Let us now suppose that only the coordinate undergoes an infinitely small change, which has the same value at all points of the field-figure. Let at the same time the system of values be shifted everywhere in the direction of over the distance . The left hand side of the equation then becomes and we have on the right hand side
After dividing the equation by we may thus, according to (74) and (75), write
By the same division we obtain from the expression occurring on the left hand side of (51), which we have represented by
where the complex is defined by (52) and (53). If therefore we introduce a new complex which differs from only by the factor , so that
The form of this equation leads us to consider as the stress-energy-complex of the gravitation field, just as is the stress-energy-tensor for the matter. We need not further explain that for the case the four equations contained in (79) express the conservation of momentum and of energy for the total system, matter and gravitation field taken together.
§ 48. To learn something about the nature of the stress-energy-complex we shall consider the stationary gravitation field caused by a quantity of matter without motion and distributed symmetrically around a point . In this problem it is convenient to introduce for the three space coordinates , ( will represent the time) "polar" coordinates. By we shall therefore denote a quantity