# 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%.

## Delay of Radar Echoes

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.

## Gravitational Radiation

Just as accelerated charges produce electromagnetic radiation traveling at the speed of light, accelerated masses produce gravitational radiation traveling at the speed of light. These *gravitational waves* are ripples through spacetime but are exceptionally weak. Only extreme events in the universe like black holes merging or supernovae produce measurable gravitational waves.

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.

An interferometer, the Laser Interferometer Gravity-Wave Observatory (LIGO) was constructed to measure gravitational waves disrupting spacetime. The laser beams inside the interferometer are usually set to destructively interfere with one another but gravitational waves can slightly change the lengths of the arms, causing an interference pattern.