Geodesics in the Generalized Schwarzschild Solution

Details Geodesics in the Generalized Schwarzschild Solution Geodesics in the Generalized Schwarzschild Solution

2003-11-12

arxiv.org/abs/gr-qc/0311038v2

Am.J.Phys. 72 (2004) 1204-1209

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

7 pages, 1 figure; revised to reflect referee comments; submitted to Am. J. of Phys

Scientific paper

10.1119/1.1701846

Since Schwarzshild discovered the point-mass solution to Einstein's equations that bears his name, many equivalent forms of the metric have been catalogued. Using an elementary coordinate transformation, we derive the most general form for the stationary, spherically-symmetric vacuum metric, which contains one free function. Different choices for the function correspond to common expressions for the line element. From the general metric, we obtain particle and photon trajectories, and use them to specify several time coordinates adapted to physical situations. The most general form of the metric is only slightly more complicated than the Schwarzschild form, which argues effectively for teaching the general line element in place of the diagonal metric.

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