Uniqueness of the Machian Solution in the Brans-Dicke Theory

TL;DR

This paper investigates the uniqueness of Machian solutions in the Brans-Dicke theory, showing that such solutions are unique for closed and open universes and exploring their properties in flat space.

Contribution

It demonstrates the uniqueness of Machian solutions in the Brans-Dicke theory for closed and open universes and derives a new solution for flat space with arbitrary coupling constant.

Findings

Machian solutions are unique for closed and open universes.

A new solution for flat space with arbitrary omega is found.

The Machian relation GM/c^2a=const is fixed to pi in the closed model.

Abstract

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.