I see a big discussion breaking out here, and although I don't like jumping on the bandwagon for these things, I'll try to say a few things to clear up the confusion. First, a mildly technical explanation of the theory:
So you know that spacetime is a Riemannian manifold (ie. a space idetitified by a set of coordinates, usually space and time), and that the metric tensor allows you to measure distances within it. Now, where the real Physics comes in is when you have derivatives of the metric:
- First order effects come from first derivatives of the metric. Acceleration and Newtonian gravity belong in this category: in Special Relativity, acceleration is simply d2xa/d2τ - the natural extension of what acceleration is in the Newtonian sense. In General Relativity, some extra terms quadratic in the velocity come in, and acceleration is then given by
d2xa/d2τ + Γabcdxb/dτdxc/dτ
The equation come from specifying that freely falling (ie. unaccelerated) particles pursue geodesics. Don't worry too much about what the Γ term means, except to bear in mind that it is a thing expressed in terms of the first derivative of the metric. Now, because of all the factors of c knocking around, at low speeds we can forget about most of these terms that have just been added, and what we're left with as an expression for the acceleration the usual d2xa/d2τ minus the acceleration due to the (Newtonian) gravitational field - and thus Newtonial gravity is recovered at small speeds in weak fields.
It is which recovers the equivalence principle: that gravitational fields behave like accelerations.
- Second order effects come from the second derivatives of the metric, which naturally characterise the curvature of spacetime (mathematically speaking, these effects come from the presence of the Riemann tensor). Thus, the Einstein field equations - which involve the presence of matter - are expressed in terms of second derivatives of the metric (which quantifies the statement 'matter bends spacetime').
It is a fundamental principle of GR that we can always find local coordinates such that the first order effects vanish - that is, a frame can be accelerated such that the effects of gravity can be anulled.
Now to resolve aozilla's problem: he states that because of the inverse square law, gravity can be distingished from uniform acceleration. The key to the problem is the crucial requirement above that (in general) you can only makes the gravity balance the acceleration locally - as soon as you start thinking about inverse square laws, you are examining the particle's behaviour in a non-local context.
Indeed, if we could make accelerations equivalent to gravitational effect everywhere, then the first derivatives of the metric would vanish everywhere, and then so to would the second order effects. If this were true, then the Einstein field equations would fall apart and there would be no gravity anywhere.
aozilla replies:Does this imply that the gravity is in essence coming from an infinitely far away source? The gravity is not within the system, but we still can measure exactly how far away the center of mass is...
What I was trying to say above is that the source is not important - the presence of matter produces seconds order effects (curvature) which are not relevent in as far as the Equivalence principle is concerned.
The fact that the gravitational field varies with position (whereas an uniform acceleration does not) is something that you must be observing on a 'larger scale' than that for which the Equivalence principle applies, so
how is varies not important inasmuch as your question is concerned.