In order to investigate the possibility of a physical system radiating gravitational waves,

(a) The metric tensor is independent of the angle .

(b) It is diagonal.

If one writes down the field equations in the empty space surrounding the system

one obtains a set of 7 equations (since
vanishes identically if one index is equal to 3) for the 4 diagonal components of
. Among these there exist 3 identities (the Bianchi identities,

The field equations are non-linear and difficult to solve. It is proposed to investigate them by the method of successive approximations. As a beginning, the first approximation can be calculated. Let us write the line element in the form

where , , , and are regarded as small of the first order. The linear approximation of the field equations has the following form (indexes denoting partial differentiation):

From these equations it is possible to derive the wave equation for ,

and to express the other unknowns in terms of the solution for .

If we represent the radiating system by a quadrupole

Here

It is planned to use this solution as the starting point for a more accurate calculation. The interesting question, of course, is whether the exact equations have a solution going over into the above for sufficiently weak fields.

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J. N. GOLDBERG

TONNELAT:

in which is the Ricci tensor built from an arbitrary affine connection . The variations , lead to the field equations. However, if one tries to apply the Einstein, Infeld, Hoffmann method to these equations, one obtains merely the results of general relativity, and one does not obtain the equations for a charged particle. This result stems from the condition

which is imposed by the theory.

To avoid this situation, one can start from an affine connection with vanishing torque

Lagrange multipliers are needed in the variations of because the 64 are not independent and, moreover, the vector related to the torque is introduced. In this case, one obtains

Introducing the metric

one obtains

with

Setting

defines a tensor whose divergence gives a Lorentz force. This tensor plays in the field equations the role of a Maxwell tensor and leads, with the use of an extension of the Einstein, Infeld, Hoffmann method, to a Coulomb force.