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