where $\eta^{im}$ is the Minkowski metric.
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$
Derive the geodesic equation for this metric.
$$ds^2 = -\left(1 - \frac{2GM}{r}\right) dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$$
Consider two clocks, one at rest at infinity and the other at rest at a distance $r$ from a massive object. Calculate the gravitational time dilation factor.
$$\frac{d^2r}{d\lambda^2} = -\frac{GM}{r^2} \left(1 - \frac{2GM}{r}\right) \left(\frac{dt}{d\lambda}\right)^2 + \frac{GM}{r^2} \left(1 - \frac{2GM}{r}\right)^{-1} \left(\frac{dr}{d\lambda}\right)^2$$
Consider the Schwarzschild metric
The geodesic equation is given by
