Schwarzschild spacetime under generalised Gullstrand-Painlev\'e slicing

TL;DR

This paper explores a generalized foliation of Schwarzschild spacetime using freely falling observers, providing new insights into the geometry and potential implications for quantum fields and gravity theories.

Contribution

It introduces a generalized Gullstrand-Painlevé coordinate system for Schwarzschild spacetime, expanding the understanding of different observer-based foliations.

Findings

Distinct spatial geometries for static and falling observers

Embedding diagrams differ between foliations

Potential implications for quantum field theory and gravity

Abstract

We investigate a foliation of Schwarzschild spacetime determined by observers freely falling in the radial direction. This is described using a generalisation of Gullstrand-Painlev\'e coordinates which allows for any possible radial velocity. This foliation provides a contrast with the usual static foliation implied by Schwarzschild coordinates. The $3$-dimensional spaces are distinct for the static and falling observers, so the embedding diagrams, spatial measurement, simultaneity, and time at infinity are also distinct, though the $4$-dimensional spacetime is unchanged. Our motivation is conceptual understanding, to counter Newton-like viewpoints. In future work, this alternate foliation may shed light on open questions regarding quantum fields, analogue gravity, entropy, energy, and other quantities. This article is aimed at experienced relativists, whereas a forthcoming series is intended for a general audience of physicists, mathematicians, and philosophers.