Talk:de Sitter space

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AdS

Can someone please unify the notation of this article and the Anti de Sitter one.200.145.112.189 (talk) 17:16, 11 November 2009 (UTC)

Plain English Summary

It would be great if there was a plain English summary of what de-Sitter space is. Or at least a link to another website where there is an explanation of what de-Sitter space is. Is de-Sitter space space of a certain shape? After reading this article I still have no idea what de-Sitter space is. It seems to me that Wikipedia articles should be written with lay people in mind. This article seems to be highly technical. Dedwarmo (talk) 15:28, 10 April 2013 (UTC)

Its a highly technical topic. Start by reading the linked articles; they provide background. User:Linas (talk) 04:30, 3 November 2013 (UTC)

I agree. Wikipedia is an encyclopedia, so the aim should be to give a description aimed at a general audience. Adding further detail at textbook level can be useful, but only as an addition. Scottwh (talk) 10:31, 7 March 2024 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:De Sitter space/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Geometry guy 21:20, 1 June 2007 (UTC) The problem I have with this article is I doubt anyone but hard core physicists can understand what is trying to be conveyed. In that this encyclopedia suppose to be for a general audience I would think a verbalized non-formulated explanation of De Sitter space would be more useful. May I suggest the follwing link is a possible alternative: =560692 ==Agreed== User:Geometry guy, I totally agree. Dedwarmo (talk) 15:29, 10 April 2013 (UTC)

Last edited at 15:30, 10 April 2013 (UTC). Substituted at 13:03, 29 April 2016 (UTC)

Defining characteristics of a de Sitter space?

Defining a de Sitter space as a submanifold of a Minkowski space using a vector space as given in the article is unambiguous, but is not ideal as a general mathematical definition, since it relies on embedding and coordinates. If there is another definition along the lines of "a real Lorentzian manifold of dimension n that is maximally symmetric and has constant positive curvature", this would be worth giving. What I have given in quotes defines it locally, and It would be worthwhile noting in the article that other spaces (with distinct topologies) exist with the same local description. For example, in the vector space definition if we identify every point in a de Sitter space with that opposite the origin, we obtain a space with the topology of a punctured projective space (e.g. a Möbius strip for n=2). Such a space is not simply connected, and is not necessarily nonphysical as a model of the universe, notwithstanding the argument that paths exist that reverse the direction of time. Are there any references that discuss these possibilities? —Quondum 20:37, 26 May 2016 (UTC)

Metric signature

Are the definitions in terms of induced metric from ambient space with the given Miknowski metric (mostly plus convention) and the definition as a homogeneous space as the quotient O(1; n)/O(1; n – 1) and the mention of the Lorentz group as being O(1; n) really in compliance? YohanN7 (talk) 10:44, 9 December 2016 (UTC)

Lead paragraph

Last paragraph in lead,

More recently it has been considered as the setting for special relativity rather than using Minkowski space, since a group contraction reduces the isometry group of de Sitter space to the Poincaré group, allowing a unification of the spacetime translation subgroup and Lorentz transformation subgroup of the Poincaré group into a simple group rather than a semi-simple group. This alternate formulation of special relativity is called de Sitter relativity.

really needs a reference. Besides, the Poincare group certainly isn't semi-simple (if that is what is hinted). YohanN7 (talk) 10:55, 9 December 2016 (UTC)

Merge with De Sitter universe

I propose that De Sitter universe be merged into this article. There doesn't seem to be any reason why the two articles should be separate. Thoughts?

I oppose the suggestion. The De Sitter space article is written for scientists who (probably already) understand the mathematics of a De Sitter space. While it has issues, the De Sitter universe article is written for ordinary people who want to understand what a De Sitter space means in the context of cosmology.Work permit (talk) 17:10, 14 April 2017 (UTC)

I'm removing the banner. There was discussion of this back in 2005, and the proposal garnered no support.Work permit (talk) 20:12, 28 April 2017 (UTC)

Does anyone actually understand this subject?

Leonard Susskind on the recent Lex podcast, says that he doesn't get De Sitter spaces. And that we live in them. Avindratalk / contribs 03:26, 27 September 2019 (UTC)

This comment is mostly irrelevant here. If I understand correctly, Susskind was talking about the quantum mechanics of a De Sitter geometry, i.e. its realization in a quantum theory of gravity. This is unrelated to understanding the properties of De Sitter as pseudo-Riemannian manifold which is the main topic of this page. Virtual Neutrino (talk) 10:41, 29 May 2020 (UTC)

Sign of curvature

Adumbrativus, I think your comment is fair. Yet, I think we technically have a problem in the existing description, which in itself is confusing once you look more deeply into it, and hence my attempt at highlighting this. And yes, we should have sourcing, and hopefully this has been discussed in the literature, but my reference (Penrose) makes no direct mention of curvature that I see. To make my point clearer, consider an embedding in R⁴, defined by x²
₁ + x²
₂ − x²
₃ − x²
₄ = ±α²
. You should agree that the sign on the right makes no difference: the two forms describe isometric manifolds (though this problem is far more acute in the article Anti-de Sitter space). Whether we consider the manifold to have positive or negative curvature is a matter of convention (and in Lorentzian manifolds, I expect that the convention is to consider only so-called "space-like" sections when speaking of sign of the curvature). In this article (for which the context is essentially mathematical rather than GR), such a convention should be made explicit, since the sign of the curvature is ill-defined when the metric tensor is indefinite.

Maybe I should approach this differently: the two articles De Sitter space and Anti-de Sitter space make uncited claims about positive and negative curvature, but without a convention, these statements are meaningless. Perhaps I should just tag these claims with {{citation needed}} – unless someone wants to discuss this first. —Quondum 12:27, 4 November 2022 (UTC)

Thanks for the thoughtful discussion. The article is consistent with Carroll, Spacetime and Geometry section 8.1, and Hawking and Ellis section 5.1. Carroll writes, "the Riemann tensor for a maximally symmetric n-dimensional manifold with metric $g_{\mu \nu }$ can be written $R_{\rho \sigma \mu \nu }=\kappa (g_{\rho \mu }g_{\sigma \nu }-g_{\rho \nu }g_{\sigma \mu })$ where... ${\textstyle \kappa ={\frac {R}{n(n-1)}}}$ and the Ricci scalar R will be a constant over the manifold.... The maximally symmetric spacetime with positive curvature ($\kappa >0$) is called de Sitter space." The current lead sentence particularly seems to echo Carroll's phrasing.

Positive scalar curvature refers to $R=g^{\mu \nu }R_{\mu \nu }$. There are matters of convention involved, but it doesn't involve choosing particular subspaces. After all, spacelike "slices" can have zero, negative, or positive curvature, depending on which way you slice, as seen in the explicit "flat slicing", "open slicing", and "closed slicing", as the article currently calls them.

The conventions involved are that $g_{\mu \nu },R_{\rho \sigma \mu \nu },$ and $R_{\mu \nu }$ have sign conventions (or index-ordering conventions) in their definitions. In principle, different author have different sign conventions leading to different signs of R. In the specific context of talking about de Sitter space, I have only seen R > 0. But that's just what I've seen; if some sources say the opposite, it would be worth mentioning. If we wanted to be more explicit about the sign convention in the article, we could specify $R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }$ and $R_{\mu \nu }=R^{\lambda }{}_{\mu \lambda \nu }$; that's the convention implicit in the Properties section. Adumbrativus (talk) 03:41, 5 November 2022 (UTC)

I am not familiar enough with the conventions of general relativity (choices of sign of metric tensor, Riemannian tensor, etc.) other than to know that there are at least three binary choices to be made in any combination (I think it was in Gravitation, MTW). I'm not sure how this relates directly to the sign of R, even. However, one can say that elliptic geometries have what we call positive curvature and hyperbolic geometries have what we call negative curvature.

"Slicing" appears to be nothing other than an arbitrary assignment of submanifolds for purposes of coordinatization. Intrinsic curvature inherited by such manifolds tells us nothing. Many assumptions/conventions are assumed by physicists, often without acknowledgement, and what they "mean" by "positive curvature" seems to be one of these. Carroll is potentially doing the same. Ideally, we should use phrasing that would make sense to a geometer as well. It does not help that physicists seem to have introduced the most material about indefinite metric tensors and de Sitter manifolds, bringing with it assumptions/conventions that are routine in GR.

In homogeneous manifolds we can define geometric k-flats (e.g. lines for k = 1, planes for k = 2, and so on). The tangent geodesics of a flat are contained in the flat, unlike for other submanifolds. In a Euclidean geometry, the intrinsic curvature of such 2-flats is zero, in elliptic/spherical geometries this a positive constant (with closed geodesics for k ≥ 2), and in hyperbolic geometries this is a negative constant (with open geodesics). This gives us some intuitive handle on curvature in homogeneous spaces. Adding dimensions does not appear to change the relationship between curvature of a k-flat and that of the complete geometry of n dimensions (e.g. an n-sphere has curvature 1/α² for all its k-flats for k ≥ 2, and similarly for the other geometries). Here is the kicker: a homogeneous geometry constructed using this quasi-sphere model with at least two dimensions of each type (which physicists might call time-like and space-like) through every point simultaneously contains flats of pure positive curvature and flats of pure negative curvature, with the associated closed/open contained geodesics. That is, positive "space-like curvature" and negative "time-like curvature" occur in the same geometry (and vice versa). Whether the curvature of the manifold is called positive or negative depends entirely on which type of coordinate you are referencing the curvature to. (Caveat: This exposition is pure OR on my part.)

TL;DR: What Carroll really seems to "mean" is "positive space-like curvature", since "positive curvature" is not amenable to definition for a general pseudo-Riemannian manifold, and this should be worth noting in the article. A section elaborating this (if it can be sourced) would be a great addition. —Quondum 17:33, 5 November 2022 (UTC)

The lead sentence says "positive scalar curvature". Scalar curvature (a.k.a. Ricci scalar) has a precise definition. It is the same notion of scalar curvature used by geometers and physicists alike. It has the same definition irrespective of the metric signature, Riemannian or pseudo-Riemannian. There are underlying sign conventions involved (yes, the three bits of information documented in MTW); the remedy for that is to state expressly what convention the article is using. I took a shot at adding it to the article just now (which can be revised). One does not have to guess at what Carroll or any other book means – they say what they mean, and it is scalar curvature R. It seems to me that the comments above are confusing scalar curvature with some other notion of curvature. Adumbrativus (talk) 02:18, 6 November 2022 (UTC)

We will probably not converge on a shared understanding any time soon. I'll leave you with this thought: dS and AdS spaces of two dimensions are the same space; only the (geometrically irrelevant) "space" and "time" labels have been switched, yet the curvature is described as being positive or negative by context. —Quondum 15:54, 6 November 2022 (UTC)

Capitalisation of "De" in De Sitter

This article frequently writes "De Sitter Space" with a lowercase D.

While the tussenvoegsel in Dutch surnames is lowercase when writing the full name, when only mentioning the surname and thus beginning the name with the tussenvoegsel, it would be capitalised. As a Dutch speaker, that seems to be the right way to write it, and Dutch wikipedia does so as well. I would suggest changing "de Sitter space" tot "De Sitter space". Opinions? Wolkenkijker (talk) 15:13, 30 October 2023 (UTC)