Page:LorentzGravitation1916.djvu/30

III.

(Communicated in the meeting of April 1916.)^[1]

§ 32. In the two preceding papers^[2] we have tried so far as possible to present the fundamental principles of the new gravitation theory in a simple form.

We shall now show how Einstein's differential equations for the gravitation field can be derived from Hamilton's principle. In this connexion we shall also have to consider the energy, the stresses, momenta and energy-currents in that field.

We shall again introduce the quantities $g_{ab}$ formerly used and we shall also use the "inverse" system of quantities for which we shall now write $g^{ab}$. It is found useful to introduce besides these the quantities

${\mathfrak {g}}^{ab}={\sqrt {-g}}g^{ab}$

Differential coefficients of all these variables with respect to the coordinates will be represented by the indices belonging to these latter, e.g.

$g_{ab,p}={\frac {\partial g_{ab}}{\partial x_{p}}},\ g_{ab,pq}={\frac {\partial ^{2}g_{ab}}{\partial x_{q}\partial x_{p}}}$

We shall use Christoffel's symbols

$\left[{\begin{array}{c}ab\\c\end{array}}\right]={\frac {1}{2}}\left(g_{ac,b}+g_{bc,a}-g_{ab,c}\right)$

and Riemann's symbol

${\begin{array}{l}(ik,lm)={\frac {1}{2}}\left(g_{im,lk}+g_{kl,im}-g_{il,km}-g_{km,il}\right)+\\\\\qquad +\sum (ab)g^{ab}\left\{\left[{\begin{array}{c}im\\a\end{array}}\right]\left[{\begin{array}{c}kl\\b\end{array}}\right]-\left[{\begin{array}{c}il\\a\end{array}}\right]\left[{\begin{array}{c}km\\b\end{array}}\right]\right\}\end{array}}$

Further we put

$G_{im}=\sum (kl)g^{kl}(ik,lm)$

(40)

$G=\sum (im)g^{im}G_{im}$

(41)

This latter quantity is a measure for the curvature of the field-figure. The principal function of the gravitation field is

1. Published September 1916, a revision having been found desirable.
2. See Proceedings Vol. XIX, p. 1341 and 1354.