Problem sheet 3

Lecture: Introduction to general relativity, 2023-2024

Deadline: Dec 9th, 2023 (3 weeks)

Topic: coordinate transformations, Christoffel symbols and connection

Please send the solutions via email to korzynski@cft.edu.pl. I prefer PDF's (for example LaTeX-generated) or common graphic formats (for example scans of your hand-written notes).

Problem 1.

Express the flat Minkowski metric

g = -dt^2 + dx^2 +dy^2 + dz^2

in the new coordinates $(\tau,x,y ,u)$ related to the old $(t,x,y,z)$ by

t = \frac{1}{\sqrt{1-u^2}}\,\tau

z = \frac{u}{\sqrt{1-u^2}}\,\tau,

while $x$ and $y$ do not change. Assume that $\tau > 0$ and $u \in {\bf R}$ .

Additional question, not evaluated: which part of the Minkowski space does this coordinate system cover?

Problem 2. Completing the proof from Lecture 7 that the Levi-Civita connection is the only metric-compatible and torsion-free connection

Show that if the connection coefficients are given by

\Gamma^\mu{}_{\alpha\beta}=\frac{1}{2}\,g^{\mu\nu}\left(g_{\nu\alpha,\beta}+g_{\nu\beta,\alpha}-g_{\alpha\beta,\nu} \right),

then:

the connection is torsion-free, i.e. $\Gamma^\mu{}_{\alpha\beta}=\Gamma^\mu{}_{\beta\alpha}$ ,
we have indeed $g_{\mu\nu;\sigma}=0$ .