Page:LorentzGravitation1916.djvu/38

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\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}.