Moore General Relativity Workbook Solutions Direct

This factor describes the difference in time measured by the two clocks.

The gravitational time dilation factor is given by

where $\eta^{im}$ is the Minkowski metric.

Derive the geodesic equation for this metric. moore general relativity workbook solutions

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

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

$$\frac{d^2r}{d\lambda^2} = -\frac{GM}{r^2} \left(1 - \frac{2GM}{r}\right) \left(\frac{dt}{d\lambda}\right)^2 + \frac{GM}{r^2} \left(1 - \frac{2GM}{r}\right)^{-1} \left(\frac{dr}{d\lambda}\right)^2$$ This factor describes the difference in time measured

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

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

The equation of motion for a radial geodesic can be derived from the geodesic equation. After some algebra, we find where $\lambda$ is a parameter along the geodesic,

After some calculations, we find that the geodesic equation becomes

which describes a straight line in flat spacetime.

Consider the Schwarzschild metric