|
(77)
|
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
|
(78)
|
we find
|
(79)
|
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