Schwarzschild Spacetime Under Generalised Gullstrand–Painlevé Slicing

  • Colin MacLaurinEmail author
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


We investigate a foliation of Schwarzschild spacetime determined by observers freely falling in the radial direction. This is described using a generalisation of Gullstrand–Painlevé 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.


Gullstrand–Painlevé Falling observer Coordinates 


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Authors and Affiliations

  1. 1.University of QueenslandBrisbaneAustralia

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