Regular Solutions in Vacuum Brans-Dicke Theory Compared to Vacuum Einstein Theory

We will confront some static spherically symmetric vacuum Brans-Dicke solutions in the Jordan and Einstein Frames with theRobertson parameters. While the regular solution in the vacuum Einstein theory is just the Schwarzschild black hole, the samein the Jordan frame Brans-Dicke theory is shown to represent not a black hole but a traversable wormhole. But, in this case, thevalid range of becomes too narrow to yield the observed weak field Robertson parameters at the positive mass mouth. Thecorresponding solution in the Einstein frame also provides a regular wormhole, and it yields the correct parametric values but onlyup to “one and half order.” We argue that a second-order contribution can in principle distinguish between the signatures of theregular wormhole and the singular Buchdahl solution in the Einstein frame. Thus, at the level of regular solutions, Brans-Dicketheory in each frame predicts effects very different from those of Einstein’s theory. To our knowledge, these theoretical distinctionsseem not to have received adequate attention so far