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.

#	As handed in
1	physics754-01-handed_in.pdf
2	physics754-02-handed_in.pdf
3	physics754-03-handed_in.pdf
4	physics754-04-handed_in.pdf
5
6	physics754-06-handed_in.pdf
7	physics754-07-handed_in.pdf
8	physics754-08-handed_in.pdf
9	physics754-09-handed_in.pdf
10	physics754-10-handed_in.pdf

Important formulas

The equations of motions:

$$m_i[{\ddot{x}}_i^{\mu }+{\mathrm{\Gamma }}_{\alpha \beta }^{\mu }{\dot{x}}_i^{\alpha }{\dot{x}}_i^{\beta }]=q_i{F}^{\mu }{}_{\nu }{\dot{x}}_i^{\nu }$$

The geodesic equation:

$${\ddot{x}}_i^{\mu }+{\mathrm{\Gamma }}_{\alpha \beta }^{\mu }{\dot{x}}_i^{\alpha }{\dot{x}}_i^{\beta }=0$$

The Christoffel symbols:

$${\mathrm{\Gamma }}_{\alpha \beta }^{\mu }=\frac{1}{2}{g}^{\mu \delta }[g_{\delta \alpha ,\beta }+g_{\delta \beta ,\alpha }-g_{\alpha \beta ,\delta }]$$

Curvature tensor:

$$R_{\nu \alpha \beta }^{\mu }={\mathrm{\Gamma }}_{\nu \beta ,\alpha }^{\mu }-{\mathrm{\Gamma }}_{\nu \alpha ,\beta }^{\mu }+{\mathrm{\Gamma }}_{\lambda \alpha }^{\mu }{\mathrm{\Gamma }}_{\nu \beta }^{\lambda }-{\mathrm{\Gamma }}_{\lambda \beta }^{\mu }{\mathrm{\Gamma }}_{\nu \alpha }^{\lambda }$$

Ricci tensor:

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

Curvature scalar

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

Full action

$$\begin{array}{rl}\mathcal{S}=& -\sum _{i=1}^N\int \mathrm{d}{\tau }_i[m_i\sqrt{g_{\alpha \beta }{\dot{x}}_i^{\alpha }{\dot{x}}_i^{\beta }}+q_i{A}_{\alpha }{\dot{x}}_i^{\alpha }]\\ & -\frac{1}{16\pi }\int {\mathrm{d}}^{4}x\sqrt{|g|}{F}_{\alpha \beta }{F}^{\alpha \beta }-\frac{1}{16\pi G}\int {\mathrm{d}}^{4}x\sqrt{|g|}\mathcal{R}\end{array}$$

Inhomogeneous Maxwell equations:

$$\frac{1}{\sqrt{|g|}}{\partial }_{\mu }\left[\sqrt{|g|}{F}^{\mu \nu }\right]=4\pi J^{\nu }$$

Einstein tensor

$$G^{\mu \nu }=R^{\mu \nu }-\frac{1}{2}\mathcal{R}$$