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

Abstract

The Machian cosmological solution satisfying $\varphi =O(\rho /\omega )$ 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 $|\dot{G}/G|\sim 10^{-13}y{r}^{-1}$ at present is derived in this model.

10 pages, LaTeX2e