De Sitter space, basic questions

[PeterDonis]

Since the different $k$ values correspond to a different space-like slicing - coordinate chart - can you conclude that the observers have relative motion as different coordinate charts give a rise to different relative velocities?

In de Sitter spacetime, observers at rest in one of the three charts are not at rest in the other two. That's all you need to conclude that observers at rest in each of the charts are in relative motion.

 

[binbagsss]

In de Sitter spacetime, observers at rest in one of the three charts are not at rest in the other two. .

But how do we know this?
Or do you have any references stating it, thanks

 

[PeterDonis]

how do we know this?

By looking at how worldlines that are at rest in one chart transform into the other charts.

 

[binbagsss]

By looking at how worldlines that are at rest in one chart transform into the other charts.

There are solutions where the "comoving" observers see k=1, solutions where they see k=-1, and solutions where they see k=0. It just happens that the solution that (as far as we can tell) describes our actual universe is the one in which comoving observers see k=0.

Do you know of any sources that state certain compositions and then the corresponding curvature/coordinate chart ? (for the obsevers that view the universe as isotropic and homoenous). Thanks.

 

[PeterDonis]

Do you know of any sources that state certain compositions

I don't know what you mean by "compositions". Many references discuss the FRW charts for all three values of k and the corresponding curvature components. For example, Carroll's lecture notes, IIRC, go into this.

 

[binbagsss]

I don't know what you mean by "compositions". Many references discuss the FRW charts for all three values of k and the corresponding curvature components. For example, Carroll's lecture notes, IIRC, go into this.

Apologies I meant different distributions of the matte/radiation/ dark-energy.

 

[PeterDonis]

I meant different distributions of the matte/radiation/ dark-energy.

Carroll's notes discuss this too, at least with respect to the most basic cases--ordinary non-relativistic matter, radiation, and dark energy.

 

[LoicM]

Another very good reference for the same topic is "V.Mukhanov : Physical foundations of Cosmology", §1.3.6. On pp.33-34 is actually a very simple and intuitive solution giving the results described by binbagss in his initial message.

http://www.astro.caltech.edu/~george/ay21/readings/Mukhanov_PhysFoundCosm.pdf

I would like to thank binbagss for his very good questions and especially PeterDonis for his detailed and clear explanations!