I'm puzzled by the implication of the Einstein's theory about time being in a way a function of speed and/or acceleration. Doesn't it mean that time must be actually different on the bodies with different speed if this speed is big enough to notice it? Might be not noticeable on Mars compared to Earth, since the rotation of the Solar System probably compensates or balances somehow and their speed isn't big anyway. But what about time in the whole of the Solar System which rotates with a really huge speed around the centre of the Galaxy compared to time around Antares? What about time in our Local Group compared to the time in the M83, for instance, ot M31?
You ask a good question; it turns out that astronomers do have to worry about the time effects implied by Einstein's theories in certain circumstances.
As you mention, the time measured by two observers in two different reference frames depends on their relative speeds; in short, an observer that is observing an event in a reference frame moving at speed v will measure the time in that frame to move slower than in her frame by a factor equal to the square root of (1-(v/c)2), where c is the speed of light. So, we only need to worry about these effects when (v/c) squared is comparable to (but never greater than) 1. To be safe, let's assume that no correction is needed if v is 20% the speed of light.
Now, the speed of light is 300000 km/s, and so only objects with velocities of 60 000 km/s relative to the Earth need to take relativity into account. The Solar System does move quickly around the galactic centre by Earth standards, averaging a speed of 220 km/s, but this is nowhere near fast enough to require a correction for relativistic effects. So, the speed with which time elapses in the Solar System and, say, in Antares are very, very nearly the same. Nevertheless, astronomers do have to worry about these relativistic effects when studying so-called "high energy phenomena" in the Universe. A good example is the study of jets that are emitted in the vicinity of black holes, both at the centre of distant galaxies and in stellar systems in the solar neighbourhood. These jets are often observed to propagate through the interstellar medium at substantial fractions of the speed of light (some are even observed to move faster than c, but this is just an illusion caused by relativity). Estimates of the propagation speed and energetics of the jets may require substantial corrections for relativistic effects. Look here for a pretty picture of a jet in a nearby galaxy.
This page was last updated on January 28, 2019.