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Relativity, Cosmology, and Time

Schwarzschild's Solution and Black Hole

The equations of gravitational field 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 space-time metric has the form:

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

This is known as the Schwarzschild solution. The integration constant 2GM/c2 appears in the formula in order to relate to Newton's inverse square law, otherwise it is rather arbitrary. It is a useful example for illustrating the effect of gravity in general relativity:

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