Moore General Relativity Workbook Solutions Access

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

where $L$ is the conserved angular momentum.

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

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

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

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

For the given metric, the non-zero Christoffel symbols are Using the conservation of energy, we can simplify

After some calculations, we find that the geodesic equation becomes

Consider a particle moving in a curved spacetime with metric

which describes a straight line in flat spacetime. Using the conservation of energy

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

$$\frac{d^2t}{d\lambda^2} = 0, \quad \frac{d^2x^i}{d\lambda^2} = 0$$

$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$

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.

The gravitational time dilation factor is given by