Problem sheet 5

Lecture: Introduction to general relativity, 2023-2024

Deadline: 04.02.2024

Topic: Gravitational light bending, gravitational redshift

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. Light deflection angle from the Sun

Calculate the deflection angle of light due to the gravitational light bending by the Sun for a distant point-like object observed on Earth. Assume that the object's position on the celestial sphere is separated from the position of the center of the Sun's image by the angle $\alpha = \frac{\pi}{2}$. Note: you cannot use the formula from the lecture, it requires a correction. Can you see why?

More difficult, optional: Derive and plot the deflection angle $\theta$ for any angular seperation $\alpha$ of the source from the center of the Sun's image, $\alpha_{\rm min} \le \alpha \le \pi$, where $\alpha_{\rm min}$ is half of the angular size of the Sun as seen from the Earth. You may use Python/Mathematica or your favourite software.

Problem 2. Gravitational frequency shift in the Earth-Moon system

Calculate the gravitational frequency shift between the surface of the Moon and the surface of Earth (ignore the Doppler effect). Take into account the gravitational potential from the Earth and from the Moon. Estimate also the variable contribution from the potential of the Sun.