Embedding spacetime via a geodesically equivalent metric of Euclidean signature

TL;DR

This paper introduces a dual Euclidean metric that shares the same geodesics as a Lorentzian spacetime, enabling visualization of gravitational effects through embedded surfaces, aiding teaching of general relativity.

Contribution

It presents a method to construct a Euclidean signature metric with identical geodesics to a Lorentzian spacetime, facilitating intuitive visualization of gravity effects.

Findings

Dual metric shares geodesics with original spacetime

Embedded surfaces illustrate particle motion in gravity

Applicable to higher dimensions and planetary models

Abstract

Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent, or dual, metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles move on the shortest path. Thus one can visualize how acceleration in a gravitational field is explained by particles moving freely in a curved spacetime. Freedom in the dual metric allows us to display, with substantial curvature, even the weak gravity of our Earth. This may provide a nice pedagogical tool for elementary lectures on general relativity. I also study extensions of the dual metric scheme to higher dimensions. In an addendum I extend the analysis concerning the shape of an embedding of the dual spacetime of a line through a planet of constant proper density.