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

Abstract

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