Problem sheet 4

Lecture: Introduction to general relativity, 2023-2024

Deadline: 22.12.2023

Topic: Christoffel symbols and connection, curvature

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. Christoffel symbols on a hyperbolic plane

Calculate the Christoffel symbols for the 2-dimensional metric of (+,+) signature:

g=d\chi^2+\sinh^2\chi\,d\phi^2,

where $\phi$ runs from 0 to $2\pi$ and $\chi$ from 0 to $\infty$ . Assume the order of coordinates to be $(\chi,\phi)$ .

Problem 2. Curvature of a hyperbolic plane

Calculate the Riemann curvature tensor, the Ricci tensor and the Ricci scalar for the metric from Problem 1.