Variation of G, ${\mathrm{\Lambda }}_{(4)}$ and Vacuum Energy From Brane-World Models

In brane-world theory in five dimensions, the bulk metric is usually written in gaussian coordinates, where $g_{4\mu }=0$ and $g_{44}=-1$. However, the choice $g_{44}=-1$ is an external condition, not a requirement of the field equations. In this paper we study the consequences of having $g_{44} = \epsilon \Phi^2$,where$\epsilon = \pm 1$and$\Phi$ is a scalar function varying with time, $\dot{\mathrm{\Phi }}\mathrm{\ne }0$. This varying field entails the possibility of variable fundamental physical “constants”. These variations are different from those predicted in scalar-tensor and multidimensional theories. We solve the five-dimensional equations for a {\em fixed} brane and use the brane-world paradigm to determine the fundamental parameters in the theory, which are the vacuum energy $\sigma$, the gravitational coupling $G$ and the cosmological term ${\mathrm{\Lambda }}_{(4)}$. We present specific models where these physical quantities are variable functions of time. Different scenarios are possible but we discuss with some detail a model for which $\dot{G}\mathrm{/}G\sim H$ and ${\mathrm{\Lambda }}_{(4)}\sim H^2$, which seems to be favored by observations. Our results are not in contradiction to previous ones in the literature. In particular, to those where the brane is described as a domain wall moving in a static $Sch-AdS$ bulk. Indeed these latter models in RS scenarios describe the same spacetime as other solutions (with fixed brane) in gaussian coordinates with $\dot{\mathrm{\Phi }}=0$. We conclude that the introduction of a time-varying $\mathrm{\Phi }$ in brane-world theory yields a number of models that show variation in the fundamental physical “constants” and exhibit reasonable physical properties.