Solutions to Einstein’s Field Equations & Breakthrough Discoveries
Gravitational Lensing
Einstein was excited to find solutions to these new equations. The first thing he did was to re-test previous experimental data that had used Newton’s laws of gravity, and found the equation’s predictions were correct. Massive objects bend space-time around them, so Einstein solved that they should bend light around themselves as well, and after observational proof, was called gravitational lensing.
http://www.cfhtlens.org/public/what-gravitational-lensing
Relativistic Red-shift
As light leaves the gravity well of a star, the light should be red-shifted, meaning the light changes to a longer wavelength over time, which was also proven experimentally. Resulting from this phenomena, the Cosmic Microwave Background (CMD) originally was a much lower wavelength of light, but as space has expanded, the wavelength increased to what we now call Microwaves.
Universal Expansion
However, there was one thing that Einstein got wrong. He predicted that the universe was static and unchanging, due to a mathematical assumption that the density of mass/energy in the universe didn’t change. Alexander Friedmann correctly predicted that, using general relativity, the universe can be static, or expanding, or contracting. However, Einstein published a paper criticizing Friedmann, since his answer did not match Einstein's. After a private letter by Friedmann asked where he was wrong, and if he wasn't, make a corrective paper, Einstein wrote a paper retracting his original statements about Friedmann being wrong.
Mercury's Precessing Orbit
Mercury’s orbit baffled astronomers, as the orbit of Mercury precessed around the sun. Before general relativity, this was not explainable by Newton’s law of universal gravitation. Einstein’s equation, however, predicted that there should be precession around the sun by Mercury. This was also a problem for neutron stars and other massive objects, but with the help of general relativity, there was now an explanation for an otherwise unexplainable phenomenon.
Apsidal precession, as this gif shows, is what Mercury's orbit does, but couldn't be explained by Newton's law of Gravitation, but came as a consequence of solving Einstein's Field Equations.
The first picture of a black hole taken by the Event Horizon Telescope array on April 10th, 2019.
https://eventhorizontelescope.org/press-release-april-10-2019-astronomers-capture-first-image-black-hole
Black Holes
Blackholes were predicted by a German physicist Karl Schwarzschild, who developed a solution to Einstein’s field equations that predicted that there should be a density of matter great enough that the space around it would be so curved that no matter or light could ever escape from it. Other solutions were later discovered that included if the black hole is spinning or had an electric charge, but had not been proven that they truly exist, as just because something exists in math does not mean that it will exist in real life. That was, until April 10th, 2019, when the first blackhole was “photographed” by the Event Horizon Telescope and the team behind it. The thing is, since black holes take in all light, they cannot photograph it per se, but they managed to take a picture of where one was because of the hot matter that was orbiting the black hole in what is known as an accretion disk.
Gravitational Waves
With the assumption of empty space, Einstein predicted gravitational waves should be able to propagate. This was also deemed theoretical, until, on September 14, 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) at Caltech announced they had made the first direct observation of gravitational waves, originated from a pair of merging black holes. Afterwards, there were two additional detections, followed by one on August 14, 2017, in which both LIGO and the European Virgo gravitational-wave detector detected gravitational waves, thereby experimentally proving the theoretical gravity waves to exist.
https://www.nature.com/articles/s41586-019-1129-z