Brans-Dicke Theory Under a Transformation of Units and the Three Tests

The Brans-Dicke (BD) gravitational field equations are written down in arbitrary units by taking $\varphi ={\varphi }_0\lambda$ in the original equations, and then scaling length, time, and reciprocal mass by ${\lambda }^{\frac{(1-\alpha )}{2}}\mathrm{}\left(x\right)\mathrm{}$. This is a generalization of Dicke's units transformation (UT). It is found that the Machian requirement that matter make a positive contribution to $\overline{\varphi }(={\varphi }_0{\lambda }^{\alpha })$ cannot be maintained under a UT. Mach's principle finds its expression, instead, in terms of matter making a positive contribution to the dimensionless scalar field $\lambda$. We find a solution to the spherically symmetric field equations in arbitrary units and generalize Schild's formula for perihelion precession to include nongeodesic equations of motion. We use these results to show that the BD field equations lead to the same predictions for the three tests in all units. This fact is consistent with the dimensionless character of the angular measurements and the interpretation of the alternate BD formalisms as being the same theory cast in different units.