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]

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

(59)

and for $b\neq c$

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

(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.