Brans-Dicke Theory on $M_4\times Z_2$ Geometry

TL;DR

This paper derives the Brans-Dicke theory using a simplified geometric approach on a space combining four-dimensional spacetime with a two-point discrete space, clarifying the role of the discrete space in the theory.

Contribution

The paper presents a more straightforward geometric derivation of the Brans-Dicke theory on a $M_4 imes Z_2$ space, avoiding the complexities of noncommutative geometry.

Findings

Clearer geometric interpretation of the $Z_2$ space in Brans-Dicke theory

Simplified derivation compared to noncommutative geometry methods

Enhanced understanding of the discrete space's effect on gravitational theory

Abstract

The gauge theory on $M_4\times Z_2$ geometry is applied to the Brans-Dicke(BD) theory, where $M_4$ is the four dimensional space-time and $Z_2$ is a discrete space with two points. This approach had been previously proposed by Konisi and Saito without recourse to noncommutative geometry(NCG). Since our approach is geometrically simpler and clearer than NCG, one can see more directly the effect of the $Z_2$ space in obtaining the BD theory.