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]
|
(59)
|
and for
|
(60)
|
The set of quantities
might be called the stress-energy-complex (comp. § 38). As for a change of the system of coordinates the transformation formulae for
are similar to those by which tensors are defined, we can also speak of the stress-energy-tensor. We have namely
§ 41. The equations for the gravitation field are now obtained (comp. §§ 13 and 14, 1915) from the condition that
|
(61)
|
for all variations
which vanish at the boundary of the field of integration together with their first derivatives. The index
in the first term indicates that in the variation of
the quantities
must be kept constant.
If we suppose
to be expressed in the quantities
and if (42), (45) and (48) are taken into consideration, we find from (61) that at each point of the field-figure
|
(62)
|
If now in the first term we put
- ↑ The notations
and
(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
and
are two of the numbers 1, 2, 3, 4,
and
denote the other two in such a way that the order
is obtained from 1 2 3 4 by an even number of permutations of two ciphers.
If
are replaced by
and if for the stresses the usual notations
, etc., are used (so that e.g. for a surface element
perpendicular to the axis of
is the first component of the force per unit of surface which the part of the system situated on the positive side of
exerts on the opposite part) then
, etc. Further
are the components of the momentum per unit of volume and
the components of the energy-current. Finally
is the energy per unit of volume.