hossi said:
Spacetime is either (globally) flat or it isn't.
And locally, to a good approximation ?
Can we say that some domain of a spacetime manifold is, to a good approximation, flat ? (eg, the Riemann curvature tensor vanishes over the domain, or is, at least, extremely small? )
Consider two such domains. One domain is relatively close to the world line of a huge black hole (so there will be SOME curvature, but you can make it as small as you want, by making the black hole as massive as you want and increasing the distance). Call it domain A.
Another domain is very far from any matter or energy. Call it domain B.
In a Newtonian limit, one would say that domain A is in a relatively homogeneous gravitational field, while domain B is free of gravity.
In both domains, there is so good as no tidal effect observable (very tiny Riemann tensor).
But in general relativity, both are rather equivalent, flat lumps of spacetime. So locally, they are equivalent concerning their metric structure. Now, let us choose coordinates over both of these domains which make this flat metrical structure explicit (in other words, take local coordinates which correspond to inertial observers). In domain A and in domain B, in such coordinates, normal particles undergo a uniform motion on a straight line.
But anti-gravity particles accelerate (strongly) in domain A, while they undergo uniform motion in domain B. (at least, I assume, that these anti-gravity particles accelerate AWAY from the black hole).
Nevertheless, BOTH domains correspond to locally FLAT pieces of spacetime.
There are no 'certain flat spacetimes'. If spacetime is globally flat, both particles do the same thing, no matter what. If it is curved, they can move on different trajectories.
And "locally" ? Do the anti-gravity particles do the same thing as normal particles in (to a good approximation) the locally flat spacetime not too far away from a huge black hole ?
It seems to me you are using the 'uniform gravity field' as a synonym for acceleration. To use the equivalence principle, you say the effect of gravity (locally) is the same as acceleration in flat space. For the anti-g particle, its just the opposite acceleration. I.e. the ratio of intertial to graviational mass in inverted. It might be helpful to forget for a moment that there are two types of particles and only to consider the second possibility.
What I don't understand, is this: from the structure of spacetime, it is always possible to find ONE frame (= coordinate set) which is "locally inertial" at point P (think it is called Riemann normal coordinate system). Normal matter is supposed, in such a frame, to behave like in free space (uniform motion on a straight line), at least, locally. This is true whether the spacetime is locally flat or curved.
Now, if some stuff starts *accelerating* in that frame, then, this is because of some *OTHER* interaction, but it can certainly NOT be due to the local structure of spacetime (= gravity), can it ? And what we have here, is that those anti-gravity particles start accelerating in the case of domain A. Of course, you can find now another frame, in which this anti-gravity particle is in uniform motion, but the point is that such a frame is not a Rieman-normal coordinate set. (= inertial frame).
So my point is that locally IDENTICAL lumps of spacetime (domains A and B) give rise to *different* behaviour for that anti-gravity particle ; which means, to me, that it cannot be the (local) structure of spacetime that describes this motion.
Of course, if ALL matter were "anti-gravity" there would not be a problem either (we just made a mistake in our assignment of the metric tensor, and it must be re-defined in order for the "co-moving frame" to be inertial now). But I don't think both can be coexisting and derive both from the metric structure of spacetime.
But of course, there can always be *another* interaction which makes these "anti-gravity" particles accelerate, for one reason or another.