# Geodesic dual spacetime

## TL;DR

This paper introduces the concept of geodesic duality between spacetime manifolds, where a transformation preserves geodesic structures, and explores its implications for spherically symmetric spacetimes like Schwarzschild and Reissner-Nordström.

## Contribution

It defines geodesic duality, establishes a general framework for diagonal metrics, and applies it to specific spherically symmetric spacetimes.

## Key findings

- Identified conditions for geodesic duality in diagonal metrics.
- Constructed geodesic dual spacetimes for Schwarzschild and Reissner-Nordström.
- Demonstrated that geodesic duality preserves geodesic structures under specific transformations.

## Abstract

A duality between spacetime manifolds, the geodesic duality, is introduced. Two manifolds are geodesic dual, if the transformation between their metrics is also the transformation between their geodesics. That is, the transformation that transforms the metric to the metric of the dual manifold is also the transformation that transforms the geodesic to the geodesic of the dual manifold. On the contrary, for nondual spacetime manifolds, a geodesic is no longer a geodesic after the transformation between the metrics. We give a general result of the duality between spacetime manifolds with diagonal metrics. The geodesic duality of spherically symmetric spacetime are discussed for illustrating the concept. The geodesic dual spacetime of the Schwarzschild spacetime and the geodesic dual spacetime of the Reissner-Nordstr\"om spacetime are presented.