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]

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

and for ${\displaystyle b\neq c}$

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

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

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

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

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

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

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

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

If now in the first term we put

1. The notations ${\displaystyle \psi _{ab},{\overline {\psi _{ab}}}}$ and ${\displaystyle \psi _{ab}^{*}}$ (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 ${\displaystyle p}$ and ${\displaystyle q}$ are two of the numbers 1, 2, 3, 4, ${\displaystyle p'}$ and ${\displaystyle q'}$ denote the other two in such a way that the order ${\displaystyle p\ q\ p'\ q'}$ is obtained from 1 2 3 4 by an even number of permutations of two ciphers.
If ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$ are replaced by ${\displaystyle x,y,z,t}$ and if for the stresses the usual notations ${\displaystyle X_{x},X_{y}}$, etc., are used (so that e.g. for a surface element ${\displaystyle d\sigma }$ perpendicular to the axis of ${\displaystyle x,X_{x}}$ is the first component of the force per unit of surface which the part of the system situated on the positive side of ${\displaystyle d\tau }$ exerts on the opposite part) then ${\displaystyle {\mathfrak {T}}_{1}^{1}=X_{x},{\mathfrak {T}}_{1}^{2}=X_{y}}$, etc. Further ${\displaystyle -{\mathfrak {T}}_{1}^{4},-{\mathfrak {T}}_{2}^{4},-{\mathfrak {T}}_{3}^{4}}$ are the components of the momentum per unit of volume and ${\displaystyle {\mathfrak {T}}_{4}^{1},{\mathfrak {T}}_{4}^{2},{\mathfrak {T}}_{4}^{3}}$ the components of the energy-current. Finally ${\displaystyle {\mathfrak {T}}_{4}^{4}}$ is the energy per unit of volume.