# 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

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

and for $b\ne c$

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