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hossi
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vanesch said: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.
Hi vanesh,
thanks for your thoughts
You mean local in a surrounding, including some infinitesimal region around this point (i.e. first derivatives included). In this infinitesimal region - as you point out - the gravitational pull will feel like an acceleration in flat space. The direction of which is inverted for the anti-gravitational particle.
From this naive pictorial point of view, I see no disagreement with the equivalence principle as I stated it above - namely that it holds for both types of particles on it's own. Both can be fooled by an angel pulling the elevator to believe that there is gravity. In the one case though, the elevator goes up, in the other case down, respecting that they feel the gravitational pull in the other direction. In the usual case, the ratio of inertial to graviational mass is 1, in the other case -1.
However, I take it, you have a very distinct problem with the mathematical formulation. I hope, I eventually get the point:
When you define the the Riemann normal coordinate system (which actually is not really a coordinate system on the whole manifold when I recall that correctly) you make a gauge requiring that the Christoffelsymbols vanish. This makes the geodesic motion look especially simple, i.e. as in flat space, 2nd derivative of x equals zero.
For this, you have used the form of the covariant derivative acting on the quantity to be parallel transported. The momentum of the anti-graviational particle behaves differently under such transport. The curve on which it remains 'parallel' therefore is a different one. Or, as you would have expected from the modified transformation behaviour of the new particle's momentum, it's covariant derivative is different.
Consequently, the Riemann normal coordinates for the anti-grav. particle are defined by a different gauge. I.e. by the requirement that the corresponding connection coefficients (which are related to the Christoffelsymbols but not identical) vanish. For details and indices, see paper.
So, you are right that the Riemann frame for the anti-g particle is not the usual Riemann frame. What is the problem with that?
Indeed, the anti-gravtational particle gets repelled from the black hole (unfortunately, the example with the motion in a Schwarzschild background dropped out of the paper because it was too long.)
B.
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