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Einstein and Hawking seem to agree that this is a four-dimensional universe, with the fourth spatial dimension being what we perceive as time. So, it's not that the fourth dimension is time, it's more like time is the shadow of the fourth dimension and it's only our perception that we're moving through it.
I realized I've touched upon this, seemingly, throughout my entire writing career. I've got stories about this sort of notion in 2000AD, there's Dr. Manhattan's view of time in Watchmen and all the stuff in From Hell.
C. Howard Hinton, one of the Victorian mathematicians who first proposed a mathematical fourth dimension, said you'd have to suppose that it's only our awareness that we're moving through time. That nothing is actually changing; the universe is a four-dimensional solid, like a great big egg with the Big Bang at one end and the Big Crunch at the other end, and every moment that has ever or will ever exist, suspended, forever, in between.