Page:LorentzGravitation1916.djvu/38



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

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

This is of the same form as '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 's formulae

We must have therefore

and for $$b\ne c$$

§ 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), 1915 we find therefore

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.

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