The gravitational time dilation factor is given by
Consider a particle moving in a curved spacetime with metric
$$\frac{d^2r}{d\lambda^2} = -\frac{GM}{r^2} + \frac{L^2}{r^3}$$ moore general relativity workbook solutions
This factor describes the difference in time measured by the two clocks.
$$\Gamma^0_{00} = 0, \quad \Gamma^i_{00} = 0, \quad \Gamma^i_{jk} = \eta^{im} \partial_m g_{jk}$$ The gravitational time dilation factor is given by
Derive the equation of motion for a radial geodesic.
$$\frac{t_{\text{proper}}}{t_{\text{coordinate}}} = \sqrt{1 - \frac{2GM}{r}}$$ \quad \Gamma^i_{00} = 0
After some calculations, we find that the geodesic equation becomes