De Sitter space, basic questions

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Discussion Overview

The discussion revolves around the properties and interpretations of de Sitter and anti-de Sitter spaces, particularly focusing on the scale factor solutions for different curvature parameters (k values) and their implications in various coordinate charts. Participants explore the dynamics of these solutions, the nature of spacetime geometry, and the relationship between different coordinate systems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that for ## \lambda > 0 ##, the solutions for ## k=-1,0,1 ## represent the same spacetime but differ by coordinate transformations.
  • There is a question about the validity of the exponential growth of the scale factor, suggesting it may depend on the coordinate chart used.
  • One participant asserts that the dynamics of the scale factor are contingent on the choice of chart, indicating that there is no single 'true' relation.
  • Participants discuss the geometric properties of de Sitter space, noting that open, closed, or flat characteristics pertain to spacelike slices rather than the entire spacetime.
  • There is a query regarding the interpretation of the ## \pm ## in the exponential solution for ## k=0 ##, with some uncertainty about its implications for contraction and expansion.
  • One participant mentions that the flat geometry corresponds to comoving observers being at rest, but clarifies that they are not at rest relative to each other.
  • Another participant reflects on the Friedmann equations and how the sign of ## \dot{a} ## does not affect the validity of solutions due to it being squared.
  • There is a discussion about the implications of negative cosmological constants in anti-de Sitter space and how it differs from de Sitter space.
  • Some participants express confusion regarding the physical expectations of different k values and their relation to comoving coordinates.

Areas of Agreement / Disagreement

Participants generally agree that the scale factor and its dynamics depend on the choice of coordinate chart, but there is no consensus on the implications of the ## \pm ## in the exponential solution or the specific characteristics of anti-de Sitter space. Multiple competing views remain regarding the interpretation of spacetime geometry and the nature of comoving observers.

Contextual Notes

Participants highlight that the properties of spacelike slices can vary based on the chosen chart, and there are unresolved questions about the implications of negative cosmological constants and the nature of different solutions in the context of Friedmann equations.

  • #31
binbagsss said:
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.
 
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  • #32
PeterDonis said:
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
 
  • #33
binbagsss said:
how do we know this?

By looking at how worldlines that are at rest in one chart transform into the other charts.
 
  • #34
PeterDonis said:
By looking at how worldlines that are at rest in one chart transform into the other charts.
PeterDonis said:
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.
 
  • #35
binbagsss said:
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.
 
  • #36
PeterDonis said:
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.
 
  • #37
binbagsss said:
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.
 
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  • #38
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!
 
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