Changes

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>

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 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 <math>P</math> 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.

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.