physics754 General Relativity and Cosmology

Lecturer

Metsch

Semester

Summer 2014

Problem sets

These are my results for the problem sets. There are several errors in there, so please be careful with them.

On problem set 5, all the intermediate results were given. I did not typeset my calculations in LaTeX since there was nothing really new in the problems.

Important formulas

The equations of motions:

$$m_i \left[ \ddot x^\mu_i + \Gamma_{\alpha\beta}^\mu \dot x_i^\alpha \dot x_i^\beta \right] = q_i F^\mu{}_\nu \dot x_i^\nu$$

The geodesic equation:

$$\ddot x^\mu_i + \Gamma_{\alpha\beta}^\mu \dot x_i^\alpha \dot x_i^\beta = 0$$

The Christoffel symbols:

$$\Gamma_{\alpha\beta}^\mu = \frac12 g^{\mu\delta} [g_{\delta\alpha,\beta} + g_{\delta\beta,\alpha} - g_{\alpha\beta,\delta}]$$

Curvature tensor:

$$ R^\mu_{\nu\alpha\beta} = \Gamma^\mu_{\nu\beta,\alpha} - \Gamma^\mu_{\nu\alpha,\beta} + \Gamma^\mu_{\lambda\alpha} \Gamma^\lambda_{\nu\beta} - \Gamma^\mu_{\lambda\beta} \Gamma^\lambda_{\nu\alpha} $$

Ricci tensor:

$$R_{\nu\beta} = R^\alpha{}_{\nu\alpha\beta}$$

Curvature scalar

$$\mathcal R = g^{\nu\beta} R_{\nu\beta}$$

Full action

$$\begin{aligned} \mathcal S = &- \sum_{i=1}^N \int \mathrm d \tau_i \, \left[ m_i \sqrt{g_{\alpha\beta} \dot x^\alpha_i \dot x^\beta_i} + q_i A_\alpha \dot x_i^\alpha \right] \ &- \frac{1}{16\pi} \int \mathrm d^4 x \sqrt{|g|} F_{\alpha\beta} F^{\alpha\beta} - \frac{1}{16\pi G} \int \mathrm d^4 x \sqrt{|g|} \mathcal R \end{aligned}$$

Inhomogeneous Maxwell equations:

$$\frac1{\sqrt{|g|}} \partial_\mu \left[ \sqrt{|g|} F^{\mu\nu} \right] = 4\pi J^\nu$$

Einstein tensor

$$G^{\mu\nu} = R^{\mu\nu} - \frac 12 \mathcal R$$