the planet (this is not earth) would still rotate on its axis, just not around anything. [spoiler]However, at the end of the game, you discover that the sun hasn't actually burnt out, like everyone believes, but is merely obstructed and might, one day, shine again on Arx [/spoiler]
Spoiler: The reason everyone is underground is that the sun gets obscured by space debris, shown in the full ending. Also there are people alive on the surface