It is a different "dimension;" it's called God. The saints in heaven share in His divinity (by grace), but outside of that we have no clue what it is, or what it means.
Physical space as such is irrelevant when it comes to God, as we would surely say that God is present in all parts of the Creation, immensly vast and it is, and all at the same time. This is a highly transcendental concept which we know of but cannot comprehend.
Truly, the term "eternity" is woefully inadequate in speaking of God because eternity is merely time without boundaries, i.e. ends (past or future).
I prefer the term timelessness, but it still doesn't capture the point that time is a part of Creation and not something in or by which the Creator exists.