According to relativity, there is no such thing as absolute rest in space. Absolute rest relative to what?? If you were floating over the Earth at "absolute rest", you'd still be flying around the sun. And the sun orbits the center of the Milky Way. So any question discussing "absolute rest" in the universe is flawed from the beginning in a nontrivial way. Any matter of time behaving differently relates to the previous comment in very complicated ways... but let's just say that time is ALWAYS behaving differently in some minor way or another. And furthermore... while I'm being picky... by "dimension" I assume you're referring to another spacial dimension. The only theoretical candidates for extra spacial dimensions would be coming from M-Theory (the new superstring deal that includes the idea of the "multiverse") is completely untestable for the time being... but assuming it's the case that it's true, then the extra dimensions would be, though VERY sub-atomically small, existing dimensions that your matter currently occupies, so YOU'RE ALREADY "ENTERED" INTO THEM... just by being here. So... the short story is: you are (in some arbitrary sense) absolutely still in the universe, time does behave differently from (respective) "moment" to "moment", and you're in (assuming M-Theory) 11 different dimensions. So... how cool are you! :)

The only absolute rest is in the mind!