Uniqueness of the Machian Solution in the Brans-Dicke Theory

Machian solutions of which the scalar field exhibits the asymptotic behavior $\phi =O(\rho /\omega)$ are generally explored for the homogeneous and isotropic universe in the Brans-Dicke theory. It is shown that the Machian solution is unique for the closed and the open space. Such a solution is restricted to one that satisfies the relation $GM/c^{2}a=const$, which is fixed to $\pi $ in the theory for the closed model. Another type of solution satisfying $\phi =O(\rho /\omega)$ with the arbitrary coupling constant $% \omega $ is obtained for the flat space. This solution has the scalar field $% \phi \propto \rho t^{2}$ and also keeps the relation $GM/c^{2}R=const$ all the time. This Machian relation and the asymptotic behavior $\phi =O(\rho /\omega)$ is equivalent to each other in the Brans-Dicke theory.