Uniqueness of the Machian Solution in the Brans-Dicke Theory

Machian solutions of which the scalar field exhibits the asymptotic behavior φ = O(ρ/ω) 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/c2a = const, which is fixed to π in the theory for the closed model. Another type of solution satisfying φ = O(ρ/ω) with the arbitrary coupling constant ω is obtained for the flat space. This solution has the scalar field φ ∝ ρ t2 and also keeps the relation GM/c2R = const all the time. This Machian relation and the asymptotic behavior φ = O(ρ/ω) is equivalent to each other in the Brans-Dicke theory. PACS numbers: 04.50.+h, 98.80.-k ∗Email: miyazaki@loyno.edu, miyazaki@nagasakipu.ac.jp 1