Schwarzschild Geodesics in Terms of Elliptic Functions and the Related Red Shift
Gunter Scharf
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DOI: 10.4236/jmp.2011.24036   PDF   HTML     5,231 Downloads   9,647 Views   Citations

Abstract

Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For example we calculate the perihelion precession and the light deflection in the post-Einsteinian approximation. The bounded orbits are computed in the post-Newtonian order. As a topical application we calculate the gravitational red shift for a star moving in the Schwarzschild field.

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G. Scharf, "Schwarzschild Geodesics in Terms of Elliptic Functions and the Related Red Shift," Journal of Modern Physics, Vol. 2 No. 4, 2011, pp. 274-283. doi: 10.4236/jmp.2011.24036.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] D. J. D’Orazio and P. Saha, “An Analytic Solution for Weak-field Schwarzschild Geodesics,” Monthly Notices of the Royal Astronomical Society, Vol. 406, pp. 2787-2792.
[2] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover Publications, Inc., New York.
[3] G. Scharf and Gen. Relativ. Gravit, “From Massive Gravity to Modified General Relativity,” General Relativity and Gravitation, Vol. 42, pp. 471-487. doi: 10.1007/s10714-009-0864-0
[4] S. Chandrasekhar, “The Mathematical Theory of Black Holes,” Oxford/New York, Clarendon Press/Oxford University Press, 1983.
[5] E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis,” Cambridge University Press, 1950.
[6] A. Erdelyi et al., “Higher Transcendental Functions,” McGraw-Hill Book Co., Inc., New York, 1953.
[7] J. Tannery, J. Molk, “Fonctions elliptiques,” Chelsea Publishing Company, Bronx, New York , 1972.
[8] Ch. Darwin, Proc. Roy. S. London A 249 (1959) 180, A 263 (1961) 39.
[9] S. Weinberg, “Gravitation and Cosmology,” John Wiley, New York, 1972.
[10] G. Scharf, “Quantum Gauge Theories-Spin One and Two,” Google-Books (2010) free access.
[11] Y. Hagihara, Japanese J. Astron. Geophys. 8 (1930) 68.

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