# Regular coordinate systems for Schwarzschild and other spherical spacetimes

- Published: 22 Jan 2000

- University of Guelph Canada

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

[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 Unpublished; posted on http://xxx.lanl.gov/abs/gr-qc/9407005.

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

[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).

[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-