Page:LorentzGravitation1916.djvu/37

 electromagnetic system; for this purpose we must use the equations (45) and (46) (1915) or in 's notation, which we shall follow here,

and for $$b\ne c$$

The set of quantities $$\mathfrak{T}_{c}^{b}$$ might be called the stress-energy-complex (comp. § 38). As for a change of the system of coordinates the transformation formulae for $$\mathfrak{T}$$ are similar to those by which tensors are defined, we can also speak of the stress-energy-tensor. We have namely

$\frac{1}{\sqrt{-g'}}\mathfrak{T}_{c}^{'b}=\frac{1}{\sqrt{-g}}\sum(kl)p_{kc}\pi lb\mathfrak{T}_{k}^{l}$|undefined

§ 41. The equations for the gravitation field are now obtained (comp. §§ 13 and 14, 1915) from the condition that

for all variations $$\delta g_{ab}$$ which vanish at the boundary of the field of integration together with their first derivatives. The index $$\psi$$ in the first term indicates that in the variation of $$\mathrm{L}$$ the quantities $$\psi_{ab}$$ must be kept constant.

If we suppose $$\mathrm{L}$$ to be expressed in the quantities $$g^{ab}$$ and if (42), (45) and (48) are taken into consideration, we find from (61) that at each point of the field-figure

If now in the first term we put