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Schwarzschild Spacetime Under Generalised Gullstrand–Painlevé Slicing

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Einstein Equations: Physical and Mathematical Aspects of General Relativity (DOMOSCHOOL 2018)

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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–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.

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Notes

  1. 1.

    Infinity can be made rigorous by conformal compactification, which produces a new manifold with a boundary consisting of timelike, null, and spacelike infinities. However, for our purposes a simple limit r →∞ is often sufficient, or at least an approximation r ≫ 2M. Physically, infinity is loosely analogous with Solar System observers far from a black hole.

  2. 2.

    Recall the intuition for combining a vector with a 1-form dT to produce a scalar, as the number of level sets T = const crossed by the vector [47, §3.3] [35, §2.5].

  3. 3.

    We write ds for both, but these should not be equated, as they are restrictions of the full spacetime metric along different 4-vectors.

  4. 4.

    By “ruler” we mean technically a vector orthogonal to u in the local tangent space, but intended as an approximation to an extended object on the manifold. This could be a hypothetical construction based on radar results, or a “resilient” physical rod [42, §2.5] whenever the rod hypothesis is justified.

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Author information

Authors and Affiliations

  1. University of Queensland, Brisbane, QLD, Australia

    Colin MacLaurin

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  1. Colin MacLaurin
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Correspondence to Colin MacLaurin .

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Editors and Affiliations

  1. Department of Science and High Technology, University Insubrien, Como, Italy

    Sergio Cacciatori

  2. Mathematisches Institut, Humboldt-Universität zu Berlin, Berlin, Germany

    Batu Güneysu

  3. Department of Science and High Technology, University Insubrien, Como, Italy

    Stefano Pigola

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MacLaurin, C. (2019). Schwarzschild Spacetime Under Generalised Gullstrand–Painlevé Slicing. In: Cacciatori, S., Güneysu, B., Pigola, S. (eds) Einstein Equations: Physical and Mathematical Aspects of General Relativity. DOMOSCHOOL 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-18061-4_9

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