Page:LorentzGravitation1916.djvu/37

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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.