Page:LorentzGravitation1916.djvu/60

because other functions of the coordinates occur in it, but which nevertheless no observation will be able to discern from it, the indefiniteness which is a necessary consequence of the covariancy of the field equations, again presenting itself.

What has been said shows that the total gravitation energy in this new system will have the same value as in the original one, as has been found already in § 60 with the restrictions then introduced.

§ 63. If ${\mathfrak {t}}'$ were a tensor, we should have for all substitutions the transformation formulae given at the end of § 40. In reality this is not the case now, but from (96) and (97) we can still deduce that those formulae hold for linear substitutions. They may likewise be applied to the stress-energy-components of the matter or of an electromagnetic system. Hence, if ${\mathfrak {T}}_{a}^{b}$ represents the total stress-energy-components, i. e. quantities in which the corresponding components for the gravitation field, the matter and the electromagnetic field are taken together, we have for any linear transformation

${\frac {1}{\sqrt {-g'}}}{\mathfrak {T}}_{c}^{'b}={\frac {1}{\sqrt {-g}}}\sum (kl)p_{kc}\pi _{lb}{\mathfrak {T}}_{k}^{l}$

(121)

We shall apply this to the case of a relativity transformation, which can be represented by the equations

$x'_{1}=ax_{1}+bcx_{4},\ x'_{2}=x_{2},\ x'_{3}=x_{3},\ x'_{4}=ax_{4}+{\frac {b}{c}}x_{1}$

(122)

with the relation

$a^{2}-b^{2}=1$

(123)

In doing so we shall assume that the system, when described in the rectangular coordinates $x_{1},x_{2},x_{3}$ and with respect to the time $x_{4}$, is in a stationary state and at rest.

Then we derive from (97)^[1]

1. We have $g_{14}=g_{24}=g_{34}=0$, while all the other quantities gab are independent of $x_{4}$. Thus we can say that the quantities $g_{ab}$ and $g_{ab,c}$ are equal to zero when among their indices the number 4 occurs an odd number of times. The same may be said of $g^{ab}$, $g^{ab,c}$, ${\tfrac {\partial Q}{\partial g_{ab,cd}}}$ (according to (116)), ${\tfrac {\partial }{\partial x_{k}}}\left({\tfrac {\partial Q}{\partial g_{ab,cd}}}\right)$ and also of products of two or more of such quantities. As in the last two terms of (97) the indices $a,b$ and $f$ occur twice, these terms will vanish when only one of the indices $e$ and $h$ has the value 4. As to the first term of (97) we remark that, according to the formulae of § 32, each of the indices $a,b$ and $e$ occurs only once in the differential coefficient of $Q$ with respect to $g_{ab,e}$, while other indices are repeated. As to the number of