Page:LorentzGravitation1916.djvu/37

electromagnetic system; for this purpose we must use the equations (45) and (46) (1915) or in Einstein's notation, which we shall follow here,[1]

 $\mathfrak{T}_{c}^{c}=-\mathrm{L}+\sum\limits _{a\ne c}(a)\psi_{ac}^{*}\psi_{a'c'}$ (59)

and for $b\ne c$

 $\mathfrak{T}_{c}^{b}=\sum\limits _{a\ne c}(a)\psi_{ac}^{*}\psi_{a'c'}$ (60)

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

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

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

 $\delta_{\psi}\int\mathrm{L}dS+\frac{1}{2\varkappa}\delta\int QdS=0$ (61)

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

 $\sum(ab)\left(\frac{\partial\mathrm{L}}{\partial g^{ab}}\right)_{\psi}\delta g^{ab}+\frac{1}{2\varkappa}\sum(ab)G_{ab}\delta\mathfrak{g}^{ab}=0$ (62)

If now in the first term we put

1. The notations $\psi_{ab},\overline{\psi_{ab}}$ and $\psi_{ab}^{*}$ (see (27), (29) and § 11, 1915), will however be preserved though they do not correspond to those of Einstein. As to formulae (59) and (60) it is to be understood that if $p$ and $q$ are two of the numbers 1, 2, 3, 4, $p'$ and $q'$ denote the other two in such a way that the order $p\ q\ p'\ q'$ is obtained from 1 2 3 4 by an even number of permutations of two ciphers.
If $x_{1},x_{2},x_{3},x_{4}$ are replaced by $x, y, z, t$ and if for the stresses the usual notations $X_{x},X_{y}$, etc., are used (so that e.g. for a surface element $d\sigma$ perpendicular to the axis of $x,X_{x}$ is the first component of the force per unit of surface which the part of the system situated on the positive side of $d\tau$ exerts on the opposite part) then $\mathfrak{T}_{1}^{1}=X_{x},\mathfrak{T}_{1}^{2}=X_{y}$, etc. Further $-\mathfrak{T}_{1}^{4},-\mathfrak{T}_{2}^{4},-\mathfrak{T}_{3}^{4}$ are the components of the momentum per unit of volume and $\mathfrak{T}_{4}^{1},\mathfrak{T}_{4}^{2},\mathfrak{T}_{4}^{3}$ the components of the energy-current. Finally $\mathfrak{T}_{4}^{4}$ is the energy per unit of volume.