Page:LorentzGravitation1915.djvu/9

759

To these equations we add the transformation formula for ${\bar {\psi '}}_{ab}$, which may be derived from (28)^[1]

${\bar {\psi '}}_{ab}=\Sigma (cd)p_{ca}p_{db}{\bar {\psi }}_{cd}$

(33)

§ 9. We shall now consider the 6 quantities (27) which we shall especially call "the quantities $\psi$" and the corresponding quantities ${\bar {\psi }}$, viz. ${\bar {\psi }}_{41}\dots {\bar {\psi }}_{12}$.

According to (30) these latter are homogeneous and linear functions of the former and as (because of (5)) the coefficient of $\psi _{cd}$ in ${\bar {\psi }}_{ab}$ is equal to the coefficient of $\psi _{ab}$ in ${\bar {\psi }}_{cd}$, there exists a homogeneous quadratic function $\mathrm {L}$ of $\psi _{41}\dots \psi _{12}$, which, when differentiated with respect to these quantities, gives $\psi _{41}\dots \psi _{12}$. Therefore

${\frac {\partial \mathrm {L} }{\partial \psi _{ab}}}={\bar {\psi }}_{ab}$

(34)

and

$\mathrm {L} ={\tfrac {1}{2}}\Sigma ({\overline {ab}})\psi _{ab}{\bar {\psi }}_{ab}$

(35)

If, as in (34), we have to consider derivatives of $\mathrm {L}$, this quantity will be regarded as a quadratic function of the quantities $\psi$.

The quantity $\mathrm {L}$ can now play the same part as the quantity that is represented by the same letter in §§ 4 — 6. Again $\mathrm {L} dS$ is invariant when the coordinates are changed.^[2]

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we have

${\sqrt {-g}}\ \Sigma (cd)\gamma _{ac}\gamma _{bd}{\bar {\psi }}_{cd}=\Sigma (cdef)\gamma _{ac}\gamma _{bd}g_{ec}g_{fa}\psi _{ef}=\Sigma (df)\gamma _{bd}g_{fd}\psi _{af}=\psi _{ab}$

The last two steps of this transformation, which rest on $(\alpha )$ and $(\beta )$, will need no further explanation. In a similar way we may proceed (comp. the following notes) in many other cases, using also the relations $\Sigma (a)p_{ba}\pi _{ba}=1$ and $\Sigma (a)p_{ba}\pi _{ca}=0$ (the latter for $b\neq c$), which are similar to (α) and (β).

1. If we start from the equation for ${\bar {\psi '}}_{ab}$ that corresponds to (29) and if we use (7) and (28), attending to ${\sqrt {-g'}}=|p|{\sqrt {-g}}$, we find

   ${\begin{array}{c}{\bar {\psi '}}_{ab}={\dfrac {1}{\sqrt {-g'}}}\Sigma (cd)g'_{ca}g'_{db}\psi '_{cd}=\\\\={\dfrac {1}{\sqrt {-g}}}\Sigma (cdefhijk)p_{ec}p_{fa}p_{hd}p_{ib}\pi _{jc}\pi _{kd}g_{ef}g_{hi}\psi _{jk}.\end{array}}$

   This may be transformed in two steps (comp. the preceding note) to

   ${\frac {1}{\sqrt {-g}}}\Sigma (efhi)p_{fa}p_{ib}g_{ef}g_{hi}\psi _{eh}.$

   In this way we may proceed further, after first expressing $\psi _{eh}$ as a function of ${\bar {\psi }}_{lm}$ by means of (32).

2. Instead of (35) we may write $\mathrm {L} ={\tfrac {1}{4}}\Sigma (ab)\psi _{ab}{\overline {\psi _{ab}}}$ and now we have according to (28) and (33)