Time-Variation of the Gravitational Constant and the Machian Solution in the Brans-Dicke Theory

Abstract

The Machian cosmological solution satisfying $ϕ=O(ρ/ω)$ for the perfect-fluid with negative pressure is discussed. When the coefficient of the equation of state $γ\to -1/3$, the gravitational constant approaches to constant. If we assume the present mass density $ρ_{0}\sim ρ_{c}$ (critical density), the parameter $ε$ ($γ=(ε-1)/3$) has a value of order $10^{-3}$ to support the present gravitational constant. The closed model is valid for $ω<-3/2ε$ and exhibits the slow accelerating expansion. We understand why the coupling parameter $| ω|$ is so large ($ω\sim -10^{3}$). The time-variation of the gravitational constant $| \dot{G}/G| \sim 10^{-13} yr^{-1}$ at present is derived in this model.

10 pages, LaTeX2e