Is acceleration absolute or relative - follow up

[PeterDonis]

any (an entire family of)

The parenthetical qualifier here is crucial. Without it, "any" just means one single geodesic, not a family of them.

 

[cianfa72]

We said that in an arbitrary spacetime there is no guarantee to find a global coordinate chart in which every timelike geodesic is "at rest" in it.

Does exist an example of curved spacetime in which there is a timelike geodesic congruence filling it all ?

In that case however, since the spacetime curvature, the geodesics deviation (i.e. tidal gravity) must be not null.

 

[PeterDonis]

We said that in an arbitrary spacetime there is no guarantee to find a global coordinate chart in which every timelike geodesic is "at rest" in it.

"Every timelike geodesic" is way too strong. You need to say "every one of some particular family of timelike geodesics". And to really have it be meaningful, the family of timelike geodesics needs to fill the entire spacetime.

For example, consider flat Minkowski spacetime. There are an infinite number of possible inertial coordinate charts on this spacetime. Each one has one family of timelike geodesics that fills the entire spacetime at rest in it; but only that one family. There are also an infinite number of other families of timelike geodesics that are not at rest in any given inertial chart. It's impossible to find a single chart in which all timelike geodesics in the entire spacetime are at rest.

Also, since we know there are global coordinate charts in flat spacetime in which a family of timelike geodesics that fill the entire spacetime is at rest, saying that there is no guarantee of this in an "arbitrary" spacetime obviously means an arbitrary curved spacetime.

Does exist an example of curved spacetime in which there is a timelike geodesic congruence filling it all ?

Of course: FRW spacetime.

In that case however, since the spacetime curvature, the geodesics deviation (i.e. tidal gravity) must be not null.

Yes.

 

[cianfa72]

"Every timelike geodesic" is way too strong. You need to say "every one of some particular family of timelike geodesics". And to really have it be meaningful, the family of timelike geodesics needs to fill the entire spacetime.

Yes that's right, I was inaccurate in describing it. As you said the main requirement is that 'that some particular family of timelike geodesics' needs actually to fill the entire spacetime.

Thanks for pointing that out.