystyle 1^{*},2^{*},3^{*},4^{*}}$ coincide with —1, —2, —3, 4, we come to the conclusion that the directions 1, 2, 3 and 4 correspond to the sets ${\displaystyle 2^{*},3^{*},4^{*};3^{*},1^{*},4^{*};1^{*},2^{*},4^{*}}$ and ${\displaystyle 1^{*},2^{*},3^{*}}$ respectively. The rule of gradual change (§11) involves that this holds also for the original case, in which 1, 2, 3, 4 were not yet mutually conjugate.
§ 23. In the neighbourhood of a point ${\displaystyle P}$ of the field-figure we may introduce as coordinates instead of ${\displaystyle x_{1},\dots x_{4}}$ the quantities ${\displaystyle \xi _{1},\dots \xi _{4}}$ defined by (19). Line-elements or finite vectors can be resolved in the directions of these coordinates, i.e. in the directions