Moore General Relativity Workbook: Solutions

$$\frac{d^2r}{d\lambda^2} = -\frac{GM}{r^2} \left(1 - \frac{2GM}{r}\right) \left(\frac{dt}{d\lambda}\right)^2 + \frac{GM}{r^2} \left(1 - \frac{2GM}{r}\right)^{-1} \left(\frac{dr}{d\lambda}\right)^2$$

The gravitational time dilation factor is given by

$$\Gamma^0_{00} = 0, \quad \Gamma^i_{00} = 0, \quad \Gamma^i_{jk} = \eta^{im} \partial_m g_{jk}$$

where $\lambda$ is a parameter along the geodesic, and $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols. moore general relativity workbook solutions

$$\frac{d^2x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda} = 0$$

Consider a particle moving in a curved spacetime with metric

This factor describes the difference in time measured by the two clocks. where $L$ is the conserved angular momentum

Consider the Schwarzschild metric

Derive the geodesic equation for this metric.

where $L$ is the conserved angular momentum. the non-zero Christoffel symbols are

$$\frac{d^2r}{d\lambda^2} = -\frac{GM}{r^2} + \frac{L^2}{r^3}$$

Using the conservation of energy, we can simplify this equation to

$$\frac{t_{\text{proper}}}{t_{\text{coordinate}}} = \sqrt{1 - \frac{2GM}{r}}$$

For the given metric, the non-zero Christoffel symbols are