publication . Preprint . Article . 2000

Regular coordinate systems for Schwarzschild and other spherical spacetimes

Karl Martel; Eric Poisson;
Open Access English
  • Published: 22 Jan 2000
Abstract
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, using these coordinates comes with several drawbacks, and we advocate the use of simpler, but equally effective, coordinate systems. One such system, introduced by Painleve and Gullstrand in the 1920's, is especially simple and pedagogically powerful; it is, however, still poorly known today. One of our purposes here is therefore to popularize these coordinates. Our other purpose is to provide genera...
Subjects
free text keywords: General Relativity and Quantum Cosmology, Classical mechanics, Deriving the Schwarzschild solution, Schwarzschild metric, Kerr metric, Reissner–Nordström metric, Apparent horizon, Schwarzschild geodesics, Physics, Coordinate time, Schwarzschild coordinates
Related Organizations

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

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[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).

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[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).

[11] This strategy was used by I.D. Novikov to construct yet another set of coordinates for Schwarzschild spacetime. The reference is I.D. Novikov, Doctoral dissertation, Shternberg Astronomical Institute, Moscow (1963). The Novikov coordinates are discussed in Sec. 31.4 of Ref. [2]. Unlike the coordinates considered in this paper, the Novikov coordinates are comoving with respect to the observers to which they are attached. This means that these observers move with a constant value of Novikov's spatial coordinates.

[12] H.P. Robertson and T.W. Noonan, Relativity and cosmology (Saunders, Philadelphia, 1968).

[13] P. Kraus and F. Wilczek, A simple stationary line element for the Schwarzschild geometry, and some applications, Mod. Phys. Lett. A 9, 3713-3719 (1994).

[14] C. Doran, A new form of the Kerr solution, Phys. Rev. D 61, 067503-067506 (2000).

[15] For p > 1, or E˜ < 1, the motion does not extend to in-

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