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Relativity, Cosmology, and Time

Schwarzschild's Solution and Black Hole (2018 Edition)

The equations of metric tensors can be solved exactly for the case of a centrally symmetric field in vacuum with mass M at the center. In terms of spherical coordinates and ct, the "world line" has the curvilinear form :

ds2 = (1 - 2GM/c2r) c2dt2 - dr2 / (1 - 2GM/c2r) - r2 (sin2 d2 + d2) ---------- (13)

This is the celebrated Schwarzschild solution. The integration constant often referred to as the Schwarzschild's radius (or event horizon) rs = 2GM/c2 appears in the formula in order to relate to Newton's inverse square law, otherwise it is rather arbitrary. Some special properties about this metric are listed in the followings :

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