Acceleration Frame Dependency - General Relativity

[kent davidge]

I was thinking about the geodesics equations and I realized that a particle will not have acceleration if the connection coefficients vanish, which (I think) is to say we are attatching a inertial frame to the particle. But if we attach a non-inertial frame to the particle, it will probably have acceleration. Nothing new to this point as it works this way even in Newtonian mechanics.

The problem seems to be this: we might be on a flat space-time but using "curved" coordinates. Then in general the particle will be accelerating. Then General Relativity would dictate that the particle is in a gravitational field. Paradox?

Edit: or... maybe this is not a problem as the equivalence principle says that the space-time is locally flat and the geodesics equation is for analysing a curve only locally?

 

[A.T.]

The problem seems to be this: we might be on a flat space-time but using "curved" coordinates. Then in general the particle will be accelerating.

Coordinate acceleration is frame dependent. Proper acceleration is frame independent.

 

[Dale]

I realized that a particle will not have acceleration if the connection coefficients vanish,

This is not generally true as stated. Do you mean “an inertial particle will not have coordinate acceleration if the connection coefficients vanish”? Please revise your OP paying careful to distinguish between an inertial particle and a non inertial particle and to distinguish between coordinate acceleration and proper acceleration.

Then General Relativity would dictate that the particle is in a gravitational field. Paradox?

No paradox. The “gravitational field” describes by the connection coefficients (aka Christoffel symbols) is frame variant.

 

[haushofer]

I'd say, "Then locally, GR dictates that the accelerated particle can also be described as if freely falling in a gravitational field".

As if. This is just the equiv.princ.