The Differential Equations of the Gravitational Field

The differential equation of a gravitational field which is due to a matter- energy tensor are

These equations have to be specialized for a static field of the type (2). By a straight forward calculation the following equations are obtained for a point on the z-axis:

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These equations have to be specialized for a static field of the type (2). By a straight forward calculation the following equations are obtained for a point on the x-axis:

For Tu and Twe have to substitute the expressions given by (7a), (7b). As m is to be considered a given constant, the only functions of the coördinates in these equations are n, a, and b. It is to be expected in the first place that n, i.e. radial distribution of matter, remains undetermined by the equations. This makes necessary the existence of an identity between the equations (9), (10), (11).

In fact such an identity exists. Its form is

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It may be obtained in the following way: We have constructed T., by consider- ing particles which satisfy the equations of motion in the field.

Therefore the covariant divergence of this tensor is bound to vanish identically. On the other hand, the divergence of RR vanishes identically on account of the Bianchi identities.

Of these 4 equations having the form of divergences only the one with the index 3 yields anything which does not already vanish identically with respect to the G., and that is (12). From the form of (12) it follows that (10) is the consequence of (9) and (11).

The problem is therefore reduced to (9) and (11), and the particle density remains undetermined, as was to be expected.

This result makes possible a further simplification of the problem. If, in (9), the quantities a = In (ra) and 8 In b are introduced, we obtain the equation

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By taking into account (13) and (7a), we obtain from (11)

This is a differential equation for a alone. When a is already known b is obtained by a simple integration from