Regular coordinate systems for Schwarzschild and other spherical spacetimes

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Martel, Karl; Poisson, Eric;

The continuation of the Schwarzschild metric across the event horizon is almost always (in textbooks) carried out using the Kruskal-Szekeres coordinates, in terms of which the areal radius r is defined only implicitly. We argue that from a pedagogical point of view, usi... View more
  • References (15)
    15 references, page 1 of 2

    [1] B.F. Schutz, A first course in general relativity (Cambridge University Press, Cambridge, 1985).

    [2] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

    [3] R.M. Wald, General relativity (University of Chicago, Chicago, 1984).

    [4] P. Painlev´e, La M´ecanique classique et la th´eorie de la relativit´e, C. R. Acad. Sci. (Paris) 173, 677-680 (1921).

    [5] A. Gullstrand, Allegemeine L¨osung des statischen Eink¨orper-problems in der Einsteinschen Gravitations theorie, Arkiv. Mat. Astron. Fys. 16(8), 1-15 (1922).

    [6] K. Lake, A class of quasi-stationary regular line elements for the Schwarzschild geometry (1994). Unpublished; posted on

    [7] R. Gautreau and B. Hoffmann, The Schwarzschild radial coordinate as a measure of proper distance, Phys. Rev. D 17, 2552-2555 (1978).

    [8] R. Gautreau, Light cones inside the Schwarzschild radius, Am. J. Phys. 63, 431-439 (1995).

    [9] The Gautreau-Hoffmann coordinates are very similar to the coordinates of Sec. III. They also constitute a oneparameter family of coordinate systems, and their parameter Ri is related to our p by Ri = 2M p/(p − 1). The difference lies with the fact that while their Ri is not meant to be negative, our p is restricted to the interval 0 < p ≤ 1. These are mutually exclusive statements. But the point remains that formally, the coordinates of Sec. III are identical to the Gautreau-Hoffmann coordinates for Ri < 0. Note that for Ri > 0 (the case considered by Gautreau and Hoffmann in Ref. [7]), the surfaces of constant time extend only up to r = Ri; they do not reach infinity.

    [10] S. Corley and T. Jacobson, Lattice Black Holes, Phys. Rev. D 57, 6269-6279 (1998).

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