← Cosmology

# General Relativity Experiments

Friday, April 22, 2022

## Deflection of Starlight

When light from stars passes by stars, it bends by some angle $\theta$. Using Newtonian gravitation, the beam of photons is assigned an effective mass $m=E/c^2$ and deflected by the Newtonian gravitational force. The deflection angle predicted is

$\theta=\frac{2GM}{Rc^2}$

where $M$ is the mass of the star and $R$ is its radius. This results in a prediction of $\theta=0.87''$.

According to general relativity, spacetime is curved around the star. The predicted angle from general relativity is exactly twice that of Newtonian gravitation: $\theta=1.74''$.

Experiments done during total solar eclipses give results within 10% of general relativity's predictions. Radio emissions from quasars give even stronger agreement: within 2%.

When a line between Earth and a planet like Venus passes right next to the sun (called a superior conjunction), general relativity predicts it will not travel in a Euclidian straight line. Sending radar signals and timing how long they take to return, there should therefore be some delay, about $10^{-4}~\text{s}$. This experiment has been done and resulted in agreement within a few percent. When done with Mars (since we know the terrain better), the result was within 0.1%. The Cassini spacecraft has provided the most sensitive test of time delay, agreeing with general relativity to within 0.002%.

## Precession of Perihelion of Mercury

In a simple planetary system, Newtonian gravitation predicts the planet travels in a perfect ellipse around the star. The equation of that ellipse is

$r=r_\text{min}\frac{1+e}{1+e\cos{\phi}}$

where $e$ is the eccentricity of the orbit. When $r=r_\text{min}$, the planet is said to be at perihelion, occurring regularly at the exact same point every $2\pi$ radians.

According to general relativity, the orbit should not be a perfect ellipse but instead precess somewhat. There is some difference in the angle $\Delta\phi$ at which it reaches perihelion. The equation then becomes

$r=r_\text{min}\frac{1+e}{1+e\cos{\left(\phi-\Delta\phi\right)}}$

where

$\Delta\phi=\frac{6\pi GM}{c^2r_\text{min}\left(1+e\right)}$

The precession of perihelion of different planets in the solar system have been measured and match extremely well to the predictions of general relativity.

In 1974, a binary pulsar system was observed to have some loss in orbital period on the level of $67~\text{ns}$ per 8 hour orbit. This observation matched very well to the predictions of general relativity.