Riemann Tensor of the Ambient Universe, the Dilaton and the Newton's Constant

TL;DR

This paper explores a 4D universe embedded in 5D spacetime, deriving the Riemann tensor and equations that relate the dilaton field to Newton's constant, offering new insights into higher-dimensional gravity models.

Contribution

It generalizes the Gauss-Codacci equations to a 5D context and establishes a relation between the dilaton field and Newton's constant, including solutions for the dilaton.

Findings

Square of the dilaton equals Newton's constant

Derived generalized equations for the embedded universe

Found plausible constant and non-constant dilaton solutions

Abstract

We investigate a four-dimensional world, embedded into a five-dimensional spacetime, and find the five-dimensional Riemann tensor via generalisation of the Gauss (--Codacci) equations. We then derive the generalised equations of the four-dimensional world and also show that the square of the dilaton field is equal to the Newton's constant. We find plausable constant and non-constant solutions for the dilaton.