# Page:LorentzGravitation1916.djvu/38

 ${\displaystyle \left({\frac {\partial \mathrm {L} }{\partial g^{ab}}}\right)_{\psi }={\frac {1}{2}}{\sqrt {-g}}T_{ab}}$ (63)

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

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

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

 ${\displaystyle T=\sum (cd)g^{cd}T_{cd}}$ (64)

is introduced,

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

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

 ${\displaystyle G_{ab}=-\varkappa \left(T_{ab}-{\frac {1}{2}}g_{ab}T\right)}$ (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 ${\displaystyle T_{ab}}$ and ${\displaystyle {\mathfrak {T}}_{c}^{b}}$ defined by (63), f59) and (60) are connected by Einstein's formulae

 ${\displaystyle {\mathfrak {T}}_{c}^{b}={\sqrt {-g}}\sum (a)g^{ab}T_{ac}}$ (66)

We must have therefore

 ${\displaystyle 2\sum (a)g^{ac}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=-\mathrm {L} +\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi _{a'c'}}$ (67)

and for ${\displaystyle b\neq c}$

 ${\displaystyle 2\sum (a)g^{ab}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=\sum \limits _{a\neq c}(a)\psi _{ab}^{*}\psi _{a'c'}}$ (68)

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

 ${\displaystyle L={\frac {1}{8}}{\sqrt {-g}}\sum (pqrs)\left(g^{pr}g^{qs}-g^{qr}g^{ps}\right){\bar {\psi }}_{pq}{\bar {\psi }}_{rs}}$ (69)

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

 ${\displaystyle \left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=-\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }}$ (70)

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

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