The gravitational time dilation factor is given by
This factor describes the difference in time measured by the two clocks.
$$\frac{d^2t}{d\lambda^2} = 0, \quad \frac{d^2x^i}{d\lambda^2} = 0$$ moore general relativity workbook solutions
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$
$$\Gamma^0_{00} = 0, \quad \Gamma^i_{00} = 0, \quad \Gamma^i_{jk} = \eta^{im} \partial_m g_{jk}$$ The gravitational time dilation factor is given by
which describes a straight line in flat spacetime.
Using the conservation of energy, we can simplify this equation to \quad \Gamma^i_{00} = 0
Consider the Schwarzschild metric