Wikipedia:Reference desk/Archives/Mathematics/2022 November 8

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November 8

dS = AdS

A de Sitter space and an anti-de Sitter space (in their more general forms: see EOM: De Sitter space and EOM: Anti-de Sitter space) may each be defined as any pseudo-Riemannian manifold that is isometric to a quasi-sphere embedded in a pseudo-Euclidean space with the induced metric tensor. The sign of this metric tensor is immaterial and is thus only a matter of convention. Yet neither WP nor EOM acknowledges this strong equivalence: they both describe these spaces as having positive and negative curvature respectively, as though this distinguishes them. If one attaches the labels "space-like" and "time-like", it is evident to me that it is the curvature from the "space-like perspective" being referred to, and that if one swaps the labels, the sign of this curvature changes. Yet from a geometric perspective, these labels are superfluous. The typical physicist might be surprised by this dS = AdS equivalence. What am I missing here? Can we find references that deal with this? —Quondum 14:31, 8 November 2022 (UTC)

If both are isometric to a quasi-sphere, they should be isometric to each other. Do they have the same light cones?  --Lambiam 15:53, 8 November 2022 (UTC)

Inherently (given they have the same metric up to a sign: the light cone is the set of null geodesics through a point), yes. What might be confusing is that if one applies "space"/"time" labels, the re-labelling would change say three "space" and two "time" dimensions to three "time" and two "space" dimensions (and the sign of the "curvature", as I see it). —Quondum 18:36, 8 November 2022 (UTC)

To assign a physical meaning, the tangent spaces should be Minkowski spaces, having three space dimensions and one time dimension, distinguished by having another sign in the signature of the Minkowski metric than the other three dimensions. It does make a difference whether the curvature of a spacelike (three-dimensional) section, an ordinary Riemannian manifold, is positive or negative; this is an intrinsic property of the space that does not depend on how the space is embedded in a higher-dimensional flat space.  --Lambiam 23:19, 8 November 2022 (UTC)

This is a mathematical question, not a physics question, so the restriction does not apply (though in physics, lower- and higher-dimensional spaces without such restriction are used and also referred to as dS and AdS). I explicitly opened with "in their more general forms", as at EOM. The concepts of "space-like" and "time-like" do not apply in a purely geometric context. When we have a manifold with a definite metric, we have only one "sort" of dimension to reference, and sign of curvature can be given a canonical definition. This is not true with indefinite spaces (in a dS or AdS space we simultaneously have "totally geodesic submanifolds" that have positive and negative curvatures – equal but for the sign – with spherical and hyperbolic geometries respectively).

For concreteness (and to show how blindingly obvious my point is), we can define two manifolds to match the definitions at EOM, each with the induced metric:

• dS: The manifold $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-x_{5}^{2}=1/c>0$ in $\mathbf {R} ^{5}$ with metric $dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-dx_{4}^{2}-dx_{5}^{2}$ ($n=2,p=2$).
• AdS: The manifold $x_{5}^{2}+x_{4}^{2}-x_{3}^{2}-x_{2}^{2}-x_{1}^{2}=-1/c<0$ in $\mathbf {R} ^{5}$ with metric $dx_{5}^{2}+dx_{4}^{2}-dx_{3}^{2}-dx_{2}^{2}-dx_{1}^{2}$ ($n=2,p=2$).

The variables have been re-indexed so that they are algebraically obviously equivalent (except that the metric differs by a sign, but we know that this is immaterial). Any choices of $n$ and $p$ work; their values swap between the two manifolds. —Quondum 23:59, 8 November 2022 (UTC)

The four-dimensional de Sitter spacetime and anti-de Sitter spacetime are different, being distinguishable by the sign of their curvature. Both are specific instances of a more general geometric object, namely a space that is isometrically embeddable as a pseudo-hypersphere in a pseudo-Euclidean space of one dimension higher. So it looks like the EOM describes this generalization twice, in different entries.  --Lambiam 20:38, 9 November 2022 (UTC)

Yes, that is what I'm saying: the two-parameter dS and AdS families are actually the same family, and I have read nothing that recognizes this.

Agreed, the 3+1 dS and AdS cases of physics are distinct members of the family. The dS case has three dimensions from which the curvature seems positive and one from which it is negative, and the AdS case has this reversed. The bigger family makes it clear that describing the curvature as categorically "positive" and "negative" is not correct. Reducing these to 1+1 as is reasonable in physics, we have the same space whether we call it dS or AdS, and the distinction is now only which direction you label as "space-like", with the sign of the curvature conventionally referring to only the space-like curvature.

Try to convince a physicist that the curvature is both negative and positive in this way (and that selecting which to describe it is only a convention), though ... Thanks – I feel affirmed in my analysis. —Quondum 23:02, 9 November 2022 (UTC)

I am not sure the difference between the four-dimensional dS and AdS cases is one of systematically reversing the signs. They appear to be embedded in different pseudo-Euclidean spaces. Quoting from: Bang-Yen Chen, “Marginally trapped surfaces and Kaluza-Klein theory”, April 2009, International Electronic Journal of Geometry 2(1):1–16^[1]:

     De Sitter space-time can be defined as a hypersurface of Minkowski space. Take Minkowski space-time $\mathbb {E} _{1}^{5}$ with the standard metric

$g=-dt^{2}+dx^{2}+dy^{2}+dz^{2},$

the de Sitter space-time is the submanifold described by the hyperboloid

$-t^{2}+x^{2}+y^{2}+z^{2}=c^{2},$

where $c$ is some positive constant. The metric on de Sitter space-time is the metric induced from the ambient Minkowski metric.

     Similarly, an anti de Sitter space-time can be realized as a hypersurface of the pseudo-Euclidean space $\mathbb {E} _{2}^{5}$ with index 2 described by

$-t^{2}-x^{2}+y^{2}+z^{2}=-c^{2}$

where $c$ is some positive constant.

 --Lambiam 03:13, 10 November 2022 (UTC)

The source has an ignorable error of one too few variables in these expressions. dS₄ and AdS₄ (resp. S⁴
₁ and H⁴
₁ in the notation of the source) are different spaces: S⁴
₁ ≠ H⁴
₁. I have not suggested trying to draw any equivalence between these two cases: I am only interested in the interpretation of the sign of the "curvature", which I am saying has no natural interpretation (possibly outside of the cases S^s
₀ and H^h
₀).

I am saying that S^s+h
_h = H^s+h
_s for all s and h (full dS–AdS equivalence), where s denotes "spherical" dimensions and h denotes "hyperbolic" dimensions, and that sources call the curvature of S^s+h
_h positive and of H^s+h
_s negative, despite them being identical geometrically. —Quondum 14:37, 10 November 2022 (UTC)