the corresponding quantities for the matter or the electro-magnetic system with the opposite sign. It is obvious that by this the condition of the conservation of momentum and energy for the whole system would be immediately fulfilled. It was in fact this circumstance that made me think of the tensor . The way in which was introduced in §§ 38 and 39 has only been chosen in order to lay stress on (58) being an identity, so that equation (85) is but another form of (79).
At first sight the relations (87) and the conception to which they have led, may look somewhat startling. According to it we should have to imagine that behind the directly observable world with its stresses, energy etc. there is hidden the gravitation field with stresses, energy etc. that are everywhere equal and opposite to the former; evidently this is in agreement with the interchange of momentum and energy which accompanies the action of gravitation. On the way of a light-beam e.g. there would be everywhere in the gravitation field an energy current equal and opposite to the one existing in the beam. If we remember that this hidden energy-current can be fully described mathematically by the quantities and that only the interchange just mentioned makes it perceptible to us, this mode of viewing the phenomena does not seem unacceptable. At all events we are forcibly led to it if we want to preserve the advantage of a stress-energy-tensor also for the gravitation field. It can namely be shown that a tensor which is transformed in the same way as the tensor defined by (57) and (86) and which in every system of coordinates has the same divergency as the latter, must coincide with .
Finally we may remark that (78), (86), (58), (87) give
so that we have, both from (79) and from (85), .
The question is this, that, so long as the gravitation field is considered as given, we may introduce "external" forces, but that in the equations for the gravitation field itself we must also take into consideration the stress-energy-tensor of the system by which those forces are exerted.