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Is it pure coincidence that if you put ##c=v_e=\sqrt{2GM/R}## in the escape velocity, you end up with the Schwarzschild radius ##R=2GM/c^2##?

The derivation of the escape velocity is purely classical mechanics. It involves ##E_{kin}=mv^2/2## which is incorrect in special relativity even for massive particles and is entirely useless for massless photons.

One could argue that with the constants ##G,M,c## at hand the form of the Schwarzschild radius follows from dimensional analysis, but that doesn't explain why the factor ##2## is present in both derivations.

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# Derivation of Schwarzschild radius from escape velocity

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