NOVEMBER , 1963 CENTRALLY-SYMMETRIC GRAVITATIONAL FIELDS

THE importance of the particular problem of determining the field in the case of central symmetry in Einstein's theory of gravitation is well known. It is most frequently necessary to consider a centrally-symmetric field in vacuum, defined by a Schwartzschild metric. As regards the problem of centrally-symmetric fields, not necessarily static ones, in vacuum, their existence has been rejected on the basis of the following statement, first expressed by Birkhoff[l]: every centrallysymmetric field in vacuum is static, and therefore, with accuracy up to a coordinate transformation, it is defined by the Schwartzschild metric. This statement is generally accepted and is ineluded in all serious monographs on the general theory of relativity [27] (with the exception of Fock' s book [B]), has previously also been accepted by the author of this paper, [9] is often used in relativistic mechanics and cosmology as a basis for crucial conclusions, and is correct under certain conditions, but only under those conditions. In this paper a rigorous analysis is presented of the conditions under which the solution of the equations of the centrally-symmetric field in vacuum is sought: (1) a physical interpretation of these conditions is given, and the general solution of Eqs. (1) is determined which will in general be nonstatic and will contain a functional arbitrariness which cannot be eliminated. It is easy to cite a formal example of a metric satisfying the field equations in vacuum (1) which is centrally-symmetric, and nonetheless nonstatic. For this purpose it is sufficient to take, for instance, the Schwartzschild metric in polar coordinates eds2 = _rdr 2 r 2 (dU2 -;-sin2 f)dcp 2) -ira dt 2 (e = ± 1 ), a-r r (2) where a is the gravitational radius, consider it in the space-time region inside the "hypersphere" ( r < a) and carry out the substitution r t. We then obtain the metric