Has time dilation been proved to occur?

Time dilation is a consequence of the theory of relativity. There are in fact two forms of time dilation.

Time dilation due to speed was first described by Hendrik Lorentz and is one of the Lorentz transforms. It is a bit strange.

If two spaceships are travelling apart at a constant velocity an observer on each spaceship will see their clocks tick at the usual rate. Both observers will also see that the clock on the other spaceship is ticking slower. This is time dilation.

Time dilation has been measured by having two highly accurate and synchronised atomic clocks. If one clock is put on an aircraft which travels around the world. When it lands the clock from the aircraft will be a fraction of a second behind the other.

Muons are very short lived particles. Muons arrive on Earth constantly from space. As they are short lived, Muons should have decayed before getting far into the atmosphere. As the muons are travelling at high speeds, time slow for them due to time dilation. This makes them decay after a longer time for us. This has been proved as muons have been detected at lower altitudes which proves the time dilation effect.

There is a second form of time dilation which is a consequence of Einstein's General Relativity. Gravity slows time. The stronger the gravity, the slower time passes. This results in a curious effect.

When an object is in orbit, time passes slower due to the orbital speed. Time also passes faster the weaker the gravity due to altitude.

The ISS is in a low orbit and the slowing of time due to speed is greater than the speeding up of time due to altitude. Astronauts age slightly slower on the ISS than on the surface.

The GPS satellites are in a high orbit and the slowing of time due to speed is less than the speeding up of time due to altitude. Clocks on GPS satellites run faster than clocks on the Earth's surface and need to be artificially slowed down to compensate.