We think it's obvious that the Earth goes around the sun, today, but back before Copernicus, detractors had the most obvious argument possible against heliocentrism. To combat this, in his argument in favor of a sun-centered universe, Galileo came up with a thought experiment about relativity now known as Galileo's Ship.
Back during the times when society scrutinized the stars as they - like all the universe - dutifully circled the Earth, some people noticed that something was amiss. Although all the stars moved, they didn't all move at the same speed. They didn't move in the same arc. They didn't even move in the same direction. Astronomers, like Copernicus and Galileo, began popping up and proposing a new theory about why this happened. The stars weren't moving. The Earth was.
The point was so brilliant that everyone listening to it laughed themselves sick. Sure, it might explain a few movements of dots in the sky, but it contained a logical gap so large they didn't understand how anyone could fail to see it. They were all standing on the Earth - and it wasn't moving. What's more, quite a few people knew what it felt like when the Earth did move. Houses collapsed. Possessions fell over. People were knocked off their feet. If the world really was moving no one should be able to get their footing.
To silence the detractors, Galileo came up with one of the early (and still valid) thought experiments - known as Galileo's Ship:
Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. . . . When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.
The concept is also called the Salvatius Ship. Salvatius is the wise narrator in Galileo's book expounding on the veracity of heliocentrism. This was not just an early thought experiment, but an early idea of relativity. Galileo took the first step towards noting that, if all the effects of constant motion are exactly the same as the effects of stillness, there isn't any way of telling what is in motion and what it not. He was constrained by the fact that this idea didn't include accelerated motion - the world would need Einstein equating acceleration and gravity for that - but he took a step towards both heliocentrism and the realization that there is no absolute frame of reference for anyone.