The study of conformal geometry and its exact solution of the geodesic deviation equation

TL;DR

This paper explores conformal geometry's properties, solves the geodesic deviation equation exactly, and analyzes the stress-energy tensor, revealing a specific equation of state and behavior in the low-energy regime.

Contribution

It provides an exact solution to the geodesic deviation equation within conformal geometry and compares the stress-energy tensor to standard models, deriving a novel equation of state.

Findings

Exact solution to geodesic deviation equation in conformal geometry

Derived equation of state P = -1/3 rho in 4D spacetime

Analyzed stress-energy tensor in low-energy regime

Abstract

In this paper, the geometric properties of the conformal metric are studied and its exact solution of the geodesic deviation equation is presented. We also find out the stress-energy tensor of this geometry and compare it with the usual prefect-fluid case, obtaining an equation of state as $P = -\frac{1}{3}\rho$ in 4D space-time dimension. Finally, the low-energy regime of the metric is studied, in which we obtain the stress-energy tensor proportional to the projection tensor.