Page:LorentzGravitation1916.djvu/38

$\left({\frac {\partial \mathrm {L} }{\partial g^{ab}}}\right)_{\psi }={\frac {1}{2}}{\sqrt {-g}}T_{ab}$

(63)

and if for $\partial g^{ab}$ the value (49) is substituted, this term becomes

${\frac {1}{2}}\sum (ab)T_{ab}\partial {\mathfrak {g}}^{ab}-{\frac {1}{4}}\sum (abcd)g^{ab}g_{cd}T_{ab}\delta {\mathfrak {g}}^{cd}$

or if in the latter summation $a,b$ is interchanged with $c,d$ and if the quantity

$T=\sum (cd)g^{cd}T_{cd}$

(64)

is introduced,

${\frac {1}{2}}\sum (ab)\left(T_{ab}-{\frac {1}{2}}g_{ab}T\right)\delta {\mathfrak {g}}^{ab}$

Finally, putting equal to zero the coefficient of each $\delta {\mathfrak {g}}^{ab}$ we find from (62) the differential equation required

$G_{ab}=-\varkappa \left(T_{ab}-{\frac {1}{2}}g_{ab}T\right)$

(65)

This is of the same form as Einstein's field equations, but to see that the formulae really correspond to each other it remains to show that the quantities $T_{ab}$ and ${\mathfrak {T}}_{c}^{b}$ defined by (63), f59) and (60) are connected by Einstein's formulae

${\mathfrak {T}}_{c}^{b}={\sqrt {-g}}\sum (a)g^{ab}T_{ac}$

(66)

We must have therefore

$2\sum (a)g^{ac}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=-\mathrm {L} +\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi _{a'c'}$

(67)

and for $b\neq c$

$2\sum (a)g^{ab}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=\sum \limits _{a\neq c}(a)\psi _{ab}^{*}\psi _{a'c'}$

(68)

§ 42. This can be tested in the following way. The function $\mathrm {L}$ (comp. § 9, 1915) is a homogeneous quadratic function of the $\psi _{ab}$'s and when differentiated with respect to these variables it gives the quantities ${\bar {\psi }}_{ab}$. It may therefore also be regarded as a homogeneous quadratic function of the ${\bar {\psi }}_{ab}$. From (35), (29) and (32)^[1], 1915 we find therefore

$L={\frac {1}{8}}{\sqrt {-g}}\sum (pqrs)\left(g^{pr}g^{qs}-g^{qr}g^{ps}\right){\bar {\psi }}_{pq}{\bar {\psi }}_{rs}$

(69)

Now we can also differentiate with respect to the $g^{ab}$'s, while not the $\psi _{ab}$'s but the quantities ${\bar {\psi }}_{ab}$ are kept constant, and we have e.g.

$\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=-\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }$

(70)

According to (69) one part of the latter differential coefficient is

1. The quantities $\gamma _{ab}$ in that equation are the same as those which are now denoted by $g^{ab}$.