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- Summary
- Why does the derivation of gravitational time dilation use classical mechanics' definition of kinetic energy as opposed to the relativistic definition of kinetic energy?

Why do we use the equation ##\frac {1}{2}mv^2 = \frac {GmM}{r}## to derive potential velocity, and then put that in the Lorentz factor in order to derive gravitational time dilation? Shouldn't we be using the relativistic definition of kinetic energy -> ##mc^2(\gamma - 1)## to derive the gravitational time dilation? Using the first approach we get $$\gamma = \frac 1 {\sqrt {1 - \frac {2GM} {rc^2}}},$$ but using the relativistic definition of kinetic energy we get $$\gamma = \frac {GM} {rc^2}+1$$, so clearly one of these approaches is wrong.