Changes
Let's assume that math works
====Some assumptions about the universe====
Before we proceed into a discussion of what curvature is and how the Einstein equation describes it, we must first pause to state some fundamental assumptions about the universe.<ref>As we will later see, these assumptions may in fact turn out not to be valid for all of spacetime. It may be more accurate, although significantly less satisfying, to say that we assume these things to be true about spacetime, ''except where they aren't.''</ref>
The first assumption we're going to make is that spacetime is ''continuous.'' In essence, this means that for any event <math>P</math> in spacetime — that is, any point in space and moment in time — there exists some local neighborhood of <math>P</math> where the intrinsic properties of spacetime differ from those at <P> by only an infinitesimal amount.
The second assumption we're going to make is that spacetime is ''differentiable'' everywhere. In other words, the geometry of spacetime doesn't have any sharp creases in it.
If we hold these two assumptions to be true, then a convenient property of spacetime emerges: Given any event <math>P</math>, there exists a local neighborhood where spacetime can be treated as ''flat,'' that is, having zero curvature. It is not necessarily true that ''all'' of spacetime be flat — in fact, it most definitely is not — but given any event in spacetime, there exists ''some neighborhood'' around it that is flat. This neighborhood may be arbitrarily small in both time and space, but it is guaranteed to exist as long as our two assumptions remain valid.
With these two assumptions and this convenient property in hand, we will now examine what it means to say that spacetime is curved.
====The metric tensor====
