A static, spherically symmetric solution of Yang's vacuum gravitational field equations is given which predicts incorrect values for experimental observations. It is argued that Yang's field equations must be supplemented by further restrictions on the class of allowable space-times.In a recent paper Yang 1 proposed vacuum gravitational field equations of the form a)where Rij is the Ricci tensor and the semicolon denotes covariant differentiation. In a recent note 2 I discussed certain aspects of the static, spherically symmetric solutions of these equations and, in addition, mentioned that seemingly unphysical solutions of (1) had been found. 3 However, these unphysical solutions appear in coordinate systems which do not lend themselves readily to simple physical interpretation*. Arguments to accept such solutions as physically meaningful are therefore not particularly strong. In contrast to these solutions, I shall now present a new solution of (1) whose physical interpretation is obvious and whose physical predictions are sufficiently incorrect to necessitate restricting allowable solutions of (1) to certain classes of space-times, Consider the metric ds 2 = -(l+M/ry 2 dr 2 -r 2 (de 2 + sin 2 edcp 2 ) + (l+M/ry 2 dt 2 .(2)This metric satisfies (1) exactly. 4 It possesses a singularity at r = 0 and has the proper Newtonian correspondence limit so that we must interpret M as the gravitational mass. In addition the metric is flat at infinity but curved for finite values of r. Thus all reasonable criteria for a physically interesting metric are satisfied. However, the equations of motion for (2) predict no light deflection about a massive body and an anomalous perihelion shift of ^ the Einsteinian value but in the retrograde direction. These predictions are in harsh contrast to observed values and imply that (2) must not be allowed as a physically acceptable solution of (1). To exclude (2) from the theory implies a restriction upon the permissible class of spacetimes. It can be shown that (2) is conformally flat and may be transformed into the metric ds 2 = -(l-M/r) 2 x[dr 2 + r 2 (d6 2 +sm 2 edcp 2 )-dt 2 ].(3) An examination of the unphysical metrics found by Thompson and Kilmister seems to confirm that conformally flat solutions of (1) have undesirable physical properties. We believe that such solutions cannot be allowed and must be eliminated from the theory by a constraint However, the elimination of conformally flat solutions does not rid the theory of all unphysical metrics. For example, ds 2 =-(l-2M/ry i dr 2 -r 2 (d0 2 + sm 2 0d

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Recently I argued that Yang's pure-space equations must be supplemented by restrictions on the class of allowable space-times. I now consider time-dependent solutions of the pure-space equations and prove the violation of Birkhoff's theorem. Time-dependent spherically symmetric solutions are displayed, as well as solutions representing plane gravitational waves. The suggestion is made that pure spaces are unphysical and Yang's theory requires the explicit presence of a source in vacuum, in contrast to general relativity.A number of recent Letters 1 " 4 have appeared, based upon the gravitational field equations proposed by Yang 5 which were given by Kilmister and Newman 6 some years earlier. The equations of the Kilmister-Yang (KY) theory are written in the form where R u is the Ricci tensor and S ijk is the source of the field which has yet to be postulated. The KY vacuum equations (called "pure spaces" by Yang) areRecently, I demonstrated that pure spaces admit static, asymptotically flat, spherically symmetric solutions which are unphysical and suggested one should restrict the classes of allowable space-times to those which are not conformally flat and/or do not possess a vanishing scalar curvature. These unphysical space-times have also been discussed by Thompson 2 who calls them "geometrically degenerate." Camenzind has argued that static, spherically symmetric solutions of the pure-space equations should be characterized by two parameters (not simply the mass). 4 If this is correct, my argu-961 ments concerning unphysical solutions are not as strong. Others have suggested that the purespace equations may not obey the geodesic equations of motion, in which case one does not have a mechanism for computing the usual physical phenomena.It is therefore necessary to examine aspects of (2) which circumvent these difficulties to determine whether pure spaces have real significance. Below I examine certain time-dependent solutions of the pure-space equations without calling upon the equations of motion. I will show that the KY equations require more restriction than simply the elimination of degenerate space-times. Indeed, it appears that pure spaces themselves allow the generation of unphysical solutions; and I suggest that one should not examine (1) unless the source S ijk is nonvanishing.The status of Birkhoff's theorem in a gravitation theory is important since its violation implies, for example, that a radially pulsating, spherically symmetric distribution of matter can affect the gravitational field outside the body. Such a notion conflicts with Newtonian theory and current observations. Thus, Birkhoff s theorem is a test of the viability of a theory.The recent Comment by Ni 3 gives conformally flat solutions of the pure-space equations in the

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colored mesons which can decay rapidly via HiO,(3) There are nine semicolored mesons which decay electromagnetically because of conservation of /" spin [ip{i, 3), where 0 ^ e < 8].